A dvanced X-ra y Analytical Metho ds fo r the Cha racterization of Buried Interfaces with Relevance fo r Energy Conversion Devices v orgelegt v on Master of Science Jonas Baumann geb oren in Berlin v on der F akult¨ at I I - Mathematik und Naturwissensc haften der T ec hnisc hen Univ ersit¨ at Berlin zur Erlangung des akademisc hen Grades Doktor der Naturwissensc haften Dr. rer. nat. genehmigte Dissertation Promotionsaussc h uss: V orsitzender: Prof. Dr. Norb ert Esser Gutac h terin: Prof. Dr. Birgit Kanngießer Gutac h ter: Prof. Dr. Klaus Rademann T ag der wissensc haftlic hen A ussprac he: 30. August 2017 Berlin 2017 Contents Abstract 1 Zusammenfassung 2 1. Intro duction 3 2. Metho dological Bases 7 2.1. X-ra y Fluorescence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1. Calculation of Fluorescence In tensities . . . . . . . . . . . . . . . . 8 2.1.2. Quan tification Approac hes . . . . . . . . . . . . . . . . . . . . . . . 9 2.2. Grazing Incidence and Grazing Emission XRF . . . . . . . . . . . . . . . 10 2.2.1. The Complex Refractiv e Index . . . . . . . . . . . . . . . . . . . . 13 2.2.2. Shallo w Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.3. Shallo w Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 . 3 . S c a n n i n g - F r e e G E X R F ............................. 1 9 2.3.1. Single Photon Coun ting with a CCD - Energy Resolution . . . . . 19 2.3.2. Geometry Considerations - Angular Resolution . . . . . . . . . . . 23 2.4. Soft w are for (AR)XRF Ev aluation . . . . . . . . . . . . . . . . . . . . . . 26 2.4.1. xrfLibrary and xrlfupa for X-ra y Fluorescence Calculations . . . . 26 2 . 4 . 2 . D e p t h P r o fi l i n g ............................. 2 8 2.5. Soft X-ra ys in Scanning-F ree GEXRF Analysis . . . . . . . . . . . . . . . 30 2.5.1. Application of Soft X-ra ys . . . . . . . . . . . . . . . . . . . . . . . 30 2.5.2. Pro duction of Soft X-ra ys . . . . . . . . . . . . . . . . . . . . . . . 31 3. Thermo electric Nanofilms 35 3.1. Sample Preparation and Surface Characterization . . . . . . . . . . . . . . 35 3.2. In v estigation of Lateral Homogeneit y b y means of Lab oratory-based XRF 37 4. Synchrotron Radiation based Analysis 43 4 . 1 . I n s t r u m e n t a t i o n ................................. 4 3 4 . 1 . 1 . B e a m l i n e s ................................ 4 3 4 . 1 . 2 . E n d s t a t i o n s............................... 4 4 I CONTENTS 4.2. Data Recording and T reatmen t . . . . . . . . . . . . . . . . . . . . . . . . 45 4.3. In tegral Quan tification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.4. Qualitativ e GIXRF-NEXAFS . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.5. Depth Profiling with GIXRF . . . . . . . . . . . . . . . . . . . . . . . . . 57 4 . 5 . 1 . 1 - L a y e r M o d e l ............................. 6 1 4.5.2. N - L a y e r M o d e l ............................. 6 3 4.5.3. N -La y er Mo del with Con tamination . . . . . . . . . . . . . . . . . 68 5. Development of Lab o rato ry Scanning-F ree Soft X-ra y GEXRF 77 5.1. Grazing Incidence vs. Grazing Emission XRF Sp ectrometer Concept . . . 78 5 . 2 . P r i n c i p l e S e t u p ................................. 8 0 5.2.1. Laser-Pro duced Plasma Source . . . . . . . . . . . . . . . . . . . . 81 5.2.2. Multila y er Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2.3. Sp ectroscop y Cham b er . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2.4. Charged Coupled Device . . . . . . . . . . . . . . . . . . . . . . . . 86 5.2.5. Alignmen t Pro cedures . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.3. Calibration of the Angular Scale . . . . . . . . . . . . . . . . . . . . . . . 94 5.3.1. Calibration with GEXRF profile . . . . . . . . . . . . . . . . . . . 95 5.3.2. Absolute Angular Calibration . . . . . . . . . . . . . . . . . . . . . 96 5.4. Single Photon Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.4.1. Split Even t Recom bination . . . . . . . . . . . . . . . . . . . . . . 102 5.4.2. Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.5. Compilation of GEXRF Profiles . . . . . . . . . . . . . . . . . . . . . . . . 116 5.5.1. Region of In terest Metho d . . . . . . . . . . . . . . . . . . . . . . . 116 5.5.2. Sp ectra Decon volution Metho d . . . . . . . . . . . . . . . . . . . . 116 6. F easibilit y Study of Lab o rato ry Scanning-F ree Soft X-Ra y GEXRF 119 6 . 1 . M u l t i l a y e r S a m p l e ............................... 1 1 9 6.1.1. Structure Analysis with Complemen tary Metho ds . . . . . . . . . 120 6.1.2. GEXRF Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.2. First Beamtime - Pro of of Principle . . . . . . . . . . . . . . . . . . . . . . 125 6.2.1. Setup Description and Alignmen t . . . . . . . . . . . . . . . . . . . 125 6.2.2. Data Recording and Image Pro cessing . . . . . . . . . . . . . . . . 129 6.2.3. In tegral XRF Sp ectrum . . . . . . . . . . . . . . . . . . . . . . . . 130 6.2.4. Angular Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6 . 2 . 5 . G E X R F P r o fi l e s ............................ 1 3 6 6.3. Second Beamtime - Setup Dev elopment and Characterization . . . . . . . 143 6.3.1. Setup Developmen t . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 I I 6.3.2. Measuremen t Stability and Repro ducibilit y . . . . . . . . . . . . . 144 6.3.3. GEXRF Profiles of the C/Ni-m ultila y er . . . . . . . . . . . . . . . 151 6.3.4. GEXRF Profiles of Thermo electric Nanofilms . . . . . . . . . . . . 159 7. Summa ry 167 7.1. Metho dological Dev elopmen t of the Lab oratory Scanning-F ree GEXRF S e t u p ...................................... 1 6 8 7.2. Thermo electric Nanofilms . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 7 . 3 . F u t u r e P e r s p e c t i v e ............................... 1 7 4 App endix 175 A. Angular Limits of the Sherman Equation . . . . . . . . . . . . . . . . . . 176 B. A tomic F orce Micrographs of the Silicon Substrate for the Thermo electric N a n o fi l m s .................................... 1 7 7 C. Estimation of Uncertainties for Fluorescence Calculations . . . . . . . . . 177 D. Effectiv e Solid Angle of Detection for an Off-Axis Sample Surface . . . . . 178 E. Estimation of Efficiency of GI- and GEXRF Sp ectrometer Concepts . . . 179 F. Effect of Tilted CCD on Distance Calculation . . . . . . . . . . . . . . . . 181 G. Study on the Influence of Noise Thresholds on CCD Sp ectra using the C l u s t e r i n g M e t h o d ............................... 1 8 2 H. Sim ulated In tensit y Distribution of SPEs on a CCD Mesh . . . . . . . . . 184 I . C h o o s e R O I s .................................. 1 8 5 J. Sample Alignment with a CCD . . . . . . . . . . . . . . . . . . . . . . . . 186 List of Figures 191 List of T ables 193 Nomenclature 195 Bibliography 210 Danksagung 213 III Abstract The efficiency of devices for no v el renew able energy sources, lik e solar cells and thermo- electric generators (TEGs), is strongly increasing since the adv ent of nanotec hnology . Sync hrotron radiation-based adv anced X-ra y fluorescence (XRF) tec hniques, lik e grazing emission (GE-) or grazing incidence (GI-) XRF, offer non-destructiv e and quan titativ e access to elemen tal depth profiles in the nanometer range. Using soft instead of hard X-ra ys not only increases the sensitivit y for ligh t elemen ts, but also enhances depth- resolving capabilities due to the stronger atten uation of the radiation. Clearly , material researc h could b e accelerated b y a b etter a v ailabilit y of those metho ds, whic h demands the dev elopmen t of an efficien t lab oratory setup. In this thesis, the first X-ra y fluorescence sp ectroscop y measuremen ts b y excitation with a laser-pro duced plasma (LPP) source are presen ted. Moreo v er, the highly bril- lian t source allo w ed to design a sp ectrometer concept, whic h enables lab oratory-based, scanning-free GEXRF measuremen ts in the soft X-ra y range for elemen tal depth profiling of nano-scaled materials. The scanning-free GEXRF approac h requires a precise calibration of the measuremen t geometry and the accurate ev aluation of single photon ev en ts to exploit the prop erties of t w o-dimensional, energy-disp ersiv e detectors. Both is ac hiev ed in this thesis, leading to excellen t angular and go o d energy resolution of the GEXRF measuremen ts. The sp ectrometer concept is v alidated b y ev aluation of the K ossel lines in the GEXRF profile of a C/Ni-m ultila y er sample. F urthermore, the setup is applied to the c haracterization of gold-dop ed copp er o xide nanofilms, whic h can b e used in TEGs. The sample structure is thoroughly c haracterized b y v arious XRF measuremen ts in the lab oratories of the Ph ysikalisc h T echnisc he Bunde- sanstalt at the sync hrotron radiation facilit y BESSY I I in Berlin. Complemen tary mea- suremen ts with the lab oratory scanning-free GEXRF setup sho w similar results obtained in comparable measuremen t times with resp ect to SR-based GIXRF. This demonstrates the feasibilit y of the sp ectrometer concept for elemen tal depth profiling of samples with relev ance for renew able energy sources. 1 Zusammenfassung Seit der An w endung v on Nanotec hnologien in Ger¨ aten zur Nutzung erneuerbarer Ener- giequellen, wie Solarzellen o der thermo elektrisc he Generatoren (TEG), k onn te deren Ef- fizienz stark gesteigert w erden. Die Nutzung ho c hen t wic k elter, auf Sync hrotronstrahlung (SR) basierender R¨ on tgenfluoreszensanalyse (RF A), wie GIXRF o der GEXRF (engl.: grazing incidence bzw. grazing emission X-ra y fluorescence analysis) erm¨ oglic ht es, Ele- men ttiefenprofile mit A ufl¨ osungen im Bereic h v on Nanometern zerst¨ orungsfrei und quan- titativ zu b estimmen. Wird w eic he statt harter R¨ ontgenstrahlung gen utzt, erh¨ oh t dies die Sensitivit¨ at f ¨ ur leic h te Elemen te und v erb essert die Tiefenaufl¨ osung der Messmetho de durc h die st¨ ark ere Absorption der Strahlung. Die En t wic klung effizien ter Lab orger¨ ate f ¨ ur diese Analysemetho den ist V oraussetzung f ¨ ur eine v erbreiterte Nutzung derselb en, w o durc h die En t wic klungsprozesse neuer Materialien b esc hleunigt w ¨ urden. In der v orliegenden Arb eit w erden die ersten RF A-Messungen un ter Anregung mit einer laserinduzierten Plasmaquelle (LPQ) v orgestellt. Zudem wird diese ho c h brillian te Quelle f ¨ ur w eic he R¨ on tgenstrahlung genutzt, um ein laborbasiertes, scanfreies GEXRF- Sp ektrometer zu en t w erfen, w elc hes die Elemen ttiefenprofilierung von nanoskaligen Ma- terialien erm¨ oglic h t. Damit die Eigensc haften des v erw endeten zw eidimensionalen, energiedisp ersiv en De- tektors im scanfreien GEXRF-A ufbau v oll ausgen utzt w erden k¨ onnen, m uss die Mess- geometrie exakt b ekann t und m ¨ ussen Einzelphotonenereignisse pr¨ azise ausw ertbar sein. Beides wird in dieser Arb eit erreic h t und f ¨ uhrt zu einer sehr guten Wink el- und einer guten Energieaufl¨ osung. Das Sp ektrometerk onzept wird durch die Messung v on K ossel- Linien in den GEXRF-Profilen einer C/Ni-Vielsc hic h tprob e v alidiert. W eiterhin wird das Sp ektrometer f ¨ ur die Untersuc h ung v on golddotierten Kupfero xid- nanosc hic h ten gen utzt, w elc he in TEGs Anw endung finden k¨ onnen. Die Prob en w erden mit v ersc hiedenen SR-basierten RF A-Metho den in den Lab oren der Ph ysikalisc h T ec h- nisc hen Bundesanstalt b ei BESSY I I in Berlin c harakterisiert. Die k omplemen t¨ aren Mes- sungen mit dem scanfreien GEXRF-Sp ektrometer ergeb en b ei v ergleic h baren Messzeiten ¨ ahnlic he Ergebnisse zu den SR-basierten GIXRF Messungen. Dies demonstriert erfolg- reic h die An w endbark eit des Sp ektrometers f ¨ ur Elemen ttiefenprofilierung von Proben, w elc he relev an t f ¨ ur die Gewinn ung v on erneuerbaren Energien sein k¨ onnen. 2 1. Intro duction The United Nations In tergo v ernmen tal P anel on Climate Change (IPCC), established in 1988, is gathering and summarizing the scien tific findings concerning climate c hange and its impacts. While clear evidence of an throp ogenic influence on global climate could not b e pro vided in the first Assessmen t Rep orts [1, 2, 3], climate mo deling and the necessary scien tific data impro v ed o v er the decades. In their F ourth Assessmen t Rep ort, published in 2007, the IPCC states that global w arming is unequiv o cal and most of the observ ed global a v erage temp erature increase is caused b y increased an throp ogenic greenhouse gas concen trations [4]. Esp ecially CO 2 from fossil fuel use pla ys a ma jor role in the global an throp ogenic greenhouse gas emissions, whic h is wh y renew able energy sources ha v e b een found to b e an imp ortan t part in greenhouse gas mitigation [5, 6]. Nano-scaled materials, like nanoparticles, nano wires, nano-structured films or nano- films, sho w significan t differences in their ph ysical and c hemical prop erties with resp ect to bulk materials on the one hand or single atoms and molecules on the other hand. F urthermore, by precisely con trolling the shape, structure or comp osition of these ma- terials, the prop erties can often b e adjusted to desired applications. Nanotec hnology led to recen t adv ancemen ts with resp ect to efficiency , cost and stabilit y of renew able energy sources lik e photo v oltaics [7] and thermo electric generators [8], th us b eing v aluable for greenhouse gas mitigation strategies. Understanding of nanomaterial prop erties m ust b e accompanied b y precise analytics, pro viding structural, comp ositional and c hemical information on the nanometer scale. F or the in v estigation of pro cessing accuracy , stabilit y or diffusion pro cesses of buried in terfaces or dopan t profiles, in v asiv e metho ds, lik e secondary ion mass-sp ectrometry (SIMS), transmission electron microscop y (TEM) or glo w-disc harge optical emission sp ectrometry (GD-OES), are frequen tly used [9, 10, 11]. One drawbac k is clearly the sample consumption. This prohibits rep eated measuremen ts on the same sample or di- rect comparison with complemen tary metho ds, whic h is “necessary for unam biguous and quan titativ e elemen tal distribution analysis of a thin film with unkno wn comp ositional in-depth distribution” [12]. X-ra y fluorescence analysis (XRF) pro vides non-destructiv e and quan titativ e access to elemen tal comp ositions. Ho w ev er, analysis of nanomaterials is c hallenging due to 3 1. INTR ODUCTION t ypical atten uation lengths in the micrometer range and high demands on efficien t X-ra y optics for b eam shaping. One p ossibility to o v ercome these limitations is the application of shallo w excitation or detection sc hemes in adv anced XRF analytical metho ds, i.e. grazing incidence (GI-) and grazing emission (GE-) XRF. Both metho ds can b e used to analyze in-depth elemen tal profiles in the nanometer range, as w as demonstrated for example b y the analysis of CIGSe absorb er la y ers of thin film solar cells [13], transparen t conductiv e m ultila y er systems [14] or b y metal thin film studies [15, 16]. T o efficien tly in v estigate ligh t elements and increase depth-sensitivit y , soft X-ra ys are preferable with resp ect to hard X-ra ys, b ecause of larger photoionization cross sections and decreased atten uation lengths. In this energy range, the in v estigation of dopan t pro- files in ultra-shallo w junctions [17, 18], sub-nanometer la y ers in transistor gate stac ks [19] or nanoparticle analysis [20] w as rendered p ossible b y using 3 rd generation sync hrotron radiation. Y et, the need to apply for b eam times dela ys the feedback time of analytical results in the dev elopmen t pro cess of no v el materials. Th us, readily a v ailable laboratory analysis impro v es researc h and dev elopmen t cycles and can reduce timescales for new tec hnological in v en tions, whic h migh t e.g. help for future greenhouse gas mitigation. As a second asp ect, the wider the a v ailabilit y of a metho dology , the faster is the ad- v ancemen t of the metho dology itself. In GIXRF and GEXRF, ev aluation strategies for quan titativ e analysis of depth profiles are still maturing and no b est pro cedure for an arbitrary sample system exists ∗ . Esp ecially , pro viding reliable uncertain ties in depth profiles is difficult, since the direct inv ersion of the measured angular profiles to depth profiles is an ill-p osed problem and mo deling strategies are applied. A b etter under- standing of quan titativ e approac hes with GIXRF and GEXRF will further impro v e the impact of the metho d. In this thesis, a lab oratory scanning-free GEXRF setup w orking in the soft X-ra y range is dev elop ed, c haracterized and applied for depth profiling applications. T o achiev e the transfer of a sync hrotron metho d to the lab oratory , the lo w er brilliance of a v ailable lab o- ratory sources requires highly efficien t concepts in the excitation and detection c hannel. The dev elop ed sp ectrometer utilizes the laser-pro duced plasma (LPP) source at the Berlin Lab oratory of Inno v ativ e X-ra y T ec hnologies (BLiX) in combination with focus- ing reflectiv e m ultila y er mirrors, whic h allows efficien t excitation in GEXRF geometry . Instead of the t ypically applied w a v elength-disp ersiv e detectors in (lab oratory) GEXRF setups [21, 22, 23], the concept of scanning-free GEXRF [24] with a t wo-dimensional, energy-disp ersiv e detector is applied to increase the o v erall solid angle of detection and ∗ Probably , a general metho d for arbitrary samples w ould scarcely b e used. The usually applied fitting routines are m uc h more robust with b oundaries applied to the parameters and b y limiting the num b er of parameters. Th us, prior kno wledge, which strongly depends on each individual sample, is frequen tly applied. 4 dynamic range of the energy scale. F urther gain in efficiency is ac hiev ed b y small sample-to-detector distances, whic h on the other hand demand an accurate angular calibration, to accoun t for geometric effects of the distribution of fluorescence emission angles on the flat detector plane. This is ac hiev ed b y accurately describing and measuring the whole detection geometry and pre- cise setup and sample alignmen t. The dev elop ed metho dologies allo w the determination of an absolute angular calibration without the need of reference samples and sho w ex- cellen t repro ducibilit y and high accuracy . The pro cedures are tested and v alidated with GEXRF measuremen ts on a w ell-defined C/Ni-m ultila y er sample. Moreo v er, a test of the applicabilit y of the setup and metho dology is p erformed on thermo electric samples as scien tific case with relev ance for renew able energy sources. In the researc h group of Prof. K. Rademann at the departmen t of c hemistry of the Hum b oldt Univ ersit y of Berlin, gold-dop ed copp er o xide nanofilms are tested for their applicabilit y as thermo electric generators. These regenerativ e energy devices con v ert temp erature gradien ts to electrical p o w er without the need of mo v able parts. They can b e applied e.g. as energy source in w earable electronics [25] or for recup eration of thermal w aste energy [26, 27]. In a first step, the thermo electric samples are thoroughly c haracterized b y sync hrotron radiation-based near edge X-ra y absorption fine structure (NEXAFS) and quan titativ e reference-free GIXRF measuremen ts in the lab oratories of the Ph ysikalisc h T ec hnisc he Bundesanstalt (PTB) at the sync hrotron radiation facilit y BESSY I I. Then, GEXRF analysis of the sample system with the dev elop ed lab oratory setup is p erformed. The results allo w direct comparison b et w een the sync hrotron radiation-based GIXRF and the lab oratory GEXRF metho dologies and sho w the feasibilit y of elemen tal depth profiling of nano-scaled materials with the lab oratory setup. Th us, the dev elop ed sp ectrometer can help in researc h for, amongst others, new materials with relev ance for greenhouse gas mitigation. 5 2. Metho dological Bases 2.1. X-ra y Fluo rescence Analysis When an X-ra y photon in teracts with an atom, it can b e scattered or absorb ed and the probabilities of eac h pro cess dep end strongly on the atom n um b er Z and the energy of the inciden t X-ra y photon. While the cross sections for Ra yleigh scattering, where the X-ra y photon energy is preserv ed (elastic scattering), are larger for soft X-ra ys, Compton scattering cross sections (inelastic scattering) increase with higher photon energy and are more prominen t in the hard X-ra y regime. How ev er, the in teraction of X-ra ys with energies up to 100 k e V with matter is dominated b y photoionization. In this case, the photon energy E pr is completely transferred to the atom and an inner shell electron with a kinetic energy E kin = E pr − E B , where E B is the binding energy of the electron in the atom, is ejected from the system. The excited atom goes now through a n um b er of relaxation pro cesses, where the initial core hole is filled with an electron from an upp er shell and th us gradually mo ving to w ards the outer shells. During this cascade, radiativ e and non-radiativ e relaxation pro cesses comp ete against eac h other. Radiativ e transitions are most efficien t for dip ole transitions. F or these, the selection rules imply that the quan tum n um b ers of the atomic system m ust c hange according to ∆ l = ± 1 and ∆ j = 0 , ± 1, with the orbital quan tum n um b er l and the total angular momen tum j . This simplifies the observ ed X-ra y sp ectrum and allo ws usually a direct assignmen t of elemen t and transition sp ecific fluorescence lines, making X-ra y fluores- cence (XRF) analysis sensitiv e to elemen tal comp ositions. Figure 2.1 exemplarily sho ws the p ossible dip ole transitions and line assignmen ts in the Siegbahn notation for copp er. It has to b e noted that also satellite lines with m uc h lo w er in tensities, which originate from quadrup ole transitions, can app ear in an X-ra y sp ectrum. In a non-radiativ e relaxation, the released energy can b e transferred to an electron, whic h again lea v es the atomic system. This is known as A uger effect and the electron is called A uger electron. In the sp ecial case that the core hole of an L-shell (or energetically higher shell) is filled with an electron from the same shell (but differen t subshell), the pro cess is referred to as Coster-Kronig transition [28]. 7 2. METHODOLOGICAL BASES Figure 2.1.: Electron configuration and X-ra y transitions for Cu. Only the allo w ed dip ole transitions, i.e. which satisfy ∆ l = ± 1 and ∆ j = 0 , ± 1, are sho wn. 2.1.1. Calculation of Fluo rescence Intensities Considering the case of a homogeneous la y er, which is irradiated with a monochroma- tized, collimated X-ra y b eam, a fluorescence in tensit y of that la y er, recorded b y a suitable detector, can b e calculated b y the Sherman equation [29]. Here, the form as it is deriv ed in [30] for primary , mono c hromatic excitation is giv en. N i,j = N pr ( E pr ) G ( E i,j ) ξ i,j,s ( E pr ) ρ C i Z d 0 exp ( − µ ∗ tot ( E pr , E i,j ) ρx ) d x , with G ( E i,j ) = det ( E i,j ) Ω 4 π sin( ψ pr ) ξ i,j,s ( E pr ) = τ i,s ( E pr ) ω i,s p i,j µ ∗ tot = µ tot ( E pr ) sin( ψ pr ) + µ tot ( E i,j ) sin( ψ fl ) (2.1.1) In Equation 2.1.1, N i,j is the count rate of detected fluorescence photons from transi- tion j of elemen t i . N pr ( E pr ) refers to the count rate of primary photons of energy E pr irradiating the sample and G ( E i,j ) accoun ts for the prop ortionalit y to most geometric parameters, com bining the detector efficiency det ( E i,j ), the normalized solid angle of detection Ω/4 π and the inciden t angle of the primary radiation with resp ect to the sam- ple surface ψ pr . The photo pro duction cross section ξ i,j,s ( E pr ) describ es the fraction of pro duced fluorescence photons on the atomic scale. It is the pro duct of the photoioniza- tion cross section τ i,s ( E pr ) of subshell s with the fluorescence yield ω i,s and the transition probabilit y p i,j . The in tegration v ariable x is the p osition along the la y er surface nor- 8 2.1. X-RA Y FLUORESCENCE ANAL YSIS mal and is v aried in the in tegral from surface ( x = 0) to d , where d is the thic kness of the la y er with densit y ρ and total mass atten uation co efficien t of the la y er µ tot . The latter sums the cross sections of the p ossible photon-matter in teraction pro cesses, i.e. µ tot = τ + σ coh + σ inc , with τ the photoionization cross section and σ coh and σ inc the cross sections for coheren t and incoheren t scattering. The in tegrand µ ∗ tot ( E pr , E i,j ) ρx describ es the atten uation due to Lam b ert-Beer’s la w of the inciden t primary photons up to a depth x and of the emitted fluorescence photons from depth x at an angle ψ fl with resp ect to the sample surface. The Sherman equation can b e extended for p olyc hromatic excitation b y in tegrating o v er E pr and for secondary fluorescence, i.e. X-ra y fluorescence whic h follo ws after absorption of an initial fluorescence photon. In this thesis, secondary fluorescence en- hancemen t is negligible, since the la y er thic knesses of the analyzed samples are w ell b elo w 100 nm [31]. Ho w ev er, details of the adapted form ulas can b e found e.g. in [31] or [30]. 2.1.2. Quantification App roaches Quan tification of the comp osition of a bulk sample can b e accomplished b y using a calibration curv e. In this case, X-ra y fluorescence intensities of standards made from reference materials with accurate kno wledge ab out the comp osition are measured. The dep endency of the fluorescence in tensities of an elemen t and its concen tration is the calibration curv e, whic h should yield a linear relation in first appro ximation. Then, the concen trations of analytes in an unkno wn sample with similar comp osition, surface prop- erties and microstructure can b e directly determined b y using the same measuremen t pro cedure as for the standards and in terp olation of the calibration curv e. Ho w ev er, whenev er (certified) reference materials are rare, whic h is e.g. the case for thin la y ers or no v el nanomaterials, quantification methods based on fundamen tal parameters (FPs) and the Sherman equation b ecome necessary . First, Equation 2.1.1 is considered again. The in tegral can actually b e solv ed, whic h yields N i,j = N pr ( E pr ) G ( E i,j ) ξ i,j,s ( E pr ) C i 1 − exp ( − µ ∗ tot ( E pr , E fl ) ρd ) µ ∗ tot ( E pr , E fl ) . (2.1.2) Th us, the num b er of detected fluorescence photons can b e correlated to the concen tration C i and the total mass dep osition ˆ m = ρd . In the case of a single elemen t la y er, the mass dep osition of that elemen t can b e directly computed with ˆ m = − 1 µ ∗ tot ( E pr , E fl ) × ln 1 − N i,j µ ∗ tot ( E pr , E fl ) N pr ( E pr ) G ( E i,j ) ξ i,j,s ( E pr ) ! . (2.1.3) 9 2. METHODOLOGICAL BASES As can b e seen, if all geometric parameters of the setup and all FPs are known, the latter tak en usually from databases lik e [32] or [33], the mass dep osition of the single la y er can b e calculated without standards or references materials. Suc h a reference-free quan tification approac h is e.g. used in Chapter 4 with the fully calibrated instrumen tation of the Ph ysikalisc h T ec hnische Bundesanstalt (PTB). In com- mercial XRF devices, lik e the one used in Chapter 3, often a FP based, standardless quan tification is implemen ted. In this case, fundamen tal parameters from data bases are also used, but the instrumental parameters are determined b y a calibration of the instrumen t with calibration samples. These calibrations samples do not need to b e stan- dards and th us do not need to b e similar to the unkno wn samples, whic h are to b e analyzed. The reference-free and standardless quan tifications can also b e applied, when analyzing samples with more than one elemen t in a single la y er (e.g. allo ys). Then, Equation 2.1.2 b ecomes a set of non-linear equations. These equations can b e solv ed with v arious influence co efficien ts metho ds [30], where deviations from the linear correlation b et w een measured coun t rate and analyte concen tration due to e.g. matrix effects or secondary excitation can b e corrected b y influence co efficien ts. These co efficien ts are determined theoretically from fundamen tal parameter equations or empirically from standards. In the case of sev eral la y ers, whic h include the same elemen ts, the ab ov e-men tioned approac hes are no longer applicable. In these cases, quan tification is carried out b y non-linear least square fitting using a mo del sample, where the mass dep ositions of eac h la y er are v aried to matc h the calculated to the measured fluorescence in tensities. Again, this approac h can b e used in a reference-free or standardless metho d and is similar to the quan tification of angular resolv ed XRF profiles describ ed in Section 2.4.2. 2.2. Grazing Incidence and Grazing Emission XRF In Equation 2.1.1, a direct influence of the inciden t and detection angles ψ pr and ψ fl on the detected coun t rate N i,j is evident. F or ev er smaller ψ pr or ψ fl , the exp onen t decreases, damp ening the in tegral. The physical meaning behind this fact is an increase of absorption either of the inciden t or the fluorescence radiation due to enlarged path lengths in the sample. Thus, the depth, from whic h fluorescence radiation is created and can reac h the detector is adjustable b y b oth angles and leads to depth resolving capabilities in angular resolv ed (AR) XRF measuremen ts. Exp erimen tally , the angle of incidence or the detection angle is v aried and for eac h angle an X-ra y fluorescence sp ectrum is recorded. A plot of the recorded fluorescence line in tensities against the inciden t and detection angle, resp ectiv ely , yields the angu- 10 2.2. GRAZING INCIDENCE AND GRAZING EMISSION XRF Figure 2.2.: Calculated Cu-L α 1 angular resolv ed XRF profiles of thin Cu lay ers (1 nm to 500 nm) excited with 1060 e V photons using Equation 2.1.2. a) sho ws the profiles for shallo w inciden t angles and a detection angle of 90 ◦ . In b) the inciden t angle is fixed at 90 ◦ and the detection angle is tuned. lar resolv ed XRF profiles. Figure 2.2 sho ws computed Cu-L α 1 ARXRF profiles using Equation 2.1.2 for a Cu la y er of v arious thic knesses irradiated with a single 1060 e V photon / s. The curv es are calculated for a fixed solid angle of detection of Ω = 1 sr and setting the detection angle in the shallo w excitation case in a) and the inciden t angle in the shallo w detection case in b) to 90 ◦ . The fundamen tal parameters are tak en from [32]. As can b e seen, in b oth cases the ARXRF profiles of the differen t samples sho w not only a v ariation of absolute in tensit y but indeed differen t shap es of the curv es. In the shallo w excitation case in a), the fluorescence in tensit y increases to w ards smaller inciden t angles b ecause of ev er more efficien t excitation in the thin la y ers. F or ψ pr → 0 ◦ , all curv es con v erge to N i,j = 1 4 π τ i,s ( E pr ) µ tot ( E pr ) ω i,s p i,j , (2.2.1) as is sho wn in App endix A. Since the atten uation co efficien t µ tot ( E pr ) is the sum of the cross sections for photoionization, coherent and incoheren t scattering, the fraction in Equation 2.2.1 is the probabilit y that the primary photon is absorb ed in the sub- shell corresp onding to the detected X-ra y fluorescence line rather than b eing absorb ed in another shell or scattered. Th us, the somewhat theoretical in terpretation of Equa- tion 2.2.1 is that in the ideal, infinite sample plane all incident photons in teract with the first atomic la y er of the sample and some fixed fraction dep ending on the atomic pro cesses alone leads to the fluorescence radiation detected in 1 sr of the whole solid an- gle of 4 π . Since only the first atomic la y er is in teracting with the primary photons, the 11 2. METHODOLOGICAL BASES fluorescence in tensit y for ψ → 0 is indep endent of the sample thic kness. The decrease of fluorescence in tensit y for steep er angles in Figure 2.2 a) strongly dep ends on the sample thic kness and is explained b y the reduced in teraction probabilit y for thin la y ers. In the real w orld, t w o factors are actually decreasing the fluorescence in tensit y to zero for ψ → 0. First, the sample plane and the excitation fo otprin t are finite and second, external reflection migh t dominate for shallo w ψ pr . Both effects lead to losses in the excitation c hannel and are discussed in Section 2.2.2. The in terpretation of the ARXRF profiles in Figure 2.2 b) is more straigh tforw ard. The excitation conditions are the same for all detection angles ψ fl but the n um b er of fluorescing atoms increases with the la y er thic kness, as long as the thic kness is in the range of the p enetration depth of the primary photons. F or ev er shallo w er ψ fl , the de- tected fluorescence radiation decreases with sin( ψ fl ) due to self-absorption in the sample, leading to zero in tensit y at ψ = 0 (also see App endix A). Despite the differences in the ARXRF profiles in Figure 2.2 a) and b), the ph ysical principles are the same and b oth t yp es of angular scans can b e used to extract infor- mation ab out the in-depth distribution of the elemen tal comp osition (elemen tal depth profiles) of the sample. Indeed, direct in v ersion of ARXRF profiles to get the elemen tal depth profile of the sample is in some sp ecial cases p ossible, but is still a sev erely ill- p osed problem [34]. That means that more than one solution migh t exist and esp ecially with the influence of exp erimen tal errors a direct in v ersion is highly unstable. Therefore, usually the elemen tal depth profiles of the sample are mo deled and the exp ected angular dep enden t fluorescence in tensities, considering the setup sp ecifications, are calculated (forw ard calculation). Then, fitting the calculated and measured ARXRF profiles is p erformed b y adjusting the sample parameters (bac k calculation). Th us, quan titativ e information ab out the sample is obtained only indirectly and the results, esp ecially with resp ect to uncertain ties, need to b e considered carefully , as will b e discussed further in Section 2.4.2. F or ev er smaller inciden t or fluorescence emission angles, b oundary effects such as reflection and refraction at in terfaces can o ccur. These effects m ust b e tak en in to accoun t to correctly predict the fluorescence radiation in tensit y of a sample. On the other hand, the b oundary effects, the o ccurrence of in terference effects and th us the dep endence on geometric length scales, mak es the metho d sensitiv e to la yer roughness, la y er thic knesses and densities. F urthermore, lo cal electrical field enhancemen ts due to an X-ray standing w a v e (XSW) field can further increase the elemen tal sensitivit y , as will b e considered in the follo wing. 12 2.2. GRAZING INCIDENCE AND GRAZING EMISSION XRF 2.2.1. The Complex Refractive Index T o describ e X-ra y scattering b y a m ulti-electron atom, the complex atomic scattering factor f is used. It relates the scattered electric field amplitude to that scattered b y a single, free electron. In the limit of short w av elengths (relativ e to the Bohr radius) or forw ard scattering, the complex atomic scattering factor f 0 = f 0 1 − if 0 2 can b e directly link ed to the complex refractiv e index n = 1 − δ − iβ via [35] δ = n a r e λ 2 2 π f 0 1 β = n a r e λ 2 2 π f 0 2 . (2.2.2) Here, n a is the a v erage densit y of atoms, r e the classical electron radius and λ the w a v e- length of the radiation. As can b e seen when considering a plane wa v e propagating through matter, δ and f 0 1 resp ectiv ely describ e the phase shift relativ e to a w a v e propa- gating in v acuum and β and accordingly f 0 2 are resp onsible for the exp onen tial deca y of the in tensit y . Indeed, f 0 2 and th us β are directly link ed to the photo electric cross section τ through [35] f 0 2 = τ 2 r e λ and β = n a λ τ 4 π . (2.2.3) In the whole X-ra y regime, δ and β are small (usually p ositiv e) n um b ers of order 10 − 2 to 10 − 6 or less and β is usually smaller than δ . Therefore, reflection and refraction at in terfaces b et w een t w o materials is negligible for non-shallo w angles and not further considered in con v en tional XRF analysis, where usually a single non-shallow inciden t and detection angle is used. 2.2.2. Shallo w Excitation Considering a plane X-ra y w a v e propagating from a material with refractiv e index n 1 to a material with refractiv e index n 2 with n 1 > n 2 and neglecting absorption ( β = 0), a critical angle for total external reflection ψ c can b e defined from Snell’s la w as ψ c = arccos( n − 1 rel ) , with n rel = n 1 n 2 . (2.2.4) F or inciden t angles b elo w ψ c , the inciden t w a v e is completely reflected and only an ev anescen t w a v e propagates along the surface with exp onen tially decreasing amplitude normal to the surface. The latter is not strictly true for lossy media ( β > 0), where also a small w a v e v ector comp onen t normal to the in terface exists and absorption in the second 13 2. METHODOLOGICAL BASES Figure 2.3.: Reflectivities for a) a v acuum-to-CuO and b) a CuO-to-Si interface and dif- feren t photon energies. In a) total external reflection o ccurs and the critical angle ψ c is indicated. Data for δ and β are tak en from [39]. media o ccurs. Suc h in terface effects at shallo w excitation conditions need to b e tak en in to accoun t when calculating fluorescence in tensities and reflectivit y at the in terface is significan t. The requiremen ts to allo w sp ecular reflection are a sample surface with small w a viness and a ro ot mean squared roughness (RMS) b elo w a few nm (dep ending also on the w a v elength and the inciden t angle) [36, 37]. An accurate description of the reflectivit y at an ideal (sharp, no roughness) in terface for p erp endicular ( R s ) and parallel ( R p ) p olarized ligh t is given b y the F resnel equations [38] R s = n rel cos( ψ ) − q 1 − n 2 rel sin 2 ( ψ ) n rel cos( ψ ) + q 1 − n 2 rel sin 2 ( ψ ) 2 R p = cos( ψ ) − n rel q 1 − n 2 rel sin 2 ( ψ ) cos( ψ ) + n rel q 1 − n 2 rel sin 2 ( ψ ) 2 (2.2.5) Figure 2.3 sho ws the calculated reflectivities (the results apply for R s and R p ) for X-ra ys propagating from a) v acuum to CuO and b) CuO to Si for 1060 e V, 3 k e V and 5 k e V. Since the real part of the refractiv e index of v acuum is greater than that of CuO ( < ( n 1 ) > < ( n 2 )), the reflectivit y for inciden t angles ψ < ψ c (with ψ c from Equation 2.2.4) in Figure 2.3 a) is increasing strongly , reac hing v alues close to 1 for 3 k e V and 5 k e V 14 2.2. GRAZING INCIDENCE AND GRAZING EMISSION XRF Figure 2.4.: XSW field in tensit y in front of and in a 30 nm thic k CuO la y er on a Si substrate. The densit y of the CuO la yer is set to 6.31 g/cm 3 , Deb y e-W aller factor is zero and the inciden t X-ra ys ha ve an energy of 1060 e V and are p olarized parallel to the plane of incidence. The atomic scattering factors for the calculation are tak en from [40]. photons. F or incident angles ab o v e ψ c the reflectivit y quic kly drops to zero. The soft X-ra ys at 1060 e V ha v e a larger critical angle ( ψ c = 2.31 ◦ ) and th us significan t reflection at larger inciden t angles compared to hard X-ra ys. Ho w ev er, the high absorption in the second material leads to losses in the reflected in tensit y , reducing the reflectivit y well b elo w 1 ev en for inciden t angles b elo w ψ c . In the case of the second in terface in b) no total reflection o ccurs, since < ( n 1 ) < < ( n 2 ). Here, the reflectivities for soft X-ra ys are usually larger than for X-ra ys with higher energies, leading to considerable v alues at steep er angles. T o correctly calculate the fluorescence in tensities of a sample irradiated at shallo w inciden t conditions, the men tioned b oundary effects bust b e considered. L.G. P arratt dev elop ed an iterativ e algorithm to calculate the electric field amplitudes at eac h depth p osition for a sample of m ultiple homogeneous la y ers, using the F resnel equations at eac h in terface [41]. The resulting electric field amplitudes can render no des and an ti-no des due to in terference of inciden t and reflected radiation (sc hematically sho wn in Figure 2.5 a) top). Th us, the mo dulated field in tensity , also referred to as X-ra y standing w a v e (XSW) field, mo dulates the excitation in tensit y with depth. As an example, the XSW field of a 30 nm CuO la y er on top of a silicon substrate is sho wn in Figure 2.4. The plot sho ws the 15 2. METHODOLOGICAL BASES Figure 2.5.: Cu-L α 1 ARXRF profiles of thin Cu la y ers on a silicon substrate as in Figure 2.2. Ho w ev er, here the profiles are shown with (solid lines) and without (dashed lines ) the b oundary effects tak en in to accoun t. On the top, the principles of GIXRF in a) and GEXRF in b) are sc hematically sho wn. depth dep enden t ( x axis) XSW field enhancemen t compared to the inciden t field in tensity for v arious inciden t angles ( y axis). Depth zero is defined as b eing 20 nm ab o v e the CuO surface, i.e. in v acuum. The CuO la yer is at depth 20 nm to 50 nm and after a depth of 50 nm the first 10 nm of the silicon substrate are sho wn. The maxim um enhancemen t close to a factor of 4 can b e found for angles b elo w 0.7 ◦ and at distances ab o v e 15 nm from the surface. In this region, the reflectivit y of the inciden t b eam at the first in terface is ab o v e 70% (see Figure 2.3) but due to the phase shift after reflection, the XSW field maxim um is a few nm ab o v e the surface. In principle, the no des and an ti-no des of the XSW field can shift through the sample with v arying inciden t angle. Th us, they can b e regarded as nano-scaled sensors for X-ra y fluorescence excitation, whic h further enhance depth resolution and sensitivit y as compared to excitation without XSW field. Figure 2.5 a) sho ws the GIXRF profiles of pure Cu la y ers on a Si substrate taking reflection and refraction in to accoun t and compares them with the results of the Sherman equation from Figure 2.2. Clearly , the effect of reflection of the primary radiation is seen for shallo w inciden t angles, where the fluorescence in tensit y no w drops to zero. But also, enhancemen t effects as compared to the case without XSW field are eviden t esp ecially for the thin la y ers. Note that the high sensitivit y in total external reflection XRF (TXRF) is partly due to this enhancemen t effect. In a grazing incidence X-ra y fluorescence (GIXRF) exp erimen t, the inciden t angle of the excitation radiation ψ pr is scanned in pro ximity of the critical angle for total 16 2.2. GRAZING INCIDENCE AND GRAZING EMISSION XRF external reflection ψ c (e.g. 0 < ψ pr < 3 ψ c ). The inciden t b eam needs to b e collimated and mono c hromatized to guaran tee a sufficien t transv ersal and lateral coherence of the b eam and p ermit the formation of an XSW field. This is easily done at sync hrotron radiation facilities, where lo w div ergence and high photon n um b ers are a v ailable. F or div ergen t lab oratory sources, e.g. m ultila y er mirrors and appropriate elemen t filters can b e applied at the exp ense of excitation in tensit y . Assuming a suitable inciden t b eam, the pro jection of that b eam on the sample surface (fo otprin t) migh t extend the sample dimensions or detector field of view, leading to further efficiency losses. This effect is esp ecially strong in the hard X-ra y regime, where ψ c , and therefore the prob ed angular range, is shallo w er than in the soft X-ra y regime. On the other hand, the (usually) energy-disp ersiv e detector can face the sample surface, which allows for the closest p ossible detector distance in an XRF exp erimen t, maximizing the solid angle of detection. In addition, the steep detection angles minimize self-absorption of the fluorescence radiation. Th us, the prob ed depth region is defined b y the absorption of the inciden t b eam, whic h leads to most efficien t excitation conditions at ev ery prob ed ψ pr . Whether the enhancing or deteriorating effects concerning efficiency dominate dep ends on the actual setup geometry and the prob ed sample. Besides the setup requiremen ts, a further analytical c hallenge in GIXRF analysis is the precise kno wledge of the effectiv e solid angle of detection. As men tioned, for short sample-to-detector distances d dist , the fo otprin t on the sample easily extends the field of view of the detector. In this case, regions of the fo otprin t exist, where the detector c hip is partly co v ered b y the detector housing. This effect dep ends on the inciden t angle and distorts the GIXRF profiles. If the setup geometry is precisely known ( ψ pr , d dist , detector configuration), the effectiv e solid angle of detection can b e calculated to correct the GIXRF profiles [42, 43]. 2.2.3. Shallo w Detection In the case of shallo w detection angles, the fluorescence radiation of the analytes in the sample can b e reflected and refracted at the existing in terfaces. Therefore, presuming smo oth in terfaces, the fluorescence at grazing emission (GE) angles is also mo dified as compared to the theoretical description b y the Sherman equation. The direct comparison is sho wn in Figure 2.5 b), where similar to the shallo w excitation case at small ψ fl an additional decrease of the fluorescence in tensit y app ears. Here, the fluorescence radiation is reflected at the sample-to-v acuum in terface and cannot reac h the detector. Less ob vious is the app earance of fluorescence enhancemen t esp ecially for the thin la y ers at ψ fl ≈ 1.7 ◦ . Considering a thin la y er on a substrate, fluorescence radiation of that thin la y er can 17 2. METHODOLOGICAL BASES reac h the detector directly or via an ev en n um b er of reflections at the sample-to-v acuum and the sample-to-substrate in terface (Figure 2.5 b) top). Due to the differen t p ossible detection paths, in terference patterns and fluorescence in tensit y enhancement can occur in the GEXRF profiles. A direct calculation of GEXRF in tensities b y applying the F res- nel equations to the in terfaces of a sample with sev eral homogeneous la y ers is p erformed b y Urbac h and de Bokx [44]. The authors started b y calculating the far field in tensit y at the detector due to a single fluorescing atom and in tegrated the result o v er a distri- bution of these atoms in depth. Th us, the interfere nce pattern observed origins from the self-in terference of the probabilit y w a v e function of ev ery fluorescence photon. This statemen t is also demonstrated in Section 6.2.5, where only single photons are detected but the predicted in terference pattern is observ ed. It has to b e emphasized again that no coheren t excitation of the sample is necessary to obtain the enhancemen t from the in terference pattern, whic h is b eneficial for lab oratory sources. Already in 1983 R.S. Bec k er, J.A. Golo v c henk o and J.R. P atel compared and discussed the similarities in GIXRF and GEXRF measuremen ts [45]. Later, in 1995, P .K. de Bokx and H.P . Urbac h sho w ed the ph ysical similarities b et w een GIXRF and GEXRF by ap- plying the recipro cit y theorem of optics in the calculation of GEXRF profiles [21]. The theorem states that the electric field from a dip ole source at a detector is the same, if the p ositions of detector and source are exc hanged. Therefore, instead of calculating the fluorescence in tensit y of a single fluorescing source in a sample directly , also the electric field at the p osition of the atom can b e calculated assuming an X-ra y b eam inciden t on the sample. Only the inciden t angle has to b e equiv alen t to the former detection angle and the photon energy needs to b e the one of the detected fluorescence line. Th us, b y calculating the X-ra y standing w a v e field for an inciden t photon energy equal to the fluorescence energy and in tegrating o v er the source strength distribution in depth, also the GEXRF profiles can b e obtained. In comparison to GIXRF, the fundamen tal excitation conditions of GEXRF measure- men ts are less efficien t and self-absorption of the fluorescence radiation in the sample (due to shallo w detection angles) is higher. T ogether with the usually better solid angle of detection due to the small sample-to-detector distance, detection limits in GIXRF are usually b etter b y 2 orders of magnitude compared to GEXRF measuremen ts [36]. On the other hand, there are sev eral adv an tages of the GEXRF metho d. First, the angular profiles are dep ending on the X-ra y fluorescence energy and not the excitation energy , whic h leads to a larger critical angel (Equation 2.2.4) and broader angular in tensit y pat- terns, reducing the requirements on the angular resolution. F urthermore, the inciden t b eam do es neither ha v e to b e mono c hromatic nor parallel and ev en particle excitation can b e used [46]. Th us, also fo cusing optics or p olyc hromatic excitation migh t b e ap- plied, whic h increase the efficiency in the excitation c hannel, esp ecially for div ergen t 18 2.3. SCANNING-FREE GEXRF lab oratory X-ra y sources. F urthermore, due to the use of fo cusing optics, the lateral res- olution in GEXRF exp erimen ts is sup erior to GIXRF, where for shallo w angle excitation fo otprin ts with extensions of mm are reac hed. The dra wbac k in the excitation efficiency of GEXRF measuremen ts can b e comp en- sated to some extend b y also applying shallo w er inciden t angles, whic h b ecomes more imp ortan t with decreasing la y er thic kness and for surface analysis. Dep ending on the sample system, a compromise b et ween efficiency and lateral resolution can be found. Ho w ev er, the requiremen ts on the angular resolution also demand small solid angles of detection, whic h is the second b ottlenec k for an efficien t GEXRF setup. This latter disadv an tage can b e exploited b y applying w a v elength disp ersiv e detectors with their inheren t relativ ely small solid angle of detection to enhance the analytical capabilities of the setup [22, 47]. Alternatively , the o v erall solid angle of detection can b e increased for an GEXRF setup, if the v arious fluorescence emission angles are not measured suc- cessiv ely , but rather sim ultaneously with a scanning-free GEXRF approac h. 2.3. Scanning-F ree GEXRF Ka yser et al. sho w ed the applicabilit y of digital energy-disp ersiv e area detectors to scanning-free GEXRF analysis [24]. The detector, a PILA TUS 100 K, op erates in a sin- gle photon coun ting mo de to use the linear correlation b et w een the n um b er of created electron-hole pairs in the CCD c hip and the photon energy , enabling energy-disp ersiv e measuremen ts. The spatial resolution of the area detector can directly b e used to de- termine the emission angle of the detected fluorescence radiation, if the setup geometry is w ell-kno wn. Indeed, every pixel can be regarded as a small energy-disp ersiv e detector with small solid angle of detection allo wing for high angular resolution. The high effi- ciency due to the large solid angle of the detection is ac hiev ed b y the thousands or ev en millions of parallel w orking pixels. Besides the sup erior efficiency of the scanning-free GEXRF approac h, also the stabilit y of the setup can b e exp ected to b e impro v ed with resp ect to an angular scan, due to the lac k of mo ving parts during the measuremen t. 2.3.1. Single Photon Counting with a CCD - Energy Resolution The k ey elemen t of the scanning-free GEXRF metho d is a t w o-dimensional, energy- disp ersiv e detector. After the late 1980s sp ecial high resolution area detectors lik e the pnCCD [48, 49], the Medipix [50] and PILA TUS [51] detector ha v e b een dev elop ed, facil- itating (semi-) energy-disp ersiv e prop erties. Ho w ev er, the dev elopmen t of these devices w as motiv ated b y space missions and large-scale facilit y measuremen ts, which enabled outstanding p erformance but up to date high cost and op erational complexit y . Con ven- 19 2. METHODOLOGICAL BASES Figure 2.6.: Sc hematic view of a) fron t-illuminated and b) back-illuminated CCD. The absorb ed photon creates a c harge cloud, which drifts to w ards the p oten tial w ell b eneath the pixel structure. tional CCDs, whic h are no w ada ys commercially distributed b y a num b er of companies, are used for scien tific X-ra y analysis since the 1970s [52, 53]. Y et, exploitation of their energy-disp ersiv e prop erties started just in the last decade [54, 55, 56]. The principle for differen tiating photon energy is similar for all men tioned devices and is based on the prop er analysis of dep osited energy of ev ery detected photon. If an X-ra y photon hits the absorb er material of the detector c hip, it can create a high energetic photo electron b y photoionization. This photo electron in teracts with the surrounding atoms and leads to electron impact ionization, creating further electron- hole pairs. The generated mean n um b er of electron-hole pairs N eh in the c harge cloud dep ends on the c hip material and the energy of the detected fluorescence photon E ph b y N eh = E ph / W , where W is the electron-hole pair creation energy , e.g. W = 3.62 e V for pure silicon [30]. This basic principle of energy discrimination can also b e used in a con v en tional c harge-coupled device (CCD) for X-ra y applications, if op erated in a single photon coun ting mo de. A general explanation of the w orking principles of CCDs can b e found for example in [57]. The follo wing description will fo cus on single photon detection and ev aluation. A CCD consists of an absorb er material with a fully depleted detector v olume and a mesh of electro des creating p oten tial minima (pixels) in the substrate (Figure 2.6). The primary c harge cloud created in the absorb er has a size in the nm range [58] but due to diffusion pro cesses on its w a y to the p oten tial minima b elo w the CCD pixel structure, it expands to diameters in the µ m range. Dep ending on the path length of the c harge cloud to the p oten tial w ell, whic h is usually larger for bac k-illuminated CCDs, the charge cloud size can reac h v alues in the order of the pixel size of the CCD [59, 60, 61, 62], with a t w o-dimensional shap e w ell-appro ximated b y a Gaussian curv e [63]. This can lead to a splitting of the total c harge to sev eral neigh b oring pixels (split ev en ts), as illustrated in Figure 2.6 b). During image recording, c harges are accum ulated and stored in the 20 2.3. SCANNING-FREE GEXRF Figure 2.7.: Darkframe corrected image of soft X-ra y single photon ev ents detected with a commercial CCD. Computer algorithms are needed to ev aluate the h undreds to thousands of ev en ts in ev ery measurement frame. pixel structure. Then, the c harges are successiv ely shifted to the read-out amplifier and subsequen t analog-to-digital con v erter. By the exact kno wledge of the shifting pro cess, the spatial information for eac h measured c harge is preserv ed. The first c hallenge in the analysis of single photon ev en ts (SPEs) is to iden tify SPEs in the image. Figure 2.7 shows a section of a dark image corrected CCD frame (a frame with the same camera settings but without illumination is subtracted from the measuremen t frame) with a n um b er of SPEs visible as brigh t sp ots. Man y of the SPEs are split ev en ts consisting of up to 4 pixels, as is exemplarily shown in the enlarged frame. Also, the pixels without SPEs sho w in tensit y v ariations due to noise, th us, the signal-to-noise ratio in the frame already giv es a lo w er b oundary for the detectable SPE in tensit y , i.e. the accessible sp ectrum to w ards lo w photon energies. F urthermore, if a relev an t n um b er of split ev en ts o ccur, they need to b e iden tified and either rejected or the actual c harge cloud in tensit y from the in tensit y distribution in the split ev ents has to b e calculated. Sev eral approac hes can b e found in the literature to analyze single photon ev en ts, the algorithms also dep ending on the purp ose of the measuremen t and applied CCD detector. SPE analysis is p erformed in w a v elength disp ersiv e measuremen ts to increase the spatial resolution to the subpixel regime, b y using cen ter of gra vit y calculations [61, 60, 64]. In [64] the algorithm is describ ed b y finding pixels with in tensit y ab o v e an empirical threshold and using the N × N area ( N = 3 or 5) cen tered at the pixel to calculate the cen ter of gra vit y . The in tensit y of SPEs can also b e facilitated to preselect v alid ev en ts. They are filtered from the whole CCD image if their in tensit y is within a region of in terest (for the exp ected measured energy range), whic h can enhance the signal-to-noise ratio of the w a v elength disp ersiv e sp ectra [65, 62]. La wrence et al. [62] calculate for that purp ose sum in tensities of ev ery 2 × 2-pixel b o x of the images and 21 2. METHODOLOGICAL BASES compare them with ev ery o v erlapping b o x. SPEs are then defined for those b o xes with maxim um in tensit y in the comparison and subsequen tly the cen troidal p osition and the SPE in tensit y are calculated for eac h b o x. The algorithm of Szlac hetk o et al. [65] w orks only for mono c hromatic radiation. Here, the SPEs without even t splitting (1-p x ev en ts) are used to get an estimate of the exp ected summed in tensit y of the split ev en ts, whic h are considered for up to four pixels. SPE ev aluation can also b e applied in full-field XRF imaging b y using a pinhole camera with a CCD as energy-disp ersiv e detector [55, 56]. While Alfeld et al. [55] just rejected ev ery photon ev en t, where less than 95% of the in tensit y is found in an y of the single pixels of an SPE, the algorithm b y Romano et al. c hec ks a monotonic decrease of the neigh b oring pixels to iden tify an ev ent. Energy- disp ersiv e sp ectra of a CCD in a space mission are obtained in [66] b y classification of the in tensit y distribution in 3 × 3 pixel b o xes in to 8 grades. Dep ending on the grade, in tensities of the corner pixels in the b o x are used or rejected. Summarizing, a large v ariet y of SPE ev aluation algorithms exists, all of them dealing in one w a y or the other with the direct shap e of eac h SPE. Ho w ev er, often the descriptions, if giv en at all, lac k details on the algorithm or on the criteria of c ho osing apparen tly applied thresholds. Esp ecially in the soft X-ra y range, where the signal-to-noise ratio (of pixel in tensities) due to the smaller photon energy compared to hard X-ra ys is lo w, SPE treatmen t is crucial. Therefore, in Section 5.4.1 an o wn algorithm is dev elop ed, ev aluated and adjusted for the p erformed soft X-ra y fluorescence measuremen ts. Finally , some notes on noise in CCD images are giv en, since it not only defines the lo w er limit of the energy sp ectrum obtainable b y the CCD, but also effects the energy resolution, as it is directly linked to the signal-to-noise ratio (see also Section 5.4.2). First, there might be inhomogeneous systematic offsets in every recorded CCD frame. These could result e.g. from insufficien t c harge transfer efficiency (probabilit y that no electrons get lost when shifting c harges during readout), unequal pixel efficiencies or hot pixels (increased dark curren t). If the effects are known, they can b e compensated for or some pixels can b e ignored during image pro cessing. F urthermore, there are t w o significan t statistical noise con tributions, which can be influenced by the user via c amera settings. On the one hand, during the recording time electron-hole pairs can b e created b y thermal excitation and accum ulate in the p oten tial w ells. This dark curren t can b e corrected for b y subtracting a dark image, i.e. an image with the same recording parameters but without illumination. Ho wev er, the dark curren t itself has a statistical deviation, whic h is wh y the dark curren t and the dark curren t shot noise is usually reduced b y co oling the CCD c hip. The second significan t noise con tribution is the on- c hip amplifier noise or readout noise. It origins from the sampling of the data and increases with the readout frequency , for whic h often differen t settings can b e c hosen b y the user. Esp ecially for the GEXRF measuremen ts with the con v en tional CCD, a 22 2.3. SCANNING-FREE GEXRF Figure 2.8.: Sc hematic illustration of p ositions on a CCD c hip corresp onding to the same fluorescence emission angle in a scanning-free GEXRF setup. compromise has to b e made b et w een fast readout (reduced o v erall measuremen t time) and lo w noise lev el. 2.3.2. Geometry Considerations - Angula r Resolution Fluorescence radiation from a sample, whic h has the same emission angle ψ fl with resp ect to the sample surface, is emitted in a cone-lik e shap e, with an ap erture of 2 × (90 ◦ − ψ fl ). The equi-angle lines on a flat CCD detector are slice planes of the cones, having the shap es of h yp erb olas (see Figure 2.8). If the p osition of the CCD detector is kno wn with resp ect to the excitation sp ot on the sample (v ertex of the cone), for eac h p osition on the CCD c hip the resp ectiv e emission angle of the fluorescence radiation can b e calculated. During his PhD thesis in the researc h group Analytical X-ray Ph ysics at the T ec hnical Univ ersit y of Berlin, C. Herzog dev elop ed an algorithm to calculate ψ fl for an arbitrary sample p osition and p oin t in space, using simple v ector calculations in 3-dimensional space. F urthermore, he implemen ted the algorithm of Asv estas et al. [67], whic h can compute the solid angle of an y p olygon with resp ect to a p oin t in space. In the Bac helor’s Thesis b y F. F¨ orste [68], b oth co des w ere applied to the calculation of corresp onding fluorescence emission angles (angle maps) and solid angles of detection (solid angle maps) for ev ery pixel on a CCD c hip, giv en an y detector geometry with resp ect to the sample. Suc h calculations can b e used to in v estigate the influence of the h yp erb olic shap e of the equi-angle lines on the CCD c hip with resp ect to the angular resolution ∆ ψ fl . F or this purp ose, angle maps are calculated for a con v en tional CCD as it is used in this thesis, i.e. 515 × 2046 pixels with 13.5 × 13.5 µ m 2 pixel size. Th e geometry used in the calculations is sk etc hed in Figure 2.9 and shortly describ ed hereafter. The CCD c hip is aligned with the larger asp ect in the horizon tal plane and the c hip is p erp endicular to 23 2. METHODOLOGICAL BASES Figure 2.9.: Sc hematic illustration of geometric effects on angular resolution. the sample plane. Also, the lo w er CCD edge is parallel to the sample plane and has zero distance to it. The closest distance of the CCD to the excitation p osition on the sample, whic h is also the sample-to-detector distance d CCD , is situated at the cen ter of that edge. No w, the num ber of pixel rows nee ded to confine an equi-angle line (equi-angle smearing, see Figure 2.9 a)) is calculated for ev ery ψ fl . The results for v arious d CCD are sho wn in Figure 2.10 a). The y axis sho ws the v arious fluorescence emission angles of the CCD in a sample-to-detector distance on the x axis. The blac k corner to the top righ t arises from the limited extension of the CCD, so that these angles cannot b e detected at large distances. The colorbar indicates the effect of the equi-angle smearing in units of pixel ro ws. As can b e seen, for larger emission angles and smaller d CCD , the equi-angle smearing increases. Already for d CCD = 10 cm, the curv ature of the equi-angle lines can b e exp ected to influence the angular resolution, but ev en smaller sample-to-detector distances are en visaged to impro v e the o v erall solid angle of detection. Ho w ev er, if the geometry is kno wn, then the angle maps can b e calculated and pixels corresp onding to the same fluorescence emission angle (or rather an angular range) can b e com bined for the ev aluation. The calculations here are p erformed for a p erp endicularly aligned CCD, whic h in an exp erimen tal setup migh t not alw a ys b e p ossible (or desired). F or a tilted CCD, the correct angle map calculation is ev en more imp ortan t, since summing pixel in tensities o v er ro ws or columns then leads to sev ere distortions of the GEXRF profile. If the angle maps on the CCD detector can b e correctly calculated, the angular resolu- tion of the scanning-free GEXRF measuremen t is still limited b y t w o more effects. First, the pixel size itself influences the angular resolution (pixel broadening), since ev ery pixel detects a small fluorescence emission angle incremen t, as is sho wn in Figure 2.9 b). In a first order appro ximation, the angular resolution ∆ ψ fl dep ends on the pixel edge length d p x and the sample-to-detector distance d CCD as ∆ ψ fl ≈ d p x / d CCD . Th us, small pixel sizes are required, esp ecially , if the sample-to-detector distance is to b e minimized. Ho w- ev er, there are tec hnological limits for the pixel size, whic h migh t b e o v ercome by using the subpixel resolution prop erties when carefully analyzing split ev en ts as is men tioned 24 2.3. SCANNING-FREE GEXRF Figure 2.10.: a) Degradation of angular resolution due to equi-angle smearing for de- tectable fluorescence emission angles and differen t sample-to-detector distances. b) Degeneration of the angular resolution due to fo otprin t broadening. F or details please refer to the text. ab o v e. The second degenerative effect on the angular resolution origins from the finite fo otprin t size, i.e. the sample area from whic h fluorescence radiation is emitted. As is illustrated in Figure 2.9 c), radiation emitted from differen t sp ots in the sample plane with the same emission angle will hit the detector at differen t p ositions. The influence of this effect is estimated in Figure 2.10 b) for a CCD with the ab o v e-men tioned sp ecifi- cations and geometry but with d CCD = 5 cm to emphasize the effect. Each grid point in the image represen ts a p osition in the sample plane with the cen ter of excitation in the middle of the image. No w, for eac h p oint of the grid, the angle maps on the CCD are calculated as if all fluorescence radiation is emitted only from that p oin t and compared to the angle map of the cen ter p osition. Then, the pixelwise absolute difference of the t w o compared angle maps is calculated. The maxim um of the angle differences is the v alue sho wn in Figure 2.10 b) for eac h grid p oin t. It yields an upp er estimate for the degradation of the angular resolution due to fo otprin t broadening. As can b e seen, the angular resolution degrades stronger with an extended fo otprin t in the x axis, whic h is the direction to w ards the CCD detector. The red diamond like shap e represen ts the fo ot- prin t extension, whic h w ould just result in a resolution degeneration similar to the pixel broadening (0.015 ◦ ). Th us, in the presen t case, the fo otprin t in the sample plane should b e confined to 100 × 500 µ m 2 to prev en t a significan t decrease in the angular resolution. Of course, for ev ery detector geometry , this effect should b e considered separately , as is done for the differen t measuremen ts in this thesis in Chapter 6. 25 2. METHODOLOGICAL BASES 2.4. Soft w a re fo r (AR)XRF Evaluation 2.4.1. xrfLib ra ry and xrlfupa fo r X-ra y Fluo rescence Calculations In the researc h group Analytical X-ra y Ph ysics of the T echnical Univ ersit y of Berlin exist t w o soft w are pac kages to handle fluorescence calculations. The “xrlfupa” allo ws access to v arious published fundamen tal parameter databases lik e the compilation of Elam et al. [32] and Eb el et al. [33] for in tegral cross sections or Chan tler [40] for atomic scattering factors. If necessary , the databases can b e adjusted and extended to use new and more reliable v alues or suc h from measuremen ts instead of theoretical calculations and in terp olation. F urthermore, the xrlfupa is implemen ted in to the “xrfLibrary”, the second soft w are pac kage, which allo ws to calculate the detected X-ra y fluorescence of an arbitrary sample. The xrfLibrary can b e used to sim ulate a complete XRF exp erimen t, from e.g. calculation of a p olyc hromatic X-ra y tub e sp ectrum, whic h can b e altered b y transmission filters or optics and used for the fluorescence excitation in the sample. The sample is mo deled b y discrete la y ers, eac h with its o wn comp osition, thic kness and densit y . The Sherman equation is applied to all la yers, considering absorption of primary and fluorescence radiation, as w ell as cascade effects and Coster-Kronig transitions in the fluorescence pro duction itself. Optionally , secondary fluorescence can b e calculated using the algorithm of de Bo er [31]. If the n um b er of fluorescence photons from a sample is to b e calculated, whic h consists of sev eral homogeneous la y ers, Equation 2.1.1 has to b e applied to all the la y ers and for ev ery la y er the influence of the top la y ers (absorption of inciden t and fluorescen t radiation) m ust b e tak en in to accoun t. Similarly , a con tin uous c hange of the sample comp osition can b e appro ximated b y sub dividing the sample in to N homogeneous la y ers, resulting in a step-wise c hange of the comp osition. F or large N , this appro ximation can b e reasonable. Recen tly , L. L ¨ uhl and C. Herzog adapted the xrfLibrary to handle grazing incidence and grazing emission XRF calculations b y taking in to accoun t refraction and reflection at in terfaces as w ell as sp ecial detector geometries. F or this purp ose, the X-ra y standing w a v e (XSW) field can b e calculated for a sample defined in the xrfLibrary framew ork, follo wing the w ork of de Bo er [37], whic h is based on the equations of P arratt [41]. F or GIXRF calculations, Equation 2.1.1 needs to b e adapted. First, the absorption term in µ ∗ tot for the inciden t radiation is remo v ed, since the absorption of the incident radiation is already included in the XSW field. Second, the XSW field in tensit y I XSW is in tro duced 26 2.4. SOFTW ARE F OR (AR)XRF EV ALUA TION as further excitation factor in the in tegral o v er depth. Both adaptions lead to N i,j = N pr ( E pr ) G ( E i,j ) ξ i,j,s ( E pr ) ρ C i Z d 0 I XSW ( E pr , ψ pr , x ) exp − µ tot ( E i,j ) sin( ψ fl ) ρx d x . (2.4.1) In practice, the in tegral is solv ed n umerically b y sub dividing the la y ers of the samples in to virtual, thin subla y ers, for which the XSW field can be assumed to b e constan t (a t ypical thic kness is 0.1 nm). F or eac h subla y er, the absorption of the fluorescence in the sup erp osed subla y ers has to b e accoun ted for. F or GIXRF exp erimen ts, the solid angle of detection is usually strongly affected b y the tuned inciden t angle. In the xrfLibrara y , the solid angle of a p olygon shaded b y another parallel p olygon can b e calculated, applying the form ulas of Asv estas and Englund [67]. This is used to compute the effectiv e solid angle of detection of e.g. an SDD detector consisting of c hip and housing in fron t of the excited sample. F urther details and a comparison to the solid angle of detection calculation of similar systems b y Bec khoff et al. [42] can b e found in [43]. The XSW field can also b e applied for the calculation of GEXRF profiles as is rendered plausible b y P .K. de Bokx and H.P . Urbac h [21]. In this case, a virtual XSW field (there is no actual formation of an X-ra y standing w a v e field, see Section 2.2.3) is calculated for ev ery fluorescence energy and the pro duced fluorescence in tensit y of ev ery virtual la y er is mo dified b y the resp ectiv e virtual XSW field in tensit y in this la y er. In con trast to the GIXRF case, no w the absorption of the primary radiation is considered and the absorption of the fluorescence radiation, whic h is already included in the virtual XSW field, is omitted. Since the calculation of the XSW field is already implemen ted in the xrfLibrary , also the GEXRF profiles calculated with the xrfLibrary use the approac h of [21]. This also allo ws a straigh t-forw ard comparison of GI- and GEXRF measuremen ts with the same sample mo del, as is applied in Section 4.5 and 6.3.4 for gold-dop ed copp er o xide nanofilms. F or scanning-free GEXRF measuremen ts, the xrfLibrary can also b e used to calculate the solid angle of detection of ev ery pixel on the CCD c hip, again with the form ulas of Asv estas et al. [67]. Esp ecially for a tilted CCD, the solid angle of detection of eac h angular region used to compute the in tensities of the GEXRF profile can v ary drastically due to the differen t n um b ers of pixels included in these regions (e.g. the highest detected angles are confined to the corner of the CCD c hip). Both the xrlfupa and the xrfLibrary are programmed in C++ using an ob jectiv e orien tated approac h, which mak es the soft w are highly adaptable. Also, an in terface to 27 2. METHODOLOGICAL BASES the high-lev el programming language Python is a v ailable for most functions, allo wing easy data handling and access to the v arious w ell do cumen ted additional pac kages for example for fitting routines. This is notably of imp ortance for depth profiling approac hes. 2.4.2. Depth Profiling So far, only the forw ard calculation of fluorescence in tensities for a sample measured with tunable excitation or detection angles w as describ ed. Depth profiling from these angular resolv ed X-ra y fluorescence (ARXRF) profiles (bac k calculation) can b e p erformed b y mo deling the measured sample, comparing the calculated fluorescence in tensity of the mo del with the measured fluorescence in tensit y and adjusting the sample parameters b y a non-linear least square fit. The calculation of a GI- or GEXRF profile with the xrfLibrary can tak e sev eral seconds up to min utes on a mo dern desktop PC, dep ending on the n um b er (virtual) la y er of the sample and the n um b er of angles used for the profile. Th us, the whole bac k calculation pro cedure can tak e sev eral min utes up to hours, dep ending on starting parameters and con v ergence of the fit. The bac k calculations in this thesis use χ 2 minimizations based on the Lev en b erg-Marquard algorithm (if b oundaries for the parameters are applied) or a T rust Region Reflectiv e algorithm (if no b oundaries are used), whic h are b oth implemen ted in the p ython function curv e fit of the pac kage scip y .optimize. They will b e further describ ed later. A χ 2 minimization, if successful, giv es the maxim um lik eliho o d estimators of param- eters, whic h are applied to a mo del that reflects the ph ysical principle b ehind a mea- suremen t. Ho w ev er, this is only true if first of all, the mo del is a go o d appro ximation of the “real” ph ysical principles resp onsible for the measuremen t results and second, if the measuremen t errors are normal distributed. This has to b e k ept in mind, esp ecially when quan titativ e information is to b e deriv ed. In the case of ARXRF analysis, the mo del parameters a 0 , a 1 , ... a M − 1 are the M parameters of the sample (lik e density , roughness, comp osition and thic kness of the la y ers in the sample) and the w eigh ted χ 2 - fit minimizes the sum of squares of the w eigh ted differences of N measured ( y i ) and calculated ( y ( x i | a 0 ...a M − 1 )) fluorescence in tensities [69] minimize χ 2 ≡ N − 1 X i =0 y i − y ( x i | a 0 ...a M − 1 ) σ i 2 . (2.4.2) Here, σ i is the normal distributed error of measuremen t y i . There are differences in algorithms and quan titativ e statemen ts, if the mo del function y ( x i | a 0 ...a M − 1 ) dep ends in a linear or non-linear w a y on the mo del parameters. Often, an estimated co v ariance matrix C for the b est fit parameters is giv en b y χ 2 fitting 28 2.4. SOFTW ARE F OR (AR)XRF EV ALUA TION algorithms. In the case of a linear χ 2 -fit and normal distributed uncertain ties of the measuremen t v alues, the diagonal elemen ts of the co v ariance matrix are directly link ed to the parameter uncertain ties ∆ a j b y ∆ a j = q C j j (2.4.3) The mo dels used for ARXRF fitting are usually non-linear. Ev en in these cases, the error estimations b y the co v ariance matrix migh t hold, if the χ 2 -distribution close to the minim um can b e appro ximated b y a quadratic form. Then, the mo del function is appro ximately linear in the parameters [70, 69]. The appro ximation needs to b e v alid in the range of the uncertain ties of the parameters giv en b y Equation 2.4.3 themselv es, whic h is already one reason wh y the results should b e treated with care. F urthermore, the precondition of the measuremen t uncertain ties b eing normal distributed needs to b e k ept in mind. If some of the conditions are not fulfilled, the b o otstrapping metho d migh t b e applied. In this metho d, v arious new syn thetic data sets are dra wn from the actual measuremen t, fitted again with the same χ 2 -fit and the distribution of b est-fit parameters is used for an uncertain t y estimation [69]. Ob viously , if a single fit needs already sev eral min utes up to hours for con v ergence, the whole b o otstrapping pro cess can need da ys and is not applied in this thesis. F or the ev aluation of GIXRF and GEXRF profiles in this thesis, the uncertain ties of the fits as defined b y Equation 2.4.3 are giv en and their analytical v alue is discussed. Of- ten, the giv en uncertain ties will b e declared unreliable due to an insufficien t mo del on the one hand, or a to o complex mo del on the other hand. In the latter case, the in tro duced parameters migh t in terfere with eac h other and lead to unreliable error estimates. Also, for the GIXRF measuremen ts in Section 4.5, an additional large uncertain t y originating from the solid angle of detection is in tro duced. This uncertain ty is clearly not normal distributed, further diminishing the quan titativ e v alues of the result. Since the solid angle of detection has a stronger influence in GIXRF measuremen ts than in GEXRF measuremen ts, there migh t b e an adv an tage for the latter concerning the analytical v alidit y . There exists a v ariet y of algorithms to minimize χ 2 , whic h can b e roughly group ed in to lo cal mo dels, whic h will find a lo cal minim um in the proximit y of the starting v alues of the parameters and global metho ds, whic h striv e to find a global minim um (and th us a probably b etter solution of the parameters). The Lev en b erg-Marquardt algorithm is a lo cal minimization metho d. It decreases χ 2 with a steep est descen t metho d if the parameters are still far from the parameters minimizing χ 2 and switc hes to a metho d 29 2. METHODOLOGICAL BASES appro ximating χ 2 b y an analytical function to directly appro ximate the minim um, if the parameters are close to the minimizing ones [70, 69]. This com bination yields a go o d robustness and quic k con v ergence, whic h is wh y the metho d has b ecome a standard algorithm for least-squares fitting. Ho w ev er, if the starting parameters are far from the b est-fit parameters, the step size (c hange of parameters) is rather small, since a steep decen t (strong c hange in χ 2 ) is exp ected. Here, a T rust Region algorithm, whic h is closely related to the Lev en b erg-Marquardt metho d, can sho w a b etter p erformance with resp ect to con v ersion time. These metho ds tend to b etter estimate the necessary step sizes b y estimating “trusted regions”, where the analytical estimation to the χ 2 - function is sufficien t. F urthermore, the T rusted Region Reflectiv e algorithm [71] can handle b oundaries for the parameters and is used, when those are giv en. F or further details also refer to the p ython do cumen tation for scip y .optimize.least squares in [72]. Global minimization metho ds as e.g. “particle swarm” ha v e a b etter c hance to find a global minim um of χ 2 . Ho w ev er, such algorithms often p erform man y more iterations than lo cal minimization metho ds and th us the total fitting time is usually increased. This is not con v enien t for the fits in this thesis, where every single iteration step already tak es sev eral tens of seconds. Ho w ev er, in for instance the particle sw arm algorithm, h undreds of the iterations are similar and indep enden t of eac h other. Thus, parallel programming and efficien t use of pro cessing unites with m ultiple cores (probably ideally the graphics pro cessing unit due to its thousands of cores), migh t decrease the fitting time drastically (ideally to the duration a single iteration tak es) and mak e the whole ev aluation pro cesses more efficien t in the future. 2.5. Soft X-ra ys in Scanning-F ree GEXRF Analysis There is no strict definition of soft X-ra ys with resp ect to exact energy limits of this electro-magnetic radiation. Here, the definition of [35] is follo w ed, roughly defining the energy of soft X-ra ys to b e in the range of 250 e V to several k e V. A ccordingly , the energy of hard X-ra ys ranges from sev eral k e V up to several 100 k e V. 2.5.1. Application of Soft X-ra ys The use of soft X-ra ys in GEXRF exp erimen ts has sev eral adv an tages in comparison to excitation with hard X-ra ys. First of all, the photoionization cross sections for ligh t elemen ts can b e 1-3 orders of magnitude higher, directly increasing the sensitivit y for thin films or con taminan ts. F urthermore, the larger photoionization cross sections lead to a decrease in p enetration depth for soft X-ra ys, which in principle incr eases depth- resolving prop erties. Also, the critical angle for total reflection t ypically increases with 30 2.5. SOFT X-RA YS IN SCANNING-FREE GEXRF ANAL YSIS the w a v elength, so that the depth sensitiv e part of the GEXRF profile is extended o v er a wider angular range. In scanning-free GEXRF, this requires the detector to b e placed closer to the sample, which is beneficial for the o verall solid angle of detection and th us setup efficiency . Of course, the angular resolution degrades with closer sample-to- detector distances, but angular in tensit y patterns are broader in the soft X-ra y regime. This is a result of the longer w a v elength compared to hard X-ra ys, whic h directly affects the width of no des and an ti-no des of the X-ra y standing w a v e field patterns in the GIXRF case and similarly the (depth dep enden t) probabilit y pattern for fluorescence detection in the GEXRF case. The main dra wbac ks of a scanning-free GEXRF sp ectrometer op erating in the soft X-ra y range are, first of all, the limited access to hea vier elemen ts. F or example, with a primary photon energy of 1 k e V, K fluorescence radiation can b e excited in elemen ts with atomic n um b ers Z ≤ 10 (Ne), while L shell fluorescence has to b e used for the analysis of elemen ts with Z ≤ 29 (Cu). The latter exhibit usually higher uncertain ties in the tabulated fundamen tal parameters, complicating fundamen tal parameter based quan tification. Secondly , the fluorescence yield is small in the soft X-ra y regime, i.e. non-radiativ e A uger deca y is dominan t and fluorescence pro duction comparativ ely lo w. Finally , soft X-ra y sources are rare compared to common X-ra y tub es used in the hard X-ra y range. 2.5.2. Pro duction of Soft X-ra ys Sev eral t yp es of sources enable the pro duction of soft X-ra ys. F or example, large-scale facilities lik e sync hrotron radiation facilities or free electron lasers on the one hand and lab oratory sources lik e soft X-ra y tub es, disc harge plasma sources and laser-based sources (e.g. high harmonic generators or laser-pro duced plasmas) on the other hand migh t b e applied. In this thesis, measuremen ts with sync hrotron radiation and a laser- pro duced plasma (LPP) source are p erformed, whic h is why both source types are shortly describ ed. The probably most suited radiation for the application of soft X-ra ys in angular re- solv ed XRF exp erimen ts is pro vided b y sync hrotron radiation facilities. Here, electron (or p ositron) bunc hes with v elo cities close to the sp eed of ligh t are circulating in a stor- age ring and emit electro-magnetic radiation when deflected b y a magnetic field [35, 73]. Due to the relativistic sp eed of the accelerated c harges, the t ypical dip ole emission c haracteristics is deformed to a narro w radiation cone. F urthermore, the high particle energies and strong magnetic fields enable sp ectral emissions from the infrared to the γ -ra y region, dep ending on the sp ecific device sp ecifications. While b ending magnets and wigglers create broadband emission, undulator radiation consists of discrete narro w 31 2. METHODOLOGICAL BASES Figure 2.11.: Photograph of the copp er cylinder in the plasma in teraction c hamber. The infrared laser, plasma p osition and emitted X-ra ys are illustrated. p eaks of high in tensit y (harmonics). Since the radiation already has small con v ergence and high mono c hromaticit y (i.e. a high degree of spatial and temp oral coherence), the in tense undulator radiation is probably b est suited for GIXRF exp erimen ts, since the relativ e losses due to the necessary b eam shaping (mono c hromatization, parallelization) are relativ ely lo w. The latter is not critical for GEXRF exp erimen ts, where restrictions on b eam con v ergence and mono c hromaticit y are less tigh t. Indeed, fo cusing of syn- c hrotron radiation to the nm range has b een sho wn [74, 75], whic h could allo w for the com bination of nanometer depth resolution and nanometer lateral resolution in future b eamlines b y applying a scanning-free GEXRF setup. Besides the high photon flux, the second ma jor adv an tage of sync hrotron radiation is its energy tunabilit y , enabling opti- mized excitation conditions for the analyzed elemen ts. Details ab out sp ecific b eamlines and endstations, whic h are used in this thesis, can b e found in Section 4.1. When X-ra y tub es are to b e op erated in the soft X-ra y regime, first of all the en trance windo w m ust b e esp ecially thin or omitted to reduce absorption. Then, either L- or M-lines of high-Z ano de material or ano des consisting of lo w-Z material ha v e to b e used. While using L- and M- lines reduces the o v erall efficiencies, the utilization of lo w Z-targets leads to insufficien t thermal conductivit y , limiting the applicable maxim um p o w er. An LPP o v ercomes this limit by in ten tionally consuming target material for the formation of a hot dense plasma, whic h can irradiate in tense soft X-radiation [35]. Indeed, the amoun t of consumed target material is lo w and regenerativ e target systems lik e gas puffs [76], liquid jets [77], metal tap es [78] or rotating metal cylinders [79] can b e applied. A t the Berlin Lab oratory of Inno v ativ e X-ra y T ec hnology (BLiX) a LPP source with 32 2.5. SOFT X-RA YS IN SCANNING-FREE GEXRF ANAL YSIS Figure 2.12.: a) X-ra y sp ectrum emitted b y the LPP source using a Cu target from [80]. The high in tensit y lines at 1.15 nm (1078 e V) of 20-fold ionized Cu are used for XRF exp erimen ts. The transmission of a 200 nm Al infrared filter is also sho wn. b) Photo pro duction cross section ξ for K α and L α fluorescence from v arious elemen ts excited with 1078 e V photons. rotating copp er cylinder is used. While a detailed description can b e found in [79] and [81], here the prop erties, whic h are imp ortan t for XRF analysis, are discussed. The LPP source is driv en b y a pulsed Yb:Y A G thin disk laser with pulse energies up to 220 mJ, a pulse length of 1.2 ns and a rep etition rate of 100 Hz. The laser is fo cused on to a rotating copp er cylinder (see Figure 2.11), where the high laser intensit y of > 2 × 10 14 W/cm 2 [82] leads to the ionization and efficien t heating of the target material, allo wing the formation of a hot dense plasma. The plasma emits p olyc hromatic radiation from the infrared to the soft X-ra y region, whic h can b e used for X-ra y absorption in the 1-5 nm range [83] and X-ra y fluorescence measuremen ts [84]. The emitted soft X-ra y sp ectrum is sho wn in Figure 2.12 a). Due to the high absorption of soft X-ra ys in matter and the small differences of refractiv e indices, refractiv e optics are not applicable. Instead, m ultila y er reflectiv e optics can efficien tly collect and refo cus the radiation on a sample. Since high reflectivities are only ac hiev ed for a single w a v elength satisfying the Bragg equation (in first diffraction order), the m ultila yer optics are adapted to the in tense plasma lines at 1078 e V (1.15 nm). Th us, the sample in the fo cus of the optics is excited with mono c hromatic radiation (the plasma in tensit y at photon energies of higher order Bragg reflexes is negligible). Figure 2.12 b) sho ws the photo pro duction cross section ξ (whic h is directly prop ortional to the exp ected fluorescence in tensit y) of the elemen ts, whic h can b e excited with 1078 e V primary radiation. As can b e seen, efficient excitation is p ossible for K α radiation of ligh t elemen ts and L α radiation of the 3d transition metals. The samples in v estigated in this thesis are comp osed of C, Ni, O, Cu and A u. Of these, Cu and Ni are esp ecially 33 2. METHODOLOGICAL BASES suited for excitation b y the BLiX LPP source. On the top righ t of Figure 2.11, a pinhole coupled to a 4-quadran t-dio de (4Q-dio de) is attac hed to monitor the source p osition. It is aligned in a w a y , that c hanges of the radius of the metal cylinder, e.g. b y out-of-roundness of the target cylinder, a displacemen t of rotation axis to cylinder axis and temp erature c hanges, can b e detected b y the 4Q-dio de Y axis. Via a real-time feedbac k lo op to a target motor con trolling the v ertical p osition of the target cylinder, these displacemen ts can b e corrected for. The 4Q-dio de X axis is in first order appro ximation sensitiv e to a laser fo cus displacemen t. The feedbac k system allo ws for con tin uous and long-term op eration with a stable source p osition, whic h is critical for scanning-free GEXRF measuremen ts. F urther details ab out op eration conditions and exemplary stabilit y measuremen ts are giv en in Section 5.2.1 and 6.3.2. 34 3. Thermo electric Nanofilms Because of their earth abundancy , lo w cost and non-to xicity , transition metal o xides are promising material systems for the application in thermo electric generators. K. Bethk e in v estigated in his Diploma Thesis thermo electric prop erties of sev eral copp er and silv er comp ound nanofilms on so da-lime-silicate glass and found promising results for copp er o xide nanofilms [85]. Therefore, researc h contin ued b y increasing the electrical conduc- tivit y of these copp er o xide nanofilms b y doping the material with gold, which indeed increased the o v erall efficiency of the device. T o impro v e the understanding of the ther- mo electric prop erties of these materials it is of in terest to obtain structural and c hemical information of the nanofilms, i.e. information ab out lateral homogeneit y and in-depth distribution of Cu and A u and depth dep enden t o xidation states of Cu with nm depth resolution. In this thesis, t w o of suc h gold dop ed copp er nanofilm samples, one as de- p osited (DM0150A) and one thermally o xidized sample (DM0149A), are in v estigated with sev eral lab oratory and sync hrotron based X-ra y fluorescence metho ds to address the men tioned analytical questions. F urthermore, the comparison of grazing incidence (GI) XRF measuremen ts p erformed at the sync hrotron radiation facilit y and measure- men ts with the herein dev elop ed scanning-free grazing emission (GE) XRF approac h in the lab oratory with the same sample allo w for a v alidation of the metho d dev elop ed in this thesis and a direct comparison with state of the art GIXRF. 3.1. Sample Prepa ration and Surface Cha racterization Sample preparation and first c haracterization measuremen ts are p erformed b y K. Bethk e of the researc h group of Prof. K. Rademann from the departmen t of ph ysical c hemistry at the Hum b oldt Univ ersit y of Berlin. In a Cressington sputter coater 108 A uto (Figure 3.1), thin films of gold and copp er are t ypically dep osited on a so da-lime glass b y magnetron sputtering. Ho wev er, to reduce in terferences from glass comp onen ts in the XRF sp ectra and reduce the la y er roughness (and th us increase the sensitivit y of GI- and GEXRF due to X-ra y standing w a v e (XSW) field effects), the samples used in this thesis are sputtered on an 800 µ m thic k silicon w afer with an area of appro ximately 18 × 18 mm 2 . As sputter target a 0.2 mm thic k 35 3. THERMOELECTRIC NANOFILMS Figure 3.1.: a) photograph and b) sc hematic view of the magnetron sputter coater used for preparation of the copp er o xide nanofilms. In an argon plasma, the ions are accelerated to w ards a sputter target, where they kno c k out atoms and ions of the target material. The ejected atoms can then condensate on the substrate surface. p erforated Cu foil with a hole densit y of 5% is placed just b eneath a 0.2 mm thic k A u foil (Figure 3.1). The target has a distance to the substrate of 30 mm and directly on top of the substrate a sputter mask is placed to confine the sputtered area to ≈ 14 × 14 mm 2 . The pro cess gas is Alphagaz TM 1 Ar from Air Liquide, the current is 40 mA and the silicon w afers are sputtered for 2 × 40 s, rotating the sample ab out 180 ◦ after the first 40 s to mak e the sputtering more homogeneous. T w o samples, DM0150A and DM0149A are prepared as describ ed one after another. Subsequen tly , only DM0149A is temp ered in a tube furnace for 360 s at 300 ◦ C, to o xidize the copp er la y er. By in v estigating b oth the non-temp ered and temp ered sample, structural and c hemical differences in the samples can b e traced to the temp ering pro cess. Figure 3.2 sho ws exemplarily an atomic force microscop y image of a non-temp ered sample prepared with the same pro cess parameters and substrate, which are used for DM0150A and DM0149A. Heigh t v ariations of some tens of nm on a lateral scale of tens of µ m are visible in the o verview (righ t image). Suc h structures are force induced deformations of the substrate and are often visible in AFM images of large areas. In the first magnification (top left), v alleys of ≈ 2 nm depth and extensions in the µ m range can b e seen. These structures probably originate from the substrate, whic h sho ws similar v alleys (App endix B). The lo cal ro ot mean squared (RMS) roughness determined in the t w o areas sho wn on the b ottom left are ab out 1 nm, similar to the substrate roughness. Th us, the copp er o xide la yer seems to be homogeneously cov ering the substrate in the micrometer range. The roughness of the sample is in the range of the w a v elength of ≈ 1 ke V photons, whic h are used in the GIXRF measuremen ts in Section 4.5 and the GEXRF measure- 36 3.2. INVESTIGA TION OF LA TERAL HOMOGENEITY BY MEANS OF LABORA TOR Y-BASED XRF Figure 3.2.: A tomic force microscopic image of a gold-dop ed copp er o xide nanofilm. The ro ot mean squared (RMS) roughness on a micrometer scale is ab out 1 nm (bottom righ t). men ts in Section 6.3.4, so that the formation of an XSW field can b e exp ected. F ur- thermore, particles or islands with a size of a few µ m do wn to 100 nm are visible on the surface. This agrees with the findings in Section 4.5, where the GIXRF measuremen ts imply the presence of carb on and o xygen ric h particles with similar size. 3.2. Investigation of Lateral Homogeneit y b y means of Lab o rato ry-based XRF XRF measuremen ts to in v estigate the absolute mass dep ositions of A u and Cu and their lateral homogeneit y are p erformed with a commercial Fisc herscop e X-Ra y XD V-SDD (Helm ut Fisc her Gm bH). The X-ra ys are pro duced b y a rho dium X-ra y tub e, op erated with a high v oltage of 50 k V and an ano de curren t of 642 µ A. 500 µ m Al are used as primary filter and the collimated X-ra y b eam has a sp ot size of ab out 1 mm diameter on the sample surface. Sp ectra are recorded with a co oled silicon drift detector with a measuremen t time of 100 s for eac h measuremen t p oin t. On b oth samples, DM0150A and DM0149A, 15 measuremen ts are recorded at the same p osition close to the cen ter of the sample to get information ab out the measuremen t repro ducibilit y . Then, a grid of 10 × 10 measuremen ts o v er the whole sample is measured once and a smaller grid of 5 × 5 measuremen t p oin ts close to the sample cen ter is measured 6 times. Mass dep ositions of the samples are quan tified with a fundamen tal parameter based, 37 3. THERMOELECTRIC NANOFILMS T able 3.1.: Quan tification results of the thermo electric nanofilms obtained from 15 single measuremen ts on the same sp ot close to the sample cen ter. The uncertainties are the ones giv en b y the device. The standard deviation σ is calculated from the 15 single measuremen ts. DM0150A DM0149A mean σ mean σ ˆ m A u / (ng × cm − 2 ) 600 ± 300 76.2 600 ± 300 72.7 ˆ m Cu / (ng × cm − 2 ) 23600 ± 300 197.4 22100 ± 300 309.4 standardless metho d (Section 2.1.2) implemen ted in the device soft w are WinFTM of the man ufacturer. The mean v alue for the mass dep ositions ˆ m for Cu and A u of the 15 measuremen ts of b oth samples are displa y ed in T able 3.1. Because of the v ery lo w quan tities of A u, the uncertain ties calculated b y WinFTM are v ery high (50%) and indicate the limits of the device sensitivit y . Assuming bulk densit y for Cu (8.96 g/cm 3 ), the quan tified mass dep ositions can b e used to roughly estimate a la y er thic kness of 26 nm and 25 nm for DM0150A and DM0149A, resp ectiv ely . As will b e seen in Chapter 4 and 6.3.4, the nanofilms are strongly o xidized, which reduces the la y er densit y and leads to an increased geometrical la y er thic kness. In Figure 3.3, the results of the 10 × 10 mapping are sho wn for sample DM0150A in a) and c) and for DM0149A in b) and d) for Cu and A u, resp ectiv ely . F urthermore, the p ositions of the 5 × 5 area scan and of the 15 single measuremen ts are indicated. T o mak e statistically relev an t deviations in the mass dep osition visible, the color scale in the plot is graded in units of 4 times the standard deviation σ of the resp ectiv e 15 measuremen ts on the same sp ot (see T able 3.1). That means, for a mass dep osition in the middle of the b order of one color, a measuremen t of the same mass dep osition will in ≈ 95% ( ± 2 σ area) b e displa y ed in the same color. As can b e seen for the 10 × 10 mapping of Cu, the dep osited material decreases sligh tly to w ards the edges. Obviously , a single rotation b y 180 ◦ is not sufficien t to accoun t for the non-uniformity of the plasma during magnetron sputtering. In b oth the 10 × 10 mappings for A u in c) and d), the statistics is not sufficien t for a statemen t concerning the homogeneit y . Only the edges of the sample are visible, similar as in a) and b). The results of the 5 × 5 mapping of b oth samples is sho wn Figure 3.4. Plotted are the mean v alues of the mass dep ositions of the 6 measuremen ts on eac h of the 5 × 5 p ositions. The color scale is graded in units of 4 times the standard error of the mean σ m = σ / √ 6, using again σ from the 15 measuremen ts on the same sp ot. Th us, the results ha v e a reduced statistical uncertain t y and the probabilit y of c hanges in the color of eac h measuremen t p oin t due to statistics alone are again < 5%. In Figures 3.4 a) and 38 3.2. INVESTIGA TION OF LA TERAL HOMOGENEITY BY MEANS OF LABORA TOR Y-BASED XRF Figure 3.3.: 10 × 10 mappings of DM0150A (left) and DM0149A (righ t). The top graphs sho w the Cu and the b ottom graphs the resp ectiv e A u distributions. In a) and b) the areas of the 5 × 5 mapping and the p osition of the 15 measuremen ts on the same sp ot are indicated. 39 3. THERMOELECTRIC NANOFILMS Figure 3.4.: 5 × 5 mapping of DM0150A (left) and DM0149A (righ t). F or eac h p osition, the mean v alues of 6 measuremen ts of the mass dep ositions for Cu (top) and Au (bottom) are displa y ed. The mean o v er the whole area is shown in the color bar. 40 3.2. INVESTIGA TION OF LA TERAL HOMOGENEITY BY MEANS OF LABORA TOR Y-BASED XRF b), the gradien t of the Cu mass dep osition is resolv ed, sho wing reduced v alues tow ards the b ottom left for DM0150A and top left for DM0149A. This is consisten t with the p ositions of the 5 × 5 grids indicated in Figure 3.3. Concerning the A u mass dep ositions in Figure 3.4 c) and d), the v alues are laterally homogeneous for b oth samples with resp ect to the statistical resolution. Only three v alues in b oth mappings are not within the ± 2 σ m confidence in terv al (deviation from mean v alue < 10%), but w ould b e included in a ± 3 σ m in terv al. Considering also the exp ected gradien t of the mass dep osition, there is no clear evidence for inhomogeneities of the A u mass dep osition with resp ect to the resolution of the metho d. T o enhance the latter, more extensiv e mappings on the sample could b e p erformed. F or example, to ac hiev e in a 5 × 5 grid a ± 2 σ m confidence in terv al, whic h refers to < 3% of the mean v alue, the measuremen t m ust b e rep eated ≈ 60 times. This leads to v ast measuremen t times of > 10 da ys, whic h is incon v enien t, but p ossible with a lab oratory setup and could b e pursued in future w ork. Also, the use of suitable standards could reduce the uncertain ties of the determined mass dep ositions, whic h migh t b e esp ecially of in terest for the amoun t of gold in the la y ers. Ho w ev er, appropriate reference materials are not readily a v ailable, yet. 41 4. Synchrotron Radiation based Analysis So far, XRF analysis for the c haracterization of the thermo electric nanofilms are p er- formed in the hard X-ra y range b y tub e excitation. T otal mass dep ositions and their lateral homogeneit y are in v estigated in the previous c hapter, but neither access to the copp er o xidation state nor to inhomogeneit y in depth with nanometer resolution can b e ac hiev ed with the a v ailable commercial equipmen t. These prop erties can b e analyzed with soft X-ra ys for the effectiv e excitation of ligh t elemen ts and b y using adv anced, an- gular and energy resolv ed XRF metho ds at sync hrotron radiation facilities. In the scop e of this thesis, grazing incidence (GI-) XRF and near edge X-ra y absorption fine structure (NEXAFS) measuremen ts are p erformed at the lab oratories of the Ph ysikalisc h T ec h- nisc he Bundesanstalt (PTB) at the sync hrotron radiation facilit y BESSY I I in Berlin. F urthermore, the dedicated, fully calibrated instrumen tation a v ailable there enables reference-free quan tification (Section 2.1.2) in the soft and hard X-ra y regime, whic h can b e compared to the prior lab oratory based approac h. After describing the applied instrumen tation and metho dology , the following c hapters will presen t quan titativ e and qualitativ e structural and c hemical differences b et w een the non-temp ered and temp ered thermo electric nanofilms DM0150A and DM0149A. 4.1. Instrumentation 4.1.1. Beamlines The analysis of the thermo electric copp er o xide nanofilms is carried out in the hard X-ra y range to get access to A u-L and Cu-K fluorescence as w ell as in the soft X-ra y range to efficien tly excite O-K fluorescence. Therefore, measuremen ts at three different beamlines at BESSY I I in Berlin are p erformed, whic h are shortly describ ed in the follo wing. The plane-grating mono c hromator (PGM) b eamline [86] is using the radiation of the undulator U49. The b eamline pro vides mono c hromatic X-ra y photons with energies from 78 e V to 1870 e V and a resolving p o w er E / ∆ E from 1000 to 9000 [42], dep ending on the angular settings of the grating. Higher order con tributions from the undulator and stra y ligh t are suppressed b y total reflection at the mirrors acting as lo w-pass filters and further optional transmission filters and ap ertures. At a typical electron curren t in the 43 4. SYNCHR OTR ON RADIA TION BASED ANAL YSIS sync hrotron storage ring of 200 mA, the photon flux reac hes from 6 × 10 9 photons/s at 1.7 k e V to 6 × 10 11 photons/s at 400 e V [87]. The fo cus, whic h is usually the place where the sample in the endstation is aligned, has a size (FWHM) of 40 µ m (v ertical) × 40- 600 µ m (horizon tal), adjustable b y means of an ap erture. F urther details can also b e found in [88, 89]. The KMC b eamline facilitates a four-crystal mono c hromator after the dip ole magnet D71, pro viding mono c hromatic radiation b et w een 1.75 k e V and 10.5 k e V [90, 91]. Due to the four Bragg reflections in InSb(111)- or Si(111) crystals, a resolving p o w er E / ∆ E of ≈ 4000 to 12000 can b e ac hiev ed and again higher orders and stra y ligh t are sup- pressed b y total reflection on v arious mirrors and transmission filters. The photon flux is ab out 10 10 photons/s o v er the whole sp ectral range and can b e monitored with a thin transmission dio de when using X-ra ys with energies ab o v e 3 k e V. Finally , the BAMline b eamline, op erated b y the Bundesanstalt f ¨ ur Materialforsc h ung (BAM), uses a sup erconducting 7 T w a v elength shifter to pro duce photons with ener- gies from 6 k e V to 60 k e V [92, 93]. The b eam is mono c hromatized either b y a double- m ultila y er mono c hromator ( E / ∆ E ≈ 40) for high photon flux applications or b y a double-crystal mono c hromator ( E / ∆ E up to 1000) if energy resolution is more im- p ortan t. In the b eamline, a fo cusing mirror with v ariable curv ature radius is applied, allo wing for adjustable fo cus size in the horizon tal direction from 185 mm do wn to 1 mm. The photon flux densit y with a fo cused b eam reac hes 5.7 × 10 10 photons/(s × mm 2 ) and can b e monitored with an ionization c ham b er. 4.1.2. Endstations All exp erimen ts are carried out under v acuum conditions to minimize absorption of primary and fluorescen t X-ra ys. T w o differen t v acuum c hambers are used to realize the v arious geometries of the applied exp erimen ts with resp ect to inciden t b eam p osition, sample alignmen t and diagnostics (see sc hematic in Figure 4.1). The measuremen ts at the BAMline are carried out with the smaller of the t w o sp ectrometer c ham b ers b ecause of spatial restrictions. It is usually applied for reference-free quantification in con v en tional XRF geometry , i.e. with an inciden t and detection angle of 45 ◦ [87, 94]. T o improv e the excitation conditions for the thermo electric nanofilms, the c ham b er is moun ted to the b eamline with a tilt of ab out 45 ◦ , enabling shallo w incident angle conditions. The sample holder can only b e mo ved in the sample plane, whic h p ermits precise angular resolv ed measuremen ts. The second, larger, spectroscopy c ham b er [95] is more flexible concerning the measure- men t geometry and is applied in the measuremen ts at the KMC and PGM b eamlines. The sample can b e aligned and p ositioned with an 8-axes goniometer pro viding all 6 44 4.2. D A T A RECORDING AND TREA TMENT Figure 4.1.: Sc hematic view of the v acuum cham b ers and possible measurement geometries for the XRF exp erimen ts carried out at the lab oratories of the PTB. degrees of freedom for the sample (3 translational and 3 rotational axes) and t w o more axes for rotation and translation of a dio de on a second goniometer arm. Therefore, not only GIXRF exp erimen ts can b e p erformed b y aligning the sample in the piv otal p oin t of the goniometer and c hanging the inciden t angle of the radiation b y a rotation of the sample. Also X-ra y reflectometry measuremen ts (XRR) can b e applied sim ultaneously b y c hanging the angular p osition of a photo dio de on the second goniometer arm t wice as fast as the sample angle. The whole v acuum c hamber can b e mo ved with respect to its base frame, which is of imp ortance for the alignmen t with resp ect to the b eamline and helpful to measure the sample-to-detector distance. The latter is imp ortan t to cal- culate the solid angle of detection applied in GIXRF measuremen ts and reference-free quan tification. 4.2. Data Reco rding and T reatment T o quan tify XRF sp ectra of a sample with a reference-free fundamen tal parameter ap- proac h, all the instrumen tal parameters in Equation 2.1.1 ha v e to b e kno wn. These are the inciden t n um b er of photons N pr ( E pr ), the detection efficiency det ( E i,j ) (usually of a silicon drift detector) for all fluorescence photon energies E i,j used in the quantification and the setup geometry , i.e. the effective solid angle of detection Ω as w ell as the inciden t and detection angles ψ pr and ψ fl . Esp ecially the geometry is of ma jor imp ortance for the analysis of GIXRF measuremen ts, whic h is obvious for ψ pr , but also Ω v aries strongly 45 4. SYNCHR OTR ON RADIA TION BASED ANAL YSIS with ψ pr (for small ψ pr ) and affects the shap e of the GIXRF profiles. During the measuremen ts, the inciden t flux on the sample is monitored b y e.g. a thin photo dio de, whic h is calibrated shortly b efore or after the measuremen t against the calibrated photo dio de b ehind the sample. Thus, the absolute photon flux on the sample can b e calculated. The fluorescence sp ectra are recorded with a calibrated SDD with kno wn efficiency and detector resp onse function. The life time of the measuremen t is deriv ed from the zero p eak of the SDD detector, which is increased b y the SDD’s electronics during the measuremen t. T o get access to Ω, ψ pr and ψ fl , a precise alignmen t of the sp ectroscop y c ham b er and of the sample is necessary . This is p erformed b y routine pro cedures dev elop ed at the PTB X-ra y sp ectrometry division. A t first, the center positions of all samples mounted on the sample holder are aligned b y horizon tal and v ertical line scans with the sync hrotron b eam. The cen ter p osition is then defined b y the middle of the edge p ositions of the line scans, whic h are indicated b y the decrease of fluorescence in tensit y measured with an SDD. Th us, the measuremen t p osition of the thermo electric copp er o xide nanofilms is indeed the cen ter of the sputtered area and not the cen ter of the somewhat larger silicon w afer. The surface of the sample is aligned b y radial scans and recording the direct sync hrotron b eam with a photo dio de b ehind the sample, similar to a knife-edge scan, but with the whole sample surface. The aligned p osition concurs with an intensit y drop b y 50%. T o align the sample surface parallel to the inciden t b eam, the shado wing of the inciden t b eam b y the sample is minimized b y rotating the sample ab out the v ertical axis. T o increase the accuracy , the t w o pro cedures (radial and rotational scan) can b e rep eated iterativ ely . The solid angle of detection is determined b y the detector geometry (c hip size, ap er- ture and distance) and the irradiated area on the sample (fo otprin t). The fo otprin t is stretc hed b y a factor 1/sin( ψ pr ) in the horizon tal direction b ecause of the pro jection of the inciden t b eam on to the sample surface. Therefore, for shallow angles and dep ending on the detector distance, the fo otprin t migh t w ell exceed the field of view of the detector or the sample dimension itself. F or example, with a t ypical horizon tal b eam diameter of 140 µ m at the PGM b eamline, b elo w 0.6 ◦ , the fo otprin t exceeds the 1.4 cm length of the thermo electric nanofilms of DM0149A and DM0150A. This needs to b e tak en in to accoun t b y calculating an effectiv e solid angle of detection [42] XRF sp ectra can b e sim ulated with ph ysical mo dels of the bac kground and delta functions for the fluorescence p eaks, con v olv ed with the measured detector resp onse function of the SDD. This allo ws to fit sim ulated sp ectra to the measured sp ectra b y adjusting the in tensities of the fluorescence lines and the bac kground functions. As bac kground mo dels, b esides the Ra yleigh scattered p eak, resonant Raman scatte ring 46 4.2. D A T A RECORDING AND TREA TMENT Figure 4.2.: Energy-disp ersiv e sp ectra of sample DM0149A excited with a) 1060 e V at the PGM b eamline and b) 13 k e V at the BAMline b eamline. Decon volution is performed with ph ysical bac kground mo dels and detector resp onse functions. (RRS) [96, 97] and bremsstrahlung [98] are calculated for sample matrices and inciden t energies as appropriate. Figure 4.2 a) and b) sho w exemplarily the energy-disp ersiv e sp ectra of sample DM0149A recorded at 1060 e V and 13 k e V, resp ectiv ely . In the sp ectra recorded at the PGM b eam- line (Figure 4.2 a)), the bac kground due to bremsstrahlung is calculated for the copp er o xide la y er and resonan t Raman scattering for the Si substrate. The fit of ph ysical back- ground and X-ra y line con tributions is in excellen t agreemen t with the measured data. Besides the prominen t con tributions from Cu-L α,β , Cu-L l,n , O-K α and Si-L l,n , minor con tributions of carb on, nitrogen (surface contamination) and probably iron (perhaps originating from secondary excitation of the detector housing) migh t b e presen t. The sp ectrum in Figure 4.2 b) is recorded at 13 k e V at an inciden t angle of ≈ 0.05 ◦ . Because of the lo w transmittance of the thermo electric nanofilm for the primary radiation at these shallo w angles ( < 10%), the bremsstrahlung bac kground is calculated for Cu. F ur- thermore, no RRS bac kground app ears, but a further detector shelf for the most intense line (Cu-K α ) has to b e considered to ac hiev e a satisfactory fit. In the lo w energy part of the sp ectrum (b elo w 2 k e V), further constan t con tributions to the background migh t im- pro v e the sp ectrum fitting in this region. They are not applied here, since only the high energy region is used for the quan tification. In the sp ectrum, Cu-K α , Cu-K β , A u-L α , A u-L l , A u-L β 2 , Si-K α and the Cu-L lines are prominen t. F urthermore, con tributions of C, Ni and F e are visible, the latter t w o probably originating from the sample holder itself or as ab o v e from the detector housing. Note that the excitation energy is b elow the A u-L 2 edge, prev en ting fluorescence originating from A u-L 1 and A u-L 2 v acancies. F or the follo wing quan tification approac hes, whic h are fully based on fundamen tal parameters (FPs), the o v erall uncertain ties are dominated b y the uncertain ties of these 47 4. SYNCHR OTR ON RADIA TION BASED ANAL YSIS fundamen tal parameters. In App endix C, a list of the applied FP uncertain ties from Krause and the compilation of Zsc hornac k [99, 100] is giv en. In principle, uncertainties are increased for ligh t elemen ts and lo w energy X-ra ys. 4.3. Integral Quantification The absolutely calibrated instrumen tation at the lab oratories of the PTB allo w to quan- tify mass dep ositions with XRF without the need of calibration standards. This is par- ticularly of in terest for sample systems, where suc h calibration standards are not readily a v ailable, as is the case for the thermo electric nanofilms in v estigated in this thesis. F or this purp ose, measuremen ts at the PTB lab oratories are p erformed using 4 differen t excitation energies to optimally excite the comp onen ts of the nanofilms, namely O-K and Cu-L lines at 1060 e V pro vided b y the PGM b eamline, Cu-K and A u-M lines at 10 k e V of the KMC b eamline and finally Cu-K and A u-L lines at 13 k e V and 17 k e V at the BAMline b eamline. All sp ectra are decon v olv ed with the PTB soft w are describ ed ab o v e (Section 4.2) to get the net p eak areas. The complete quan tification is then realized in the follo wing successiv e steps. The measuremen ts at the BAMline b eamline are p erformed at 13 k e V and 17 k e V at shallo w inciden t angles of ab out 0.05 ◦ to efficiently excite the thermoelectric nanofilm. Th us, the detection angle ψ fl is almost 90 ◦ , minimizing self-absorption of the detected fluorescence radiation. F or this b eam time, no absolutely calibrated instrumen tation is applied. Therefore, the measuremen ts cannot b e used to quan tify the gold and copp er mass dep ositions directly . Ho w ev er, if A u is homogeneously distributed in depth in the copp er o xide nanofilm and if self-absorption of the fluorescence radiation can b e neglected, the A u to Cu atom ratio can b e obtained from the measuremen ts. The first condition is at least roughly fulfilled, as will b e seen in the GIXRF measuremen ts in Section 4.5. The latter condition is also true for b oth fluorescence energies (Cu-K α and A u-L α ), since the transmission through ev en a solid 50 nm thic k copp er la y er (whic h is a conserv ativ e estimate) is ab out 99% (calculated with FPs from [39]). By using initially the second condition, the absorption term of the fluorescence radiation in µ ∗ can b e 48 4.3. INTEGRAL QUANTIFICA TION neglected in Equation 2.1.1 in Section 2.1.1, resulting in I : N Au, j 1 = N pr ( E pr ) G ( E A u, j 1 ) ξ A u, j 1 ,s 1 ( E pr ) × Z d 0 ρ ( x ) C A u ( x ) exp − Z x 0 µ tot ( E pr , x 0 ) ρ ( x 0 ) sin( ψ pr ) d x 0 ! d x I I : N Cu, j 2 = N pr ( E pr ) G ( E Cu, j 2 ) ξ Cu, j 2 ,s 2 ( E pr ) × Z d 0 ρ ( x ) C Cu ( x ) exp − Z x 0 µ tot ( E pr , x 0 ) ρ ( x 0 ) sin( ψ pr ) d x 0 ! d x (4.3.1) Note the second in tegral app earing in the factor resp onsible for the atten uation of the primary radiation due to Lam b ert-Beer’s la w, whic h is no w v alid for an (in depth) inhomogeneous la y er. If gold and copp er are similarly distributed in depth, i.e. C i ( x ) = C i × f ( x ), then the depth indep enden t concen tration factor C i can b e written in fron t of the in tegral, whic h is then the same for (I) and (I I), resulting in I I I : N A u, j 1 N Cu, j 2 = det ( E A u, j 1 ) det ( E Cu, j 2 ) × ξ A u, j 1 ,s 1 ( E pr ) ξ Cu, j 2 ,s 2 ( E pr ) × C A u C Cu ⇔ C A u C Cu = N A u, j 1 det ( E A u, j 1 ) ξ A u, j 1 ,s 1 ( E pr ) × N Cu, j 2 det ( E A u, j 1 ) ξ Cu, j 2 ,s 2 ( E pr ) ! − 1 (4.3.2) The last ro w in Equation 4.3.2 sho ws that the A u-to-Cu mass ratio can b e calculated b y the ratio of the detected net p eak areas N i,j corrected b y the detector efficiency det ( E i,j ) and normalized to the fluorescence pro duction cross sections ξ i,j,s . Dividing the A u-to-Cu mass ratio b y the Cu-to-A u atomic w eigh t ratio yields the relativ e n um b er of atoms for gold and copp er in T able 4.1. It can b e stated that there is 1 gold atom for ev ery 100 copp er atoms in the sample. The absolutely calibrated measuremen ts at the KMC b eamline at 10 k e V can b e used to quan tify the absolute copp er mass dep osition directly and th us, with the informa- tion of the A u-to-Cu mass ratios, also the absolute mass dep osition for gold. F or the quan tification, the Cu-K α line is used and its net p eak area calculated from the de- con v olv ed sp ectrum. The in tensit y is normalized with resp ect to incoming photon flux, measuremen t time, detector efficiency and solid angle of detection. Then, the xrfLibrary is used to calculate the fluorescence in tensit y of a single la y ered, single element sample for the fluorescence lines, which are not separable in the measured sp ectrum (Cu-K α 1 and Cu-K α 2 ). A fit of the mass dep osition of that la y er, so that the calculated and measured fluorescence in tensities are the same, yields the quan tification results for Cu 49 4. SYNCHR OTR ON RADIA TION BASED ANAL YSIS T able 4.1.: Quan tification results for the relativ e atomic fractions of Au to Cu atoms obtained from the measuremen ts on the thermo electric nanofilms DM0150A and DM0149A (temp ered) at the BAMline (BAM) and absolute mass dep ositions ˆ m mea- sured at the KMC b eamline (KMC). ˆ m A u is not measured directly , but calculated from ˆ m Cu and the relativ e atomic fractions. m energy DM0150A DM0149A C at A u / C at Cu 13 k e V (BAM) 1.04% ± 0.12% 0.99% ± 0.11% C at A u / C at Cu 17 k e V (BAM) 1.07% ± 0.12% 0.98% ± 0.11% ˆ m A u / (ng × cm − 2 ) 10 k e V (KMC) 580 ± 80 560 ± 80 ˆ m Cu / (ng × cm − 2 ) 10 k e V (KMC) 18900 ± 1400 18500 ± 1300 in T able 4.1 ∗ . The mass dep osition for A u is calculated with the ev aluated A u-to-Cu mass ratio. The quan tification for A u and Cu obtained from the measuremen ts with the con- v en tional lab oratory setup in Chapter 3 resulted in (23600 ± 300) ng/cm 2 and (22100 ± 300) ng/cm 2 for Cu in the cen ter of DM0150A and DM0149A, resp ectiv ely . The gold mass dep osition is for b oth samples (600 ± 300) ng/cm 2 . Thus, there are small but sig- nifican t discrepancies b et w een the quan tified Cu mass dep ositions with the lab oratory setup and the fully FP based sync hrotron measuremen ts. Probably the uncertain ties are sligh tly underestimated either for the FPs used in the reference-free approac h and or the uncertain ties of the commercial device. The latter is usually used for the determination of la y er thic knesses in the 100 nm to µ m range and probably w orking at its limit, when analyzing films with tens of nm thic knesses. † The quan tification results for A u of b oth approac hes (commercial setup and reference-free) agree within their uncertain ties. Here, the commercial device giv es h uge uncertain ties of up to 50%, indicating the limitation with resp ect to sensitivit y . The results migh t b e impro v ed, if reference materials were a v ailable. The next quan tification step concerns the o xygen con ten t in the t wo samples. Due to the temp ering pro cess applied to DM0149A, a difference in the o xygen concen tration in b oth samples is exp ected. T o get access to the o xygen concen tration, XRF measuremen ts need to b e p erformed in the soft X-ra y range. The thermo electric nanofilms DM0150A and DM0149A are dep osited on silicon w afers. ∗ As fitting algorithm, the Lev en b erg-Marquardt least-squares fit implemented in the p ython function curv e fit of the pac kage scipy .optimize is used. † In terestingly , and as will b e seen later in Section 4.5.2, the quan tification approach of the Cu mass depo- sition with the GIXRF measuremen ts ((22600 ± 800) ng/cm 2 for DM0150A and (20500 ± 1200) ng/cm 2 for DM0149A), whic h is less dep enden t on correct FPs, rather supp orts the v alues obtained b y the commercial device. 50 4.3. INTEGRAL QUANTIFICA TION T able 4.2.: Quan tification results for atomic fractions and mass dep ositions of A u, Cu and O in the thermo electric thin films DM0150A and DM0149A obtained at the PGM b eamline with 1060 e V primary photons. Carb on ric h con taminants with a mass depo- sition ˆ m C are presumably found on top of the thermo electric la y er. The quan tification results for the o xygen mass dep osition ˆ m O (SiO 2 ) of the SiO 2 nativ e la y er on top of the Si w afer, measured next to the thermo electric la y ers, are also shown . DM0150A DM0149A ˆ m C / (ng × cm − 2 ) 1600 ± 200 970 ± 120 ˆ m A u / (ng × cm − 2 ) 600 ± 170 560 ± 160 ˆ m Cu / (ng × cm − 2 ) 18000 ± 5000 18000 ± 5000 ˆ m O / (ng × cm − 2 ) 3800 ± 800 4300 ± 900 C at A u / at.% 0.57 ± 0.17 0.51 ± 0.15 C at Cu / at.% 50 ± 20 50 ± 20 C at O / at.% 45 ± 9 48 ± 10 ˆ m O (SiO 2 ) / (ng × cm − 2 ) 200 ± 40 230 ± 50 Since the w afers ha v e b een stored in am bien t conditions, a nativ e silicon dioxide surface la y er with nm thic kness can b e exp ected. T o accoun t for the o xygen signal originating from the SiO 2 , XRF measuremen ts are p erformed just next to the sputtered sample area on the same w afer material for b oth samples DM0149A and DM0150A. The quan tifica- tion is p erformed as ab o v e but for a single SiO 2 la y er and using the mass dep osition of o xygen as fitting parameter for the adjustmen t of the O-K α fluorescence in tensit y . The results are sho wn in T able 4.2. The mass dep ositions can b e calculated to more descrip- tiv e la y er thic knesses, if assuming pure SiO 2 with a densit y of 2.65 g/cm 3 . In that case, the w afers of DM0149A and DM0150A ha v e SiO 2 capping la y ers of (1.6 ± 0.5) nm and (1.4 ± 0.5) nm, indicating no significan t increase in the nativ e oxide la y er from the tem- p ering pro cess. The quan tified SiO 2 la y ers will b e included in the mo del for the o xygen quan tification of the thermo electric nanofilms. F urthermore, the sp ectra of the PGM b eamline measuremen ts sho w con tributions from a carb on signal. This signal originates probably from some surface con tamination (thin organic la y er or dust particles) and is also quan tified in a similar approac h as the SiO 2 la y ers. The results (also shown in T a- ble 4.2) sho w that there is appro ximately ab out 50% more carb on con tamination presen t on DM0150A as compared to DM0149A. Assuming a homogeneous la y er with a densit y of amorphous carb on of 2 g/cm 3 , the measured mass dep ositions refer to con tamination thic knesses of (4.9 ± 0.6) nm on DM0149A and (8.0 ± 1.0) nm on DM0150A. Because of the nativ e silicon dio xide la y er and non-negligible self-absorption in the 51 4. SYNCHR OTR ON RADIA TION BASED ANAL YSIS Figure 4.3.: Sample mo del used for the in tegral quan tification of copp er mass dep osition ˆ m Cu and atomic fraction x . sample matrix, a more complex mo del than a single lay ered, single elemen t sample has to b e applied for the quan tification of the in tegral comp osition of the thermo electric nanofilms. It consists of a carb on con tamination la yer, the gold dop ed copp er o xide la y er (Cu x O y A u z ) itself with a fixed A u-to-Cu ratio ( z = x × C at A u /C at Cu ) and the measured nativ e silicon dio xide la y er on the silicon w afer (Figure 4.3). Since y = 1 − x − z , the atomic concen tration of copp er x and the mass dep osition of copp er ˆ m Cu are the only indep enden t parameters. They can b e calculated b y a fit of the O-K α and Cu-L α,β in tensities. The results of the quan tification are also sho wn in T able 4.2. First of all, it can b e stated that the quan tified mass dep osition of Cu is in agreemen t with the v alues obtained at 10 k e V (T able 4.1). Ho w ever, due to the high uncertain ties of the FPs used in the quan tification, uncertain ties for the absolute quan tified mass dep ositions reac h up to almost 30%. Nev ertheless, when comparing the results of b oth samples to eac h other, the similar excitation conditions and sample comp osition will also ha v e similar effects on the actual FP v alues, so that uncertain ties in the direct comparison can b e exp ected to b e m uc h less. Th us, the increased measured o xygen concen tration from 45 at.% to 48 at.% in the temp ered sample DM0149A can b e exp ected to b e significan t and is most lik ely resulting from ongoing o xidation during the temp ering pro cess. The results also sho w that ev en for the non-temp ered sample a high o xygen concen tration is found, indicating the o xidation of the Cu nanofilm due to the exp osure to air, whic h is also describ ed in [101, 16]. 4.4. Qualitative GIXRF-NEXAFS Already the results of the quan titativ e XRF analysis sho w an induced o xidation in the thin copp er film of sample DM0149A due to the temp ering pro cess. This o xidation is further analyzed with qualitativ e near edge X-ra y absorption fine structure (NEXAFS) measuremen ts at the Cu-L 2 and Cu-L 3 edges with different shallo w angles of incidence to tune the information depth. Since the Cu-L edges are prob ed, the measuremen ts are p erformed at the PGM b eamline. All samples are measured at their cen ter p ositions. F or eac h NEXAFS scan, the energy of the excitation radiation is scanned from 925 e V to 52 4.4. QUALIT A TIVE GIXRF-NEXAFS 970 e V with energy steps of 0.25 e V. Measuremen t time is 15 s p er p oin t, resulting in a total measuremen t time of ab out 45 min utes. The fluorescence radiation is detected with an SDD detector in 90 ◦ geometry to the excitation b eam. The recorded XRF sp ectra are decon v olv ed with the PTB soft w are using a ph ysical bac kground (bremsstrahlung and resonan t Raman scattering) and the detector resp onse function. No normalization to detector efficiency or solid angle of detection need to b e applied, since only relativ e fluorescence in tensities are of in terest. T o discuss and in terpret the NEXAFS sp ectra, t w o reference samples, pure CuO and Cu 2 O, are measured. F or the references, p o wders of pure Cu 2 O (from Sigma Aldric h) and CuO (from storage of the researc h group of Prof. Rademann) are pressed with p otassium bromide to pallets, applying 5 tons pressure b y a h ydraulic lev er press. They are moun ted and aligned along with the t w o thermo electric samples DM0150A and DM0149A on the sample holder. The references are measured with a shallo w incidence angle of 5 ◦ to apply similar measuremen t conditions as for the thermo electric nanofilms. The latter are also measured at sev eral shallo w incidence angles (0.8 ◦ , 3.4 ◦ and 15.0 ◦ for DM0150A and 0.8 ◦ , 1.5 ◦ , 3.4 ◦ and 15.0 ◦ for DM0149A) to tune the information depth. By c hanging the energy of the inciden t radiation o v er the absorption edge of the main comp onen t of the thermo electric nanofilms, the p enetration depth of the radiation is v aried. This leads to a v ariation of the n um b er of excited Cu atoms during the energy scan. Therefore, the in tensit y of the Cu fluorescence is not purely dep enden t on the c hange of the absorption co efficien t, but also on the n um b er of Cu atoms in the excitation v olume. This self-absorption effect of the inciden t radiation can deform the NEXAFS sp ectra and sev eral approac hes to correct for it can b e found in the literature [102, 103, 104, 105]. Usually , the self-absorption correction is p erformed b y using tabulated v alues or reference measuremen ts of the atten uation co efficien ts and applying them to fluorescence calculations (using Equation 2.1.1) of a kno wn sample comp osition. In the follo wing, a less complex self-absorption correction is applied to the measured copp er o xide nanofilms and shortly motiv ated. A simple appro ximation for a self-absorption correction is p erformed b y normalizing the fluorescence in tensities of the Cu signal to the O-K α in tensities. Since in the scanned energy range the absorption co efficien t for o xygen is almost constan t, the normalization to the fluorescence in tensit y of O-K α is effectiv ely a normalization to the n um b er of excited o xygen atoms. In the case of constan t concen trations of O and Cu throughout the whole thermo electric la y er, this normalization w ould accoun t for the v ariation of p enetration depth of the incoming X-ra y b eam and th us correct for self-absorption, as can b e seen b y applying the assumptions to the Sherman equation. Before in tegration, the Sherman equation (Equation 2.1.1) in Section 2.1.1 yields 53 4. SYNCHR OTR ON RADIA TION BASED ANAL YSIS N i,j = N pr ( E pr ) G ( E i,j ) ξ i,j,s ( E pr ) × Z d 0 ρ ( x ) C i ( x ) exp − Z x 0 µ ∗ tot ( E pr , E i,j , x 0 ) ρ ( x 0 )d x 0 d x , with µ ∗ tot ( E pr , E i,j , x 0 ) = µ tot ( E pr , x 0 ) sin( ψ pr ) + µ tot ( E i,j , x 0 ) sin( ψ fl ) , (4.4.1) if assuming a depth dep enden t sample comp osition for no w. Th us, b ecause of the dep endence of µ ∗ tot ( E pr , E i,j , x 0 ) on the primary photon energy , the detected n um b er of fluorescence photons N i,j is not directly prop ortional to the absorption co efficien t τ i,s ( E pr ) in ξ i,j,s ( E pr ). In the follo wing, the effect of the nor- malization to the o xygen fluorescence in tensit y is considered. First, self-absorption of the fluorescence radiation is small compared to the absorption of primary radiation, due to shallo w inciden t and steep detection angles. If therefore appro ximating µ ∗ tot ≈ µ tot ( E pr , x 0 ) / sin( ψ pr ), the ratio of the copp er and o xygen fluorescence is N Cu ,j 1 N O ,j 2 = G ( E Cu ,j 1 ) G ( E O ,j 2 ) ξ Cu ,j 1 ,s 1 ( E pr ) ξ O ,j 2 ,s 2 ( E pr ) × Z d 0 ρ ( x ) C Cu ( x ) exp − Z x 0 µ tot ( E pr , x 0 ) ρ ( x 0 ) sin( ψ pr ) d x 0 ! d x × Z d 0 ρ ( x ) C O ( x ) exp − Z x 0 µ tot ( E pr , x 0 ) ρ ( x 0 ) sin( ψ pr ) d x 0 ! d x ! − 1 (4.4.2) As can b e seen, the outer in tegral cancels out, but only if indeed the concen trations C Cu,O ( x ) are constan t for 0 <x<d . This is surely true for the t w o reference samples, whic h are made of a single comp osition CuO and Cu 2 O, resp ectiv ely . During the NEX- AFS exp erimen t, only the primary radiation energy E pr is c hanged. So with ξ O ,j 2 ,s 2 ( E pr ) ≈ constan t for o xygen in the considered energy range, it follo ws that the d etected coun t rate ratio is prop ortional to the photoionization cross section of the Cu-L 3 shell. N Cu ,j 1 N O ,j 2 ∝ ξ Cu ,j 1 ,s 1 ( E pr ) ∝ τ Cu ,s 1 ( E pr ) (4.4.3) The results of the corrected and not corrected NEXAFS measuremen ts are sho wn in Figure 4.4 and compared to data from literature [106]. The latter is obtained by X-ra y absorption sp ectroscop y in the total electron yield, th us self-absorption effects should 54 4.4. QUALIT A TIVE GIXRF-NEXAFS Figure 4.4.: Near edge X-ra y absorption fine structure (NEXAFS) sp ectra at the Cu-L 3 edge of a) CuO and b) Cu 2 O. After the applied self-absorption correction, the NEXAFS sp ectra measured at the PGM b eamline (dashed) agree m uc h b etter to literature [106] (solid) than the NEXAFS profiles without correction (dotted). The literature data is shifted b y 0.85 e V to o v erlap the p eaks. b e small [107] ∗ . The energy axes of the literature sp ectra are b oth shifted b y 0.85 e V to o v erla y the p eak p ositions to those of the measured data. The energy resolution of the literature data is giv en with 0.4 e V, whic h implies that the absolute uncertain t y is probably similar or ev en higher. F or the measuremen ts p erformed at the PGM b eamline at BESSY I I the energy resolution is similar, so that the energy offset in total migh t originate from the differen t energy calibrations. F or b oth materials, the applied self- absorption correction leads to a b etter agreemen t of the measured NEXAFS sp ectra to the literature data, whic h indicates the v alidit y of the metho d. The main p eak in b oth, the CuO and Cu 2 O sp ectra, is a transition from Cu 2p 3 / 2 in to empt y d-states [108, 106]. In the sp ectrum of CuO (Figure 4.4 a)), the p eak is at 930.4 e V. Th us, also the small p eak at 930 e V in the sp ectra of Cu 2 O (Figure 4.4 b)) migh t originate from some CuO con tributions. This is lik ewise argued in [106], where also changes in in tensit y of this p eak w ere detected for differen t samples. The main p eak in the Cu 2 O sp ectra is at 932.8 e V and has a broad shoulder on the high energy side, whic h is probably caused b y a large Cu 2 p core-hole p oten tial [109]. Both sp ectra are w ell distinguishable and can b e used as fingerprin t for the t w o copp er comp ounds in the follo wing in v estigation of the thermo electric nanofilms. Here, the normalization to the o xygen K α fluorescence is applied, to o. Ev en if the assumptions of negligible self-absorption of the fluorescence ∗ Self-absorption could app ear, if shallo w inciden t angles are used such that the excitation depth is com- parable to the depth, where electrons are detected from. Ho w ev er, no glancing excitation conditions are men tioned in the resp ectiv e pap er, indicating no sp ecial effort b y the authors to realize those. 55 4. SYNCHR OTR ON RADIA TION BASED ANAL YSIS Figure 4.5.: NEXAFS sp ectra at the Cu-L 3 edge of a) DM0150A and b) DM0149A for v arious inciden t angles (solid lines). Also shown are the measured reference spectra for CuO and Cu 2 O (dashed lines). All sp ectra are corrected for self-absorption (see text) and normalized to the in tensit y of the main p eak. radiation and constan t copp er and o xygen concen trations are not strictly v alid, the normalization leads to a reduction of p eak shift and damping induced b y self-absorption, similar to the effect on the reference sp ectra in Figure 4.4. The NEXAFS profiles of the Cu-L 3 edge for differen t angles of the inciden t b eam are sho wn in Figure 4.5 a) for DM0150A and in b) for DM0149A. They are plotted together with the t w o reference measuremen ts of CuO and Cu 2 O with a differen t offset for eac h sp ectrum for reasons of clarit y . The 3 sp ectra of DM0150A are clearly dominated b y con tributions from Cu 2 O, showing that ev en the non-temp ered sample is strongly o xidized, whic h is also found in the integral quan tification in the previous Section 4.3. Only at shallo w angles the sligh tly increasing p eak at 930 e V indicates some minor con tributions of CuO close to the surface. In all 4 sp ectra of sample DM0149A, t w o p eaks, referring to the strong resonances of CuO and Cu 2 O, are visible. Ho w ev er, the p eak ratio c hanges, indicating a CuO-ric h phase close to the surface (shallo w angles) and an increase of Cu 2 O con tributions with depth (increasing angles). Th us, also the NEXAFS measuremen ts supp ort the findings of the in tegral quan tification (Section 4.3) that the temp ering pro cess induced further o xidation. Note that the app earance of b oth p eaks (CuO and Cu 2 O) at 15 ◦ do es not necessarily mean that b oth o xides are presen t close to the silicon substrate, since the signal is an in tegrated information of the differen t comp ounds o v er the whole la y er. As exp ected, due to the c hanging concen trations of copp er and o xygen in the ther- mo electric la y ers, the applied self-absorption correction is not accurate enough to mo del 56 4.5. DEPTH PR OFILING WITH GIXRF the v arious sp ectra b y linear com bination of references from Cu, Cu 2 O and CuO. Es- p ecially for the sample DM0149A, where a strong comp ositional inhomogeneit y is seen in the qualitativ e NEXAFS sp ectra, attempts of fitting are unsatisfactory . Also, all the other self-absorption correction algorithms men tioned ab o v e, can probably not lead to more reliable NEXAFS sp ectra whic h could b e used for a more quan titativ e analysis, since they all require a kno wn sample comp osition. In [89], a differential algorithm for the analysis of t w o subsequen tly buried titan o xide la y ers is v alidated. The basis of the analysis are NEXAFS measuremen ts with v arying inciden t angles, similar to the mea- suremen ts p erformed here. It migh t b e p ossible to apply an adapted algorithm for the presen t sample system, but the smo oth v ariation of the comp osition in con trast to t w o w ell defined la y ers as used in [89] will increase the complexit y of the analysis. Su c h an approac h, whic h migh t b e pursued in future w ork, is not in the scop e of this thesis. The NEXAFS measuremen ts first of all indicate the progressed o xidation of the surface near region due to the temp ering pro cess in DM0149A. Secondly , they sho w the imp or- tance of sync hrotron radiation based analysis, since c hemical information with a depth resolution in the nm range is not a v ailable with a lab oratory setup. Ho w ev er, a second approac h to similar information is gained b y analyzing stoic hiometric c hanges with depth b y angular resolv ed XRF. The p oten tials and limitations of the metho d, whic h is more easily adapted to a lab oratory setup, are demonstrated by GIXRF in v estigations with sync hrotron radiation in the follo wing Section 4.5. Again, the thermo electric nanofilms DM0150A and DM0149A are used for demonstration exp erimen ts. 4.5. Depth Profiling with GIXRF Grazing incidence XRF measuremen ts are p erformed during the b eam time at the KMC b eamline at 2700 e V to in v estigate the angular profiles of A u-M α , Cu-L α,β and O-K α qualitativ ely . The incident angle of the radiation is v aried b y tilting the sample. The v ertical rotational axis in tersects the measuremen t p osition and ensures a stationary excitation. The measuremen t angles range from -0.1 ◦ to 1.5 ◦ in steps of 0.025 ◦ and additionally from 1.6 ◦ to 4 ◦ in 0.1 ◦ steps. With a measu remen t time of 45 s, the total measuremen t time is a bit more than 1 h. Figure 4.6 sho ws the normalized GIXRF pro- files for a) DM0150A and b) DM0149A. With increasing inciden t angle, the fluorescence in tensit y of all elemen ts increases, passes the inflection p oin t at ab out 0.9 ◦ and reac hes an in tensit y maxim um at ab out 1.1 ◦ . Then, the intensit y drops, whic h is t ypical for a thin film. As can b e seen, the GIXRF profiles of Cu, O and A u ha v e a v ery similar shap e and a similar p osition of the inflection p oin t (Figure 4.6 c) and d)), whic h indi- cates a similar in-depth distribution of these elemen ts. Ho w ever, the o xygen signal in the 57 4. SYNCHR OTR ON RADIA TION BASED ANAL YSIS Figure 4.6.: Grazing Incidence X-ra y Fluorescence (GIXRF) measuremen ts of O, Cu and A u recorded at the KMC b eamline with 2700 e V incident photon energy . The normal- ized GIXRF profiles and their deriv ativ es are shown for the thermoelectric thin films DM0150A (not temp ered) in a) and c) and for DM0149A (temp ered) in b) and d). 58 4.5. DEPTH PR OFILING WITH GIXRF GIXRF profile of DM0150A is increased for lo w angles. This signal migh t originate from organic con tamination on the sample surface, whic h is clearly detected in the GIXRF profiles of the measuremen ts at the PGM b eamline, as will b e sho wn in Section 4.5.3. A ctually , differences of the in-depth distribution of Cu and O are exp ected from the qualitativ e NEXAFS measuremen ts in Section 4.4. Esp ecially in sample DM0149A, concen tration gradien ts for Cu and O should b e presen t due to the induced o xidation b y temp ering. Therefore, measuremen ts at the PGM b eamline with an excitation energy of 1060 e V and th us increased depth sensitivit y and sensitivit y for ligh t elements are used for a quan titativ e depth profiling approac h. Both samples DM0150A and DM0149A are measured at the same sp ot as is used for the quan tification with con v en tional 45 ◦ geometry (Section 4.3). The fo cus size (FWHM) of the sync hrotron b eam at the PGM b eamline is 40 × 140 µ m 2 (vertical × horizon tal) and the horizon tal fo otprin t size is enlarged b y 1 / sin( ψ pr ). This leads to a horizon tal fo otprin t size of ab out 8 mm at an inciden t angle ψ pr = 1 ◦ and ≈ 0.4 mm at ψ pr = 20 ◦ . F or ev ery GIXRF scan, angles of the sample are v aried from -0.4 ◦ to 8 ◦ in steps of 0.1 ◦ and subsequen tly from 8.5 ◦ to 20 ◦ in steps of 0.5 ◦ . The measuremen t time for eac h angle is 60 s resulting in a total measuremen t time of almost 2 h for ev ery GIXRF scan. Figure 4.7 sho ws the normalized GIXRF profiles for a) DM0150A and b)-d) DM0149A. The plotted uncertain ties refer to coun ting statistics and an additional 1% uncertain t y from the calculated inciden t flux determined with a calibrated photo dio de [110]. In general, the GIXRF profiles of the main comp onen ts of the thermo electric la y ers, Cu and O, follo w the t ypical trend of a thin la y er. After reac hing an in tensit y maxim um at shallo w angles (b elo w 5 ◦ ), the in tensit y drops b ecause of the reduced absorption of the inciden t radiation in the thin la y er. Ho w ev er, also some rather unusual features are presen t at shallo w angles for the GIXRF profiles of O and C. In Figure 4.7 a), the high in tensit y for O and C b elo w 2 ◦ indicates a strong surface con tamination with probably organic comp ounds and w ater. Also, the rather constan t C signal, ev en for high angles, is somewhat un usual. This b eha vior, as will b e demonstrated in some detail in Section 4.5.3, can b e explained b y a large particle, whose fluorescence con tributes with a constan t signal to the GIXRF profiles sho wn here. A less strong signal from organic con taminan ts (C and O) is seen for the measuremen ts on sample DM0149A (Figures 4.7 b)-d)) and also for steep er angles the C fluorescence seems to follo w the exp ected trend of a thin la y er. The 2 measuremen ts in c) and d) are obtained at +2 mm and -2 mm v ertical distance from the cen ter p osition of DM0149A. The Cu and O signals are rather similar and indicate a homogeneous structure within at least 2 mm distance from the cen ter. Only the GIXRF profiles for O and C at shallo w angles differ significantly . Clearly , surface con taminan ts are not ev enly distributed on the sample. 59 4. SYNCHR OTR ON RADIA TION BASED ANAL YSIS Figure 4.7.: Normalized GIXRF measuremen ts of C, O and Cu p erformed at the PGM b eamline with 1060 e V incident photon energy . a) sho ws the GIXRF profiles of the copp er o xide nanofilm sample DM0150A (not temp ered) and b) the GIXRF profile of DM0149A measured at the cen ter p osition. c) and d) sho w the GIXRF profiles of DM0149A (temp ered) measured in 2 mm horizon tal distance to the cen ter p osition. 60 4.5. DEPTH PR OFILING WITH GIXRF Figure 4.8.: a) Sc hematic of the 1-la yer model used in the fitting pro cedure of the GIXRF profiles. b) Applied mo del for the densit y in the Cu x O y la y er. Sev eral further differences b et w een the GIXRF profiles of the non-temp ered sample DM0150A in a) and temp ered sample DM0149A in b) are visible. While the inflection p oin t (0.6 ◦ ± 0.1 ◦ ) and the maxim um p osition (3.2 ◦ ± 0.1 ◦ ) are the same for the Cu GIXRF profiles of b oth samples, the slop e at higher angles is sligh tly steep er for DM0150A, indicating a lo w er mass dep osition of Cu. Ho w ev er, the main differences are visible in the o xygen signal (apart from the con tamination con tributions b elo w 2 ◦ ), whic h displa ys a broader p eak and a shallo w er decrease to w ards steep er angles for sample DM0150A. If these differences in the GIXRF profiles of the temp ered and not temp ered sample are indeed significan t with resp ect to the sample depth dep enden t o xidation state, is analyzed in three subsequen t steps. In these steps, the complexit y of the applied mo del for the thermo electric nanofilms is subsequen tly increased to b etter describ e the GIXRF profiles and the analytical results concerning the sample structure are discussed. 4.5.1. 1-La y er Mo del The easiest mo del of the sample consists of one single la y er con taining O and Cu on a Si substrate (Figure 4.8 a)). The small gold con ten t in the thermo electric nanofilms will b e neglected for no w and tak en in to accoun t, when more quantitativ e results are en visaged. F rom XRF measurements on the substrate just next to the sputtered sample area, an o xygen mass dep osition is quan tified and a la y er thic kness for the nativ e SiO 2 calculated (Section 4.3). This la y er is added b et w een copp er o xide film and substrate for the mo dels of sample DM0150A and DM0149A, resp ectiv ely . Figure 4.9 sho ws the sim ulated GIXRF profiles for the 1-la y ered mo del with a comp o- sition of pure Cu, Cu 2 O, CuO and Cu x O y with x and y = 1 − x b eing the atomic fractions of Cu and O as determined in Section 4.3. F or all sim ulations here and hereafter, the influence of the samples optical prop erties (reflection and refraction at the in terfaces) is tak en in to accoun t b y the XSW mo del describ ed in Section 2.2.2. The densit y of the la y er is set to bulk densit y of 8.94 g/cm 3 for Cu, 6.0 g/cm 3 for Cu 2 O and 6.31 g/cm 3 for CuO. As densit y for the Cu x O y la yer, a linearly in terp olated v alue b et w een the former pure comp ounds is used (Figure 4.8 b)). The densities of thin la y ers can indeed differ 61 4. SYNCHR OTR ON RADIA TION BASED ANAL YSIS Figure 4.9.: Sim ulated and measured GIXRF profiles of Cu and O for the copp er o xide nanofilms DM0150A (Cu-L α,β in a) and O-K α in c)) and DM0149A (Cu-L α,β in b) and O-K α in d)). The inciden t photon energy is 1060 e V. 62 4.5. DEPTH PR OFILING WITH GIXRF from bulk densities b y sev eral p ercen t. Suc h a difference affects here only sligh tly the inflection p oin t of the curv es at shallo w angles and is neglected for the follo wing sim u- lations. La y er roughness, mo deled by Deb y e-W aller factors, also sho w only small effects on the sim ulated curv es and are set to zero for the presen t purp ose. The fluorescence calculations are carried out for eac h measuremen t angle and normalized to the resp ectiv e solid angle of detection. The sim ulated curv es in Figure 4.9 are fitted to the measuremen t at the steep est angle ( ≈ 20 ◦ ) b y adjusting the la y er thic kness. A t steep inciden t angles, geometrical and XSW effects are smallest and so are the uncertain ties. F or the fitting, here and hereafter, a weigh ted least square fit is used within a Python program (curv e fit from the pac kage scip y .optimize, using a T rust Region Reflectiv e algorithm for minimization [72]). As can b e seen, a go o d agreemen t b et w een measured and sim ulated v alue at 20 ◦ is only ac hiev ed for sample DM0149A with the sim ulation of Cu x O y , where the la y er thic kness is determined to (36.0 ± 0.6) nm. The giv en uncertain ties of the fitting algorithm ha v e to b e used with some care and should probably b e understo o d as lo w er limit, as will b e discussed more detailed in Section 4.5.3. In all other curv es, the Cu signal is o v erestimated and the O signal underestimated. This b eha vior is exp ected for all sim ulations but the sim ulation of Cu x O y with x and y tak en from the in tegral quan tification (Section 4.3), since the relative concen tration of Cu to O do not fit. F or the sim ulation of Cu x O y for sample DM0150A, an additional O signal from con tamination is presen t, as is discussed in Section 4.5.3. Ov erall, the results of the 1-la y er mo del, when comparing the whole range of the GIXRF profiles, are not satisfactory . The maximum of the Cu GIXRF profiles for b oth samples is o v erestimated b y all sim ulations. The same is true for the shallow angular regions of the O GIXRF profiles, ev en for the sim ulations of CuO and Cu x O y of DM0149A, where at least the slop e of the curv e ab o v e 4 ◦ seems to fit reasonably . The sim ulated curv es for CuO are closer to the measuremen ts than the curv es for Cu 2 O or pure Cu, indicating a strong progression of the o xidation pro cess. How ev er, this information migh t b e misleading and actually originating from a wrong mo del of the GIXRF sim ulations. Indications for this can b e found in the GIXRF-NEXAFS sp ectra (Section 4.4), where the main con tribution originates from Cu 2 O. 4.5.2. N -La y er Mo del A c hange in the Cu o xidation state with depth in the copp er o xide nanofilms is exp ected from the GIXRF-NEXAFS measuremen ts. This implies a change of the major chemical comp ound to b e found in a sp ecific depth region and th us a c hange of the concen trations of the elemen ts in depth. T o accoun t for suc h concen tration c hanges in the presen t 63 4. SYNCHR OTR ON RADIA TION BASED ANAL YSIS Figure 4.10.: Sc hematic illustration of the N -lay er mo del with increasing subla y er thic k- ness. mo del of the copp er o xide nanofilms, an N -lay er mo del is tested (Figure 4.10). Instead of a single la y er with v ariable thic kness, no w a sample mo del consisting of N discrete la y ers is used. Eac h la yer consists of a Cu x ( n ) O y ( n ) A u z ( n ) comp ound, where x ( n ), y ( n ) and z ( n ) are the atomic fractions of eac h elemen t in la y er n . Ho w ev er, only x ( n ) is used as v ariable, since z ( n ) = x ( n ) × ( C at A u / C at Cu ) is calculated from the quan tification in Section 4.3 and y ( n ) = 1 − x ( n ) − z ( n ). The densit y of eac h la y er is again interpolated b et w een the bulk densities of the pure comp ounds Cu, Cu 2 O and CuO (Figure 4.8 b)). The thic kness of eac h la y er d ( n ) dep ends on the la y er n umber n and the summed thic kness of all N la y ers d tot according to d ( n ) = d tot × n P N i =1 i (4.5.1) E.g. for N = 5 the la y ers ha v e an increasing thic kness d (1) = d tot × 1 / 15, d (2) = d tot × 2 / 15 etc., whic h is partly accounting for reduced information from lo w er la y ers due to the absorption in the la y ers on top. A similar approach is motiv ated in [43]. In the fitting pro cedure, only the total la yer thic kness d tot is used as free parameter. Besides the N + 1 free parameters, x ( n ) and d tot , t w o more scaling parameters for the fluorescence in tensities of copp er and o xygen, s Cu and s O , are applied. They should accoun t for the uncertain ties of the absolute v alues of the fundamen tal parameters used in the photo pro duction cross section ξ i,j,s ( E pr ) in Equation 2.1.1 in Section 2.1.1. T able 4.3 summarizes the fitting parameters, their starting v alues and the applied b oundaries. Sim ultaneous fitting of the Cu and O GIXRF profiles is carried out with the describ ed mo del for v arious la y er n um b ers N ≤ 10 for b oth samples. Apparen tly , during the GIXRF measuremen ts, the sample surface w as radially misaligned b y up to 60 µ m from the piv otal p oin t of the goniometer. This has a strong effect on the effectiv e solid angle of detection for shallo w inciden t angles, as is sho wn in detail in App endix D. T o considered this during the fitting pro cedure, an additional uncertain t y , b eing the relativ e difference b et w een the effectiv e solid angle of detection with and without sample misalignmen t, is applied to the measured data. Figure 4.11 sho ws the results of the fitted GIXRF profiles for N = 1 , 2 , 5 and 10. The 64 4.5. DEPTH PR OFILING WITH GIXRF T able 4.3.: P arameters and settings for the non-linear least square fit applied to the GIXRF profiles with the N -la yer model. Besides in tensit y factors s Cu,O and total lay er thic k- ness d tot , for eac h of the N la y ers an atomic fraction x ( n ) for Cu is fitted. starting v alue b oundary N × x ( n ) / at.% 75 50 - 100 d tot / nm 30 0 - 100 s Cu 1.0 0.50 - 1.5 s O 1.0 0.50 - 1.5 Figure 4.11.: Fitted and measured GIXRF profiles for Cu and O of the thermo electric nanofilms DM0150A (a) and c)) and DM0149A (b) and d)), applying the N -la yer mo del with N = 1, 2, 5 and 10. The inciden t photon energy is set to 1060 e V. The fit is impro v ed compared to the 1-la yer model mainly b ecause of the introduced intensit y factors. Ho w ev er, it shows still some deviations in the shallo w angular regime b elo w 3 ◦ and o v erall for the O GIXRF profile of DM0150A (c)). 65 4. SYNCHR OTR ON RADIA TION BASED ANAL YSIS T able 4.4.: Results of the fitted parameters for the N -lay er mo del with N = 1 for the ther- mo electric nanofilm samples DM0150A (not temp ered) and DM0149A (temp ered). The uncertain ties are deriv ed from Equation 2.4.3 in Section 2.4.2 and should b e un- dersto o d as lo w er b oundary b ecause of the imp erfect model and the non-normal errors applied to the measuremen t. DM0150A DM0149A s Cu / % 70.5 ± 0.5 86.2 ± 1.5 s O / % 127 ± 14 130 ± 40 d tot / nm 43.5 ± 1.4 40 ± 3 ˆ m Cu / (ng × cm − 2 ) 22600 ± 800 20500 ± 1200 ˆ m O / (ng × cm − 2 ) 3490 ± 110 3350 ± 200 ˆ m A u / (ng × cm − 2 ) 740 ± 90 620 ± 80 sim ulated curv es appro ximate the measuremen ts visibly b etter than with the 1-la y er mo del. Ho w ev er, since ev en the curv es for N = 1 agree b etter with the measurements, the effect is mainly caused b y the additional factors for the o v erall fluorescence in tensit y , s Cu and s O . They allo w for an adjustmen t of the absolute fluorescence in tensit y , when c hanging the total la y er thic kness d tot . The latter is similar to a c hange in the total mass dep osition of the elemen ts and dominates the shap e of the slop e ab o v e 3 ◦ . The fitting v alues for s Cu and s O together with d tot and the corresp onding mass dep osition for N = 1 can b e found in T able 4.4. A few p oin ts of discussion arise from the results. First, the giv en uncertain ties ha ve to b e discussed. They are deriv ed from Equation 2.4.3 in Section 2.4.2. T o b e reliable, the mo del needs to b e appropriate (see the discussion in Section 2.4.2) and the mea- suremen t uncertain ties need to b e normal distributed. The latter is not true b ecause of the unkno wn, systematic uncertain ties of the effectiv e solid angle of detection due to misalignmen t. Ho w ev er, the uncertain ties are given here nev ertheless, but they should b e understo o d as lo w er limits. They can b e used to deriv e hin ts ab out the correctness of the mo del and are discussed in detail. The Cu fluorescence in tensit y is reduced b y 15-30% ( s Cu ), not only for the fit with N = 1, but also for all tested la y er n umbers N = 1 , 2 , 3 , 5 , 7 , 10. This seems reasonable with resp ect to the estimated uncertain ties for L-shell fluorescence, whic h are in this w ork estimated with 26% (see App endix C). F or o xygen on the other hand, the signal is increased b y 30% for N = 1, but also v alues at the b oundary limit of 50% are reac hed in some fits. In addition to the high uncertainties giv en b y the fit, the results for s O are not accurate. As will b e seen in the fitting pro cedure with the next mo del, a large part 66 4.5. DEPTH PR OFILING WITH GIXRF of the o xygen fluorescence seems to originate from surface con tamination and is th us not co v ered in the presen t mo del. Therefore, also all the v alues based on the o xygen signal ( ˆ m O directly and d tot via the applied densit y mo del) are not trustw orth y and should b e in terpreted with care. Surely , the uncertain ties of these v alues are strongly underestimated. Nev ertheless, the mass dep osition results for Cu (and th us also A u) should b e rather trust w orth y , since they mainly dep end on the slop e ab o v e 3 ◦ , where neither large un- certain ties in the solid angle of detection, nor effects due to con tamination app ear. The v alues supp ort the quan tification results of the commercial setup in Chapter 3 and are ab out 15% higher than the quan tification results obtained at the KMC b eamline with Cu-K α . This indicates an underestimation of the FP or instrumental uncertain ties in the latter approac h. By in tro ducing a factor for the fluorescence in tensit y , the determi- nation of the mass dep ositions do es not rely on the precise kno wledge of the solid angle of detection (whic h do es not c hange drastically ab o v e 2 ◦ for the presen t setup), the in- ciden t photon flux or most fundamen tal parameters, leading to decreased uncertain ties. Esp ecially in the soft X-ra y range, the fluorescence yield has high uncertainties, whic h directly affect the uncertain ties of la y er quan tification, as is illustrated in Section 4.3. Th us, the sho wn results nicely demonstrate the adv an tages of reference-free, FP based quan tification, using a range of inciden t (or emission) angles instead of a single measure- men t. This already indicates that even without a fully calibrated setup and with rather high uncertain ties of the FPs in the soft X-ra y range, single la y ered nanofilm quan tifica- tion is feasible with the scanning-free GEXRF setup describ ed later in Chapter 5. But ev en for sync hrotron applications, the describ ed quan tification approac h could b e useful to decrease uncertain ties. The main sensitivit y for in-depth comp ositional c hanges of the thermo electric nanofilms is found in the GIXRF profiles at shallo w angles. Figure 4.12 sho ws exemplarily an en- larged view of the GIXRF profiles of Cu and O for the temp ered sample DM0149A. As can b e seen in a), the mo del already fails to precisely describ e the measured Cu GIXRF profile. This is ev en more pronounced for the O GIXRF profile in b), ev en though there is some impro v emen t for fits with N > 1. The main reason is an additional con tribution from con tamination, whic h has to b e describ ed in the mo del accordingly . Otherwise, the con tribution of the con tamination in terferes with the O GIXRF profile of the ther- mo electric nanofilms and leads to wrong depth profiling results (also affecting the Cu GIXRF profile). 67 4. SYNCHR OTR ON RADIA TION BASED ANAL YSIS Figure 4.12.: Zo om to the shallo w angular regime for a) the Cu and b) the O GIXRF profiles in Figure 4.11 b) and d), resp ectiv ely . 4.5.3. N -La y er Mo del with Contamination F rom the measuremen ts sho wn in Figure 4.7 on page 60, the o xygen signal and esp e- cially the carb on signal at shallo w angles strongly indicate surface con tamination. T o accoun t for this in the sim ulated GIXRF profiles, sev eral p ossibilities to in tro duce suc h a con tamination are tested. First, the XSW field ab o v e the copp er oxide la y er, whic h can excite the con tamination la y er, exhibits only low in tensities at the first few nm abov e the surface for inciden t angles b elo w 0.3 ◦ (Figure 2.4 in Section 2.2). Thus, the high carb on signal at ab out 0.1 ◦ cannot b e explained with a smo oth con tamination la y er on a reflecting surface. Therefore, some fluorescence in tensit y has to originate from con tamination, whic h is not affected b y the destructiv e in terference of the XSW field in this angular and spatial region. This could b e the case, either if the XSW field is disturb ed b y rough interfaces, or if fluorescence in tensit y of the con tamination originates from a larger distance to the reflecting sample surface than a few tens of nanometers ∗ . Therefore, the calculated fluorescence of a single la y er without XSW field mo dification is added to the fluorescence of the mo del sample. The summed in tensities are again w eigh ted with the effectiv e solid angle of detection of eac h inciden t angle. Note that the absorption of primary and fluorescence radiation in the con tamination la y er, whic h migh t affect the signal from the thermo electric thin films, is negligible with resp ect to the o v erall uncertain ties and not implied in the presen t ∗ The XSW field v anishes ab o ve the surface because of the finite temp oral coherence of the synchrotron b eam. Assuming a resolving p o wer of E/∆ E =5000, the longitudinal coherence length l coh = λ 2 / (2 × ∆ λ ) w ould b e ≈ 3 µ m for 1060 e V photons. That means, at a shallo w incident angle of 1 ◦ , the XSW field should v anish at ab out 50 nm ab o v e the reflecting surface. 68 4.5. DEPTH PR OFILING WITH GIXRF calculations. A second p ossibilit y for the in tro duction of con taminan ts is a nm or sub-nm thic k con tamination la y er, whic h is excited by the XSW field. The XSW field enhancemen t w ould b e largest at ab out 2.2 ◦ (Figure 2.4 in Section 2.2) and indeed some p eak at ab out 2 ◦ is visible for sample DM0149A (in Figure 4.7 on page 60). How ev er, in the fitting attempts for sample DM0150A and DM0149A, this la y er usually con v erged to a thic kness b elo w 0.5 nm and the con tribution to the GIXRF profiles is small. T o ke ep the n um b ers of free parameters small, this la yer is not included in the final fitting pro cedure. Since unexp ectedly high in tensities at v ery shallo w angles are found for b oth, C-K α and O-K α , the comp osition of the con tamination la y ers in the mo del is set to C x O 1 − x with atomic fractions x for carb on. The carb on con ten t migh t originate from dust and o xygen probably from w ater, whic h adheres on the surface if the sample is stored in am bien t conditions. Ho w ev er, ev en with the additional con tamination mo del describ ed so far, the GIXRF profiles cannot b e repro duced completely . Mainly the rather slo w decrease of fluorescence in tensit y ab o v e 3 ◦ for C and O in the GIXRF profiles of DM0150A and for C in the GIXRF profiles of DM0149A cannot b e sim ulated in this w a y . Regarding the C GIXRF profile for sample DM0150A, the almost constan t in tensit y signal with increasing angles is at ypical for la y ered structures. Because of the decreasing path length in a thin la y er with increasing inciden t angle, the absorption probabilit y of the primary radiation should also decrease, leading to the t ypical decline of fluorescence in tensities. Ho w ev er, if a spherical particle on a (nonreflecting) surface is fully irradiated for all inciden t angles, the fluorescence signal of that particle is indeed exp ected to b e constan t. Therefore, for the sim ulation of sample DM0150A, a constan t con tribution to the fluorescence in tensities of C is added as fitting parameter, to sim ulate suc h particles (or particle). Since the con tamination seems to consist of C and O (b oth elemen ts sho wing high in tensities b elo w 3 ◦ ) and also the O signal exhibits a v ery flat gradien t at steep er angles (ab o v e 5 ◦ ) an offset as fitting parameter is also added to the O GIXRF profile. While the fitted sim ulation of the Cu GIXRF profile do es not c hange m uc h (not sho wn), the added parameters lead to m uc h b etter agreemen t b et w een sim ulated and measured C and O GIXRF profiles (Figures 4.13 a) and c)). The flat decline of the fluorescence in tensities with steep er angles can no w b e describ ed b y the sim ulation, even with a 1-la y er mo del for the copp er o xide nanofilm. Also, at shallo w inciden t angles b elo w 3 ◦ (Figures 4.13 b) and d)), the sim ulated GIXRF profiles agree muc h b etter with the measuremen t than without the additional parameters for the con tamination. This indicates that the con tamination mo del applied is a sufficien t estimate of the real 69 4. SYNCHR OTR ON RADIA TION BASED ANAL YSIS Figure 4.13.: Measured and fitted GIXRF profiles for a) C (close-up in b)) and c) O (close- up in d)) of the thermo electric nanofilm sample DM0150A (not temp ered) irradiated with 1060 e V photons and applying contamination con tributions to the N -lay er mo del. 70 4.5. DEPTH PR OFILING WITH GIXRF Figure 4.14.: Sk etc h of the N -la y er mo del with con tamination. A constant con tribution to the fluorescence signal of O and C in the GIXRF profile of DM0150A migh t originate from one or sev eral large particles. Small particles could giv e rise to a signal whic h w ould b e exp ected without XSW field mo difications of the excitation conditions. The measuremen ts also indicate a thin surface con tamination la yer, whic h is not mo deled (see text). In the temp ered sample DM0149A, evidence for a C signal from the copp er o xide matrix is presen t. sample con tamination, whic h is sk etched in Figure 4.14. Here, the constan t fluorescence in tensit y con tribution is originating from the large particles and some fluorescence is emitted b y smaller particles (extensions from surface should b e larger than tens of nm), mainly unmo dified b y the XSW field. The latter is mo deled as a la yer and the X-ra y fluorescence in tensit y is calculated without XSW field. Ev en though not mo deled, the measuremen ts sho w some C and O con tribution to the GIXRF profiles from con taminan ts affected b y the XSW field. This could b e small particles of v arious shap e and size in the nm or sub-nm range, forming a more or less closed, surface lay er. The presence of particles with v arious diameters up to the µ m range is also detected b y atomic force microscop y (AFM) measuremen ts sho wn in Figure 3.2 in Section 3.1, further supp orting the sample mo del dev elop ed here. In terestingly , the applied mo del for DM0150A do es not lead to satisfactory results for the temp ered sample DM0149A. The con tamination la y er thic kness is alw a ys o v eresti- mated b y the fit, to ac hiev e a b etter agreemen t b et w een measuremen t and sim ulation in the high angular regime. Ho w ev er, this leads to ov erv alued fluorescence in tensities in the lo w angular range (red curv e in Figure 4.15 a)). T o get a b etter agreemen t of the in tensit y decrease ab o v e 3 ◦ in the C GIXRF profile, the signal m ust originate from carb on em b edded in a matrix of a somewhat thic k er la y er than a few nm. Therefore, a constan t carb on con tamination is added in to the copp er o xide nanofilm matrix and its atomic fraction k is also fitted. The improv ed agreemen t b et w een measuremen t and sim ulation is sho wn in Figure 4.15 T able 4.5 summarizes the resulting parameters of the final fitting approac h with N = 1. The giv en uncertain ties are again estimated b y the fitting pro cedure and ha v e to b e 71 4. SYNCHR OTR ON RADIA TION BASED ANAL YSIS Figure 4.15.: Fit results of the GIXRF profiles for a) C and c) O with the N -la yer model with con tamination of the temp ered thermo electric nanofilm DM0149A. The red curv e in a) is a fit ( N =1) using only con taminants on top of the surface. The primary photon energy is 1060 e V. b) and d) show close-ups of a) and b), resp ectiv ely . 72 4.5. DEPTH PR OFILING WITH GIXRF T able 4.5.: Fit results of the approac h with the N -la yered model with contamination and N = 1. ∗ in units of (10 -9 photons. × inc. photon − 1 ) DM0150A DM0149A s Cu / % 71.0 ± 0.2 83 ± 4 s O / % 77 ± 4 107 ± 110 s C / % 100 ± 700 100 ± 300 x (con t.) / % 60 ± 120 50 ± 90 d (con t.) / nm 2 ± 7 1.8 ± 0.8 O-K α offset ∗ 8.5 ± 0.5 0.0 ± 0.4 C-K α offset ∗ 2 ± 12 not used d tot 41.0 ± 1.2 58 ± 2 x (1) / at.% 65.7 ± 1.4 50 ± 2 k / at.% not used 20 ± 70 regarded with care. As exp ected, the fitted factor s Cu for the Cu fluorescence in tensit y is similar to the v alues obtained without con tamination (T able 4.4 on page 66), since the Cu signal itself is not strongly affected b y the added con tamination la y er. This is not the case for the O GIXRF profile, whic h explains the c hange in s O as compared to the results of the N -la y er mo del. Ho w ev er, the v alue for s O in the case of DM0150A (77%) is still reasonable with resp ect to the fundamen tal parameter uncertain ties of 20%. Also, the thic kness of the con tamination lay er with a couple of nm is plausible. F or DM0150A, the offset in the O-K α in tensit y seems indeed to b e necessary to ac hiev e a b etter fit of the sim ulated to the measured GIXRF profile, as is indicated by the rather lo w uncertain t y of < 10%. On the other hand, the huge uncertain ties for s C , x (con t.), d (con t.) and the carb on offset probably indicate the somewhat strong coupling of these parameters, eac h v ariation leading to partly similar effects on the GIXRF profile for C. Also, the C GIXRF profile has the lo w est coun ting statistics, whic h also con tributes to the uncertain ties giv en b y the fit. F or the same reasons, uncertain ties for k in the mo del for DM0149A are also h uge. Consequently , since k directly affects the comp osition of the thermo electric thin film and th us also the atten uation co efficien t of the matrix, the uncertain ties of x (1) and d tot giv en by the fit are largely underestimated. In conclusion, the measured GIXRF profiles can b e describ ed b y sim ulations with a mo del of the sample including con tributions of con tamination. F urthermore, not only the presence of con tamination can b e sho wn, but also the high sensitivit y of GIXRF esp ecially in the surface-near region yields information ab out the nature of the con tam- 73 4. SYNCHR OTR ON RADIA TION BASED ANAL YSIS ination. The tested mo dels strongly indicate that carb on and o xygen ric h particles with in termediate size ( > few tens of nm) are present on the surface. Their fluorescence signal can partly b e mo deled b y a thin homogeneous la y er without XSW field mo difica- tion of the excitation. Also, a smo oth film or v ery small particles con taining C and O could b e on the surface (this can b e mo deled b y a thin la y er b eing sub ject to the XSW field), but the signal is probably co v ered b y the fluorescence of the inter mediate parti- cles. Then, also ma jor differences of the C and O GIXRF profiles b et w een DM0150A and DM0149A can b e explained b y further con tributions from con taminan ts. In the measured sp ot on sample DM0150A, a large roughly round or cylindrical particle seems to b e presen t, whic h is irradiated for the whole angular range up to 20 ◦ . In the sp ectrum used for quan tification in Section 4.3, the C-K α in tensit y is m uc h lo w er than exp ected from the in terp olated GIXRF measuremen ts, suggesting that the large particle or most of the large particles are outside of the smaller fo otprin t at 45 ◦ . The GIXRF profiles of DM0149A on the other hand sho w no suc h b eha vior, but here the C GIXRF profile can b e repro duced, if adding C to the copp er o xide nanofilm matrix. Ma yb e the temp ering pro cess led to an in tegration of C in to the copp er thin film. The NEXAFS measuremen ts in Section 4.4 sho w ed qualitativ ely a m uc h stronger c hange of the Cu o xidation state close to the sample surface for the temp ered sam- ple DM0149A, than for the non-temp ered sample DM0150A. This finding cannot b e confirmed b y the presen t GIXRF measuremen ts. The main sensitivit y to a c hange of o xygen concen tration with depth is exp ected from the O-K α GIXRF profile. But since this profile is strongly distorted b y the fluorescence originating from con tamination, the sensitivit y for comp ositional c hanges of the copp er o xide is lost. In principle, also the Cu GIXRF profile is sensitiv e to the somewhat lo w er relativ e c hanges of the Cu con- cen tration. The strongest effect is exp ected in the shallo w angular range, but there the uncertain ties of the effectiv e solid angle of detection are highest. T o comp ensate for these, GIXRF profiles of quotien ts of fluorescence lines from differen t elemen ts in the matrix can b e considered, as is sho wn in [43]. Ho w ev er, this approach cannot be used here, since only copp er as undisturb ed signal is presen t. Ev en though the depth profiles of o xygen and copp er cannot b e determined with sufficien t accuracy with the presen t measuremen ts, the ev aluation of the data surely illustrates the analytical p ossibilities of angular resolv ed XRF measuremen ts and moti- v ates the need for a lab oratory setup. Often, unexp ected results obtained from angular resolv ed XRF measuremen ts are originating from the sample preparation pro cess, mak- ing a quic k feedbac k lo op b et w een analysis and sample preparation imp ortan t. In the case of large facilit y measuremen ts, this feedbac k lo op is usually dela y ed b y the need of applying for b eam time. In the next Chapter 5, the dev elopment of a laboratory setup for GEXRF measuremen ts in the soft X-ra y range is presen ted. After c haracterizing the 74 4.5. DEPTH PR OFILING WITH GIXRF setup with a m ultila y er test sample in Sections 6.2 and 6.3, it will also b e used to test the applicabilit y with one of the thermo electric thin film samples, namely DM0149A. F or this purp ose, the ab o ve dev elop ed N -la y er mo del will b e applied to the GEXRF case. 75 5. Development of Lab o rato ry Scanning-F ree Soft X-ra y GEXRF In the previous c hapter, it has b een sho wn that grazing incidence X-ra y fluorescence (GIXRF) can giv e access to nanometer scaled structural information for complex sam- ples. How ev er, the sho wn measuremen ts w ere p erformed at a sync hrotron radiation facilit y to pro vide a highly collimated and in tense excitation b eam in the soft X-ra y range and use w as made of the absolutely calibrated instrumen tation of the PTB. Suc h measuremen t conditions are difficult to ac hiev e with a lab oratory setup. On the other hand, a lab oratory setup has the adv an tage of b eing readily a v ailable without the need of b eam time prop osals and the related long latencies. And, probably more imp ortan tly , with a larger user and scien tific comm unit y at hand, ev aluation pro cedures for the in- v erse problem of GI- and grazing emission (GE) XRF measuremen ts migh t ev olve more rapidly , making the metho d ev en more p o w erful in general. In the follo wing, the dev elopmen t of a lab oratory soft X-ra y GEXRF sp ectrometer is presen ted with a short in tro duction dealing with the applicabilit y of GI- and GEXRF in the lab oratory . After describing the principle setup, sp ecial emphasis is placed on alignmen t pro cedures to enable precise measuremen ts as w ell as an absolute angular calibration. Then, the ev aluation of single photon ev ents detected with a c harge-coupled device (CCD) camera is in v estigated to mak e use of the energy-disp ersiv e prop erties of a con v en tional CCD camera. Finally , angular calibration and energy-disp ersiv e CCD images are b oth used to obtain GEXRF profiles. Please note that setup design, adjustmen t, calibration and ev aluation pro cedures are dev elop ed for soft X-ra y GEXRF measuremen ts with a laser-pro duced plasma source. Nev ertheless, most of the ideas can b e readily transferred to the hard X-ra y range, where the sample can b e excited e.g. b y an X-ra y tub e equipp ed with a p olycapillary lens, whic h further increases the impact of this w ork. 77 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF 5.1. Grazing Incidence vs. Grazing Emission XRF Sp ectrometer Concept Grazing incidence XRF and grazing emission XRF are related measuremen ts, whic h giv e access to similar information. While the former allo ws for ideal excitation condi- tions with resp ect to excitation depth and self-absorption of the fluorescence radiation (Section 2.2), clear dra wbac ks are the high requiremen ts regarding the excitation b eam and the extended fo otprin t on the sample for shallo w angles. GEXRF measuremen ts on the other hand can b e applied b y an y , also con v ergen t, ionizing radiation (X-ra ys, elec- trons, ions, etc.). This is not only preferable in terms of lab oratory (mainly div ergen t) excitation sources, but can also b e used to guaran tee a small fo otprin t and enhanced lat- eral resolution. Ho w ev er, the fluorescence radiation needs to b e detected under shallo w emission angles. This leads to self-absorption losses in the detected in tensit y and th us, a t ypically lo w er efficiency compared to GIXRF, whic h has to b e comp ensated for. A t the Berlin Lab oratory for Inno v ativ e X-ra y tec hnologies (BLiX), a laser-pro duced plasma (LPP) source is a v ailable for the generation of div ergen t soft X-ra ys [79]. F ur- thermore, a sp ectroscop y c ham b er designed for sample manipulation and realization of v arious sp ectrometer geometries can b e used [111]. If the plasma in teraction c ham b er and the sp ectroscop y c ham b er are com bined in a sp ectrometer, the minim um distance b et w een source and sample p osition is almost 1 m. This mak es the application of X-ra y optics necessary to increase the photon flux in the sample plane. F or a grazing incidence XRF sp ectrometer, parallelizing optics are necessary . Due to the long w orking distance (the optics needs shielding from plasma debris), m ultila y er optics are sup erior to e.g. p olycapillary lenses. Suc h an optics migh t achiev e a solid angle of acceptance of Ω GI opt = 3.1 × 10 − 5 sr (see App endix E). What is lost in the excitation c hannel is comp ensated for in the detection c hannel of t ypical GIXRF sp ectrometers. The detector, usually a silicon drift detector (SDD), can b e aligned with less than 1 cm distance to the sample surface, resulting in ac hiev able solid angles of detection of Ω GI det ≥ 0.3 sr. Ho w ev er, the electronics can only compute and differen tiate photons in the microsecond range. Therefore, the pulsed radiation of the LPP source with pulse length of 1.2 ns leads to sev ere pile-up in the detector, limiting the usable coun t rate of the detector to only ≈ 100 coun ts/s (the rep etition rate of the LPP source is 100 Hz). In a grazing emission XRF sp ectrometer, the m ultilay er optics can ha v e fo cusing prop- erties. This allo ws for a higher angle of acceptance of Ω GE opt = 2.3 × 10 − 3 sr (App endix E), whic h means a factor 75 more flux in the excitation c hannel. Also, more than one optics migh t b e used, further increasing the excitation in tensity (as is applied in Sec- tion 5.2.2). This gain in in tensit y is diminished by the small solid angles of detection 78 5.1. GRAZING INCIDENCE VS. GRAZING EMISSION XRF SPECTR OMETER CONCEPT (Ω GE det ≈ 6 × 10 − 7 sr) in a GEXRF sp ectrometer, whic h t ypically has to b e applied to guaran tee sufficien t angular resolution. Only a scanning-free GEXRF setup can o v er- come b oth the limitations concerning pile-up (with an SDD) and insufficien t solid angle of detection. The o v erall solid angle of detection with a CCD can easily b e of the order of 0.05 sr and the pile-up is prev en ted b y separating the photon ev ents not only temporally (as in the SDD) but also spatially . Th us, a scanning-free GEXRF approac h is clearly fa v orable compared to a GIXRF setup. 79 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF Figure 5.1.: Sc hematic setup for the lab oratory scanning-free soft X-ra y GEXRF measure- men ts. 5.2. Principle Setup The principle setup used for the GEXRF measuremen ts is sho wn in Figure 5.1. The div ergen t radiation of the LPP source is fo cused on to the sample plane b y means of a m ultila y er mirror optics. The optics w ere in a first b eam time t w o one-dimensionally fo cusing mirrors in Kirkpatric k-Baez (KB) geometry and in a second b eam time with impro v ed setup and metho dology t w o t w o-dimensionally fo cusing, toroidal shap ed mir- rors. The optics are shielded from debris by a thin p olymer foil, sputtered with 100 nm of alumin um. Visible ligh t is additionally prev en ted to reach the spectroscopy c ham b er b y an alumin um filter with a thic kness of 1.5 µ m . The sample, p ositioned and aligned with a 7-axes goniometer in the sp ectroscop y c ham b er, is irradiated at ab out 90 ◦ and the fluorescence radiation at shallo w emission angles is detected with an area detector op erated in a single photon coun ting mo de to enable energy-disp ersiv e measuremen ts. 80 5.2. PRINCIPLE SETUP Figure 5.2.: Laser and plasma in tensit y (top) as well as X and Y v alues of the plasma p osition monitoring of the 4Q-dio de (b ottom) for 3 hours of LPP op eration. 5.2.1. Laser-Pro duced Plasma Source A short description of the laser-pro duced plasma (LPP) source is giv en in Section 2.5.2. F or GEXRF measuremen ts, the LPP source is op erated in a con tin uous mo de, pro viding 1.2 ns X-ra y pulses ev ery 10 ms. The pulse energy of the Yb:Y AG laser is t ypically set to 150 mJ for alignmen t and 200 mJ for the GEXRF measuremen ts, to effectively heat the plasma and generate X-ra ys with energies at 1078 e V from transitions in the 20-fold ionized copp er. After up to 9 hours of op eration, the copp er target is replaced b y a freshly p olished target. This ensures high surface qualit y and impro v es p osition and in tensit y stabilit y of the source. F urthermore, the debris protection foil in fron t of the b eamline is sputtered with copp er during op eration, resulting in a reduced transmission of ab out 50% for the 1078 e V radiation after ≈ 9 hours. Therefore, after eac h target replacemen t, the debris-protection foil is renew ed. The GEXRF measuremen ts sho wn in Chapter 6 are the first measuremen ts, where the LPP source is op erated con tin uously o v er sev eral hours. Ov er the whole time, a high stabilit y in the plasma p osition and a mo derate stabilit y with resp ect to X-ra y in tensit y need to b e guaran teed. While the former affects the fo otprin t p osition on the sample and th us the angular resolution of the GEXRF measuremen ts, the latter is less critical, since 81 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF in the scanning-free GEXRF approac h, the shap e of the GEXRF profile is not affected b y in tensit y instabilities. The four-quadran t-dio de (4Q-dio de) pro v es to b e a useful to ol for stabilit y monitoring. Figure 5.2 sho ws the plasma in tensit y measured with the 4Q-dio de and the resp ectiv e laser in tensit y for 3 hours of con tin uous op eration. The laser in tensit y displa ys a v ery stable mean v alue and a standard deviation of 10% o v er almost the whole time. The summed in tensit y of the four quadran ts of the 4Q-dio de, measuring the plasma intensit y , has a larger standard deviation of up to 40% and also the mean in tensit y is decreasing with time. The former can b e explained b y the high non-linearit y and complexit y in the plasma formation and the resulting sensitivit y for c hanges of target p osition and roughness. Moreo v er, the 4Q-dio de has insensitiv e areas b et w een the quadran ts, so that p ositional c hanges of irradiated area (influenced b y p osition c hanges of the plasma) can affect the summed in tensit y . The decrease of mean in tensit y originates from debris, whic h is sputtered on to a protectiv e glass disk in fron t of the laser en trance window. This leads to reduced laser in tensit y in the laser fo cus on the copp er target and consequen tly reduced plasma temp erature and X-ra y in tensit y . After 1 h to 1.5 h of op eration, the LPP source is shortly stopp ed for c hanging the glass disk in v acuum b y means of a mec hanical con v ey er, to coun ter this effect. Also, the p osition of the plasma source is monitored with the 4Q-dio de. In the b ottom panel of Figure 5.2 the X and Y v alues are display ed (see Section 2.5.2). While Y is used for the feedbac k lo op and therefore sho ws a constan t mean v alue, X drifts sligh tly in one direction during op eration but is reset after c hanging the protectiv e glass disk of the laser en trance windo w. This indicates a slight deflection of the laser beam dep ending on the sputtered copp er on the protectiv e glass disk. Ho w ev er, in other measuremen ts (not sho wn here), X can drift rather arbitrarily , revealing a more complex b eha vior of the plasma p osition. Nev ertheless, the absolute drift of the plasma p osition is negligible for the GEXRF measuremen ts, as will b e sho wn in Sections 6.2 and 6.3. As is men tioned in Section 2.3.1, the noise level in eac h CCD frame of a GEXRF mea- suremen t needs to b e as lo w as p ossible to detect and discriminate single photon ev en ts. Therefore, sp ecial care has to b e tak en to minimize stra y light in the spectroscopy c ham- b er. Besides visible ligh t en tering the sp ectroscop y cham ber through window flanges, con tributions from the LPP source itself migh t b e detected. The former can b e reduced b y carefully sealing the windo w flanges. The latter, whic h is infrared an d visible ligh t from the LPP plasma and from the plasma reflected laser ligh t, is t ypically blo c k ed b y a 1.5 µ m alumin um filter (Go o dfello w) in the X-ra y b eam path. Ho w ev er, a further significan t reduction of stra y ligh t can b e ac hiev ed b y using a debris protection foil with an additional alumin um la y er just at the exit of the plasma in teraction c ham b er (see 82 5.2. PRINCIPLE SETUP Figure 5.3.: V acuum setup and positioning stages (photograph) and schematic ra y path for the Kirkpatric k-Baez (left) and toroidal (righ t) optics. A single toroidal multila y er reflector is sho wn in the inset (b ottom righ t of photograph). Figure 5.1), e.g. 900 nm Mylar coated with 100 nm alumin um. 5.2.2. Multila y er Optics The div ergen t and p olyc hromatic X-radiation emitted from the LPP source needs to b e efficien tly transp orted to the sample plane, whic h has a distance of more than 100 cm from the source b ecause of the size of the v acuum c ham b ers, i.e. the plasma in teraction c ham b er and the sp ectroscop y c ham b er. In the suggested setup, m ultila y er mirrors, of- fering a relativ ely high solid angle of acceptance, are used to collect, mono c hromatize and fo cus the radiation on to the sample plane. In a first approac h (measuremen ts in Sec- tion 6.2), tw o one-dimensionally fo cusing W/Si-m ultila y er mirrors, esp ecially designed for the LPP source, are used in Kirkpatric k-Baez geometry (Figure 5.3 left panel). In a second dev elopmen t step (measuremen ts in Section 6.3), they are replaced b y t w o t w o-dimensionally fo cusing, toroidal-shap ed m ultila y er mirrors (Figure 5.3 righ t panel), increasing the ac hiev able photon flux on the sample. 83 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF Kirkpatrick-Baez Optics In the Master’s Thesis of A. Jonas the Kirkpatric k-Baez optics are c haracterized and aligned [112]. Here, the most imp ortan t results for the GEXRF setup are shortly summa- rized. In a first c haracterization step, b oth m ultila y er mirrors are in v estigated separately . The fo cus size (measured as FWHM of the thin line fo cus) for b oth mirrors is (101 ± 14) µ m o v er a fo cal length of ab out 1 cm. Sp ectra of the LPP source after reflection at a single KB mirror result in a Gaussian p eak at 1078 e V (1.15 nm) with a FWHM of 66 e V, i.e. a bandwidth of 3%. Note that the Gaussian p eak consists of three plasma lines at 1088 e V, 1069 e V, 1060 e V (i.e. 1.14 nm, 1.16 nm and 1.17 nm wa v elength), whic h are not resolv ed in the measuremen t. The alignmen t of the com bined m ultila y er mirrors in KB geometry is p erformed with motorized stages in v acuum (Figure 5.3, left panel). Since there are only t w o axes (one rotational and one translational) for the alignmen t for eac h of the t wo mirrors, the desired fo cus size of 100 × 100 µ m 2 (FWHM) could not b e ac hiev ed. The fo cus is minimized in the more critical direction with resp ect to angular resolution, i.e. th e horizon tal axis p oin ting to the CCD (Section 2.3.2). With this alignmen t, the fo cus has a size of 100 × 500 µ m 2 (FWHM). The in tensit y in the sample plane with KB optics is increased b y a factor (gain) of 1470, if compared with the direct radiation of the LPP source in a pinhole with 500 µ m diameter in the sample plane. T o roidal Optics In the second GEXRF b eam time (Section 6.3), toroidal m ultila y er optics instead of the KB optics are applied. These optics appro ximate the geometrically ideal ellipsoidal shap e for a t w o dimensional fo cusing optics based on reflection. Since a single m ultila y er reflector is already fo cusing the radiation in b oth dimensions, the X-ra ys are reflected only once (instead of t wice as in the KB geometry), leading to an increased efficiency . F urthermore, t w o optics are applied in the setup to increase the solid angle of acceptance, eac h aligned with 3 translational and 2 rotational stages, allo wing to align b oth mirror fo ci at the same p osition in the sample plane. The optics are characterized in the Bac helor’s Thesis of R. Reusc h [113]. There it is sho wn that a single optics facilitates a fo cus size (FWHM) of 70 × 80 µ m 2 . With a bandwidth of 4% and a maxim um reflectance at 1078 e V, the sp ectral reflection prop erties of the t w o toroidal mirrors are similar to eac h mirror of the KB optics. The gain, as with the KB optics compared to direct radiation of the LPP in a pinhole with 500 µ m diameter, is 27100 for one of the mirrors. Th us, the total n um b er of photons in the sample plane is increased b y a factor of 18 with one toroidal optics compared to 84 5.2. PRINCIPLE SETUP Figure 5.4.: a) Sc hematic view of the sp ectroscop y c hamber with the sample manipulator (enlarged in b)) in the cen ter. The sample can b e mo v ed in x , y and z direction and rotated against the axes θ and φ [111]. the KB optics, while the sp ot size is decreased. The h uge in tensity increase is mainly originating from a m uc h easier optics alignmen t, enhanced coating prop erties b ecause of a b etter substrate and adapted pro cessing tec hniques, and a sligh tly larger optical area. In principle, a ring-shap ed ellipsoidal optics could further increase the solid angle of acceptance, but setup and alignmen t requiremen ts as w ell as the high cost mak e that concept unfa v orable at the momen t. 5.2.3. Sp ectroscop y Chamb er The sp ectroscop y c ham b er used for the GEXRF exp erimen ts is describ ed in detail in [111]. It is ev acuated with a turb omolecular pump with resulting pressures do wn to 5 × 10 − 8 m bar. The high-v acuum guaran tees clean measuremen t conditions and negli- gible absorption of soft X-ra ys. In the cen ter of the sp ectroscop y c ham b er, the sample manipulator is lo cated (Figure 5.4). It allows to mo v e the sample in 3 translational di- rections in a Cartesian co ordinate system ( x and y for mo v emen t in the v ertical plane of the sample holder, z for mo v ement radial to a v ertical axis), whic h can b e rotated ab out a v ertical axis in the cen ter of the sp ectroscop y c ham b er ( θ ) and a horizon tal axis normal to the sample holder ( φ ). The latter is facing the θ -axis, suc h that the in tersection of the θ -axis and the φ -axis defines the piv otal p oin t of the goniometer and sample manip- ulator. F urthermore, another linear stage (“dio de”) is moun ted on a second rotational stage (“2 θ ”) to moun t and align e.g. an X-ra y sensitiv e dio de in the setup. If the piv otal p oin t of the goniometer is lo cated on the sample surface, and the sample 85 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF is irradiated in this p oin t, the excitation p osition will b e fixed, ev en if the sample is rotated. The piv otal p oin t will also b e essen tial for the angular calibration pro cedures and serv es as origin of the Cartesian co ordinate system, in whic h the measuremen t geometry is defined (LAB system in Section 5.2.5). Therefore, also the sample needs to b e excited in the piv otal p oin t and the sp ectroscop y c ham b er needs to b e aligned precisely with resp ect to the optics’ fo cus, as is describ ed in Section 5.2.5. 5.2.4. Cha rged Coupled Device The CCD is in tegrated in to the sp ectroscop y c ham b er with an adjustable p ositioning system at the 300-mm flange (see Figure 5.4). The system allo ws to tilt the CCD via a trip o d and to tra v erse the CCD in the plane facing the flange. F urthermore, it is p ossible to mo v e the CCD radially to w ards the cen ter of the sp ectroscop y cham ber. With the distance, also the angular resolution and the total solid angle of detection of the GEXRF measuremen ts are adjusted. Due to the small pixel sizes in mo dern CCD c hips and the more relaxed restrictions on angular resolution in soft X-ra y GEXRF compared to hard X-ra y GEXRF (Section 2.2.3), it is usually b eneficial to mo v e the detector as close as p ossible to the excitation sp ot on the sample. The smallest ac hiev able distance with the sample surface p erp endicular to the c hip surface is ab out 15 cm, due to obstructiv e parts of the sample holder itself. Ho w ev er, this distance can b e reduced further, if the sample is moun ted on a 20 ◦ w edge, allo wing to mo v e the goniometer axis θ further a w a y b y 20 ◦ and giving w ay to the CCD positioning system. With this adaption, the distance of the pnCCD c hip to the piv otal p oin t in the first b eam time is 6.5 cm and in the second b eam time the CCD has a distance of 11.1 cm. The reason for the difference is an additional adapter flange and a differen t CCD housing of the con v en tional CCD. 5.2.5. Alignment Pro cedures V arious alignmen t pro cedures are dev elop ed during this thesis, whic h are crucial for the angular calibration of the GEXRF measuremen ts. In principle, for ev ery pixel of the CCD, the resp ectiv e emission angle of the fluorescence radiation and solid angle of detection can b e calculated (Section 2.3.2), if the p ositioning of the CCD with resp ect to the excitation p osition and sample surface direction is w ell-kno wn. The excitation fo cus of the m ultila y er optics, the sample surface and the CCD camera are th us aligned with resp ect to the piv otal p oin t of the goniometer, whic h defines the origin of the co ordinate system used for the angle calculations. 86 5.2. PRINCIPLE SETUP Figure 5.5.: Sc hematic of the alignmen t steps for the precise adjustment of the toroidal optics fo ci in to the piv otal p oin t of the goniometer. The dotted blue X-ra y b eam is not used at the alignmen t step. It can b e easily aligned to a p osition, where it do es not in terfere with the other b eam. F or details ab out the alignmen t refer to text. 87 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF Goniometer Alignment F or the description of the goniometer alignmen t, the setup equipp ed with the sup erior toroidal mirrors is used. As requisite for the alignmen t of the goniometer with resp ect to the optics’ fo cus, the t w o optics should b e already adjusted and their fo ci o v erlapp ed. F or b eam diagnostics, a pinhole (diameter 500 µ m) and a CMOS sensor (768 × 576 pixel of size 6.33 × 6.33 µ m 2 ) are moun ted on the sample holder, facing the direction of the con v ergen t X-ra ys of the toroidal optics. The relativ e distance of the y p osition b et w een CMOS c hip surface and pinhole is roughly kno wn (e.g. measured with slide gauge), as w ell as the y p osition of the pinhole, if it is roughly in the piv otal p oin t. In a first step, the v ertical (here, the x axis is used, it could b e the y axis, dep ending on the φ p osition), radial ( z axis) and rotational ( θ and φ ) p ositions of the pinhole, when its normal in tersects with the piv otal p oin t of the goniometer, hav e to b e found. This is done b y scanning the X-ra y b eam, whic h is reflected b y one of the optics, with the pinhole and an X-ra y sensitiv e dio de attac hed to the 2 θ stage. Then, pinhole p ositions with maxim um in tensit y transmitting through the pinhole ha v e to b e found b efore and after rotating θ and subsequen tly φ b y 180 ◦ . The piv otal p osition is then the mean of the maxim um p ositions for eac h axis. F or further details ab out these scans see [114]. When the pinhole is in its piv otal p osition (Figure 5.5 a)), the optics fo cus m ust b e shifted a distance ∆ z and ∆ x (i.e. v ertical axis, whic h is p erp endicular to the plane of pro jection in Figure 5.5) un til it in tersects with the pinhole’s piv otal p osition. F or this purp ose, the X-ra y fo cus on the CMOS detector is mark ed (Figure 5.5 b)) and the CMOS detector shifted b y ∆ z and ∆ x . No w the optics can b e fo cused again on the mark ed sp ot on the CMOS detector (Figure 5.5 c)). If the new fo cus p osition of the toroidal optics is to o far from the optim um (i.e. the optics cannot b e fo cused on the mark), the whole sp ectroscop y c ham b er, whic h is moun ted on a motorized base frame, m ust b e mo v ed in the corresp onding direction and the pro cess rep eated. No w the setup is roughly aligned. Dep ending on the accuracy of the pivotal position of the pinhole in y direction and the measuremen t of the distance b et w een CMOS c hip and pinhole, the fo cus is probably as close as 0.5 mm to the piv otal axis. Fine adjustmen t is no w p erformed b y scanning θ (e.g. b y 20 ◦ ) at sev eral y p ositions of the CMOS detector and recording the fo cus shift of one of the optics in the CMOS image (Figure 5.6 d)). The y p osition, at whic h the fo cus do es not shift with c hanging θ is the p osition for whic h the CMOS surface is in the piv otal p oin t of the goniometer. This p osition is the in tersection p oin t of t w o linear fits to the p osition dep enden t z shifts (Figure 5.6 d), righ t). If the b eam sp ot is not stationary for θ scans at the determined z p osition, the alignmen t steps starting at a) need to b e rep eated. Now the whole sp ectroscop y c ham b er can b e shifted in the direction of the y axis of the 88 5.2. PRINCIPLE SETUP Figure 5.6.: Figure 5.5 con tin ued. 89 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF goniometer b y the distance b et w een the piv otal y p osition of the CMOS detector and the y p osition when the t w o toroidal fo ci in tersect (Figure 5.6 e)). The whole pro cedure can b e p erformed with a φ scan to further adjust the heigh t of the optics fo cus. How ev er, since the b eam in the setup is propagating almost in a horizon tal plane, the alignmen t pro cedures p erformed so far should b e sufficien t for the x p osition of the optics’ fo cus. Finally , the fo ci of b oth optics can b e fine adjusted to the mark ed p osition on the CMOS c hip (Figure 5.6 f )). The fo ci distance to the piv otal axis is no w b etter than 50 µ m in all directions. The exact piv otal p osition of the pinhole in b eam direction ( y ) can no w b e found b y scanning the z axis at v arious y p ositions (Figure 5.5 g)) and at each marking the maxim um in tensit y p ositions for b oth X-ra y b eams (of b oth toroidal optics). At the y p osition, where the t w o fitted linear curves in tersect, the pinhole is in the pivotal p osition for y (Figure 5.6 g), righ t). Scanning the remaining axes yields then the piv otal pinhole p osition, also with an accuracy of ab out 50 µ m. No w, the fo cus p osition of the toroidal optics is w ell-defined. A sample with its surface in the piv otal p oin t can b e rotated, while the excitation sp ot on the sample is stationary . F or the calculation of the angular scale on the CCD during the GEXRF measuremen ts, the CCD needs to b e aligned relativ e to that excitation p osition. T o describ e the CCD p ositioning, an optical axis m ust b e defined, as is sho wn in the follo wing c hapter. Defining the LAB System and CCD Alignment F or the alignmen t of sample and CCD, a Cartesian co ordinate system (LAB system) is defined. Its origin is in the piv otal p oin t of the goniometer, so that after the alignmen t of the sp ectroscop y c ham b er, this is o v erlapp ed with the toroidal optics fo cus p osition. An optical axis is defined and used for sample and CCD alignmen t b y an optical adjustmen t laser ((Figure 5.7). Since the direct laser sp ot will b e imaged on the CCD c hip for sample alignmen t and angular calibration purp oses later, gra y glasses and a mec hanical sh utter are deplo y ed in the b eam path outside the sp ectroscop y c ham b er to adjust the in tensit y and define exp osure times. The laser is aligned through the pinhole in its piv otal p osition (determined in the previous c hapter) rotated ab out θ b y 90 ◦ . Indeed, the angle b et w een excitation b eam and optical laser do es not ha v e to b e exactly 90 ◦ and migh t v ary to enable an easier CCD p ositioning. The laser b eam defines the x LAB axis and the θ axis of the goniometer is parallel to the z LAB axis of the LAB system. After the CCD is in its measuremen t distance, the tilt of the CCD and the translational direction are aligned with resp ect to the optical laser. The laser sp ot, which will mark the angle line where the emission angle of the fluorescence radiation in the GEXRF measuremen ts is 0 ◦ , should hit the CCD at the edge of the c hip (but still fully on 90 5.2. PRINCIPLE SETUP Figure 5.7.: Sc hematic top view of the sp ectroscop y c hamber with the optical alignment laser (1), gra y glasses (2), mec hanical sh utter (3) and the CCD p ositioning system (4). The adjustmen t laser is aligned through the piv otal p oin t of the goniometer on to the edge of the CCD c hip. In this final arrangemen t, it defines the LAB system. Also note the 20 ◦ w edge (5), which allo ws to mo v e the sample holder further a w a y from the CCD p ositioning system, while keeping the sample surface (here pinhole) aligned to the optical laser. 91 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF the c hip), so that the angular scale can extend o v er most of the remaining c hip area. F urthermore, it is imp ortan t to measure the p osition of the laser b eam reflected from the c hip surface, to calculate the tilt angles of the CCD c hip with resp ect to the optical axis, as will b e sho wn in Section 5.3.2. Alternativ ely , the laser reflection can b e adjusted bac k on the laser to align the CCD c hip in the y LAB z LAB plane. Sample Alignment The samples are moun ted on a 20 ◦ w edge on the sample holder, allo wing to set the θ motor of the goniometer 20 ◦ further a w ay from the CCD detector, while k eeping the p osition of the samples (see Figure 5.7). Th us, the detector can b e aligned closer to the sample holder without touc hing the motors of the translational stage of the goniometer, increasing the solid angle of detection. The alignmen t of the excitation p osition on the sample surface is carried out b y mo v e- men ts relativ e to a fixp oin t on the sample holder (pinhole p osition or a fluorescen t screen). The relativ e p ositions can b e measured with a slide gauge or a photograph con- taining a scale b efore load lo c king the sample holder and are accurate to ab out 0.5 mm. No w, the sample surface has to b e aligned in to the piv otal p oin t of the goniometer with high accuracy , to assure that the excitation sp ot and th us the origin of the fluorescence radiation is in the origin of the LAB system. F or this purp ose, iterativ e scans with the θ and z motors of the goniometer can b e used. Th us, the sample surface is aligned b y mo ving it in to the laser b eam and monitoring the laser b eam in tensit y on the CCD c hip. If the shado wing leads to half of the maxim um in tensit y , θ scans are used to maximize the laser in tensit y . Details can b e found in the Bac helor’s thesis of L. Bauer [114]. The application of the 20 ◦ w edge on the sample holder c hanges the alignment sligh tly . No w, the tilt of the sample surface with resp ect to the transitional axes y and z of the goniometer has to b e considered. This results in com bined y - z scans with step sizes ∆ y = ∆ z tan(20 ◦ ) instead of the pure z scans for the radial adjustmen t of the sample surface (see Figure 5.7). If this is not considered, the excitation sp ot on the sample surface w ould shift due to the alignmen t. Slightly depending on the sample extensions, this alignment procedure assures a p ositioning of the sample surface with resp ect to the piv otal p oin t within ab out 50 µ m . The zero angle of the sample, whic h is the θ p osition of the goniometer when the sample surface is parallel to the x LAB axis of the LAB system, can b e determined with an accuracy of b etter than 0.01 ◦ , as is demonstrated in Section 6.3. Of course, this v alue v aries with the CCD and sample dimensions and p ositions. If necessary , further scans with the goniometer stages in the plane of the sample surface can b e used to adjust the lateral excitation p osition on the sample with higher precision. Ho w ev er, it migh t b e necessary to rep eat the surface 92 5.2. PRINCIPLE SETUP adjustmen t afterw ard. After the alignmen t of the sample is completed and the sample is set in its measure- men t p osition, the fluorescence emission angle to pixel assignmen t is defined and can b e calculated b y an angular calibration pro cedure (Section 5.3). 93 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF 5.3. Calib ration of the Angula r Scale Since GEXRF profiles are plots of fluorescence in tensit y with resp ect to the fluorescence emission angle, the con trol of the emission angle is crucial for the GEXRF exp erimen t. If the fluorescence emission angle is scanned, for example b y rotating the sample, the angular axis is defined b y the motor steps of the rotational stage except for an offset of the angular scale. This offset is t ypically found b y measuring a calibration GEXRF profile with a w ell-kno wn feature, e.g. the inflection p oin t in the GEXRF profile of Si-K α originating from a silicon w afer [23, 115]. The w afer can either b e a differen t sample, or the substrate of the sample to b e analyzed. Ka yser et al. also use a calibration GEXRF profile (signal of the Ge w afer) in their scanning-free GEXRF exp erimen ts in [24]. The relativ e angular scale is calculated from the measured sample-to-detector distance and detector extension. The use of a calibration GEXRF profile can lead to systematic uncertain ties in the angular scale. First, if the substrate signal is used for calibration, the analyzed struc- ture (thin la y ers or dopan t profiles) can ha v e an influence on the propagation of the fluorescence radiation, whic h migh t ha v e to b e tak en in to accoun t. Since the prop erties of the analyzed structure, whic h affect the optical resp onse (e.g. roughness, densit y or stoic hiometry), migh t not b e kno wn a priori, this approac h can lead to inconsistencies in the analysis. Second, if a w ell-kno wn indep enden t sample is applied for the angular calibration, there is an uncertain t y in the alignment of the measuremen t samples with resp ect to the calibration sample. The follo wing considerations will fo cus on the angular calibration of a scanning-free GEXRF setup. If the sample-to-detector distance is large compared to the detector width, the detector can b e aligned suc h, that each pixel column (or ro w) detects fluo- rescence of a w ell-defined emission angle (see Section 2.3.2, Figure 2.10 a)). Ho w ev er, esp ecially in the soft X-ra y regime, the restrictions to the angular resolution are more relaxed, allo wing for a shorter sample-to-detector distance to increase the solid angle of detection and the analyzed angular range. In this range of large solid angles of detec- tion, the actual hyperb olic shap e of the equi-angle lines has to b e tak en in to accoun t. F urthermore, it is not sufficien t to assume an a v erage solid angle of detection for eac h pixel (or pixel ro w), but rather the op ening angle of eac h pixel needs to b e calculated, as is sho wn in Section 2.3.2. F or b oth GEXRF exp erimen ts sho wn in this thesis (Sections 6.2 and 6.3), the corre- sp onding fluorescence emission angles and solid angles of detection for all pixels of the detectors are calculated as describ ed in Section 2.3.2. T o do so, the exact geometry of the detector with resp ect to the sample, defined b y all the geometric parameters (GPs) 94 5.3. CALIBRA TION OF THE ANGULAR SCALE GP distance to / rotation of CCD d CCD along x LAB l CCD along y LAB h CCD along z LAB φ CCD ab out x LAB θ CCD ab out y LAB ω CCD ab out z LAB ω sample ab out z LAB Figure 5.8.: Sc hematic view of the GEXRF measuremen t geometry and definition of the geometric parameters in the LAB system (see table). The x LAB y LAB plane is appro x- imately a horizon tal plane. Indicated is also a hyperb olic equi-angle line on the CCD c hip for a fluorescence emission angle ψ fl . in Figure 5.8, needs to b e kno wn. As men tioned in Section 5.2.5, x LAB and z LAB are parallel to an adjustmen t laser and the θ axis of the goniometer, resp ectiv ely . The ori- gin of the LAB system is the piv otal p oin t of the goniometer and the sample surface is adjusted in the x LAB z LAB plane. Then, 6 GPs (3 displacemen ts of the edge cen ter of the CCD and 3 rotations) need to b e kno wn to calculate the corresp onding fluorescence emission angles and solid angles of detection of eac h pixel of the t w o-dimensional detec- tor. T o determine these parameters, a calibration sample is used in the first b eam time (Section 6.2) and in the second b eam time (Section 6.3) an absolute angular calibration is applied. The latter is achiev ed b y measuring all the GPs with high precision b y means of an optical laser setup. 5.3.1. Calib ration with GEXRF p rofile In principle, it is p ossible to fit all geometric parameters with an algorithm, whic h compares a measured GEXRF profile with a theoretical or w ell-kno wn GEXRF profile. Eac h v ariation of the GPs leads to a c hange in the angular calibration of the measured GEXRF profile and therefore to a differen t form of that profile. If the measured and the theoretical GEXRF profile matc h, the fitted GPs describ e the angular calibration correctly . Ho w ev er, the calculation of the whole angular distribution on the CCD c hip is time consuming (tens of seconds to min utes on a mo dern desktop PC), since angle and solid angle are calculated for ev ery pixel. This can b e critical in a pro cedure with tens or h undreds of iterations if all 6 GPs are tak en in to accoun t. Therefore, it is 95 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF Figure 5.9.: Sc hematic view of the sp ectroscop y c hamber. The aligned optical laser b eam in tersects the piv otal p oin t of the goniometer and defines the LAB system (left). It is reflected at the CCD c hip (cen ter) and exits the v acuum cham b er through the same windo w flange where it en ters. Outside the v acuum c ham b er, the b eam path can b e measured to determine the tilt of the CCD camera (righ t). helpful to com bine the direct measuremen t of some of the GPs with a fitting routine, whic h determines for example only the GPs with the strongest influence on the angular scale. It has to b e k ept in mind that the fitting algorithm can only b e as precise as the assumptions on the theoretical GEXRF profile. That means a wrong theoretical curv e migh t still lead to a go o d fitting result, but induces a systematic error in the angular scale. Ev en more, a wrong geometry leads to a wrong normalization to the solid angle of detection for eac h angle, whic h could alter the shap e of the GEXRF profile and lead to wrong depth profiling results. 5.3.2. Absolute Angula r Calib ration The basic principle of measuring all relev an t GPs to ac hiev e an absolute angular cali- bration is w ork ed out within the scop e of the Bac helor’s thesis of L. Bauer [114]. F or the b eam time in Section 6.3, the ev aluation pro cedure is automatized and refined. T o apply the ev aluation pro cedure, the setup needs to b e aligned according to the description in Section 5.2.5. Then, all GPs can b e deriv ed in the follo wing w a y . The angles ω CCD (see Figure 5.9 left) and θ CCD (lies in x LAB z LAB plane and is not sho wn in the image) migh t deviate from zero to ac hiev e a smaller sample-to-detector distance in the alignmen t pro cedure. Both can b e measured outside the sp ectroscop y c ham b er. F or this purp ose, the p ositions ( y i,r , z i,r ) of the inciden t and of the reflected optical laser b eam need to b e measured at t w o x 0 p ositions x 0 1 and x 0 2 (Figure 5.9 righ t). 96 5.3. CALIBRA TION OF THE ANGULAR SCALE Figure 5.10.: a) Section of the maxim um pixel in tensities of 9 images of a θ scan. Besides the direct laser b eam, the reflected b eam p ositions are sho wn. In b) the in tensities of the image in a) are summed along the equi-angle lines computed with the angular calibration. Then, the tilt of the CCD with resp ect to x LAB can b e calculated with tan( ω CCD ) = | y i 1 − y r 1 |−| y i 2 − y r 2 | 2 × | x i 1 − x i 2 | (5.3.1) tan( θ CCD ) = | z i 1 − z r 1 |−| z i 2 − z r 2 | 2 × | x i 1 − x i 2 | . The GPs l CCD and h CCD (displacemen t of edge cen ter of the CCD with resp ect to y LAB and z LAB ) are determined b y a single recording with the CCD camera of the direct adjustmen t laser b eam (sho wn in Figure 5.10 a)). The p osition (CCD 0 x , CCD 0 y ) can b e measured with a t w o-dimensional Gaussian fit with sub-pixel accuracy . This, together with the kno wn tilt angles ω CCD and θ CCD , yields the p osition of the CCD camera in the y LAB z LAB plane. No w, d CCD and φ CCD are extracted from the final θ scan of the goniometer from the sample alignmen t ∗ (Figure 5.10 a)). F or eac h scan p osition, an image of the ad- justmen t laser b eam reflected at the sample surface is recorded with the CCD cam- era. The p ositions of the reflected b eam on the CCD c hip can again b e fitted with t w o-dimensional Gaussian functions, yielding the cen ter p ositions (CCD x ( θ ), CCD y ( θ )). ∗ It is imp ortan t that the radial p osition of the sample ( z axis) is already aligned 97 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF Figure 5.11.: a) Linear fit to the CCD p ositions of the adjustmen t laser reflected at the sample surface for v arious θ p ositions of the goniometer. b) Distances d ( θ ) of the reflected laser b eam p ositions to the direct laser b eam position on the CCD for v arious θ p ositions of the goniometer and the corresp onding fit. Plotting CCD x ( θ ) o v er CCD y ( θ ) and a linear fit of m and n in CCD y ( θ ) = m × CCD x ( θ ) + n yields φ CCD = 1/tan( m ) (Figure 5.11 a)). The distance d CCD can b e calculated with the relation d ( θ ) /d CCD = tan(2 θ ), d ( θ ) b eing the distances of eac h reflected laser b eam p osition to the direct laser b eam on the CCD c hip for a tilt angle θ of the sample. In the case that the unobstructed adjustmen t laser b eam is p erp endicular to the CCD c hip surface ( ω CCD = θ CCD = 0), d ( θ ) can b e calcu- lated b y the Pythagorean theorem d ( θ ) 2 = CCD x ( θ ) − CCD 0 x 2 + CCD y ( θ ) − CCD 0 y 2 . F or a more general case, the tilt of the CCD has to b e tak en in to accoun t. The surface of the sample is in go o d appro ximation parallel to the θ axis of the goniometer, whic h is parallel to the z LAB axis of the LAB system. Therefore, the plane of all (at the sample surface) reflected b eams is parallel to the horizon tal x LAB y LAB plane of the LAB system and a tilt of θ CCD do es not affect the p ositions of the reflected laser sp ots. Ho w ev er, a tilt of ω CCD needs to b e considered with a correction factor K ( ω CCD , θ ) (see App endix F), so that d K ( θ ) = d ( θ ) × ( K ( ω CCD , θ )) − 1 , with K ( ω CCD , θ ) = cos( ω CCD ) + sin( ω CCD ) tan(2 θ ) . (5.3.2) With θ 0 b eing the p osition of the goniometer when the sample surface is aligned 98 5.3. CALIBRA TION OF THE ANGULAR SCALE parallel to the x LAB direction of the LAB system, the follo wing relation is v alid: d ( θ − θ 0 ) = tan(2 | θ − θ 0 | ) × d CCD × K − 1 ( ω CCD , θ − θ 0 ) (5.3.3) Figure 5.11 b) sho ws a fit of equation 5.3.3 to the measured distances d ( θ − θ 0 ) on the CCD with θ 0 and d CCD as free parameters. Consequen tly , b oth, the alignmen t angle θ 0 and the sample-to-detector distance d CCD can b e obtained. The kno wledge of all 6 GPs and θ 0 allo ws no w to calculate the corresp onding fluores- cence emission angle and the solid angles of detection for ev ery pixel on the CCD c hip for ev ery goniometer p osition θ adjusted during a measuremen t. As pro of of consistency , the image of the maxim um pixel in tensities of the θ scan is used and the in tensities along the equi-angle lines from the angular calibration are summed. Figure 5.10 b) sho ws the resulting plot. If GEXRF data w ere used, the x axis w ould corresp onds to fluorescence emission angles. The exp ected reflection angles with resp ect to the x LAB axis (2 | θ − θ 0 | ) are indicated b y the v ertical lines for eac h θ v alue. They match with the maxim um p ositions of the plot, demonstrating the v alidit y of the calibration. The p erformance of this pro cedure with resp ect to repro ducibilit y and stabilit y will b e presen ted along with the measuremen ts of the second b eam time in Section 6.3. 99 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF 5.4. Single Photon Analysis After a careful setup and sample alignmen t, the recording of the ra w data of the GEXRF exp erimen t is straigh t forw ard. While the source is in op eration and fluorescence radia- tion from the sample is excited, the CCD acquires images (GEXRF frames). The source in tensit y and camera parameters ha v e to b e adjusted in a w a y that the total in tensit y on eac h frame is lo w enough to detect and distinguish single photon ev en ts (SPEs) and that dark curren t and readout noise are minimized. Due to the t ypically short recording times ( < 1 s) and lo w c hip temp eratures ( < -50 ◦ C), the readout noise is dominan t and can b e regulated b y the readout frequency of the CCD, leading to a compromise b et w een long readout times and lo w noise lev els. While for the first b eam time a pnCCD is used (Section 6.2), measurements in the second b eam time are p erformed with a con v en tional CCD (Section 6.3). The former is optimized to detect SPEs and the ra w data treatmen t is carried out b y PNSensor Gm bH. The treated data pro vides already information ab out p osition and in tensit y of eac h detected photon. F or the measurements with the con v en tional CCD, this information is not readily a v ailable. Therefore, the metho ds dev elop ed to analyze the ra w data from the second b eam time are explicitly describ ed in the follo wing. F or the measuremen ts with the con v en tional CCD, it is useful to record e.g. n dark = 3 dark images ev ery n meas = 50 GEXRF frames to monitor the noise lev el in the dark frames (whic h is a function of c hip temp erature due to dark curren t, see Section 2.3.1). The GEXRF frames are dark frame corrected with a “master” dark (MD) image, whic h can b e e.g. the median image of the n dark dark images prior to the n meas measuremen ts. Th us, long-term instabilities of the camera temp erature can b e monitored and accoun ted for. F or the measuremen ts used in this thesis, the standard deviation of the dark frames σ dark c hanged b y less than 0.5% o v er a whole GEXRF measuremen t. In Figure 5.12, a clean GEXRF frame (after subtraction of a MD frame) is shown. The detected fluorescence radiation originates from a C/Ni-m ultila y er irradiated with 1078 e V photons from the LPP source, so that mainly Ni-L α radiation reac hes the detector. SPEs can b e seen in the magnified image on the b ottom left panel. Different shap es of the photon ev en ts can b e differen tiated. On the one hand, one-photon ev en ts are sho wn consisting of up to four pixels (lab eled 1-p x, 2-p x etc.). The SPEs, whic h consists of more than one pixel are split ev en ts (see Section 2.3.1). On the other hand, t w o examples for photon ev en ts whic h consist of more than four pixels are lab eled as pile-up. These ev en ts are usually created b y more than one photon, as will b e seen in the next ev aluation steps. Of eac h photon ev en t, the p osition, where the photon w as absorb ed in the CCD c hip, 100 5.4. SINGLE PHOTON ANAL YSIS Figure 5.12.: Single photon ev en ts detected with a CCD camera. T op: Whole dark frame corrected CCD frame. Bottom: Enlarged view of indicated image section. Different patterns of single photon ev en ts can b e discriminated. Figure 5.13.: Histogram of master dark frame (top axis) and clean GEXRF frame (b ottom axis). A region dominated b y noise and a high pixel intensit y shoulder originating from photon ev en ts can b e iden tified in the plot. 101 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF and the energy of the detected photon are of in terest. If all photon ev ents w ere 1-p x ev en ts (i.e. no split even ts are presen t), b oth prop erties are retriev ed straigh t forw ard b y the pixel co ordinate and the pixel in tensit y . In this case, an energy-disp ersiv e sp ectrum is iden tical to an (energy calibrated) plot of the n um b er of detected SPEs against their in tensit y (histogram). F or the CCD image in Figure 5.12, suc h a histogram is sho wn in Figure 5.13. A t zero in tensit y , a Gaussian-shap ed p eak is visible in the histogram of the clean GEXRF frame (or “SPE frame” hereafter, purple curv e). Its heigh t and width dep end on the actual noise con tribution. Th e Gaussian-shaped noise in the MD frame (black curv e) has a standard deviation of σ MD =3.09 ADU. The Gaussian-shap ed noise in a single dark frame yields a standard deviation of σ dark =4.67 ADU and is assumed to b e similar to the noise in the SPE frame ∗ . Th us, from error propagation it follo ws that the noise in the clean SPE frame is exp ected to b e σ meas = q σ 2 dark + σ 2 MD = 5 . 63 ADU . (5.4.1) Ho w ev er, the measured v alue of σ meas = 6 . 05 ADU (from Gaussian fit in Figure 5.13) is sligh tly higher, indicating a further noise con tribution, ma yb e due to visible ligh t from the LPP source. The noise lev el in the SPE frames is crucial to the split ev en t recom bination, influencing the final energy resolution and the analyzable lo w energy region, as will b e in v estigated in Section 5.4.2. A t higher pixel in tensities ( > 30 ADU), the histogram of the clean SPE frame sho ws con tributions, whic h originate from the detected fluorescence radiation. Ho w ever, no clear fluorescence p eak of the exp ected Ni-L α radiation from the C/Ni-multila y er sample can b e differen tiated in the histogram, since the total fluorescence photon in tensit y is most often randomly distributed o v er more than 1 pixel. This indicates the need for precise split ev en t recom bination. 5.4.1. Split Event Recombination The main c hallenge in SPE detection in the soft X-ra y range with the con v en tional CCD applied in this thesis is the lo w signal-to-noise ratio. Firstly , the total n um b er of created c harge carriers is relativ ely small compared to the case with hard X-ra y photons and secondly , these c harge carriers are also distributed o ve r more than one pixel, further reducing the signal of a single pixel. In addition, the applied CCD camera is not optimized for single photon detection concerning gain and dynamic range. Therefore, a ∗ The noise in the SPE frame cannot b e measured b y simply taking the standard deviation of the image, since the SPEs influence that v alue. 102 5.4. SINGLE PHOTON ANAL YSIS Ni-L α photon with an energy of 849 e V, whic h creates on a v erage 849 e V / 3.62 e V = 234.5 electron-hole pairs in pure silicon, is only detected with a signal of ≈ 120 ADU, although the dynamic range is ab out 50000 ADU. In the case of split ev en ts, the discrimination of pixels dominated b y noise and those giving information ab out the detected photon is not alw a ys p ossible, esp ecially for fluorescence radiation with y et lo w er energy . Clustering The first approac h to recom bine split ev en ts is a clustering metho d. The algorithm to iden tify SPEs and to find the corresp onding pixels in the clean SPE frame is illustrated in Figure 5.14. In a first step (top ro w), all pixels in a clean SPE frame are searc hed for, whose in tensit y is ab o v e a certain threshold. The threshold is defined by a m ultiple n 0 σ of the estimated noise lev el σ meas in the clean SPE frame according to equation 5.4.1. T ypical v alues of n 0 σ are 3 to 8, in the example n 0 σ = 3 is c hosen. If pixel p ositions x and y are found suc h that the pixel in tensit y I ( x, y ) > n 0 σ × σ meas , it is c hec k ed if this pixel has the highest in tensit y in a 3 × 3 area cen tered at ( x , y ). If not, the conditions are c hec k ed for the next pixel. Otherwise (2nd ro w), the neigh b oring pixels P ( x − 1 , y ), P ( x + 1 , y ), P ( x, y − 1) and P ( x, y + 1) are in v estigated. If their v alue is ab o v e a second threshold n 00 σ × σ meas (1 ≤ n 00 σ ≤ n 0 σ ), these pixels are assigned to the same cluster N ev and the neigh b oring pixels are again in v estigated in the same w a y iterativ ely , leading finally to an n -p x ev en t, n b eing the n um b er of pixels in the cluster. In the end, a total n um b er N t ev of clusters is found in the image, eac h consisting of n ( N ev ) ≥ 1 pixels. The total in tensit y of eac h cluster is then I ( N ev ) = X I ( x, y ) , P ( x, y ) ∈ N ev (5.4.2) F urthermore, a cen ter of gra vit y can b e determined with the co ordinates X ( N ev ) = P x × I ( x, y ) I ( N ev ) , P ( x, y ) ∈ N ev Y ( N ev ) = P y × I ( x, y ) I ( N ev ) , P ( x, y ) ∈ N ev . (5.4.3) The cen ter of gra vit y in principle yields a b etter spatial resolution than the pixel size, i.e. a b etter angular resolution for the scanning-free GEXRF measuremen ts. Ho w ev er, for the measuremen ts sho wn in Section 6.3, the angular resolution is not critical and therefore no sub-pixel resolution will b e in tro duced. In v estigations on the p ossible sub- pixel resolutions can b e found in [64, 59]. The calculated in tensit y can b e used to 103 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF Figure 5.14.: Exemplary description of the three in v estigated split ev ent recom bination metho ds: Clustering, 4p x-Area and 4p x-Area Clustering. The identification of the most in tense pixel of an SPE is the same for all metho ds (top ro w). The threshold factors are n 0 σ = n 00 σ = 3 and σ meas = 5 . 63 ADU. F or a complete description, please refer to the text. 104 5.4. SINGLE PHOTON ANAL YSIS Figure 5.15.: Energy-disp ersiv e sp ectra (uncalibrated) created with the three SPE recom- bination metho ds from the clean SPE frame and its histogram for comparison. In the sp ectra of all three metho ds a p eak for Ni-L α fluorescence is visible. The t wo clustering metho ds also handle pile-up. create an energy-disp ersiv e sp ectrum from the detected SPEs, if the n um b er of SPEs is plotted with resp ect to their in tensit y . Suc h a sp ectrum for the Clustering metho d (and subsequen t metho ds) is compared in Figure 5.15 to the plain histogram of the clean SPE frame. F or the present and the follo wing analysis in the c hapter, the t w o threshold factors for the analysis are set to n 0 σ = n 00 σ = 3. The high n um b er of ev en ts at lo w even t in tensities ( ≈ 25 ADU) can b e attributed to noisy pixels with high noise in tensities, as sho ws the comparison to the Gaussian p eak fitted to the noise p eak in the histogram. A t ab out 35 ADU, a small, narro w p eak app ears. This p eak can also b e attributed to noise ev en ts, as will b e shown later. The actual fluorescence p eak of the Ni-L α is lo cated at x 0 = 109.7 ADU with a full width at half maxim um of ∆ x FWHM = 57 ADU. This yields a resolving p o w er E /∆ E = x 0 /∆ x FWHM = 1.9 at Ni-L α . Note that at 243 ADU a second broad p eak with m uc h lo w er in tensit y is visible, originating from pile-up ev en ts (also sho wn later). The energy-disp ersiv e sp ectrum can also b e analyzed with resp ect to the t yp e of cluster the sp ectrum is comp osed of. Figure 5.16 a) sho ws the con tributions of n -p x ev en ts to the total n um b er of detected clusters. F or n > 4 the n umber of photon even ts drastically decreases, but also n as large as 14 is detected. In Figure 5.16 b), the con tribution of the 1-p x to 6-p x ev en ts (filled curv es) to the total sp ectrum (blue curv e) is sho wn. It 105 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF Figure 5.16.: a) Histogram of SPE t yp es (1-p x, 2-p x ...). The probabilit y decreases strongly for the app earance of 5-p x ev en ts and larger ones. b) Share of the n -p x ev ents in the total sp ectrum sho wn for n ≤ 6. The histogram in the inset is the same as in a). can b e seen that the high in tensit y p eak in the lo w energy range, starting at 16.9 ADU ≈ 3 × σ meas , is only con taining 1-p x ev en ts, supp orting the explanation that these are pure noise pixels. The second p eak at ab out 35 ADU ≈ 2 × 3 × σ meas , consists of mainly 2-p x ev en ts. This in addition to the shap e and p osition of that p eak indicate that it is also an artifact in the sp ectrum originating from t w o pixels with high noise lev els. The actual photon p eak consists of mainly 2-p x, 3-px, 4-p x and 5-p x ev en ts, eac h displa ying a m uc h narro w er but shifted p eak in the sp ectrum. This shift in the n -p x sp ectra results in the large width of the com bined sp ectrum and is resp onsible for the lo w energy resolution of the Clustering metho d. Theoretically , for a photon ev en t split in e.g. 2 pixels, on a v erage it should mak e no difference for the calculated in tensit y (i.e. p osition in the sp ectrum) if further noise pixels are con tributing to the cluster of the SPE. Y et, this is only true if the mean in tensit y of the added noise pixels is zero, whic h is not v alid since the pixels ha v e to b e ab o v e the threshold n 00 σ × σ meas to b e accepted. A more detailed study on the influence of the t w o thresholds n 0 σ and n 00 σ on the n -p x sp ectra is presen ted in App endix G. It is sho wn and rendered plausible that the sp ectra can b e shifted to o v erla y the p eaks of all the n -p x sp ectra, resulting in a sligh tly b etter resolving p o w er for the p eak in the com bined sp ectrum. Ho w ever, since the effect on the linearit y of the energy scale is not clear (to sim ultaneously o v erla y the pile-up p eaks the abscissa w ould need to b e stretc hed, to o), this approac h will not b e follo w ed further. Instead, a second approac h is pursued for SPE recom bination in the follo wing section. Finally , please note that the p eak at ab out 243 ADU is comp osed of n -p x even ts with n > 4, whic h supp orts the in terpretation as pile-up. 106 5.4. SINGLE PHOTON ANAL YSIS 4p x-Area The main idea in the 4p x-Area metho d is that the c harge cloud of an SPE has a finite size. Indeed, already by visual inspection of Figure 5.12 it can b e seen that a great part of the detected photon ev en ts consist of more than one pixel, but usually of less than 2 × 2 pixels. This is exp ected, if the detected c harge cloud is of the size of the pixels, whic h is true for the sho wn measuremen ts, as will b e in v estigated in Section 5.4.2. Therefore, an approac h is tested in whic h alw a ys 4 pixels are regarded as the whole photon ev ent. Similar to the Clustering metho d, all pixels with an in tensit y greater then n 0 σ × σ meas are searc hed for, whic h ha v e also the highest in tensity compared to their neigh boring pixels (Figure 5.14, top ro w). Then, all 8 surrounding pixels are regarded further and the summed in tensit y of the four 2 × 2-pixel areas are compared (Figure 5.14, third ro w). The 2 × 2 pixel area with the highest summed in tensit y is assigned to the photon ev en t. SPE in tensit y and p osition can b e calculated according to equations 5.4.2 and 5.4.3. The sp ectrum obtained with the 4p x-Area metho d is prin ted as red curv e in Fig- ure 5.15. The first p eak is cen tered at 35 ADU and can b e assigned to noise ev en ts. Its slop e is m uc h shallo w er compared to the noise p eak in the Clustering metho d, since alw a ys four pixels con tribute to the noise ev en ts (i.e. a corresp onding Gaussian-shap ed noise p eak w ould ha v e t wice the width). As exp ected, the fluorescence p eak is nar- ro w er compared to the Clustering metho d, resulting in a resolving p o w er of E / ∆ E = x 0 / ∆ x FWHM = 3.4 at Ni-L α . The p eak is also sligh tly shifted to higher ev ent in ten- sities compared to the Clustering metho d b ecause of the on a v erage higher n um b er of pixels assigned to the SPEs. Since all ev aluated SPEs consist of four pixels, pile-up ev en ts migh t b e coun ted in a wrong w a y . If the c harge cloud of t w o photons strongly (but not completely) o v erlap, the t w o ev en ts will b e coun ted only as one with a photon ev en t in tensit y of less than the sum of the t w o SPEs. This leads to a broadened and shifted pile-up bac kground as compared to the Clustering metho d (high energy tail in Figure 5.15), making it difficult to use the pile-up ev en ts in the ev aluation. Ho w ev er, with resp ect to measuremen t time it could b e useful to allo w a sp ecific lev el of pile-up ev en ts b e recorded during the measuremen t, if these ev en ts can b e ev aluated correctly . Therefore, in a third SPE recom bination metho d, the sup erior energy resolution of the 4p x-Area metho d is coupled to the b etter pile-up detection of the Clustering metho d. 4p x-Area-Clustering Figure 5.14 sho ws the principle of split ev en t recom bination with the 4p x-Area-Clustering metho d in ro ws 3 and 4. The start is the same as in the 4p x-Area metho d. Then, for all 107 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF T able 5.1.: Ev aluation of the fluorescence p eak p osition x 0 , its width ∆ x FWHM and the resulting resolving p o w er x 0 / ∆ x FWHM for the Ni-L α p eak. The 4p x-Area metho d yields the b est resolving p o w er, but pile-up ev en ts cannot b e handled correctly . x 0 / ADU ∆ x FWHM / ADU x 0 / ∆ x FWHM Clustering 109.7 57 1.9 4p x-Area 117.1 34.5 3.4 4p x-Area-Clustering 120.2 41.4 2.9 four pixels, the adjacen t pixels are in v estigated and assigned to the same cluster if their in tensit y is ab o v e a second threshold n 00 σ × σ meas , just as in the Clustering metho d. This pro cess is iterated un til no more pixels ab o v e the threshold are found. The example in Figure 5.14 sho ws the difference in the resulting pixel assignmen t for all three metho ds (last column). Here, the 4p x-Area-Clustering metho d seems to lead to the most complete ev en t recom bination. As can b e seen in the resulting sp ectrum in Figure 5.15 and in T able 5.1, the resolving p o w er of E /∆ E = 2.9 is b etter than for the Clustering metho d while the sp ectrum still allo ws to iden tify pile-up ev en ts. The broadening of the Ni-L α fluorescence p eak as compared to the p eak in the 4p x-Area metho d is caused b y the on a v erage higher n um b er of pixels con tributing to the recom bined SPEs. Figure 5.17 summarizes the results of the 3 differen t split ev en t recom bination meth- o ds. The first column illustrates the differences of the pixel-to-cluster assignmen t for all three metho ds, the green circle indicating the example of Figure 5.14. All three metho ds detect the same photon ev en ts, but assign differen t pixels to eac h even t. The sp ectra for the 4p x-Area and 4p x-Area-Clustering metho d are sho wn in Figure 5.17 d) and e), resp ectiv ely . Of course, the former is comp osed of only 4-p x ev en ts, resulting in the b est energy resolution. The decreased energy resolution of the 4p x-Area-Clustering metho d compared to the 4p x-Area metho d originates from a con tribution of 5-p x ev en ts, as can b e seen in Figure 5.17 e). The c hoice of the threshold factors n 0 σ and n 00 σ affects the resolution and dep ends on the actual data. This will b e shortly in v estigated for one of the measuremen ts in Section 6.3. In the final part of this c hapter, the influence of v arious camera and charge cloud prop erties are n umerically sim ulated to ev aluate optimal prop erties for soft X-ra y SPE ev aluation. 5.4.2. Numerical Simulations Dark frame corrected CCD frames with single photon ev en ts (SPEs) can b e sim ulated b y a 2-dimensional arra y , the en tries represen ting the pixel in tensities. SPEs are sim- 108 5.4. SINGLE PHOTON ANAL YSIS Figure 5.17.: Comparison of the ev aluation with the three SPE recom bination metho ds. The left column (figures a), c) and d)) sho ws the resulting pixel to cluster assignmen t of the metho ds. Circled is the area of Figure 5.14. The righ t column (figures b), d) and f )) compares the calculated sp ectra. The con tribution from the v arious split ev en t t yp es is sho wn. Of course, no 1-p x to 3-px ev en ts app ear in the 4p x-Area and 4p x-Area-Clustering metho d. 109 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF ulated with a Mon te Carlo approac h. Eac h sim ulated photon N ph creates a Gaussian- distributed total in tensit y I ph ( N ph ) in the CCD c hip with an a v erage in tensit y of I ph and a standard deviation of σ ph ( I ph , E ph ) giv en b y σ ph ( I ph , E ph ) = I ph × q F × E ph 3 . 62 e V E ph 3 . 62 e V . (5.4.4) In equation 5.4.4, E ph is the energy of the detected photon, E ph /3.62 e V the n um b er of electron-hole pairs created b y suc h a photon in pure silicon [30], and F the F ano factor, whic h is for silicon 0.115. The latter tak es in to accoun t the deviation of the statistical c harge carrier creation from a pure P oisson pro cess. Th us, the energy resolution is assumed to follo w the b eha vior of an ideal semiconductor radiation detector [116]. The algorithm randomly distributes a n um b er n ( N ph ) of suc h photons N ph homo- geneously on an initially empt y CCD mesh with 2046 × 515 pixels and a pixel size of 13.5 × 13.5 µ m 2 , similar to the conv en tional CCD used for the measuremen ts sho wn ab o v e. The total in tensities I ph ( N ph ) are split to sev eral adjacen t pixels, if the c harge cloud created b y the photon is of the size of the pixels or the hit p osition close to a pixel b order. The c harge cloud is mo deled b y an axially symmetric tw o-dimensional Gaussian distribution with a standard deviation of σ cc and centered at the exact photon hit p osition. The in tensity for the individual pixels is calculated b y an in tegration of the Gaussian distribution o v er the pixel dimensions of eac h pixel (App endix H). Finally , a random, Gaussian-distributed noise intensit y with a width of σ dark is added to eac h pixel of the sim ulated CCD image. The first sim ulations shall matc h the single photon ev en t measuremen ts in Section 5.4, where mainly fluorescence radiation of Ni-L α is detected. The size of the c harge cloud is deriv ed from these measuremen ts, using the plateau heigh t of the difference-ratio histogram as prop osed b y La wrence et al. [62]. This metho d yields a v alue of σ cc = 7.6 µ m (FWHM = 17.8 µ m ), as is sho wn in the Master’s thesis of S. Staec k [117]. F urthermore, it is found that the c harge cloud size calculated with this approac h is consisten t with sim ulated data represen ting the presen t measuremen t conditions. The noise v alue of the dark images σ dark = 4 . 67 ADU is also c hosen according to the v alues of the actual measuremen t and a total mean in tensit y of I ph = 145 ADU is set for the mono c hromatic 849 e V photons. Three dark images (without single photon ev en ts) and a single SPE frame are created. A master dark is calculated b y taking the median image of the three dark frames. The master dark is subtracted from the SPE frame to get a clean SPE frame, which is then ev aluated b y the 4p x-Area-Clustering metho d using noise thresholds n 0 σ = n 00 σ = 3. 110 5.4. SINGLE PHOTON ANAL YSIS Figure 5.18.: Sp ectrum of SPE ev aluation with the 4p x-Area-Clustering metho d for a) sim ulated and b) measured data. In the sp ectrum a noise p eak, a fluorescence p eak of Ni-L lines and a pile-up p eak are visible. The p eaks in the measuremen t are broader than in the sim ulation. Figure 5.18 compares the sp ectra of the SPE ev aluation for a single frame of measured and sim ulated data, the latter p erformed with 10000 SPEs in the SPE frame. Clearly , the main features, a noise p eak b elo w 50 ADU, the fluorescence p eak at ≈ 120 ADU and a pile-up p eak at ≈ 250 ADU are visible. F urthermore, the ratio of 4-p x even ts to 5-p x ev en ts is similar, indicating a correctly simulated c harge cloud size. Indeed, the main difference in the t w o sp ectra is a broader p eak width and a more in tense noise p eak in the measured data, whic h seems to originate from an additional noise contribution to σ dark in the SPE frame, as is already susp ected ab o v e. Therefore, in the next step, additional noise is in tro duced b y increasing σ dark in the SPE frame only . Figure 5.19 a) sho ws ho w the sp ectrum ev olv es with increasing noise in the SPE frame only . The noise is increased b y 10% from b ottom to top, resulting in a go o d matc h of sim ulated and measured data for the Ni-L α fluorescence p eak and pile-up p eak at an increase of ab out 30% (red curv e). Ho w ev er, it b ecomes ob vious that some additional bac kground con tribution b elo w 80 ADU is presen t in the measuremen t, whic h cannot b e explained b y the sim ulation. This con tribution cannot b e assigned to further fluorescence lines, since they w ould ha v e b een detected in the measuremen ts with the pnCCD, where similar excitation conditions and a m uc h b etter energy resolution are applied (Figure 6.7 in Section 6.2.3). C-K α and O-K α are presen t in the sp ectrum, but with in tensities ab out 2 orders of magnitude lo w er than the Ni-L α in tensit y . P ossible bac kground con tributions migh t b e in tro duced due to stra y ligh t or detector artifacts, but no satisfactory explanation has b een found, y et. Indeed, an enhanced energy resolution w ould not only b e b eneficial in terms of photon discrimination, but w ould also help to b etter understand the recorded sp ectra and iden tify noise con tributions. T o study the parameters that migh t influence the energy resolution, SPE frames are 111 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF Figure 5.19.: Sp ectrum ev olution for v ariation of several parameters. The sp ectra are separated b y an offset for reasons of clarit y . In a) the noise σ dark is increased in the SPE frame only (and not in the master dark image) in 10% steps with resp ect to σ dark = 4.67 ADU in the dark frames. The red curv e with an increase of 30% is the starting p oin t for all further sim ulations. This curve is also presen t in b)-d) pain ted in red. In b) the c harge cloud size σ cc and in c) the noise level σ dark is decreased in 20% steps. d) sho ws sp ectra with differen t num b ers of SPEs sim ulated on the CCD c hip to in v estigate pile-up. 112 5.4. SINGLE PHOTON ANAL YSIS sim ulated with v arying c harge cloud size, reduced CCD noise and differen t photon num- b ers. The starting p oin t for the v ariation of the parameters is alw a ys the red curv e in Figure 5.19 a), where the noise in the SPE frame is increased b y 30% with resp ect to the dark frames. Figure 5.19 b) and c) sho w the ev olution of the sp ectrum with de- creasing c harge cloud size ( σ cc ) and decreasing CCD noise ( σ dark ) in steps of 20%. The reduction of σ cc leads to a blue shift and narro wing of the fluorescence (and pile-up) p eak. The reason for this is that split ev en ts b ecome less probable and the total charges I ph ( N ph ) are collected more thoroughly , since less pixels b elo w the noise lev el app ear. The noise p eak is rather unaffected. When reducing σ dark , as is sho wn in Figure 5.19 c), the fluorescence and pile-up p eak also get narro w er and blue shifted. Besides the more complete c harge cloud collection, also noise for ev ery pixel con tributing to the SPE is reduced and therefore the width of the lines. F urthermore, also the width of the noise p eak decreases and the noise p eak is red shifted. This leads to a reduced background in the lo w energy part of the sp ectrum, whic h is helpful for the analysis of ligh t elemen ts. Finally , in Figure 5.19 d) it can b e seen that a change of photon n um b ers and th us of pile-up probabilit y do es not affect the energy resolution in the sp ectra after split ev en t recom bination. Ho w ev er, for sp ectra with sev eral fluorescence lines pile-up will increase the bac kground and lead to in tensit y losses. This is esp ecially critical for GEXRF pro- files, where the pile-up probabilit y dep ends on the fluorescence emission angle and th us migh t distort the profile. The enhancemen t in resolving p o w er is no w in v estigated in a more quan titativ e w ay . F or all curv es in Figure 5.19 b) and c) the fluorescence p eak is fitted with a Gaussian function and a v alue for the resolving p o w er at this p eak is calculated b y E / ∆ E = x 0 / ∆ x FWHM . Figure 5.19 sho ws the results not only for the 4p x-Area-Clustering metho d, whic h is used ab o v e, but also for similar in v estigations p erformed with the Clustering metho d. In Figure 5.19 a), the c hange of the resolving p o w er with decreased c harge cloud size in units of the measured c harge cloud size σ cc = 7.6 µ m is sho wn. F or the 4p x-Area-Clustering metho d, the resolving p o wer seems to increase linear up to a maxim um for negligible σ cc , the v alue of the maximum resolving pow er b eing influenced b y equation 5.4.4 and the noise in the images. The Clustering metho d sho ws also a linear b eha vior for small σ cc , but the curv e flattens for v alues approac hing the measured c harge cloud size. In teresting to note is the in tersection of b oth curv es at ab out 0.4. F or smaller c harge cloud sizes, the split ev en t recombination of al wa ys at least 4 pixels (as in 4p x-Area-Clustering) is no longer b eneficial, since most of the c harge cloud is more lik ely con tained in a smaller n um b er of pixels. How ev er, for large c harge cloud sizes, the 4p x-Area-Clustering metho d is sup erior with resp ect to resolving p o w er. In Figure 5.19 b), the dep endency of the resolving p o w er on the noise in the image is illustrated. It is strongly nonlinear with a strong increase of resolving p o w er for 113 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF Figure 5.20.: Increase of resolving p o w er when a) σ cc or b) σ dark are reduced in sim ulated data. Split ev en t recombination is performed with the 4px-Area-Clustering and the Clustering metho d. ev er smaller noise lev els, reac hing almost the theoretical limit of E / ∆ E = 19.2 giv en b y Equation 5.4.4 (a con v ersion factor for the FWHM width of 2.35 has to b e applied). The determined resolving p o w er of the sim ulations is sligh tly reduced b ecause only natural n um b ers are allo w ed in the calculation of pixel in tensities in the split ev en ts. This leads to some discrimination noise in the sim ulation, whic h limits the resolving p o w er to 18.2 but hardly affects the energy resolution when other noise con tributions dominate. The o v erall b eha vior for the 4p x-Area-Clustering and the Clustering metho d is v ery similar for noise lev els reduced b y more than 40% with resp ect to the noise of the measuremen t. In this regime, the c harges can b e differen tiated from noise similarly w ell, while for higher noise lev els, the Clustering metho d results in a less complete c harge collection. Summarizing, the b enefits of the 4p x-Area-Clustering metho d o v er the Clustering metho d with resp ect to c harge collection for the measuremen t parameters are demon- strated. Therefore, this metho d is used in Section 6.3 for split ev en t recom bination. F urthermore, it has b een found that the noise lev el, e.g. readout noise and thermal noise of the CCD system, more drastically affects the energy resolution than the c harge cloud size, or more precisely the c harge cloud size to pixel size ratio. Since angular resolution pla ys a ma jor role in GEXRF analysis, a rather extended charge cloud size to pixel ratio and the use of sub-pixel resolution metho ds (e.g. b y determining the cen ter of gra vit y [64, 59]) migh t b e b eneficial. Instead of absolute noise level, rather the signal-to-noise ratio is of imp ortance, i.e. I ph / σ meas . This also explains why SPE recom bination is m uc h easier for hard X-ra ys, where the photons create more c harges in the detector, then for soft X-ra ys. Ho w ev er, in [61] an Electron Multiplying CCD is 114 5.4. SINGLE PHOTON ANAL YSIS presen ted, whic h is pre-amplifying the charges b efore readout, so b efore readout noise is added to the signal. Th us, not only the o v erall signal-to-noise ratio is increased, but also high readout frequencies can b e preserv ed, efficien tly reducing measuremen t time. 115 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF 5.5. Compilation of GEXRF Profiles The measured data of a full GEXRF measuremen t at one sample p osition consists of t ypically sev eral h undred CCD images, where single photon ev en ts can b e discriminated. T o eac h SPE, an in tensit y (corresp onding to the photon energy) and a hit p osition on the CCD mesh can b e attributed with the algorithms describ ed in the previous Section 5.4. F urthermore, the angular calibration (Section 5.3) can b e used to assign a fluorescence emission angle and a solid angle of detection to eac h pixel of the CCD. All this information can no w b e com bined to obtain GEXRF profiles, i.e. the angular dep enden t fluorescence in tensit y normalized to the solid angle of detection for eac h angle, for distinctiv e (elemen t sp ecific) fluorescence lines (Figure 5.21). T o ac hiev e this, t w o ev aluation options are describ ed in the follo wing and applied later for the compilation of the GEXRF profiles of the t w o b eam times in Sections 6.2 and 6.3. 5.5.1. Region of Interest Metho d With this metho d, the in tegral sp ectrum of the GEXRF measuremen t is used to define regions of in terest (R OIs) of SPEs that are ev aluated for a GEXRF profile. This could b e for example all SPEs with in tensities within the FWHM (or a m ultiple) cen tered around a p eak p osition. Then, according to the angular calibration, regions on the CCD grid are defined, where the corresp onding fluorescence emission angle is ψ fl ± ∆ ψ fl . Note that since the equi-angle lines are not parallel to the CCD pixel ro ws or columns, ∆ ψ fl can b e c hosen smaller than the angular resolution corresp onding to a single pixel and still pixel p ositions migh t b e found b elonging to the (small) angular area. Ho w ev er, only with the determination of the SPE hit p osition with sub-pixel resolution (using the cen ter of gra vit y) a real gain in angular resolution is ac hiev ed. No w, the num b er of SPEs of a R OI are summed in eac h angular region for eac h GEXRF image, yielding the detected photon n um b ers (of the sp ecific R OI) for all fluorescence emission angles ψ fl . Finally , this n um b er can b e normalized to the solid angle of detection of eac h angular area, whic h is also giv en b y the angular calibration. A plot of the (normalized) photon n um b ers with resp ect to ψ fl results in the GEXRF profiles. 5.5.2. Sp ectra Deconvolution Metho d While the region of in terest metho d is computationally quic k and straigh t forw ard, sp ec- tral bac kground con tributions or line o v erlaps can lead to distortions of the GEXRF profile. In this case, it might be necessary to deconv olv e eac h sp ectrum b elonging to angular regions on the CCD c hip, whic h corresp ond to fluorescence emission angles 116 5.5. COMPILA TION OF GEXRF PR OFILES Figure 5.21.: Ev aluation chart describing the steps necessary to compute GEXRF profiles from the clean GEXRF frames and the angular calibration. 117 5. DEVELOPMENT OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF ψ fl ± ∆ ψ fl . A t first, the sp ectra are created by calculating the n um b er of SPEs with equal SPE in tensities for eac h angular region. Then, these sp ectra are decon v olv ed with, e.g., Gaussian functions for the noise, fluorescence and pile-up p eaks and a suitable bac kground mo del. This is applied and compared to the R OI metho d for the ev aluation of the second b eam time. Here, only few sp ectral features are present, but these features are somewhat distorted due to high noise lev els and small energy resolution (see Sec- tion 6.3 for details). In general, it is also p ossible at this step to use external soft w are for sp ectra decon v olution, e.g. PyMca [118]. The result of the sp ectral decon v olution should b e the n um b er of SPEs in the fluorescence lines of in terest. This n um b ers can again b e normalized to the solid angle of detection of the angular region and displa y ed as GEXRF profile. 118 6. F easibilit y Study of Lab o rato ry Scanning-F ree Soft X-Ra y GEXRF The first test of the suggested lab oratory , scanning-free, soft X-ra y GEXRF setup de- scrib ed in the previous Chapter 5 is p erformed with a C/Ni-m ultila y er sample, t ypi- cally used as an X-ra y mirror. The w ell-con trolled fabrication pro cess and a standard c haracterization of the m ultila y er b y X-ra y reflectometry (XRR) is p erformed b y AXO DRESDEN Gm bH. Th us, the sample structure is well-kno wn, whic h mak es it applicable for pro of of principle measuremen ts. Moreo v er, b esides as X-ra y optics, multila y ers are e.g. used as protectiv e coatings or as ceramic capacitors and dev elopmen t of no v el m ultila y er structures and adapted material systems is ongoing [119, 120, 121, 122]. Amongst others, this dev elopmen t is accompa- nied b y in v estigations of diffusion pro cesses, la yer roughness and la y er thic knesses using GI- or GEXRF metho ds at sync hrotron radiation facilities [123, 124], indicating exem- plarily the approac hable analytical questions with these tec hniques. Since m ultila y ers are a represen tativ e of tec hnical stratified materials, suc h as thin-film solar cells [43], ther- mo electric devices, transistor gate stac ks [19] or gas sensors [125, 126], in v estigations on m ultila y ers further indicate the analytical p oten tials of a lab oratory scanning-free GEXRF setup for the ab o v e-men tioned material systems. After describing in detail the structure of the m ultila y er sample and its mo deling for the GEXRF forw ard calculations in Section 6.1, the results of t w o b eam times with the lab oratory scanning-free GEXRF setup are presen ted. While in Section 6.2 the descrip- tion and ev aluation of the v ery first measuremen ts are sho wn, the second b eam time (Section 6.3) fo cuses on the c haracterization of setup and metho dological impro v emen ts. Most of the findings of Sections 6.1 and 6.2 are published in a pap er b y this author and co-authors in [84]. 6.1. Multila y er Sample The m ultila y er structure under in v estigation is fabricated b y Dual Ion Beam Dep osition (DIBD) and consists of 15 bila y er pairs of carb on and nic k el on a silicon w afer. Its struc- 119 6. FEASIBILITY STUD Y OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF Figure 6.1.: a) T ransmission electron microscop y (TEM) image of the perio dic C/Ni-m ultila y er structure used in the GEXRF analysis. b) Analytical results of the bi- la y er thic kness of the multila y er obtained with con v en tional XRF, X-ra y reflectometry (XRR) and TEM measuremen ts. A sc hematic of the sample is sho wn in the inset. ture has a w edge-shap ed form with a bila y er thic kness c hanging from 5 nm to 6 nm o v er a sample length of 40 mm. Since suc h structures are fabricated to b e used as X-ra y mirrors, in terfaces need to b e sharp and with lo w roughness and the in-depth thic kness v ariation needs to b e less than a few p er mill to ac hiev e go o d reflectivities. The qualit y of the m ultila y er structure is also confirmed b y transmission electron microscop y (TEM) mea- suremen ts (Section 6.1.1) carried out at the Zen traleinric h tung Elektronenmikrosk opie (Zelmi) of the T ec hnical Univ ersit y of Berlin. The size of the whole m ultila y er strip e is 13 cm × 5 mm. T o moun t it prop erly on the sample holder, the sample is cut at +3 mm distance to the cen ter in to t w o parts. In the follo wing, zero mm marks the middle of the m ultila y er, the bila y er thic kness is increasing at the p ositiv e scale. A schematic of the m ultila y er structure is sho wn in the inset of Figure 6.1 a). 6.1.1. Structure Analysis with Complementa ry Metho ds Besides GEXRF, the m ultila y er sample is also c haracterized with X-ra y reflectometry , con v en tional XRF and transmission electron microscop y , the results of whic h are sho wn in Figure 6.1 a). The XRR measuremen ts are p erformed b y AX O DRESDEN Gm bH with a t win mirror arrangemen t (TMA) using the c haracteristic Cu-K α radiation of an X-ra y tub e. The sample is irradiated at 5 p ositions (-30 mm, -15 mm, ... 30 mm) with the plane of 120 6.1. MUL TILA YER SAMPLE incidence parallel to the thic kness gradien t. T o confine the fo otprin t on the sample, i.e. the prob ed thic kness range, a slit system is used, resulting in an effectiv e fo otprin t length of 1.2 mm at ab out 1 ◦ incidence angle. The probing width p erp endicular to the thic kness gradien t is of the size of the sample width (5 mm). The measured reflectivity curv es are fitted with sim ulated curv es of a mo del of the sample using the soft w are IMD. In the fit, densit y , roughness and thic knesses of the la yers are used as free parameters. The thic kness of the bila y ers in the fit is mainly dep enden t on the p osition of the Bragg p eak at ab out 1 ◦ incidence angle. Because of the fairly linear thic kness gradien t, mainly a broadening of the Bragg p eak in the reflectivit y measuremen ts is observ ed due to the extended fo otprin t length, instead of a p eak shift. This ensures the accuracy of the determined bila y er thic kness for a giv en p osition on the m ultila y er. Con v en tional XRF analysis is carried out with a Fisc herscop e X-Ra y XD V-SDD (Hel- m ut Fisc her Gm bH), using a tungsten ano de and a 100 µ m Al filter in the excitation c hannel. The high v oltage is set to 50 k V. The sample is measured in 3 parallel lines with a distance of 1.5 mm and a sp ot size of ab out 1 mm diameter. Eac h line scan con- sists of 135 p oin ts o v er a total of 12 cm length of the m ultila yer. In eac h p oin t, sp ectra are recorded (the liv e-time is 120 s p er measuremen t), decon v olved and ev aluated with the WinFTM soft w are implemen ted in the Fisc herscop e. F or the calculation of bila y er thic knesses, a sample mo del consisting of a single Ni la yer on top of a Si substrate is assumed, since absorption of primary and fluorescence radiation in the C la y ers is negli- gible for hard X-ra ys ∗ . The la y er thickness is then divided b y the n um b er of bila y ers of the m ultila y er and the relativ e thic kness ratio for nic k el Γ = d Ni /d bilay er = 0 . 45 to deduce a bila y er thic kness for ev ery measuremen t p oin t. Inheren tly , the Fisc herscop e measures a mass dep osition, whic h is con v erted to a la y er thic kness b y the soft ware, giv en a den- sit y of the la y er. In Figure 6.1 a), an extended area for the con v en tional XRF results is plotted rather than precise measuremen t p oin ts. The lo wer border indicates the results, if bulk densit y is assumed for the Ni la y ers. Ho w ev er, the XRR measuremen ts sho w ed that the la y er densit y is reduced b y up to 10%. T o illustrate the influence of this effect, the upp er b order of the area corresp onds to a Ni la y er in the sample mo del with suc h a densit y reduction. Statistical uncertain ties giv en b y the Fisc herscop e ev aluation are negligible. As can b e seen, the XRF v alues un derestimate the XRR measuremen ts, ev en with the reduced densit y tak en in to accoun t. Therefore, probably a further uncertain t y of the XRF analysis due to the fundamen tal parameter based ev aluation of the WinFTM soft w are has to b e considered. Alternativ ely , also the Ni lay er densit y could b e smaller than is found with the XRR measuremen ts. Since la yer densit y and la y er roughness ha v e similar effects on the calculated XRR curv e, b oth v alues can to some extend comp ensate ∗ F or example, the transmission of Ni-K α through 1 µ m C is larger than 99.9% [39]. 121 6. FEASIBILITY STUD Y OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF eac h other. Nev ertheless, the bila yer thic kness gradien t (i.e. the slop e of the curv e) is in go o d agreemen t with the XRR measuremen ts and further extends the curv e to w ards the edges of the m ultila y er sample. There, strong deviations from the linear profile are measured, whic h originate from the pro duction pro cess. TEM measuremen ts are p erformed at t w o sample p ositions at the Zen traleinric h tung Elektronenmikrosk opie (ZELMI) of the T ec hnical Univ ersit y of Berlin. F or this purp ose, cross-sections ha v e to b e prepared, making a sample consumption of ab out 5 mm length at the p ositions to b e measured una v oidable. F or the preparation of the cross-sections the 5-mm long sample piece is cut in t w o halv es at the measuremen t p osition and glued with a hot resin (130 ◦ for 30 min.) face to face. A thin slice is cut from the stack and is further thinned b y p olishing and ion milling parallel to the direction of the in terface, to create a dimpled shap e. A t the cen ter, the sample pro vides p ositions with thic knesses in the range of 10 nm, sufficien t for TEM microscop y . The TEM micrographs are recorded with a FEI T ecnai G 2 20 S-TWIN at up to ten p ositions of the t wo sample cross-sections. An example of a micrograph is sho wn in Figure 6.1 b). Due to the mass con trast, the nic k el la y ers are visible as dark lines with the brigh ter carb on la y ers in b et w een and the silicon substrate on the b ottom of the picture. The la y ers app ear smo oth, homogeneous and equidistan t. Only at the silicon substrate in terface a 16th nic k el la y er with less homogeneous thic kness is visible. The origin of this la y er is probably diffusion from the 15th nic k el la y er (whic h also app ears a bit brigh ter than the lay ers on top) through the thin carb on la y er on the substrate. Whether this diffusion is induced during the sample fabrication or during the preparation of TEM cross-sections cannot b e v erified. Ho w ev er, the bila y er thic kness results should not b e affected, since the signal in GEXRF and XRR measuremen ts of the top la y ers is dominan t and for the con ven tional XRF analysis the total mass dep osition is measured. F or the TEM measuremen ts, the bila y er thic kness is deduced from the in tensit y profiles along the normal of the la y ers and the magnification factor. Uncertain ties are estimated from statistical deviations of the bilay er thic kness measuremen ts. The results are sho wn in Figure 6.1 a). 6.1.2. GEXRF Simulations F or the sim ulations of the GEXRF profiles a mo del of the m ultila yer sample is used in the xrfLibrary framew ork (Section 2.4.1), whic h is in accordance to X-ra y reflectometry measuremen ts. In the ev aluation of these measuremen ts it is found that the thic kness ratio of nic k el is Γ = d Ni /d bila yer = 0 . 45. The la y er densities are 8.21 g/cm 3 for nick el, whic h is a reduced densit y of 7.5% compared to bulk and 2.75 g/cm 3 for carb on, whic h fits rather to amorphous carb on than diamond-lik e carb on (b oth can in principle b e dep osited with DIBD). F urthermore, for the GEXRF sim ulations with the xrfLibrary , 122 6.1. MUL TILA YER SAMPLE Figure 6.2.: a) Ni-L α,β and b) Ni-L l,n GEXRF profiles for tw o bila y er thic knesses of the C/Ni-m ultila y er sample. Also sho wn are the in tensities of eac h con tributing line (dashed lines) and the p osition of the resp ectiv e Bragg angle (v ertical line). atomic scattering factors are used from Chan tler et al. [40]. Because of the energy resolution of the measuremen ts, GEXRF profiles for Ni-L α,β and Ni-L l ,n can b e separated in the first b eam time (Section 6.2). These t w o profiles are sim ulated with summed con tributions from L α 1 , L α 2 and L β 1 for Ni-L α,β and L l and L n for Ni-L l,n . The relativ e in tensities are calculated using subshell photoionization cross sections from Eb el et al. [33] and transition probabilities and fluorescence yields from Elam et al. [32]. The sim ulations of the Ni-L α,β and Ni-L l,n GEXRF profiles for t w o differen t bila y er thic knesses d = 5 nm and d = 6 nm of the C/Ni-m ultila y er are sho wn in Figure 6.2 a) and b). Up to 2 ◦ , the in tensit y of the fluorescence line increases strongly b ecause of the reduced reflection at the in terfaces of the sample, prev en ting the radiation from leaving the sample. The further, shallo w er in tensit y increase results from reduced self-absorption when the path length in the sample is reduced. Then, near the first order Bragg angle α = arcsin( λ/ 2 d ) of the m ultila y er sample with the resp ectiv e photon w a v elength λ and bila y er thic kness d , a lo cal in tensit y maxim um follo w ed b y a minim um (vice v ersa in b)) app ears. F or the resp ectiv e angles, in tensity maxima (and minima) of the XSW field used for the calculations emerge in the nic k el la y ers. This means that for fluorescence photons from atoms lo cated at these p ositions, the probabilit y to reac h the detector is increased (reduced) due to self-in terference of the probabilit y w a v e functions of each photon (Section 2.2.3). This in terference effect in m ultila y ers is similar to the K ossel effect, whic h describ es the diffraction pattern of X-ra ys originating from sources inside a crystal [127, 128]. Therefore, the features in the GEXRF profiles of m ultila y ers are also called K ossel lines. They are used to get information ab out the structure factor of 123 6. FEASIBILITY STUD Y OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF crystals [129]. The sim ulated GEXRF profiles will b e used in the next c hapters for the ev aluation of the measured GEXRF profiles and in the angular calibration for the measuremen ts of the first b eam time. 124 6.2. FIRST BEAMTIME - PR OOF OF PRINCIPLE 6.2. First Beamtime - Pro of of Principle The first b eam time serv es as pro of of principle for the suggested lab oratory scanning-free GEXRF setup. F or this purp ose, a pnCCD of PNSensor Gm bH is applied, pro viding reliable single photon analysis. As test sample, the m ultila y er describ ed in the previous Section 6.1 is used. The follo wing c hapters fo cus on the setup alignmen t (Section 6.2.1), the first measured in tegral XRF sp ectrum with the setup (Section 6.2.3), the angular calibration (Section 6.2.4) and finally the compilation and ev aluation of the measured GEXRF profiles (Sections 6.2.5). The ra w data handling, p erformed by PNSensor Gm bH is describ ed shortly in Section 6.2.2. 6.2.1. Setup Description and Alignment The principle setup and setup alignmen t is discussed in Section 5.2. In the follo wing, only deviations to this setup and alignmen t pro cedures are describ ed, whic h are applied in the first b eam time. As optics, the Kirkpatrick-Baez (KB) mirrors are used and the detector is a pnCCD of PNSensor Gm bH. The CMOS sensor, b eneficial for alignmen t and b eam diagnostics, and the optical laser for precise definition of the measuremen t geometry and sample alignmen t are only a v ailable in the second b eam time. Therefore, goniometer and sample alignmen t here is carried out relativ e to the b eam of the KB optics with a pinhole and an X-ra y sensitiv e dio de and the angular calibration is p erformed with a reference measuremen t. Goniometer Alignment Goniometer alignmen t is carried out relativ e to the b eam of the KB optics with a pinhole on the sample holder and an X-ra y sensitiv e dio de on the 2 θ axis b ehind the pinhole. Firstly , the dio de is p ositioned in the direct b eam of the LPP source. Then, the p osition of the KB fo cus is found b y scanning the radiation with the pinhole. The pivotal point p osition for the pinhole is kno wn from former measuremen ts. Therefore, the t w o axes of the piv otal p oin t p erp endicular to the b eam direction can b e adjusted b y mo ving the motorized base frame of the sp ectroscop y c ham b er in the calculated p osition. Ho w ev er, the mo v emen t of the sp ectroscop y c ham b er c hanges the forces pulling on the v acuum c ham b er con taining the KB optics and induces a shift in the fo cus p osition due to a sligh t misalignmen t of the KB optics. Therefore, the pro cedure needs to b e rep eated iterativ ely . The p osition of the piv otal p oin t of the goniometer parallel to the b eam of the KB optics is defined b y the setup, whic h is p erformed in accordance to a CAD mo del of the whole setup. Since the fo cal length of the fo cus of the KB optics is in the range of 1 cm, the fo cus is not further adjusted parallel to the LPP radiation. After all adjustmen t 125 6. FEASIBILITY STUD Y OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF Figure 6.3.: a) In tensit y scans of a 500 µ m pinhole through the pivotal point of the goniometer. The width (FWHM) of the b eam profile is less than 0.5 mm. b) Impact of the fo otprin t size on the angular resolution of the GEXRF measuremen ts for the applied geometry . The red line indicates an upp er limit of the allo w ed fo otprin t size not to affect the angular resolution. The y ello w disc illustrates the maxim um fo otprin t size during the measuremen ts. steps are completed, the alignmen t of the goniometer to the KB fo cus and the alignmen t of the KB fo cus itself are c hec k ed. F or this purp ose, a pinhole with 500 µ m diameter is scanned through the fo cus sp ot p osition and the resp ectiv e in tensit y is recorded (the CMOS is not a v ailable during this b eam time). As can b e seen in Figure 6.3 a), the profiles sho w a Gaussian-lik e shap e with full widths at half maxim um (FWHM) of less than 0.5 mm in the v ertical and horizon tal axis and th us the FWHM of the fo cus sp ot is less than 0.5 mm. Note that the sp ot size can b e ev en smaller but is unresolv ed b y the applied pinhole. Figure 6.3 b) sho ws the impact of the fo otprin t size in the sample plane to the angular resolution of the GEXRF measuremen ts for the actual setup geometry (from Section 6.2.4). A detailed description of the computation of the graph and its in terpretation can b e found in Section 2.3.2 for a general case. Here, the results sho w that ev en with an upp er limit of the fo otprin t size of 0.5 × 0.5 mm 2 , the influence of the fo otprin t size on the angular resolution is negligible. T o c hec k the p osition of the KB fo cus at the b eginning and at the end of eac h measure- men t da y , a fluorescen t screen is applied. The screen has a quadratic shap e of < 1x1 mm 2 and its p osition on the sample holder is determined just after the alignmen t of the sp ec- troscop y c ham b er. F or this purp ose, the sample holder is mo v ed while the LPP source is in op eration, un til the fluorescence light from the screen is visible and the spot size is minimized. F or the daily con trol of the p osition of the KB focus, the corresp onding motor p ositions of the fluorescence screen are adjusted and the fluorescence ligh t c hec k ed 126 6.2. FIRST BEAMTIME - PR OOF OF PRINCIPLE Figure 6.4.: Photograph of the sp ecimen holder with the fluorescen t screen and the m ul- tila y er sample part P . Indicated are also the measuremen t p ositions for the GEXRF analysis. visually . During the whole measuremen t p erio d, no misalignment of the KB optics is detected. Sample Holder The sp ecimen holder is equipp ed with the 20 ◦ wedge (Section 5.2.5), on whic h the samples are moun ted (Figure 6.4). On part P , which is the half of the m ultila y er sample with larger bila y er thic kness, GEXRF measurements are p erformed at positions P1, P2, ... P6. On part Q, a single GEXRF measuremen t is p erformed at p osition Q1. The lateral p ositions of the measuremen ts are giv en in T able 6.2.1. Sample alignmen t is p erformed b y measuring the sample-induced shado wing similar as in Section 5.2.5, but making use of the direct b eam and the X-ra y sensitive diode moun ted on the 2 θ axis. The relativ e distances b et w een the first measurement p oin t on the m ultila y er sample and the fluorescence screen are determined with a slide gauge. Th us, the p osition can b e set in to the fo cus of the KB optics b y mov emen t of the sp ecimen holder with the x and y motors. Then, alignmen t of the zero angle ( θ scan) and radial p osition ( x - z scan) are carried out. F or the final GEXRF geometry , the sample is rotated b y θ = 84 . 2 ◦ to get the full angular scale on the pnCCD chip. F or the measuremen ts of p ositions P1 to P6 on the same part of the m ultila y er sample, the sample surface is shifted through the piv otal p oin t of the goniometer b y using only the y motor. F or these measuremen ts, the whole geometry with resp ect to the pnCCD c hip sta ys the same, whic h allows to use the same angular calibration for these measuremen ts 127 6. FEASIBILITY STUD Y OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF T able 6.1.: C/Ni-bila y er thickness on sev eral GEXRF measuremen t p ositions in terp olated from XRR measuremen ts. measuremen t p osition / mm XRR bila y er thic kness / nm P1 36 ± 1 out of range P2 21 ± 1 5.63 ± 0.05 P3 31 ± 1 5.90 ± 0.05 P4 26 ± 1 5.78 ± 0.05 P5 37 ± 1 out of range P6 16 ± 1 5.48 ± 0.05 Q1 -3 ± 2 4.89 ± 0.05 (see section 6.2.4). As can b e seen in Figure 6.5, the data for the determination of the zero angle and the alignmen t of the sample surface in to the piv otal p oin t of the goniometer are noisy . F or the x - z scan (Figure 6.5 a)) the dio de is read-out 50 times p er measuremen t p oin t, resulting in a total scan time of 165 s. F or the θ scan, the mean of 150 read-outs is used p er measuremen t p oin t, resulting in a total scan time of o v er 15 min utes but still high statistical uncertain ties. Since longer scan times are incon v enien t, the uncertain ties for the z and θ adjustmen t, whic h are estimated to b e ∆ z = 0 . 1 mm and ∆ θ = 0 . 5 ◦ cannot b e reduced further. The high noise lev el of the measurement origi nates from an amplifier not optimized for the short pulses of the LPP source and the slo w readout times of the X-ra y sensitiv e dio de from the applied LabVIEW soft w are used for the readout. The rather high uncertain ties in the sample alignmen t are comp ensated b y the angular calibration with a reference and in the second b eam time (Section 6.3.1) circum v en ted b y using an optical laser instead of the direct LPP b eam. pnCCD In con trast to con v en tional CCDs, the pnCCD facilitates a fully depleted w afer v ol- ume, a frame store area and on-c hip amplifiers for ev ery pixel column [130, 116]. This allo ws for fast read-out (up to 1 kHz) while preserving lo w noise lev els, making the detector suitable for single photon detection. The pnCCD used in this b eam time has 264 × 264 pixels with a size of 48 × 48 µ m 2 . It is moun ted to the sp ectroscop y c ham b er via the CCD p ositioning system as describ ed in Section 5.2.4. Since no optical laser is used for the alignmen t pro cedures, only the distance of the pnCCD is adjusted to the sp ecimen holder. In the final p osition, the camera is fixed and not mo v ed for the whole 128 6.2. FIRST BEAMTIME - PR OOF OF PRINCIPLE Figure 6.5.: a) x - z scan of the sample through the incoming b eam with the surface parallel to the b eam. A t the half of the maxim um intensit y the sample shado ws half of the b eam. b) θ scan of the sample after its surface is adjusted to shado w half of the incoming b eam. At maxim um in tensit y , the surface is parallel to the b eam. measuremen t p erio d. A t the end of the b eam time, some parameters to describ e the p osition of the camera are measured, while the other parameters are determined in an angular calibration pro cedure (see section 6.2.4). Using a cathetometer, the heigh t of the 4 corners of the pnCCD c hip and the heigh t of the piv otal p oin t of the goniometer (using the fluorescen t screen aligned with the kno wn motor p ositions, section 6.2.1) are determined. With this information, the tilt angle of the pnCCD relative to a v ertical axis can b e calculated to φ CCD = (3 . 2 ± 0 . 5) ◦ and the relativ e heigh t of the pnCCD to h CCD = (4 . 1 ± 0 . 2) mm. F urthermore, from the CAD mo del and the distance b et w een flange and piv otal p oin t, measured with slip gauges, the distance of the c hip to the piv otal p oin t of the goniometer is calculated to d CCD = (63 ± 1) mm. F or the GEXRF measuremen ts, the pnCCD is co oled do wn to -70 ◦ to minimize thermal noise. The readout of the camera is triggered with the laser trigger of the LPP source. This allo ws to tak e a frame with the pnCCD for ev ery shot of the LPP source (1.2 ns pulse ev ery 10 ms). Due to the quic k readout times, a shutter, prev en ting radiation to reac h the detector during readout, is not necessary . 6.2.2. Data Reco rding and Image Pro cessing After setup alignmen t and p ositioning of the θ axis of the sample to the GEXRF mea- suremen t v alue (whic h is 84.2 ◦ to use the full detector area) the LPP source is put in to op eration and the pnCCD starts recording images. A measuremen t for one measure- men t p osition consists of sev eral h undred thousand frames in 1-2 hours. Eac h frame is bac kground corrected b y subtracting a master dark frame (calculated from some tens 129 6. FEASIBILITY STUD Y OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF Figure 6.6.: Energy calibration function for the measuremen ts p erformed with the pnCCD. to h undreds of dark frames, whic h are recorded without the LPP source running), and b y correction of gain and c harge transfer efficiency [130, 131]. Noise is discriminated b y setting all pixel v alues with in tensities of less than a m ultiple of the standard deviation of a single dark frame to zero. Split even ts o ccur with a maxim um of 4 pixels b elonging to the same photon ev en t, whic h means that the charge cloud is m uc h smaller than the pixel size ∗ . These split even ts are recom bined b y adding the pixel in tensit y of the adjacen t pixels, pro viding the ev ent in tensit y . The ev en t p osition is set to the p osition of the pixel with maxim um in tensit y . Th us, information ab out energy and p osition of eac h detected photon are obtained. 6.2.3. Integral XRF Sp ectrum An in tegrated sp ectrum o v er the whole angle range (in tegral sp ectrum) can b e created b y plotting the n um b er of photon ev en ts as a function of their in tensit y . F rom measuremen ts on the C/Ni-m ultila y er and on a solid Cu pallet, p eak p ositions of the fluorescence lines are ev aluated to obtain an energy calibration for the pnCCD measuremen ts. Figure 6.6 sho ws in first appro ximation a linear resp onse for the energy calibration, but the detected c harge in tensit y seems to b e underestimated for the lo w energy fluorescence photons of C-K α and O-K α . This effect migh t originate from incomplete c harge collection due to lo w signal to noise ratios of the single photon ev en ts in this energy region. Nevertheless, the sho wn energy calibration will b e used for the further illustrations of the sp ectra. In Figure 6.7 a) the in tegral sp ectrum of measuremen t P2 of the m ultila y er sample is ∗ In [130] a c harge cloud radius of ab out 10 µ m is giv en for a pnCCD 130 6.2. FIRST BEAMTIME - PR OOF OF PRINCIPLE Figure 6.7.: a) In tegral X-ra y fluorescence sp ectrum of the C/Ni-m ultilay er sample. 1.4 mil- lion Ni-L photon ev en ts are detected in a measuremen t time of 140 minutes. Reprin ted with p ermission from [84]. Cop yright 2017 American Chemical Society . b) XRF sp ec- trum for shallo w angle regions as indicated ± 0.6 ◦ . The o xygen p eak is mark ed with a red arro w. sho wn. The sp ectrum in tensit y increases sharply at ab out 38.7 e V (105 ADU), whic h is the cut-off in tensit y for the noise discrimination. All c hannels up to 184.5 e V (500 ADU) are dominated b y noise ev en ts. Fluorescence lines that can b e attributed to C-K α , Ni-L l,n , Ni-L α,β and elastically scattered radiation at 1078 e V are exp ected from the C/Ni-m ultila y er sample and clearly visible. O-K α originates mos t lik ely from sample con tamination or surface o xidation, as the sp ectra in Figure 6.7 b) indicate. Here, the angular calibration (see section 6.2.4) is already used to discriminate photons, whic h are emitted at shallo w angles, only . It can b e seen that the o xygen signal relativ e to the nic k el signal is strongest at ab out 3 ◦ , whic h is an indication that the signal originates from a surface near region. The in tegral sp ectrum sho ws a p ossibilit y to use the setup in a con v en tional XRF mo de. Indeed, b y irradiating the sample at e.g. 45 ◦ and scanning the sample in the surface plane, the CCD detector can b e used as efficien t energy-disp ersiv e detector for soft X-ra y fluorescence analysis with the LPP source (note that a single SDD detector suffers from pile-up effects b ecause of the short X-ra y pulses). In the follo wing, the in tegral sp ectrum in Figure 6.7 a) is used to estimate lo w er limits of detection (LLD) for carb on and nic k el. P eak in tensities are retriev ed from the sp ectrum b y subtracting a bac kground based on linear in terp olation b et w een supp orting p oin ts and using Gaussian fits for the fluorescence p eaks (sho wn in Figure 6.7 a)). The energy resolution at the Ni-L α,β line obtained from the Gaussian fit is 113 e V. The data ev aluated for the in tegral sp ectrum is recorded at P2 of the m ultila y er sample. F or this p osition, the bila y er 131 6. FEASIBILITY STUD Y OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF T able 6.2.: Estimated lo w er limit of detection (LLD) from an in tegral sp ectrum of the C/Ni- m ultila y er recorded with the pnCCD. The sample is excited with 1078 e V photons from the laser-pro duced plasma source and a reference time of 1000 s is used. elemen t LLD / ng cm − 2 LLD / atoms cm − 2 LLD / g nic k el 200 2 × 10 15 1 × 10 − 10 carb on 1000 6 × 10 16 6 × 10 − 10 thic kness is d = 5 . 63 nm with a thic kness ratio of Γ= d Ni /d bilay er = 0 . 45 and the densities for nic k el and carb on are ρ Ni = 8 . 21 g/cm 3 and ρ C = 2 . 75 g/cm 3 according to the XRR measuremen ts. This yields a total mass dep osition for nic k el and carb on of ˆ m Ni = 15 d Γ ρ Ni = 31 . 2 µ g / cm 2 and ˆ m C = 15 d (1 − Γ) ρ C = 12 . 8 µ g / cm 2 . The LLDs are no w calculated b y LLD i = ˆ m i 3 q N det i,j N det i,j × s t t 0 , (6.2.1) with the measuremen t time t = 140 min., a reference time of t 0 = 1000 s and N det i,j b eing the n um b er of detected photons of Ni-L α,β and C-K α , resp ectiv ely . The results are shown in T able 6.2 in v arious units for reason of comparison. F or the last column, the size of the excitation fo otprin t is m ultiplied to the relativ e LLDs, yielding absolute detection limits. T ypical detection limits for con v en tional XRF with an X-ra y tub e based sp ectrometer are in the range of 10 − 6 to 10 − 8 g [132]. In TXRF geometry , usually the analytes are dep osited on a reflecting w afer surface and th us the scattering bac kground pro duced in the w afer is drastically reduced. This leads to detection limits of 10 − 9 g to 10 − 11 g [133] or ev en further to the fg range [42] when applying mono c hromatic sync hrotron radiation. The estimated absolute LLDs for the lab oratory GEXRF setup are w ell in the range of t ypical TXRF detection limits, even though the angular range up to 12 ◦ is used, whic h exceeds the critical angle for total external reflection. Indeed, when exciting with mono c hromatic soft X-ra ys, the larger photoionization cross section as compared to hard X-ra y excitation and the lo w er resonan t scattering bac kground in con trast to p olyc hromatic excitation strongly enhance the LLDs. In the future, it migh t b e p ossible to monitor the inciden t flux on the sample (since it cannot b e assumed to 132 6.2. FIRST BEAMTIME - PR OOF OF PRINCIPLE T able 6.3.: Determination of the geometric parameters with v arious metho ds as indicated in thetable. l CCD is not measured directly . parameter determination determined v alue d CCD slip gauge and CAD (63 ± 1) mm l CCD - - h CCD cathetometer (4.1 ± 0.2) mm φ CCD cathetometer (3 ± 1) ◦ θ CCD defined b y setup (0 ± 2) ◦ ω CCD defined b y setup (0 ± 2) ◦ ω sample motor p osition (5.800 ± 0.001) ◦ b e constan t o v er time b ecause of accum ulation of debris on filters in the b eam path and th us increasing absorption) and use standard samples to prop erly calibrate the setup and p erform quan titativ e trace elemen t analysis. 6.2.4. Angula r Calib ration F or the presen t measuremen ts of the first b eam time, the optical laser to define the co or- dinate system and the geometry of the setup, is not applied. Therefore, the co ordinate system to describ e the geometry (LAB system) has to b e defined differen tly . It pro v es reasonable to define the LAB system suc h, that the surface normal of the sample after sample alignmen t p oin ts in the x direction of the LAB system (to w ards the pnCCD) and the origin is still defined b y the piv otal p oin t of the goniometer. After sample alignmen t (sample surface is parallel to the inciden t X-ra y b eam, ω sample = 90 ◦ ), the θ motor of the goniometer is mo v ed b y 84.2 ◦ , to shift the zero-angle of the fluorescence emission to one edge of the pnCCD and th us mak e use of the whole detector c hip for the measuremen ts. Ho wev er, this tilt of the sample surface, ω sample = 5 . 8 ◦ with resp ect to the x axis of the LAB system, needs to b e considered in the angular calibration. T able 6.3 summarizes the kno wledge of the v arious geometric parameters (GPs, defined in Figure 5.8 on page 95) determined b y indep enden t measuremen ts using e.g. slide gauge or cathetometer. Note that φ CCD is measured with resp ect to a horizon tal plane (the cathetometer is aligned with a spirit lev el) and not with resp ect to the co ordinate system describ ed ab o v e. Due to the una v oidable tilt of the sample normal on the sp ecimen holder with resp ect to the horizon tal plane, the uncertain t y of φ CCD is increased. The GPs φ CCD , d CCD and l CCD ha v e the strongest influence on the angular axis and 133 6. FEASIBILITY STUD Y OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF l CCD is not directly measured. This prohibits the use of an absolute angular calibration and mak es a reference measuremen t (see Section 5.3.1) and the application of a fitting algorithm necessary . The other GPs are fixed according to their measured v alue in T able 6.3. The GEXRF profile for the calibration is pro vided b y one of the GEXRF measuremen ts, namely the measuremen t at p osition P2. The sim ulated GEXRF profile of the resp ectiv e measuremen t is based on the sample structure deriv ed from X-ra y reflectometry measuremen ts. The in terp olated v alues of the XRR measuremen ts at P2 yield a bila y er thic kness of 5.63 nm, densities for the Ni and C la y ers of 8.21 g/cm 3 and 2.75 g/cm 3 , resp ectiv ely and a thic kness ratio Γ = d Ni /d bila y er = 0 . 45. F or the GEXRF profiles of Ni-L α,β , the intensities of Ni-L α 1 , Ni-L α 2 and Ni-L β 1 are calculated and summed. The GEXRF profiles of Ni-L l,n consist of the summed GEXRF profiles of Ni-L l and Ni-L n . Measured and sim ulated GEXRF profiles are used in the fitting algorithm for the angular calibration describ ed in the follo wing. In a first step of the fitting algorithm, φ CCD is optimized by maximizing the feature con trast in the measured Ni-L α,β GEXRF profile (Figure 6.8 a)). F or this purp ose, v al- ues for d CCD and l CCD are tak en from T able 6.3 as starting parameters and (measured) Ni-L α,β GEXRF profiles are compiled for a set of φ CCD v alues from 0 ◦ to 6 ◦ . Eac h profile is smo othed using a Sa vitzky-Gola y filter and the con trast is defined as the difference b et w een the maxim um in tensit y close to the Bragg angle and the lo cal minim um in ten- sit y . F or the next fitting step, the φ CCD v alue which giv es the maxim um con trast is used (Figure 6.8 b)). In the second step of the fitting algorithm, the angular scale of the measured GEXRF profile is stretc hed and shifted b y adapting d CCD and l CCD , to iden tify the b est matc h with the sim ulated profile. It is found that the b est result in terms of con v ergence and stabilit y is ac hiev ed b y using a brute force algorithm. Therefore, (measured) GEXRF profiles are compiled for a grid of 15 × 15 v alues (15 v alues for d CCD and l CCD , eac h) and compared to the sim ulated GEXRF profile. The shap es of measured and sim ulated curv es sho w significan t deviations (see section 6.2.5), prev en ting to use the whole curv es for the ev aluation. Ho w ev er, at least t w o criteria are needed for the unam biguousness of the algorithm (due to the t w o fit v alues). Th ese criteria are c hosen to b e on the one hand the difference of the lo cal in tensit y minim um p osition in the measured and sim ulated Ni-L α,β GEXRF profile (Figure 6.8 a)) and on the other hand the angle difference of the inflection p oin t of the measured and sim ulated Ni-L l,n GEXRF profiles (Figure 6.8 c)). T o find now the b est d CCD and l CCD v alues in the brute force grid, the couple is c hosen, where the maxim um of the t w o angle differences (inflection p oin t and minim um p osition) is minimized (Figure 6.8 d)). Finally , the pro cess is rep eated iterativ ely 4 times, while alw ays the grids for φ CCD in 134 6.2. FIRST BEAMTIME - PR OOF OF PRINCIPLE Figure 6.8.: Results of final iteration step for the angular calibration. a) Compiled Ni-L α,β GEXRF profile and comparison of feature minim um p osition of measured and sim u- lated data. b) Change of the feature con trast (see a)) with resp ect to φ CCD . c) Compiled Ni-L l,n GEXRF profile and comparison of inflection p oin t p osition of mea- sured and sim ulated data. d) Maxim um of angle differences for the inflection p oin t and the minim um p osition for a giv en set of d CCD and l CCD . The red line indicates the angular resolution due to pixel size (0.05 ◦ ), whic h is used to estimate uncertain ties for d CCD and l CCD . 135 6. FEASIBILITY STUD Y OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF T able 6.4.: Result of the angular calibration pro cedure. In addition to the actual v alue of eac h GP , its (estimated) uncertain t y and influence on the shift of the minim um p osition are giv en. parameter determination determined v alue estimated uncertain t y shift of in t. min. d CCD fit algorithm 65.0 mm 1.3 mm 0.02 ◦ l CCD fit algorithm 7.70 mm 0.06 mm 0.04 ◦ h CCD measured 4.1 mm 0.2 mm 0.04 ◦ φ CCD fit algorithm 3.6 ◦ 1.2 ◦ 0.06 ◦ θ CCD defined b y setup 0 ◦ 2 ◦ 0.02 ◦ ω CCD defined b y setup 0 ◦ 2 ◦ 0.03 ◦ ω sample motor p osition 5.80 ◦ 0.001 ◦ 0.03 ◦ the first step and d CCD and l CCD in the second step are refined. The four frames in Figure 6.8 sho w the final fitting steps. The uncertain ties of the fitted GPs are estimated to b e for φ CCD of the length of the plateau in Figure 6.8 b) and for d CCD and l CCD ab out size of the area in Figure 6.8 d), where the maxim um feature shift is less than 0.05 ◦ , whic h is ab out the pixel resolution of the measurement. T able 6.4 summarizes the results of the geometric parameters used in the angular calibration and also sho ws their influence on the angular scale. The latter is determined b y calculating GEXRF profiles with an angular calibration where all but one GP are k ept at the v alues sho wn in T able 6.4. The giv en influence is then the difference in the minim um p osition of the Ni L α,β GEXRF profile created with t w o angular calibrations, one with the parameter as in the table and the other one with the estimated uncertain t y added. Figures 6.9 a) and b) sho w the result of the angular calibration in terms of fluorescence emission angles and solid angles of detection for eac h pixel on the pnCCD. An angular range of ab out 10 ◦ will b e measured with decreasing pixel n um b ers con tributing to the high angular range ab o v e 10 ◦ due to the tilt of the CCD. The calculated solid angles of detection will b e used to normalize eac h angular region to its total solid angle of detection. The o v erall solid angle of detection is Ω Det =0.038 sr. 6.2.5. GEXRF Profiles GEXRF measuremen ts are p erformed at 7 p ositions of the m ultila y er sample, corre- sp onding to 7 differen t bila y er thic knesses (T able 6.2.1 in Section 6.2.1). On eac h p osi- 136 6.2. FIRST BEAMTIME - PR OOF OF PRINCIPLE Figure 6.9.: Distribution of a) detected fluorescence emission angles and b) solid angles of detection for eac h pixel on the pnCCD c hip according to the angular calibration. tion, sev eral thousand to a few millions of frames are recorded in up to a couple of hours measuremen t time. In ev ery frame on av erage 3 to 5 single photon ev en ts (SPEs) are detected and ev aluated with resp ect to ev en t p osition and in tensit y . The photon ev en t p osition can b e con v erted to an emission angle using the angular calibration describ ed in the previous Section 6.2.4. The photon even t in tensit y can b e used to further ev alu- ate the photon ev en ts of a sp ecific bandwidth of the fluorescence sp ectrum, enabling to generate GEXRF profiles of sp ecific fluorescence lines, only . F or this purp ose, regions of in terest (R OI) are defined in the in tegral sp ectrum (Figure 6.7 in Section 6.2.3). Using the decon v olution of the in tegral sp ectrum, the lo w er and upp er limits of the defined R OIs for Ni-L α,β and for Ni-L l ,n are c hosen to maximize the in tegral in tensit y on the one hand and on the other hand guaran tee the sp ectral purit y of the resp ectiv e lines (App endix I). The GEXRF profiles are compiled b y coun ting the n um b er of photons in the sp ecific R OIs and angle incremen ts, for all frames of one measuremen t. The angle incremen t for ev ery emission angle is set to 0.05 ◦ and corresponds roughly to the pixel size resolution. Plotting the photon n um b er against the emission angle pro vides the GEXRF profiles. The results of measuremen t P1 and P2 are sho wn together with corresp onding sim ula- tions in Figure 6.10. The general shap e of the GEXRF profiles follo ws the simulation, featuring a stronger in tensit y increase at shallo w angles compared to steep er angles and also a lo cal in tensit y minim um close to the Bragg angle is visible. Ho wev er, a direct comparison sho ws also significan t deviations from the sim ulation. First, the steep in- crease at shallo w angles is o v erestimated in the sim ulation and, second, the predicted in tensit y maxim um close to the Bragg angle is strongly damp ed. T ypically , in GI- and 137 6. FEASIBILITY STUD Y OF LABORA TOR Y SCANNING-FREE SOFT X-RA Y GEXRF Figure 6.10.: Sim ulated and measured GEXRF profiles for measuremen t p oin ts P1 and P2. While the sim ulations with the tabulated atomic scattering factors sho w strong deviations from the measuremen ts, f 0 1 and f 0 2 can b e c hanged to fit the sim ulation to the measured data. GEXRF analysis, suc h deviations are comp ensated b y fitting the mo del of the sample, allo wing then to determine the sample structure, lay er densit y , roughness or to detect diffusion pro cesses. In the presen t case, changing all these parameters cannot satisfacto- rily explain the deviations. Y et, if the atomic scattering factors f 0 1 and f 0 2 for nic k el are used as fit parameters, the sim ulation can w ell describ e the measured GEXRF profiles (see 6.10, purple curve). In the fit result, f 0 1 is reduced b y a factor 0.6 and f 0 2 is increased b y a factor 3.2 compared to the data for f 0 1 and f 0 2 tak en from the Chan tler database [40] for nic k el at the Ni-L α 1 fluorescence line (849 e V). As is describ ed in section 2.2.1, f 0 2 is directly link ed to the absorption b eha vior of the material. Therefore, an increase of f 0 2 means increased absorption, damping the features in the GEXRF profile, as can b e observ ed in the measuremen t. The applied adaptions to f 0 1 and f 0 2 are quite drastic. Chan tler states in [40] uncer- tain ties for the atomic scattering factors of ab out 1% in the high energy range, whic h can increase to 10% to 20% for energies b et w een 30 e V and 1 k e V. How ev er, these v alues do not explain the strong adaptions necessary in the presen t study . In Figure 6.11, database v alues of the atomic scattering factors of nic k el are sho wn from compilations of Chan tler [40] and Henk e et al. [39]. Clearly , the L resonances strongly affect the atomic scatter- ing factors. F or the GEXRF profiles with adapted atomic scattering factors, photons of 138 6.2. FIRST BEAMTIME - PR OOF OF PRINCIPLE Figure 6.11.: Database v alues for the atomic scattering factors a) f 0 1 and b) f 0 2 tak en from Chan tler [40] and Henk e et al. [39]. the Ni-L α 1 fluorescence are sim ulated with an energy of 849 e V (according to the Elam database [32]), whic h is near the L3 absorption edge of nic k el (853 e V). Consequen tly , a small uncertain t y in the energy scale of the databases can lead to c hanges in the atomic scattering factors b y e.g. a factor of 5 for f 0 2 . Also, the differen t databases sho w strong deviations b et w een eac h other, indicating high uncertain ties for the v alues close to res- onances. This supp orts the v alidit y of the fitted atomic scattering factors. Since the shap e of the GEXRF profiles can b e altered so strongly with the v ariation of atomic scattering factors, and the uncertain ties of these are so drastic, any further ev aluation on diffusion or roughness is doubtful and is not p erformed for the presen ted pro of of principle measuremen ts. Nev ertheless, the difference of the minimum p osition of the sim ulated GEXRF profiles with database v alues and fitted v alues for f 0 1 and f 0 2 is less than 0.03 ◦ . Also, mo derate c hanges in the la y er densities, Deb y e-W aller factors (mo deling in terla y er roughness [134]) or implemen ting surface con tamination do not c hange the minim um p osition significan tly T able 6.5.: Effect of parameter v ariation in sim ulated Ni-L α 1 GEXRF profiles on the minim um p osition close to the Bragg angle of the C/Ni-m ultila yer sample. parameter v ariation shift of minim um carb on la y er densit y ± 10% < 0.005 ◦ nic k el la y er densit y ± 10% < 0.005 ◦ Deb y e-W aller factor 0 nm - 2 nm 0.02 ◦ carb on con tamination la y er 0 nm - 10 nm 0.015 ◦ Ni thic kness ratio Γ ± 10% 0.065 ◦ 139 [Document text truncated for crawler view.] Why institutions use Plag.ai for originality review, entry 67 Plag.ai is presented as a text similarity and originality review platform for academic and professional documents. 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