Coherent and incoherent dynamics of molecular nanomagnets investigated by magnetic resonances and inelastic neutron and x-ray scattering

UNIVERSIT` A DEGLI STUDI DI PARMA DOTTORATO DI RICERCA IN FISICA CICLO XXXV Cohe en and incohe en dynamics o molecula nanomagne s in es iga ed by magne ic esonances and inelas ic neu on and x- ay sca e ing Coo dina o e: Chia .mo P o . S e ano Ca e a Tu o e: Chia .mo P o . S e ano Ca e a Co- u o e: D .ssa Elena Ga la i Chia .mo P o . Giuseppe Allodi Do o ando: Simone Chicco Anni Accademici 2019/20 - 2022/23 Abs ac The ocus o his hesis is he cha ac e iza ion o he cohe en and incohe en dynam- ics o molecula nanomagne s. Fo his pu pose, we will exploi se e al s a e-o - he- a expe imen al echniques, such as magne ic esonances and inelas ic sca e ing o neu ons and X- ays. By means o nuclea magne ic esonances we will cha ac e - ize he nuclea elaxa ion imes and he pa ame e s o he spin Hamil onian o wo V-based molecula qudi s, demons a ing, in addi ion, he capabili y o cohe en ly manipula e hei nuclea s a es. This p oo -o -concep s expe imen s ep esen an impo an i s s ep owa ds he implemen a ion o molecula qudi s in quan um in o ma ion p ocessing. Mo eo e , by X- ay inelas ic sca e ing, we will in es iga e he phonon dispe sions o one benchma k molecula qudi and hei ole in i s elax- a ion dynamics. We will also ocus on he key ac o s go e ning he phonon-induced elaxa ion in Dy-based single molecule magne s by s udying, h ough inelas ic neu- on sca e ing, he changes induced in hei phonon densi y o s a es by chemical subs i u ions o s uc u al de o ma ions. Finally, a syne gis ic app oach combining elec on and nuclea magne ic esonance will gi e us insigh s on he elec onic e- laxa ion dynamics o a sup amolecula assembly linking an isola ed nuclea qudi and an elec onic spin qubi . 1 Lis o Collabo a o s Uni e si y o Pa ma (IT) –Supe iso and Co-Supe iso s: E. Ga la i, G. Allodi, S. Ca e a. –Collabo a o s: A. Chiesa, P. Bon `a, I.J. Onuo ah, R. De Renzi, P. San ini. Uni e si y o Flo ence (IT) –F. San anni, A. Albino, F. To i, L. So ace, R. Sessoli. Uni e si y o Tu in (IT) –E. Sal ado i, M. Chiesa. Lab.Na . Champs Magn´e iques In enses, G enoble (FR) –M. A zo i. Uni e si y o Manches e (UK) –F. S. J. Lockye , A. B ook ield, J. Skel on, F. Tuna, D. Mills, N. Chil on, E. McInnes, R. Winpenny. T ini y College Dublin (IE) –A. Nguyen, A. Lunghi. Uni e si y o Glasgow (UK) –A. Ma omagoulos, A. B. Canaj, M. Mu ie. Uni e si y o Came ino, (IT) and ISIS Neu on and Muon sou ce, Didco , OX (UK) –T. Guidi. Eu opean Synch o on Radia ion Facili y, G enoble (FR) –L. Paolasini. Ins i ue Laue Lange in, G enoble (FR) –M. Jimenez Ruiz, A. Pio ano, A. I ano . B ookha en Na ional Labo a o y, Up on, NY (USA) –C. Mazzoli, C. Yong. 2 Con en s 1 In oduc ion 15 1.1 Molecula nanomagne s . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2 Di Vincenzo c i e ia o Molecula Magne s . . . . . . . . . . . . . . . 18 1.3 Spin Hamil onian o malism . . . . . . . . . . . . . . . . . . . . . . . 23 1.3.1 Elec onic and Nuclea Zeeman in e ac ions . . . . . . . . . . 24 1.3.2 The Hype ine in e ac ion . . . . . . . . . . . . . . . . . . . . 25 1.3.3 The Nuclea quad upola coupling . . . . . . . . . . . . . . . 28 1.3.4 Second o de e ec s . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4 Lindblad Mas e equa ion . . . . . . . . . . . . . . . . . . . . . . . . 30 2 Expe imen al echniques 33 2.1 Resonance echniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1.1 Nuclea Magne ic Resonance . . . . . . . . . . . . . . . . . . . 33 2.1.2 Elec on Pa amagne ic esonance . . . . . . . . . . . . . . . . 37 2.2 Sca e ing echniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2.1 Inelas ic sca e ing . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2.2 Inelas ic Neu on Sca e ing . . . . . . . . . . . . . . . . . . . 41 2.2.3 Inelas ic X ays Sca e ing . . . . . . . . . . . . . . . . . . . . 46 3 Ex ensi e b oadband NMR s udy o Vanadium-based molecula qudi s 50 3.1 Vanadium-based qudi s . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2 NMR in es iga ion o he [VO(TPP)] molecula qudi . . . . . . . . . 51 3.3 Comp ehensi e s udy o an O ganome allic spin Qudi wi h Radio e- quency and Mic owa e echniques . . . . . . . . . . . . . . . . . . . . 61 3.4 Encoding a QEC algo i hm on he [VO(TPP)] molecula qudi . . . . 75 3.5 Conclusions ................................ 77 4 Un eiling phonons in a p o o ypical molecula qubi by IXS 79 4.1 S a eo a ................................ 79 4.2 Inelas ic X- ay sca e ing expe imen . . . . . . . . . . . . . . . . . . 80 4.2.1 Resul s............................... 83 4.3 The c i ical ole o ul a-low ene gy ib a ions . . . . . . . . . . . . . 86 4.4 Conclusions ................................ 89 5 E ec o chemical and s uc u al modi ica ions on he phonon in- duced magne iza ion dynamics o Dysp osium-based Single-Molecule 3 Chap e 0 Magne s 90 5.1 The e ec o chemical composi ion on he pDOS o new Dy-based SMMS................................... 91 5.1.1 Inelas ic neu on sca e ing expe imen . . . . . . . . . . . . . 96 5.1.2 Resul s............................... 98 5.2 Measu ing phonon modes o a Dy-based SMM in applied p essu e . . 99 5.2.1 Inelas ic neu on sca e ing expe imen . . . . . . . . . . . . . 101 5.2.2 Resul s...............................103 5.3 Conclusions ................................105 6 Combined EPR-NMR s udy o he Sup amolecula {C 7Ni}-Cu dy- namics 106 6.1 Elec onic spin-la ice and spin-cohe ence imes . . . . . . . . . . . . 107 6.2 Spin dynamics p obed by 1H-NMR ...................110 6.3 Elec onic spin Manipula ions . . . . . . . . . . . . . . . . . . . . . . 113 6.4 Conclusions ................................114 Gene al Conclusion 115 4 Simone Chicco Lis o Figu es 1.1 On he le and on he igh side o he igu e, he chemical s uc u e o wo iconic single molecule magne s. On he le he i s SMMs o exhibi magne ic hys e esis a he molecula le el and slow elaxa ion o magne iza ion, Mn12 [1] ( o mally {Mn12O12[O2CCD2C(CD3)3]16 (CD3OD)4}). On he igh , he i s a e-ea h based SMMs o display magne ic hys e esis up o liquid ni ogen empe a u e, Dysp osoce- nium [2] ([(Cp )2Dy][B(C6F5)4]. In he middle, a ske ch o a double well po en ial o he e e sal o he magne iza ion o a high-spin S complex, wi h a high-spin g ound s a e mS=±S............ 16 1.2 S uc u e o some o he mos ep esen a i e molecula qubi eme ged o each class desc ibed in he main ex . (a) {C 7Ni} o d-block clus e molecules. [3] (b) Vanadium and Vanadyl-based molecula qubi [4,5,6] (c) Te bium and Y e bium based single ion qubi o a e-ea h based class. [7,8] (d) Two example o s uc u e linking oge he mo e han one molecula qubi : a a e-ea h based dime [CeE ] [9] and a sup amolecula chain deco a ed wi h wo {C 7Ni} qubi linked wi h h ee coppe based qudi [10] ............ 18 1.3 Bloch sphe e showing he |0⟩and |1⟩basis o a qubi and all he possible linea combina ion a|0⟩+b|1⟩ illing he sphe e su ace. . . . 20 1.4 Scheme o wo easible cooling me hods: he mal ini ializa ion (le ), based on he slow cooling o he sys em o a empe a u e low enough o isola e he g ound s a e om he exci ed ones; p ojec ion measu e- men s ( igh ), based on esonan e.m. pulses o apidly p ojec he sys em o a desi ed s a e (when he he mal popula ion is negligible). 21 1.5 In a o a ing ame sys em, p ecessing spins a e ep esen ed by s a ic Bloch sphe e s a es. On he le , he sys em e olu ion om a pe - u bed in-plane s a e owa d he he modynamical equilib ium s a e |0⟩, imed by T1 elaxa ion ime. On he igh , he spin dephasing p ocess o e ime (blue and g een a ow ep esen s spins wi h in- c eased o dec eased nu a ing equency espec i ely), imed by T2 dephasing ime............................... 22 1.6 On he le , he S=1/2 elec onic g ound s a e degene acy spli ing induced by he elec onic Zeeman in e ac ion HZ, as a unc ion o he ield. On he igh , he |−1/2⟩→|1/2⟩elec onic ansi ion enabled by his Hamil onian con ibu ion obse ed in a canonical spec al analysis .................................. 25 5 Chap e 0 1.7 On he le , he nuclea degene acy spli ing induced by nuclea Zee- man in e ac ion HZn, as a unc ion o he ield in a S= 1/2, I= 3/2 sys em. On he igh , he mIdegene a e |3/2⟩ → |1/2⟩,|1/2⟩ → | − 1/2⟩,| − 1/2⟩ → | − 3/2⟩nuclea ansi ion enabled wi hin he same elec onic mul iple ∆mS= 0. The spec a is ex ended o dis- play bo h elec onic and nuclea ansi ions. . . . . . . . . . . . . . . 26 1.8 On he le he spli ing e ec induced by he pa allel Hype ine com- ponen A∥when he ex e nal ield is applied pa allel o i ; On he igh he expec ed spec a showing he degene a e nuclea and he esol ed elec onic ansi ions, a ising om he Hype ine enso exp essed in inse ..................................... 27 1.9 On he le he spli ing e ec induced by an axial Hype ine en- so (inse ) when he ex e nal ield is applied pe pendicula o he s onges componen ; On he igh he expec ed spec a whe e all he nuclea and elec onic ansi ions a e esol ed. . . . . . . . . . . . 27 1.10 On he le he spli ing o he s a e wi h di e en |mI|induced by a Quad upola coupling Pwi h componen s only along ˆzwhen he ex e nal ield is applied pa allel o i ; on he igh he esul ing spec- al componen s o he spin Hamil onian wi h only Quad upola , elec onic and nuclea Zeeman couplings. The nuclea ansi ions a e pa ially esol ed, while he in e ac ion lea es he elec onic gaps un- pe u bed.................................. 28 2.1 Ske ch o he ”HyReSpec ” spec ome e ha dwa e and o he LC- Resonan p obe. T ansmi e and Recei e s ages ha dwa e a e be e de ailed in [11]. .............................. 37 2.2 Sca e ing p ocess wi h incoming and sca e ed beam ene gies Ei, E and wa e ec o s  ki, k , and geome ical ep esen a ion o he ans e momen um  Qand he sca e ing angle θ................. 40 2.3 The layou o he IN8 spec ome e a ILL. The pic u e is ep oduced om he acili y webpage. ........................ 44 2.4 Top (le ) and side ( igh ) iews o he IN1 spec ome e layou . The pic u e is ep oduced om he ILL webpage............... 45 2.5 Schema ic ep esen a ion o he ID28 spec ome e layou . The pic- u e is ep oduced om he ESRF websi e ............... 48 3.1 [VO(TPP)] molecula s uc u e. The p incipal o hogonal symme y di ec ions o he molecula c ys al a e shown independen ly: he po - phy in lying in he ab plane and he oxido anadium bond di ec ion along he caxis.............................. 51 3.2 Spec a collec ed a di e en applied ield B0along he molecule sym- me y di ec ions: he ab plane (a), and he caxis (b). In inse , a spu- ious peak is iden i ied om he absence o any ime-domain spin-echo signals. Indeed black line shows he echo signal in quad a u e o a eal nuclea exci a ion in co espondence o τ(dashed line), whe eas a spu ious appea s as he pe sis en g ay oscilla ions. Rep oduced om Re . [4] wi h pe mission om he Royal Socie y o Chemis y. . 52 6 Simone Chicco Chap e 0 3.3 Red do s co espond o he measu ed ansi ion equencies in he b oadband NMR spec a o 51V. Black lines ep esen he calcula ed e olu ion o he ansi ion equencies as a unc ion o he ield B0 applied (a) in he ab-plane, and (b) along c-axis. Wi h capi al le e s AB# and C# we label he ∆mI=±1 ansi ions o each di ec ion. Shaded a eas we e no expe imen ally explo ed. Rep oduced om Re . [4] wi h pe mission om he Royal Socie y o Chemis y. . . . . 53 3.4 Ene gy le els as a unc ion o he applied ield om he diagonal- iza ion o he spin Hamil onian wi h s a ic ield B0applied (a) in plane ab and (b) along c-axis. The ed and blue shades highligh he mS=±1/2 mul iple s, espec i ely. On he le o each plo , wo inse s show a zoom in o he nuclea le el spli ing o he lowe and up- pe mul iple s, wi h he le els ma ked by he nuclea spin componen s mIalong he ield. Ve ical ma ks highligh he nuclea ansi ions iden i ied in he spec a o igu e 3.2. Rep oduced om Re . [4] wi h pe mission om he Royal Socie y o Chemis y. . . . . . . . . . . . . 54 3.5 Di e ence in subsequen nuclea ansi ions ene gies δ(mI)=(EmI+1− EmI)−(EmI−EmI−1) in he mS= 1/2 mul iple (c osses), as a unc- ion o he applied ield Bˆ x. These alues a e in sound ag eemen wi h wha expec ed om he diagonaliza ion o he pseudo-quad upola Hamil onian (black line). Rep oduced om Re . [4] wi h pe mission om he Royal Socie y o Chemis y. . . . . . . . . . . . . . . . . . . 55 3.6 Echo in ensi y decay as a unc ion o he delay be ween he Hahn echo exci ing and e ocusing pulses. The expe imen al da a o a selec ed nuclea ansi ion ((a) AB2, (b) C1) a e plo ed oge he wi h he elaxa ion a e T2single exponen ial i o di e en applied ields. Rep oduced om Re . [4] wi h pe mission om he Royal Socie y o Chemis y.................................. 56 3.7 Nuclea phase memo y ime o di e en nuclea ansi ions, mea- su ed wi h a e ocusing Hahn-echo sequence, as a unc ion o he applied ield B0(a) in he ab-plane and (b) along c-axis. Rep oduced om Re . [4] wi h pe mission om he Royal Socie y o Chemis y. . 56 3.8 Nuclea Rabi oscilla ions o e a wide ime in e al, o ansi ions AB1 and AB7, co esponding o he ansi ion mI=−3/2→ −1/2 o bo h he lowe and highe -ene gy elec onic mul iple s mS= ±1/2, espec i ely. Rep oduced om Re . [4] wi h pe mission om he Royal Socie y o Chemis y. . . . . . . . . . . . . . . . . . . . . . 57 3.9 Rabi oscilla ions as a unc ion o pulse B1a enua ion, on he an- si ion labeled AB1 be ween he s a es mI=−3/2→ −1/2 o he lowes elec onic mul iple mS= 1/2. The powe is de ined wi h an inc easing a enua ion om a e e ence alue (0 dB). Rep oduced om Re . [4] wi h pe mission om he Royal Socie y o Chemis y. . 58 7 Simone Chicco Chap e 0 3.10 Simula ed e ec o a pulse esonan wi h a nuclea ansi ion ha inc ease he mI alue by ∆mI= +1, when he sys em is ini ialized in mI= 1/2,−1/2,−3/2 (a,d; b,e; c, espec i ely). On he le (a-c) he di e ence be ween a ge le el popula ion as a unc ion o ime. On he igh (d- ) he colo map ep esen s he popula ion e olu ion wi h ime o all he nuclea s a es. The pulse is applied along he molecule ˆxaxis wi h in ensi y B1= 5 G. The a ge ed ansi ions a e: (a,d) mI= 1/2→3/2, (b,e) mI=−1/2→1/2, (c, ) mI=−3/2→ −1/2, wi h ∆mS= 0. The ocus he e is on he mS= 1/2 spin mul iple , which is he only one popula ed because o he s a es ini ializa ion. Rep oduced om Re . [4] wi h pe mission om he Royal Socie y o Chemis y.................................. 60 3.11 Simula ed e ec o a pulse esonan wi h a nuclea ansi ion ∆mI= +1, on he he mally popula ed sys em. On he le (a-c) he di e - ence be ween he a ge le els popula ions as a unc ion o ime. On he igh (d-i) he colo map ep esen s he popula ion e olu ion o e ime o all he nuclea s a es o bo h he elec onic spin mul iple s mS=±1/2. The pulse is applied along he molecule ˆxaxis wi h in en- si y B1= 5 G. The a ge ed ansi ions a e: (a-d) mI= 1/2→3/2, (b-e) mI=−1/2→1/2, (c- ) mI=−3/2→ −1/2, wi h ∆mS= 0. Rep oduced om Re . [4] wi h pe mission om he Royal Socie y o Chemis y.................................. 61 3.12 On he le , he [V(Cp)2Cl2] ben -me allocene s uc u e. The molec- ula e e ence ame is shown, wi h he y-axis bisec ing he Cl-V-Cl angle, he z-axis pe pendicula o he Cl-V-Cl plane (axial di ec ion) and he x-axis unequi ocally de ined. On he igh he c ys allo- g aphic uni cell wi h he wo inequi alen molecules labeled as V1 and V2. Rep in ed wi h pe mission om Re . [12]. Copy igh 2021 Ame ican Chemical Socie y. . . . . . . . . . . . . . . . . . . . . . . . 62 3.13 Pic u e o he c ys al collec ed om mic oscopy (le ) and a ske ch o he c ys al shape ( igh ). The wo a e compa ed o show he edge leng h hie a chy and he ela i e indexing o c ys al aces wi h he co - esponding c ys allog aphic planes. Rep in ed wi h pe mission om Re . [12]. Copy igh 2021 Ame ican Chemical Socie y. . . . . . . . . . 63 3.14 The c ys allog aphic planes (111), (010) and (10-1), ha co espond o he c ys al aces, a e highligh ed in he uni cell. This ske ch assis in he isualiza ion o he applied ield di ec ions wi h espec o he molecula e e ence ame. Rep in ed wi h pe mission om Re . [12]. Copy igh 2021 Ame ican Chemical Socie y. . . . . . . . . . . . . . . 63 8 Simone Chicco Chap e 1 In oduc ion Molecula magne ism has ep esen ed a e y ac i e esea ch opic o e he las ew decades. The ui ul collabo a ion o scien is s belonging o di e en backg ounds, such as Physics and Chemis y, has con ibu ed in pa ing he way o he imple- men a ion o hese sys ems in quan um in o ma ion p ocessing. This hesis ocuses on se e al aspec s o he expe imen al cha ac e iza ion o molecula magne s spin dynamics. By exploi ing di e en echniques we deepen he unde s anding o e- laxa ion dynamics mechanisms in a ious molecula magne s, we ully cha ac e ize he model spin Hamil onian o some p o o ypical sys ems, we de ec he e ec o physical and chemical s imuli on hei phonon-induced elaxa ion and inally we implemen a ge ed manipula ion o he sys em spin s a es by adio equency and mic owa e pulses, demons a ing he possibili y o exploi ing hese complexes o quan um in o ma ion p ocesses. 1.1 Molecula nanomagne s The disco e y, a he beginning o he 1990s, o magne ic hys e esis o igina ing a he single molecule le el by R. Sessoli and cowo ke s [1] igge ed he de elopmen o a b and new esea ch ield [19] which nowadays pe mea es in o se e al b anches o science, om quan um in o ma ion heo y o supe -dense in o ma ion s o age, om nanoscopic sensing and nanoscopic p obing o elec ic anspo e c: Molecu- la magne ism. The p ogeni o o his compounds class, displaying magne ic hys e esis and slow elaxa ion o he magne iza ion, was he so-called Mn12 (aka [Mn12O12(CH3COO)16 (H2O)4]). [1,20] This sys em, and he class o compounds ha eme ged in he ol- lowing yea s (as Fe8o Ni10), a e clus e s o exchange-coupled d-block me als, whose magne ic beha io can be app oxima ed o a single gigan ic o al spin Sslowly e- laxing because o an e ec i e double well po en ial a ising om he la ge g ound s a e spin and a s ong magne ic aniso opy. [21,22] The p esence o an ene gy ba ie o he eo ien a ion o he magne ic momen leads o he bis abili y o he g ound s a e o al spin S( ig. 1.1). This e ec made hese molecules in e es ing candida es o he ul ima e minia u iza ion o he memo ies, whe e each molecule can encode a single ”bi ” o memo y ( om he e he name Single-Molecule-Magne s). Howe e , he magne ic bis abili y o he gian spin g ound s a es wi h mS=±Sin unde mined by di e en elaxa ion p ocesses: some a e empe a u e independen , 15 Chap e 1 such as he quan um unneling o he magne iza ion (QTM) below he ene gy ba - ie , o he s a e ins ead he mally ac i a ed, due o he coupling o he spin wi h molecula ib a ions, gene a ing o he elaxa ion pa hs such as he O bach mul i- s ep mechanisms o e he ene gy ba ie and non- esonan Raman. Consequen ly, he magne ic hys e esis in hese i s SMMs can be obse ed only when QTM is no ac i a ed/ine ec i e and a ex emely low empe a u es, whe e he mal luc ua ions a e supp essed. [19,23,24,25] Figu e 1.1: On he le and on he igh side o he igu e, he chemical s uc u e o wo iconic single molecule magne s. On he le he i s SMMs o exhibi magne ic hys e esis a he molecula le el and slow elaxa ion o magne iza ion, Mn12 [1] ( o mally {Mn12O12[O2CCD2C(CD3)3]16 (CD3OD)4}). On he igh , he i s a e- ea h based SMMs o display magne ic hys e esis up o liquid ni ogen empe a u e, Dysp osocenium [2] ([(Cp )2Dy][B(C6F5)4]. In he middle, a ske ch o a double well po en ial o he e e sal o he magne iza ion o a high-spin Scomplex, wi h a high-spin g ound s a e mS=±S. A ema kable s ep in he magne iza ion blocking empe a u e o SMMs ha e been achie ed in he ollowing yea s wi h a second gene a ion o molecula nanomagne s, based on a single -block magne ic ion. [26] He e indeed, he s ong spin-o bi cou- pling o he a e-ea h elemen s (e.g. Lan hanides) combined wi h a p ominen axial ligands c ys al ield, p oduces a la ge aniso opy ba ie , ha s abilize he g ound s a e spin e en a high empe a u e ( ig. 1.1). The ene gy ba ie s o se e al hun- d eds o Kel in deg ees high o mos o hese single molecule magne s is howe e no he only ac o ha go e ns he pe sis ence o he hys e esis a high empe a u es. Indeed, he complex in e play o se e al elaxa ion p ocesses in e e es wi h he magne ic s abili y and an ex ensi e knowledge on hese mechanisms is undamen al o he imp o emen o he pe o mances. A e y low empe a u e, he mos e ec- i e elaxa ion mechanism is he quan um unneling o he magne iza ion (QTM), [19] which is empe a u e independen bu can be made ine icien in p onounced axial ligands ield en i onmen . [27] Mo eo e , a highe empe a u e, whe e la ice and molecula ib a ions a e ac i a ed, he coupling o phonons wi h he magne ic momen s induce an incohe en elaxa ion dynamics in hese sys ems. As a conse- quence, he enginee ing o hese ib a ions and he s udy o hei coupling wi h he spin is a key s ep o he imp o emen o he pe o mances o molecula sys ems. [28,29,30] Taking in o conside a ion all hese ac o s, a eco d b eaking magne ic blocking empe a u es o 80 K ha e been ecen ly epo ed in Dysp osium-based molecules ([(Cp )2Dy][B(C6F5)4], wi h Cp = 1,2,4- i- e -bu ylcyclopen adienyl) wi h double-decke axial ligands en i onmen (see ig. 1.1). [2,15] He e, he p omi- 16 Simone Chicco Chap e 1 nen axiali y o he ligands ield, besides inducing a g ea isola ion o he g ound s a e K ame double s om he exci ed c ys al ield s a es, con ibu e signi ican ly o he supp ession o he quan um unneling o he magne iza ion unde he ba ie . Mo eo e , he s ong bonds in he ligands en i onmen o he a e-ea h ion p e en s e icien spin coupling wi h molecula ib a ions, hus enhancing he pe sis ence o he magne ic bis abili y a highe empe a u e.[15] Ve y ecen ly a cos -e ec i e semi-ab ini io app oach ha e been de eloped o un- de s and he elaxa ion dynamics o hese sys ems. Physical insigh s on he o igin o hese elaxa ion mechanism can indeed gi e undamen al in o ma ion in o de o de elop he nex class o eco d b eaking molecula complexes, wi h blocking empe a u es abo e he liquid ni ogen limi . [17,31] In pa icula non- esonan wo phonon Raman p ocesses we e disco e ed o play he key ole in igge ing he magne iza ion elaxa ion o hese SMM class in he ange o empe a u es c ucial o he de e mina ion o TB(see also sec . 5). Because o he long spin li e ime and he ema kable cohe ence in insic o se e al molecula magne s, hese sys ems o e s also an a ac i e op ion o he ealiza ion o quan um bi s. Indeed, he wo quan um basis s a e |0⟩and |1⟩can be easily encoded in o he wo s a es o a single spin S= 1/2 molecule. The majo obs a- cle o he exploi a ion o hese sys ems in quan um in o ma ion p ocessing is he pe sis ence o he cohe ence o a gi en supe posi ion o s a es in each qubi uni , de ined by he decohe ence ime T2(see sec . 1.2). The e o e, since la ice ib a- ions, oge he wi h elec ic and magne ic in e ac ions wi h he en i onmen , make quan um supe posi ions o s a es e y agile, lo o e o ha e been o ien ed in he las decades o he syn hesis o molecula complexes wi h long cohe ence imes and o he in es iga ion o he decohe ence mechanisms.[5,28,30,32,33,34,35,36,37] The i s p oposed molecula qubi in he 2000s wi h a cohe ence ime T2>1µs was again a clus e wi h an e en numbe o d-block a oms an i e omagne ically (AF) coupled: a [C 7NiF6Pi 16] ing. Indeed, he subs i u ion o a C magne ic cen e in an Homome allic S= 0 ing [38] wi h a di alen Ni ca ion causes he unbalance o he AF coupling and he eme gence o a S= 1/2 double g ound s a e. [3,39] As well as long cohe ence ime (T2), hese clus e ed sys ems we e also capable o implemen he i s one-qubi ope a ions, wi h clea Rabi oscilla ions o spin popula- ions demons a ed in he benchma k C 7Ni ing and in Vanadium-based molecula clus e s.[40,41] Recen ly, complexes wi h a single ansi ion-me al ion a e in ac eme ging as highly cohe en spin qubi s (see ig. 1.2). In pa icula , se e al Vanadium-based sys ems display cohe en oscilla ions o spin s a e popula ions also app oaching oom em- pe a u e, [35] as well as empe a u e- esilien cohe ence imes, o he o de o he milliseconds in he op imal condi ions.[5] Vanadium complexes ha e also been e- cen ly s udied as p omising coupled qubi -qudi sys ems. Indeed, he coupling o he elec onic Sand nuclea Ispin deg ees o eedom p o ides a 2I+ 1 mul ile el s uc u e called qudi (i.e., quan um digi s ea u ing mo e han wo le els d > 2) ha can be exploi ed o expand he compu a ional space o o implemen quan um e o co ec ion algo i hms on single molecula objec s. [4,12,42] Ra e-ea h ions play a ole also in he de elopmen o molecula qubi s. Indeed, he ones ea u ing a g ound s a e K ame double , well sepa a ed om he exci ed s a es by a signi ican c ys al ield spli ing, can be desc ibed as sys ems wi h an e ec i e spin Se = 1/2 [8,33,43] o qubi encoding (see ig. 1.2). In o he cases, owing 17 Simone Chicco Chap e 1 Figu e 1.2: S uc u e o some o he mos ep esen a i e molecula qubi eme ged o each class desc ibed in he main ex . (a) {C 7Ni} o d-block clus e molecules. [3] (b) Vanadium and Vanadyl-based molecula qubi [4,5,6] (c) Te bium and Y e bium based single ion qubi o a e-ea h based class. [7,8] (d) Two example o s uc u e linking oge he mo e han one molecula qubi : a a e-ea h based dime [CeE ] [9] and a sup amolecula chain deco a ed wi h wo {C 7Ni}qubi linked wi h h ee coppe based qudi [10] an elec onic spin S > 1/2, a e-ea h based molecula qubi s can open new possi- bili ies o he expansion o he a ailable compu a ional space. This can be done by exploi ing all he elec onic spin s a es o encode mo e han a single qubi in each molecule [44] o a single qudi wi h se e al ds a es. [7,45] In addi ion, a e-ea h based sys ems embedding mo e han one asymme ic magne ic uni , as dime s [46] o ime s [47] ha e been p oposed as single-molecule quan um ha dwa es wi h chem- ically coupled qubi s. A simila app oach has been pu sued o chemically-linking oge he se e al ansi ion me al clus e s (i.e. ings) in sup amolecula assembly (see examples in ig. 1.2). [10,48,49] The inal s ep o he ac ual implemen a ion o molecula sys ems o quan um compu a ion is he deposi ion o single objec s and hei wi e-up in ”p ocesso - like” s uc u es. [42,50,51] Consequen ly, se e al s udies ha e ecen ly ocused on he deposi ion o magne ic molecules on su aces,[52,53,54] and on he coupling wi h sys ems ha enable he manipula ion and ead-ou o hese single quan um objec s,[55] e.g. wi h supe conduc ing esona o s o nanos uc u ed junc ions.[7] 1.2 Di Vincenzo c i e ia o Molecula Magne s Molecula spin sys ems ha e been iden i ied as a p omising e olu ion o Noisy In e media e-Size Quan um de ices (NISQs) in o de o push o wa d he capa- bili ies o mode n quan um echnologies. [56,57,58] Indeed, among a a ie y o possible sys ems s udied in he pas decades, as ul a- cold a oms,[59] pho ons,[60] supe conduc ing ci cui s, [61,62] Ni ogen and o he 18 Simone Chicco Chap e 1 acancies in solid c ys als [63,64,65,66] e c., molecula sys ems appea o be pa ic- ula ly sui able o he ealiza ion o quan um compu a ion, [50,51,67,68,69]. This because hey po en ially ul ill he equi emen s se by Di Vincenzo and cowo ke s a he beginning o he millenium o he ealiza ion o a quan um compu ing pla - o m. [70] Indeed, despi e sho e wi h espec o o he pla o ms (e.g. see [71]), he cohe ence imes o many molecula qubi s ha e been demons a ed o be long enough o he spin s a e manipula ion wi hou signi ican losses in cohe ence. [39] In he ollowing, we will deepen each o hese basic c i e ia o he ac ual ealiza- ion o quan um in o ma ion p ocessing wi h physical pla o ms, highligh ing how molecula spin sys ems sa is y hese equi emen s and which aspec s a e cu en ly he mos c i ical. [42] Scalable de ined Qubi Any sys em, o being exploi ed as a quan um bi , mus be well cha ac e ized. This means ha he pa ame e o i s Hamil onian, oge he wi h he coupling wi h o he nea by qubi s and wi h physical (pho ons, phonons, ex e nal ields, e c.) s imuli exploi ed o s a e manipula ion, mus be known wi h accu acy. He e, magne ic molecules ha e he ad an ages o be pe ec ly ep oducible quan um objec s, chem- ically syn he ized and ”enginee able”, and easily cha ac e ized by bulk spec oscopic esonan echniques as Nuclea Magne ic Resonance and Elec on Spin Resonance o single molecula -c ys als (as de ailed in sec . 2.1.1 and 2.1.2). [45,72] In- deed, e en o a la ge ensamble o indi idual molecules, such as a single c ys al a angemen , he s ain o he single molecule pa ame e s (g-s ain and D-s ain as de ined in [73]) is gene ally limi ed and esul s in a mode a e b oadening o spec al linewid h (cen e ed a ound he esonan equency o he mean pa ame e alues). Thus, despi e s ain e ec s, o he pu pose o elec omagne ic ield manipula ions, molecules in a c ys al can be conside ed p ac ically iden ical. In addi ion, some o hese molecules na u ally posses a eal (o e ec i e) S= 1/2 g ound s a e ha ou line a e y simple ealiza ion o a qubi , since he wo mS=±1/2 spin s a es na u ally ep esen he |0⟩and |1⟩s a e ha de ine he qubi s a e. [3,5,8] Mo eo e , besides being well de ined, a physical sys em o being applied as a qubi in a quan um ha dwa e mus be scalable. The scaling up, o expand he com- pu a ional space, can be done in se e al ways. [50] By chemical design, o example, i is possible o c ea e molecula s uc u e ha hos s mo e ha one magne ic cen e , as dime s o ime s o qubi s. [9,47] Simila ly, AF ings, such as [C 7Ni], a e well known and ho oughly s udied sys ems ha ea u e he possibili y o be bonded oge he in sup amolecula s uc u es, combined wi h o he spin 1/2 complexes, [10,48,49,74,75,76] and o be deposi ed on su aces, e aining hei magne ic p ope ies. [53,77,78] Ano he op ion, mo e ecen ly de eloped, is o exploi he in e nal deg ees o ee- dom o he sys em, o expand he compu a ional space. The esul ing quan um sys em can be seen as a qubi ha ea u e mo e han wo disc e e s a es (qudi ) o as a collec ion o e ec i e wo-le el qubi embedded wi hin he same molecula uni . This happen o example in S > 1/2 complexes, as he S= 7/2 o Gd3+ ions whe e d= 8 elec onic spin le els a e a ailable. [43,44] Ra he han he elec onic spins, he nuclea spin deg ees o eedom o he molecule me al ion can be exploi ed as 19 Simone Chicco Chap e 1 Figu e 1.3: Bloch sphe e showing he |0⟩and |1⟩basis o a qubi and all he possible linea combina ion a|0⟩+b|1⟩ illing he sphe e su ace. ano he powe ul esou ce o quan um compu a ion, because o hei isola ion om he en i onmen . The nuclea spin is indeed e icien ly coupled o he elec ons by Hype ine in e ac ion, esul ing in a spli o he le el degene acy and in a conside - able speed up o he equi ed manipula ion ime, ega dless o i s isola ion om he en i onmen . [4,12,45] Howe e , he ad an ages gained by inc easing he numbe o elec onic and nuclea s a es o he qudi is no su icien o co e complex algo i hm a single molecule le el. The e o e, i is necessa y o design mul i-qubi pla o ms ha enables he ealiza ion o wo- o many-qubi s ga es. As p oposed in his e iew [50] by S. Ca - e a e al., a possibili y p o ided by molecula qubi s, hanks o hei s abili y (e en isola ed) and chemical unabili y, is o link he molecules chemically, by in oducing swi chable molecula linke s [76,79] wi h ul a- as swi ching- ime (well below he hund ed o ns, depending on he sys em) [76,80], o physically by pe iodic deposi- ion on ac i e subs a es, as supe conduc ing on-chip esona o . [51,81] Bo h hese app oaches allows o con ol cohe en ly and indi idually he spin s a es. [50] Qubi Ini ializa ion F om a compu a ional poin o iew i is undamen al o ha e a clea pic u e o he qubi s s a e a he beginning o he quan um in o ma ion p ocessing. This e- qui emen is ul illed i he physical sys em chosen o he compu a ion can be easily ini ialized o a well known s a e. [70] Fo he case s udy o molecula magne s, ini- ializa ion o he s a e can be easily achie ed by cooling s a egies. Gi en a molecula sys em wi h an e ec i e spin S= 1/2 g ound s a e, by na u ally cooling a he mK empe a u e ( u inely achie ed in NISQ pla o ms) we ini ialize he sys em by pop- ula ing only he lowes ene gy spin s a e ( ig. 1.4). [82] Fas ini ializa ion du ing compu a ion p ocesses can also be implemen ed by a i icial cooling wi h p ojec ing measu emen s in o he desi ed s a e (e.g. wi h elec omagne ic pulses). Ano he s a egy sui able o molecula sys ems is he exploi a ion o cooling algo i hms, [83] ha allows o each ini ialized s a es wi h pulse sequences, e en a empe a u e in which all he s a es a e he mally popula ed. 20 Simone Chicco Chap e 1 Figu e 1.4: Scheme o wo easible cooling me hods: he mal ini ializa ion (le ), based on he slow cooling o he sys em o a empe a u e low enough o isola e he g ound s a e om he exci ed ones; p ojec ion measu emen s ( igh ), based on esonan e.m. pulses o apidly p ojec he sys em o a desi ed s a e (when he he mal popula ion is negligible). Qubi Cohe ence imes When he physical quan um sys em is placed in con ac wi h he en i onmen , he gene ic qubi s a e |ψ⟩=a|0⟩+b|1⟩degene a es, in a ime we de ine as de- cohe ence ime, in o he classical mix u e desc ibed by he sys em densi y ma ix ρ=|a|2|0⟩⟨0|+|b|2|1⟩⟨1|. The e o e, his decohe ence ime ep esen s he li e ime o he quan um supe posi ion s a e in which he qubi is placed o compu a ional pu poses. Fo a p o icien implemen a ion o a quan um algo i hm, he decohe ence ime o he physical pla o m on which i is implemen ed is no supposed o las longe han he en i e compu a ion i sel . Indeed, he e o induced on quan um s a es by de- cohe ence can be co ec ed by e o co ec ion p o ocols, as p oposed by Sho and S eane in 1995 and 1996 espec i ely. [84,85] As a consequence, a good comp omise o he decohe ence o a sys em is se by he ga e ime (”Clock ime” in [70]) i Quan um E o Co ec ion (QEC) is applied. In o de o implemen p o icien ly any quan um ga e, wi h negligible cohe ence losses due o elaxa ion, he sys em decohe ence ime mus be longe han he ime needed o implemen a single ga e (a leas 104 imes he ”clock” o comple e aul - olle an compu a ion [70]). In his ega ds, magne ic molecules display unique ad an ages in uning he elax- a ion mechanisms. The p incipal sou ces o decohe ence come om he in e ac ion o molecula spins wi h molecula ib a ions (phonons), ligands nuclea spins and nea by elec onic spins. [51] In he las decades, a e he iden i ica ion o hese elax- a ion mechanisms, se e al s a egies ha e been p oposed o ema kably imp o e he decohe ence imes o molecula spins. The iden i ica ion o he speci ic ib a ional modes mos s ongly coupled wi h he spin (de imen al o quan um cohe ence) enables he supp ession o hose elaxa ion pa h by chemical op imiza ion o he s uc u e, e.g. by s i ening o ligands o g oups emo al (see sec . 4). [29,30,86] Mo eo e , decohe ence e ec s induced by magne ic dipola in e ac ion ha e been ema kably educed by exploi ing s ong magne ic dilu ion o molecula magne s in diamagne ic hos s uc u es. Finally, he in e ac ion o elec onic spins wi h he nuclei hos ed in o he molecule ligands s uc u e canno be easily supp essed, since 21 Simone Chicco Chap e 1 Hyd ogen is he one ha in e e e he mos and also he mos abundan elemen in his class o molecules. Howe e , he molecula s uc u e can be op imized in o de o educe his con ibu ion o elaxa ion by sepa a e as much as possible he sou ces o decohe ence and educing hei abundance (e.g. by deu e a ion o luo ina ion). [87] Two di e en cha ac e is ic imes a e used o quan i y he decohe ence o a molec- ula qubi : T1and T2. The i s desc ibe he in e ac ion o he spin wi h he c ys al s uc u e (phonons) and i is hen called spin-la ice elaxa ion ime, while he sec- ond is called spin-spin elaxa ion ime (o phase memo y ime) and accoun s o he in e ac ion wi h nea by spins, deno ing he cohe ence ime o he qubi . I we ep esen he spin mo ion o he sys em in he o a ing ame ( ha allows us o ge id o nu a ion mo ion) as a Bloch sphe e, he wo elaxa ion mechanisms can be dis inguished in wo di e en mo ions as shown in igu e 1.5. Figu e 1.5: In a o a ing ame sys em, p ecessing spins a e ep esen ed by s a ic Bloch sphe e s a es. On he le , he sys em e olu ion om a pe u bed in-plane s a e owa d he he modynamical equilib ium s a e |0⟩, imed by T1 elaxa ion ime. On he igh , he spin dephasing p ocess o e ime (blue and g een a ow ep esen s spins wi h inc eased o dec eased nu a ing equency espec i ely), imed by T2dephasing ime. T1indica e he ime needed o he sys em o e-es ablish he he modynamical equi- lib ium a e a pe u bing s imulus, hus p obing i s coupling wi h he en i onmen wi h which i exchanges ene gy. T2ins ead, indica es he decay a e o spin phase cohe ence wi hin he sys em. The possibili y o implemen ing QEC, o p o ec he in insic agili y o quan um in o ma ion, on a single mul ile el quan um objec ep esen s ano he undamen al ad an age o exploi ing magne ic molecules as building blocks o quan um com- pu a ion pla o ms. Indeed he qudi a chi ec u e, ea u ing mul iple elec onic o nuclea le els in a single molecule, enables he encoding o e o p o ec ed ”logical” qubi s in o a single quan um objec . [88] The ap i ude o he implemen a ion o QEC pe mi s o elease he s ic equi emen s o ex emely long elaxa ion ime o molecula qubi s, since he e o s a ising om dephasing can be mi iga ed by a i icial co ec ions. 22 Simone Chicco Chap e 1 Uni e sali y The ou h equi emen se by Di Vincenzo o a p o o ypical quan um pla o m is he capabili y o implemen ing a Uni e sal se o single and mul i-qubi quan um ga es. Since any quan um algo i hm can be decomposed as a sequence o uni a y ans o ma ions Ui,[89] he equi emen is ansla ed in iden i ying sys em in which is possible o implemen any uni a y ans o ma ion. When e e ing o a physical sys em, his condi ion is ul illed i all he manipula ion o he sys em can be iden i- ied by a se o Hamil onians, ha gene a e all he equi ed uni a y ans o ma ions Ui=eiHi /¯h. Implemen ing he uni e sal se o uni a y ans o ma ion enable he simula ion o any quan um ime e olu ion o a sys em. In his ega d, molecula magne s a e easily manipulable sys ems, whe e he molecule spin is pushed in in e ac ion wi h an ex e nal elec omagne ic (e.m.) s imulus o manipula e he elec onic and nuclea s a es, wi hou any signi ican cons ain . The e o e, we can easily assume ha gi en a magne ic molecule, i s spin s a e can be manipula ed in o de o co e uni o mly he whole Bloch sphe e ( ig. 1.3). The expansion o he compu a ional space o mul i-qubi ga es, mus go h ough he coupling o se e al molecula uni s. The e o e, as discussed o he scalabili y, o e- alize unc ional wo and mul i-qubi en angling ga es, a ious molecula uni s mus be wi ed up by coupling wi h ex e nal supe conduc ing on-chip esona o s [81] o by in oducing swi chable molecula couple s o linke s, sensible o ex e nal s imuli. [76,79] Qubi measu emen s The inal s ep o a quan um compu a ion p ocess is he eadou o he esul . The ideal physical pla o m mus allow he measu emen o he s a e o each speci ic qubi , wi hou al e ing he s a e o he o he s. This aspec is cu en ly he mos c i ical, since i po en ially equi e he ealiza ion o a single molecule ead ou mechanism. Rega ding he wo possible a chi ec u e p oposed abo e, ha ing single-molecule qubi s spa ially sepa a ed and selec i ely coupled by plana esona o s can allow he indi idual eadou o a desi ed p o- cesso egion wi h e.m. s imuli, wi hou causing he collapse o he ull sys em wa e unc ion. [50,55] Howe e , up o da e his app oach poses s ic expe imen al limi a ion, due o he e y weak spin-pho on coupling (see [90]). Mo eo e , o he pla o ms pe mi s single qubi eadou by elec ic anspo measu emen s [42] o ligh i adia ion. [91] Elec ic and magne ic ield a e also exploi ed o he eadou o single molecule qudi , when he popula ion o a single couple o le els can be ex ac ed by esonan exci a ions, wi hou a ec ing nea by s a es. 1.3 Spin Hamil onian o malism The Spin Hamil onian app oach enables he desc ip ion o he s a ic p ope ies and cohe en dynamics o magne ic molecules and i ep esen s he mos e ec i e o - malism o he in e p e a ion o expe imen al da a om spec oscopic echniques. Gi en a molecula sys em i pe mi s o desc ibe accu a ely he ene gy le els o he sys em and he hie a chy o magne ic in e ac ions. [19,92,93] This app oach pe mi s o desc ibe all he con ibu ion o he sys em Hamil onian 23 Simone Chicco Chap e 1 in e ms o spin-only ope a o s, s a ing om he assump ion ha each ion in he molecule can be desc ibed as an e ec i e spin ˆsi. The la es assump ion is accu a e o sys ems wi h quenched o bi al angula momen um (as in almos all 3d sys ems), o when he o bi al angula momen um is no quenched (as in 4 sys ems), bu he c ys al ield spli ing o he Jquan um numbe yields a well isola ed g ound s a e mul iple . The mos gene al spin Hamil onian desc ibing molecula nanomagne s can be w i - en adding se e al con ibu ions: [19,92] H=Hex +HCF +Hdip +HZ+Hn.(1.1) He e, Hex ep esen s he exchange in e ac ion be ween mangne ic ions in he molecule, HCF he e ec o he local c ys al ield, and Hdip he h ough-space and h ough- bond dipola coupling. Finally, HZmodels he Zeeman in e ac ion wi h an ex e nal magne ic ield and Hnde ined as ” ine s uc u e” Hamil onian, accoun s o he in e ac ion wi h he nuclea deg ees o eedom. In his hesis we will mainly ocus on single ion molecules wi h e ec i e o in insic spin S= 1/2, in which he ex- change in e ac ion Hex be ween ions in he molecule is absen and he c ys al ield exci ed s a es a e igno ed because o he conside able isola ion om he g ound s a e K ame double . Mo eo e , by d as ic dilu ion ( up o 2%) o he MNM in i s c ys al s uc u e wi h i s diamagne ic analogue, he in e -molecula dipola con ibu ion o he spin Hamil onian in 1.1 is also s ongly supp essed. The e o e, o hese sys ems, he spin Hamil onian in 1.1 can be uni e sally exp essed in i s educed o m: H=HZ+Hn=HZ+HHyp +Hq+HZn,(1.2) whe e in he ine s uc u e Hamil onian we made explici all he con ibu ion due o he in e ac ion wi h he nuclea momen um: he Hype ine coupling HHyp, he nu- clea quad upola in e ac ion Hqand he nuclea Zeeman e ms HZn. These e ms a e exp essed in e ms o p oduc o spin ope a o s, as de ailed in he ollowing sec ions. 1.3.1 Elec onic and Nuclea Zeeman in e ac ions The g ound s a e K ame degene acy ( o bo h sys ems cha ac e ized by eal o e ec i e spin 1/2 g ound s a es) is emo ed by applying an ex e nal magne ic ield B, as a esul s o i s Zeeman coupling wi h he elec onic spin S. The Hamil onian o he elec onic Zeeman e ec is hus gi en by he scala p oduc : HZ=µBˆ S·gS·B,(1.3) whe e µB= 1.3996 ×104MHz/T is he Boh magne on and gS he spec oscopic spli ing enso , ha a ies o di e en ions and s uc u es. I we ocus on he coo dina e e e ence ame o he molecule, he enso ial p oduc is educed as: HZ=µB(gxSxBx+gySyBy+gzSzBz),(1.4) wi h he ield componen s Bx,y,z de ined in pola coo dina es wi h espec o he molecule ame (Bx=Bsin θcos ϕ,By=Bsin θsin ϕ,Bz=Bcos θ). When he 24 Simone Chicco Chap e 1 con enien o desc ibe he sys em by ocusing on he educed Hilbe space o he spin Ssys em, which con ains all he spin dynamics in o ma ion. The ope a o s ˆ Xac ing on his educed Hilbe space a e de ined by means o he pa ial ace o malism: ⟨ˆ X⟩= Snˆ XˆρSo,(1.20) whe e he educed densi y ma ix ˆρS= B(ˆρ) is ob ained om he pa ial ace o he o al densi y ma ix, o e he en i onmen deg ees o eedom. The equa ion o mo ion o he open sys em becomes hen: d d ˆρS( ) = −i ¯h Bhˆ H o ,ˆρ( )i.(1.21) I we assume ha he co ela ion ime o he en i onmen dynamics a e negligibly sho e hen he ime scale o he sys em co ela ion (Ma ko app oxima ion), and ha he ime dependen coupling ˆ Vis a small pe u ba ion o he sys em (weak- coupling app oxima ion), we can hen o mula e he quan um mas e equa ion in in e ac ion pic u e as he Lindblad equa ion: d d ˆρS( ) = −i ¯hhˆ H o ,ˆρS( )i+D(ˆρS( )),(1.22) whe e he i s e m o he equa ion ep esen s he uni a y cohe en dynamics gen- e a ed by he sys em Hamil onian ˆ Hunpe u bed by B, while he second e m D(ˆρS( )) (Dissipa o ) accoun s o he non-uni a y e ec o he in e ac ion wi h he en i onmen and is w i en as: [94] D(ˆρS( )) = X m,n Lm,n ˆρS( )L† m,n −1 2(L† m,nLm,n ˆρS( ) + ˆρS( )L† m,nLm,n),(1.23) wi h Lm, n de ined as Lindblad ope a o s. Spin sys ems a e subjec o he pu e dephasing mechanism, due o he coupling wi h he en i onmen . Fo aking in o accoun he non-uni a y e ec o dephasing, he Lindblad ope a o s Lm, n mus be w i en as: [94,95] Lm,n =δm,n γm 2|m⟩⟨m|,(1.24) and he Lindblad equa ion becomes: [94] d d ˆρS( ) = −ihˆ H o ,ˆρS( )i+ X m,n γmδm,n |m⟩⟨m|ˆρS( )|m⟩⟨m|− 1 2(|m⟩⟨m|ˆρS( ) + ˆρS( )|m⟩⟨m|), (1.25) whe e |m⟩a e he eigens a e o he sys em. This dissipa o causes he decohe ence o he o -diagonal elemen s o he densi y ma ix ρm,n( ) wi h a a e τm,n = (γm+ γn)/2), whils he diagonal na u e o he Lindblad ope a o keeps unchanged he diagonal popula ion ρm,m( ) (no ene gy exhange be ween sys em and en i onmen ). 31 Simone Chicco Chap e 1 Fo he nuclea spin sys em ha will be discussed in sec ion 3, he eigens a es a e labelled by he co esponding componen o he nuclea spin |mI⟩and he dephasing a e co esponds o he nuclea phase memo y ime τm,n = 1/TmI,m′ I 2. [4] Mo eo e , o he case-s udy o molecula qubi s i could be use ul o simula e he e ec o an ex e nal s imulus on a spin sys em al eady coupled wi h he en i onmen . This can be simply achie ed by adding a ime dependen e m ˆ H1( ) o he Hamil onian o he mas e equa ion ha ules he uni a y e olu ion o he sys em densi y ma ix. The Lindblad mas e equa ion becomes: d d ˆρS( ) = −ihˆ H0+ˆ H1( ),ˆρS( )i+ X m,n γmδm,n |m⟩⟨m|ˆρS( )|m⟩⟨m|− 1 2(|m⟩⟨m|ˆρS( ) + ˆρS( )|m⟩⟨m|). (1.26) By nume ically sol ing his mas e equa ion by using ealis ic pa ame e s o he spin Hamil onian, he ime dependen pe u ba ion and he dephasing a es gi es undamen al insigh in o he dynamics o p o o ypical molecula qubi s. 32 Simone Chicco Chap e 2 Expe imen al echniques The expe imen al s udy o molecula nanomagne s in his hesis has been done by exploi ing se e al echniques: esonan echniques, such as b oadband Nuclea Magne ic Resonance, Elec on Pa amagne ic Resonance, and sca e ing echniques, such as neu on and X- ays inelas ic sca e ing. In he ollowing sec ions we will gi e a b ie in oduc ion o all o hem. 2.1 Resonance echniques In his sec ion we will ocus on wo esonance echniques which ha e se e al simila - i ies, om he poin o iew o he heo e ical backg ound, and one impo an di e - ence, he physical a ge . Bo h Nuclea Magne ic Resonance (NMR) and Elec on Pa amagne ic Resonance (EPR) a e in ac esonan expe imen al echniques ex- ploi ing he in e ac ion o magne ic momen s wi h an ex e nal elec omagne ic i a- dia ion, pe u bing he sys em. In he i s case, he magne ic momen s µI=µNgNI a ise om he nuclea spin Io he sys em, while, o he second echnique, he o- cus is on he magne ic momen s µe=µsgs·Sa ising om he unpai ed elec ons spin S. Since we a e dealing wi h esonan echniques, his implies ha he elec o- magne ic i adia ion mus be uned o a na u al equency o he sys em in o de o be e ec i e, which is, depending on he di e en physical a ge , he equency o he magne ic momen s gy oscopic p ecession unde an ex e nal s a ic magne ic ield. These magne ic esonance equencies alls ypically in he adio- equency ( ) egion o he nuclei, while o he elec ons in he mic owa e (mw) one, hus, we will e e o and mw i adia ion depending i we a e desc ibing an NMR o an EPR expe imen . In he ollowing chap e we will discuss in de ails he B oadband NMR (sec . 2.1.1) and EPR (sec . 2.1.2) echniques. 2.1.1 Nuclea Magne ic Resonance In he nuclea magne ic esonance echnique we a e dealing wi h nuclea mag- ne ic momen s µI, a ising om he a omic g ound s a es o se e al nuclei ea u ing I= 0. Because o he small magne ic momen µN(compa ed o he elec onic one µe/µI≈103÷4), and he absence o a s ong exchange mechanism, nuclea e omagne ism is limi ed o he mK empe a u es. Thus, in a NMR expe imen unde applied s a ic ield, he a ge sys em will be a spin pa amagne s ea u ing se e al s a es Em(depending on he spin I) co esponding o he di e en alues 33 Chap e 2 o he quan um numbe mI=Izo he spin quan ized along he applied ield. The popula ion o hese nuclea s a es will be go e ned by he Bol zmann s a is ic Pm∝e−Em/kT . Nuclea magne ic esonance is he e o e he spec oscopic echnique ha enables o s udy hese ene gy le els and he ansi ions induced among hem by a esonan adia ion. The e exis wo possible esonan exci a ion me hods: ield-sweep and equency-sweep. He e we ocus on he equency-sweep me hod, used o he NMR in es iga ions o his hesis. I consis s in a ge ing he sys em wi h a - ield o a iable equency. When his equency ν ma ches one o he cha ac e is ic gaps o he nuclea s a es s uc u e ∆ ≈hν , he ansi ion p ob- abili y is enhanced and he p ocess de ec ed. The de ec ion o hese ene gy gaps and he measu emen o he spin elaxa ion p ocesses owa ds equilib ium a e he i adia ion a e he in o ma ion commonly accessible om an NMR expe imen on a magne ic nucleus. The NMR expe imen can be in e p e ed in wo di e en ways: as a o ced p ecession o he nuclea magne ic momen in a ield ( om F. Bloch [96,97,98]) o as he esul o he compe i ion be ween he elaxa ion o he he mal equilib ium (Bol zmann popula ions) and he exci a ion induced by he elec omagne ic i adia ion. [92,99] In a quan um mechanical app oach, he eigen alues o he angula momen um ˆ I= ¯hIcomponen Izalong he ex e nal ield di ec ion ˆ z, coincide wi h all he 2I+ 1 possible con igu a ions m=−I, −I+ 1, ...I −1, I. Thus, gi en he ela- ionship be ween he magne ic momen s and he angula momen um componen s ⟨Im|µz|Im′⟩=µNgN⟨Im|Iz|Im′⟩, he e ec o an applied ex e nal s a ic ield H0ˆ z is he Zeeman in e ac ion ene gy: H=µ·B=γ¯hBIz=µNgNBIz,(2.1) ha spli s he nuclea spin s a es, cha ac e ized by he eigen alues Em o each m s a e: E=−µNgNBm. (2.2) To induce exci a ions be ween hese Zeeman le els and hus pe o ming he NMR expe imen , he sys em mus be a ge ed wi h a pe u ba ion ¯hω esonan o one o he gaps wi hin he Ems a es. ¯hω =Em′−Em.(2.3) In his case, he pe u ba ion consis s in a - ield B1⊥B0 ha induces an addi ional in e ac ion e m B1=−µNgNB0,xIxcos(ω ) ha d i es he ansi ion be ween close nuclea s a es m→m±1 wi h ∆m=±1. By conside ing he case o a spin I= 1/2 sys em, he ansi ion p obabili y be ween he wo mInuclea s a es is de ined as |⟨+|Ix|−⟩|2, whe e |+⟩and |−⟩ co espond o he s a es o componen mI=±1/2. Fo s imula ing he ansi ion i is necessa y o ha e a popula ion di e ence be ween he wo a ge s a es. Indeed, by de ining he numbe o spin pe le el as N±( ) and he ansi ion a e as W, he exci ed spin pe uni ime is de ined by he p oduc N±( )Wand he s a es popula ion a ia ion is desc ibed by he a e equa ion: dN± d =WN∓( )−WN±( ),(2.4) whose solu ion in e ms o popula ion di e ence n( ) = N+( )−N−( ) = n0e−2W gi es an exponen ial decay owa d a condi ion in which he e is no popula ion di e - ence. Indeed, he ime dependen ene gy o he sys em E( ) = N−E−+N+(E++¯hω) 34 Simone Chicco Chap e 2 sa u a es wi h an abso p ion a e: dE/d = ¯hωWn( ) = n0¯hω(We−2W ).(2.5) Howe e , a e he -s imulus is emo ed, he nuclea spin sys em in e ac s wi h he he mal ba h en i onmen and elaxes owa ds he ini ial he mal equilib ium s a e, ollowing a Bol zmann dis ibu ion: N− N+ =e−∆E+,− kT .(2.6) Thus, he elaxa ion o he popula ion di e ence induced by he ield, which is go e ned by he in e ac ion wi h he en i onmen al he mal ba h (la ice), is modeled by a elaxa ion a e 1/T1called spin-la ice elaxa ion a e: n( ) = n0(1 −e− /T1).(2.7) This, combined wi h he a e equa ion 2.4, gi es an ene gy abso p ion a e om he nuclea spin sys em: dE d =n0¯hω W 1+2WT1.(2.8) The nuclea spin ime e olu ion du ing and a e he -manipula ion can be de- sc ibed by he phenomenological Bloch equa ions o he nuclea magne iza ion. [96,97,98] This app oach enables he de ini ion o he elaxa ion cons an s ha a e usually employed o model spin elaxa ion. We s a om a nuclea spin sys em in i s he modynamical equilib ium when a s a ic ield B=Bˆ zis applied. This condi ion co esponds o a null ans e se magne iza ion Mx,y = 0 and a sa u a ed longi udinal magne iza ion Mz=M0=χnH0, wi h χnnuclea suscep ibili y. The classical p ecession o he nuclea spins in his ield, plus a elaxa ion e m, is de- sc ibed by he Bloch equa ions: (dMz d =µNgN(M×B)z+M0−Mz T1 dMx,y d =µNgN(M×B)x,y +M0−Mx,y T2 ,(2.9) whe e T1and T2accoun o he elaxa ions o he longi udinal and ans e se mag- ne iza ion, espec i ely. When a exci ing ield B1⊥Bis added o he spin sys em, in a e e ence ame ˆx ˆy ˆz o a ing a he same equency o B1, he nuclea spins ime e olu ion is s ill desc ibed by a p ecession, bu in his con igu a ion he p ecession is a ound an e - ec i e ield Be =B1ˆ x + (B−ω/γ)ˆ z, sum o he wo applied ields. The Bloch p ecession equa ions become:      dMz d =µNgNB1My−Mz−M0 T1 dMx d = (µNgNB0−ω)My−Mx T2 dMy d =−(µNgNB0−ω)Mx+µNgNB1Mz−Mx T2 .(2.10) As al eady ou lined abo e, he elaxa ions imes o he longi udinal ˆzand ans- e se ˆx, ˆymagne iza ion componen s a e kep dis inc . This because hese cons an s ep esen wo di e en p ocesses. As seen o he de ini ion o he abso p ion a es 35 Simone Chicco Chap e 2 o a he mal-like elaxa ion p ocess, T1is ela ed o he eco e y o he he mal equilib ium o he nuclea s a e popula ion, induced by he coupling wi h a he - mal ba h cons i u ed by he elec onic and c ys al en i onmen o he nuclea spins (spin-la ice elaxa ion). Con e sely, he ans e se elaxa ion consis s in a dephas- ing ha occu s du ing spin p ecession in he xy-plane (spin-spin elaxa ion). I he ield is a sho pulse o du a ion P(sho enough o a oid simul aneous decohe ence) esonan wi h he sys em na u al equency ω=µNgNB, he e ec o he pe u ba ion is a nu a ion o an angle θ=µNgNB1 Pa ound he e ec i e ield B1ˆ x. Consequen ly, by uning P, i is possible o ealize π/2 o a ions on o he xy plane o πin e sion o he nuclea magne iza ion. In pa icula , a e a π/2 pulse he nuclea magne iza ion e ol es ollowing Bloch equa ions: Mxy( ) = Mxy(0)e− /T2(2.11) which is he so-called ee induc ion decay ha ollows he esonan pe u ba ion. The decohe ence induced by s a ic ields once he nuclea magne iza ion is on he xy plane can be e ocused by applying a second πpulse o he spins sys em ha e e se he dephasing, gene a ing an echo-like signal om he e ocusing o he spin p ecession (dynamical con ibu ions a e i e e sible phenomena ha de ines he in insic T2). Con e sely, he longi udinal elaxa ion consis in a eco e y o he ini ial s a e M0by he exchange o ene gy wi h he he mal ba h. F om he Bloch equa ions he eco e y ollows he end: Mz( ) = M0+ (Mz(0) −M0)e− /T1(2.12) T2and T1can be measu ed expe imen ally by p ope pulse sequence on well de ined spin ansi ions (a ixed ω). By e ocusing he echo wi h a π/2−τ−πpulse sequence o example, as a unc ion o he delay τwe can measu e he decay o he ans e se magne iza ion (T2spin elaxa ion). By ”hea ing” he nuclea spin sys em wi h a pulse ain o he mos diso de ed s a e and hen e ocusing a e a delay τ, we measu e he eco e y o he longi udinal sa u a ion in a ”cooling-down”-like p ocess (P ain − −π/2−τ−π). Mo eo e , by collec ing he esonance signals o e a wide equency ange, om he collec ed NMR spec a we can link he ene gy gaps o he exci ed ansi ions o he le els s uc u e o he spin Hamil onian, and i i s pa ame e s, as ou lined in sec ion 1.3. HyReSpec b oadband spec ome e The pa amagne ic s a e o an o dina y magne ic ma e ial is cha ac e ized by a spin dynamics ha ea u es as Hype ine ield luc ua ions. Being hese luc ua ions as e hen he nuclea La mo ωL equency, he esul ing nuclea elaxa ion, co e- la ed o he hype ine ield luc ua ions, is ul a- as and p e en he de ec ion o a esonance signal. Molecula NanoMagne s, howe e , a low empe a u es a e cha - ac e ized by spin-la ice elaxa ion a es much longe han he a he sho ime needed o he sys em manipula ion wi h pulses ( as ened by s ong hype ine couplings). Thus, du ing he sho ime-scales o a spin-echo sequence, he a ge nucleus expe iences a quasi-s a ic elec onic en i onmen . [100] In his con ex , he NMR appa a us mus be sui able o he exci a ion and de ec ion o as elaxing signals. Fo he expe imen s epo ed in his hesis, we exploi ed a pulsed b oadband NMR spec ome e named ”HyReSpec ” (”Hype ine Resonance 36 Simone Chicco Chap e 2 Spec ome e ”), en i ely designed a he Uni e si y o Pa ma and op imized o he in es iga ion o magne ic ma e ials, spanning bo h equency and ield. [11] This Figu e 2.1: Ske ch o he ”HyReSpec ” spec ome e ha dwa e and o he LC- Resonan p obe. T ansmi e and Recei e s ages ha dwa e a e be e de ailed in [11]. expe imen al appa a us (ske ched in ig. 2.1) ea u es a wide equency span wi h a la esponse in he 8-800 MHz ange, co e ing he na u al equency ange o se e al nuclei, such as ansi ion me als (V, Cu, Mn, e c.). I is also equipped wi h a as swi ching, wi h pulses ha a e p og ammable in s eps o 12 ns, b oadband ecei e ±3 MHz and an e ec i e expe imen al i adia ing bandwid h limi ed o 1-2 MHz due o he LC esona o ini e passband. The ecei e bandwid h, oge he wi h a sho ins umen al dead ime and a high speed digi al pulse , enables he gene a ion o pulse sequences cha ac e ized by sho pulses and delays, necessa y o he de ec- ion o sho li ing signals wi h a ine ime esolu ion. Fas e signal a e aging wi h espec o comme cial NMR spec ome e s enables also he collec ion o op imal signal o noise a io on sho e expe imen imescales. The used p obehead is a LC esonan ci cui in which he sample is placed wi hin he coil and inse ed in o a Maglab EXA supe conduc ing c yomagne . 2.1.2 Elec on Pa amagne ic esonance As al eady highligh ed in he in oduc ion, Elec on Pa amagne ic Resonance (EPR) is complemen a y o NMR, since i gi es undamen al and non edundan in o ma- ion on he coupled elec o-nuclea spin sys em. Elec onic esonances indeed in es- iga e he elec onic magne ic momen s a ising in sys ems wi h unpai ed elec ons. [101,102] The EPR expe imen can be desc ibed exac ly in he same heo e ical amewo k used o he NMR expe imen , wi h he only di e ence o he magne ic momen µe=µBgS·S, which o igina es om he unpai ed elec ons o he sys em. Thus, in he case o an elec onic spin mul iple S= 1/2, o a s a ic ield Bˆ zaligned wi h he ˆ zaxis o he g enso , he Zeeman in e ac ion: H=µe·B=µBgSZBSz,(2.13) 37 Simone Chicco Chap e 2 causes he spli ing o he ene gy le els, labeled by he wo elec onic spin componen along he s a ic ield di ec ion mS=±1/2 EmS=−µBgSZBmS.(2.14) Consequen ly, o his simple model sys em, he ene gy o he mw-EPR s imulus inducing he ansi ion o he esonan expe imen be ween he wo mSs a es mus be: ¯hω =gSZµBB. (2.15) EPR expe imen s a e usually pe o med a ixed mw- equency and he ansi ion ene gies a e a ied by sweeping he s a ic ield Bun il he Zeeman gaps ma ch he i adia ion ene gy. This because he esona o quali y ac o s ongly depends on he a ge equency and hus limi s he equency span ange. Fo his eason, EPR can be pe o med wi h di e en cha ac e is ic equencies epo ed in able 2.1, each o hem pa icula ly sui ed o speci ic expe imen s/samples. EPR equency bands L-band 0.8-1.2 GHz S-band 3.4-3.8 GHz X-band 9-10 GHz Q-band 34 GHz W-band 94 GHz Table 2.1: EPR equency bands. X and Q bands a e he mos commonly exploi ed o he s udy o molecula magne s spin Hamil onians and decohe ences. In EPR spec oscopy he spin sys em can be a ge ed bo h by a con inuous mw- ield (con inuous wa e EPR) o by a pulsed mw- ield (pulsed EPR). In his sec ion we will ocus only on he pulsed EPR echnique, which is he one used o he mea- su emen s o spin elaxa ion epo ed in sec ion 6. When he sys em is a ge ed by he mw- ield B1o a pulsed EPR expe imen , in analogy wi h wha shown in sec ion 2.1.1 o NMR, he elec onic spin magne ic momen s p ecess ollowing he Bloch equa ions 2.1.1. Thus, i is possible o manipula e, h ough mw-pulses he elec onic spins, inducing uned nu a ions o angles θ=µBgxB1 P. This enables he cha ac e iza ion o he sys em ene gy spec a, oge he wi h he elec onic spin la ice eT1and spin-cohe ence eT2 elaxa ion imes. The o me is de ec ed wi h a sligh ly di e ence pulse sequence wi h espec o NMR. He e he pulse ain is subs i u ed by a simple in e sion πpulse and he esul ing sequence is known as in e sion- eco e y sequence π−π/2−π. The de ec ion is done by e ocusing he elec onic spins p e iously pushed in o he ”in e ed” non equilib ium s a e −MZ, as a unc ion o he ime equi ed o eco e he equilib ium condi ion M0=MZ. Pulsed-EPR spec a as a unc ion o he s a ic ield Bin ixed equency egime en- ables he i ing o he spin Hamil onian model. This analysis is complemen a y o he i ing o he NMR spec a, since he wo echniques ea u e di e en sensi i i y o he Hamil onian pa ame e s o elec onic o nuclea o igin espec i ely (e.g., he bes i o he spec oscopic spli ing enso gis achie ed wi h EPR, while NMR ea u es be e sensi i i y on he hype ine enso componen s). 38 Simone Chicco Chap e 2 Elexsys-E580 Pulsed spec ome e In o de o pe o m he measu emen s epo ed in he p e ious sec ion, he EPR appa a us mus ea u e high sensi i i y and high speed manipula ions. Fo his pu pose, we exploi ed a B uke Elexsys-E580 pulsed-EPR spec ome e om he EPSRC Na ional Se ice o Elec on Pa amagne ic Resonance Spec oscopy Labo- a o y a he Uni e si y o Manches e . This pulsed-EPR spec ome e is equipped wi h a Supe Q-FT mic owa e b idge o pulsed Q-band (νEPR = 34 GHz) expe i- men s combined wi h a dedica ed Q-band esona o . The spec ome e is supplied wi h a 1.8 T elec omagne and a c yogen- ee c yos a inco po a ing a closed he- lium ci cui . The sample is placed in o a modula B uke Flexline esona o , sui able o he mul i- esonance Elexsys pla o m, ea u ing high ca i y illing ac o and sho dead imes. The ca i y quali y Q- ac o is a iable and enables expe imen op imiza ion. I could be c i ically coupled o con inuous wa e (CW) in es iga ions, o kep low o sho dead ime and la ge i adia ion bandwid h o ma ched o he sample elaxa ion ime o inc ease he signal o noise a io. The wide de ec ion bandwid h (800 MHz) and he as a e aging signal p ocessing a e gua an eed by he combina ion o he Pa e nJe -II pulse p og amme , which deli e s a high ime esolu ion (up o 1 ns), and he as SpecJe -III digi ize , which enables high speed acquisi ion wi h 1 ns esolu ion, ze o dead ime be ween sho s and on boa d signal p ocessing. This wo digi al uni s in pa icula allowed o a e age FIDs and as -decaying spin echoes wi hin he sample T1limi . The SpinJe -AWG (a bi a y Wa e o m Gene a o ) addi ional modulus, enables high speed manipula- ion and de ec ion o pulses wi h non- i ial shaping o e a mul iple channel a chi- ec u e. 2.2 Sca e ing echniques S udying he sca e ing o a cohe en adia ion on a sample and he ene gy ex- changed du ing he p ocess gi es unique insigh s in o he a omic s uc u e and cohe en dynamics o he sys em. Indeed, sca e ing echniques a e o undamen- al in e es in molecula magne ism, since he adia ion in e ac ion wi h he nuclei allows o di ec ly p obe he molecula and la ice ib a ions igge ing hei phonon- induced elaxa ion dynamics. [30,31] Mo eo e , o p obes ea u ing a non-ze o spin, i.e. neu ons, he magne ic in e ac ion wi h spin and o bi al momen a enables also he cha ac e iza ion o he sys em spin Hamil onian and i s cohe en spin dynamics. [103] In his hesis, X- ay and neu on sca e ing expe imen s a e used o p obing he phonon spec a and dispe sions in Molecula nanomagne s (MNMs). Thus, in he ollowing sec ions, we will gi e a b ie in oduc ion o he Inelas ic Neu on Sca e - ing (INS) and he Inelas ic X- ay Sca e ing (IXS) echniques, highligh ing he im- po an in o ma ion ha can be ex ac ed om he expe imen s. [104,105,106,107] 39 Simone Chicco Chap e 2 2.2.1 Inelas ic sca e ing An inelas ic sca e ing p ocess, as ske ched in igu e 2.2, is de ined by he wo conse a ion laws o momen um and ene gy:  Q= k − ki,(2.16) ¯hω =E −Ei,(2.17) wi h he squa ed momen um depending on he sca e ing angle: Q2=k2 i+k2 −2kik cos(θ).(2.18) To desc ibe he sca e ing p ocess we s a de ining an inciden lux o he incoming Figu e 2.2: Sca e ing p ocess wi h incoming and sca e ed beam ene gies Ei, E and wa e ec o s  ki, k , and geome ical ep esen a ion o he ans e momen um  Q and he sca e ing angle θ. beam pe uni o pe pendicula sample su ace and pe second I0. The sca e ing c oss sec ion σsis hen de ined as he numbe o sca e ed pa icle pe lux uni : σs=numbe o sca e ed pa icles I0 .(2.19) I we now es ic o a gi en di ec ion iden i ied by a small solid angle dΩ and o a inal ene gy o he sca e ed pa icles in a na ow ange (E , E +dE ), he local double di e en ial c oss sec ion becomes: ∂2σs ∂Ω∂E =sca e ed pa icles in o solid angle dΩ wi h ene gy (E ,E +dE ) I0dΩdE . (2.20) The double di e en ial c oss sec ion includes all he physical in o ma ion ha can be ex ac ed om he sca e ing expe imen . All he sca e ing p ocesses can be desc ibed wi hin he Fe mi Golden Rule o - malism, de ining a ansi ion a e be ween wo iden i ied s a es, by means o he in e ac ion o he p obe wi h he sample exp essed by a po en ial V. Le ’s con- side a sca e ing e en in a dΩ solid angle, de ining a inal wa e ec o  k , in which he ene gy o he sca e ing sys em changes om Eλ o Eλ′. The p obe-sample s a es changes consequen ly om | ki, λ⟩ o | k , λ′⟩. The esul ing double-di e en ial c oss-sec ion can be w i en as: ∂2σs ∂Ω∂E =1 I0dΩdE X k in dΩ W| ki,λ⟩→| k ,λ′⟩.(2.21) 40 Simone Chicco Chap e 2 in o he sca e ing unc ion: S( Q, ω) =Scoh( Q, ω) + Sincoh( Q, ω) = 1 2π¯hX k,k′ 2Z∞ −∞⟨e−i Q· Rk(0)ei Q· Rk′( )⟩e−iω d + 1 2π¯hX k 2Z∞ −∞⟨e−i Q· Rk(0)ei Q· Rk( )⟩e−iω d . (2.42) In oducing a small displacemen uk( ) o each ka oms, we de ine he same op- e a o s U=−i Q·uk(0) and V=i Q·uk′( ) o equa ion 2.2.2 and we de i e he one-phonon sca e ing unc ion om he i s o de Taylo expansion o he he - mal a e aged exponen ial exp(⟨UV ⟩). The cohe en one phonon X- ays sca e ing unc ion hus becomes: S( Q, ω)∝X s,q 1 ωs(q)|Fs 1−ph( Q)|2 [(ns(q) + 1)δ(ω−ωs(q))δ( Q−(τ +q)) + ns(q)δ(ω+ωs(q))δ( Q−(τ −q))], (2.43) wi h he s uc u e ac o Fs 1−ph(Q) de ined as: Fs 1−ph( Q) = X k k( Q) √2Mk e−2Wk( Q)( Q·es(q))ei Q· k,(2.44) whe e we no e he explici dependence on he phonon modes ene gies ωs(q) (eigen- alue) and pola iza ion es(q) (eigen ec o s), which can be di ec ly p obed in he sca e ing expe imen . We inally no e ha , di e en ly om a neu on sca e ing expe imen , he e he en- e gy o he exci ed phonon modes a e much smalle han he inciden pho on ene gy (¯hω ≪Ei), hus he exchanged momen um Qand inciden pho on momen um ki a io depends only on he sca e ing angle θ(in ig. 2.2). Thus, he e a e no kine- ma ic es ic ion on he ene gy ans e depending on he Q alues. Mo eo e , he IXS c oss sec ion is mainly composed by he cohe en sca e ing unc ion ex ac ed abo e, wi h he incohe en sca e ing ha con ibu es a highe ene gies and, in he mode a e low ene gy phonon egime, can be neglec ed. Finally, elemen s con- ibu e di e enl y o he IXS and INS c oss sec ions. This is due o he in insic di e ence be ween he X- ays o m ac o and he neu on sca e ing leng h, which ea u e dis inc dependence on he momen um ans e Qand on he sca e ing sys- em elec on numbe Z. A mo e de ailed analysis o he di e ences and simila i ies be ween INS and IXS is epo ed below, as a subsec ion o his chap e . ID28 The Inelas ic X- ay sca e ing measu emen s shown in sec ion 4 we e collec ed a he Eu opean Sync o h on Radia ion Facili y (ESRF) on he beamline ID28. ID28 is a iple axis spec ome e dedica ed o he s udy o phonon dispe sions in con- densed ma e in a wide ange o ans e ed momen um  Qand exchanged ene gy 47 Simone Chicco Chap e 2 ¯hω, e en o samples o e y small dimensions. Measu emen s can also be pe o med in a a ie y o sample en i onmen s, such as high- acuum, c yogenic empe a u es, high-p essu e, e c. This beamline is a one-o -a-kind because, hanks o i s unique ca- pabili ies in e ms o lineshape and esolu ion, enables he cha ac e iza ion o op ical and acous ic phonons o molecula c ys als. The beamline layou is shown in igu e 2.5. The spec ome e ea u es an almos exac backsca e ing Si monoch oma o Figu e 2.5: Schema ic ep esen a ion o he ID28 spec ome e layou . The pic u e is ep oduced om he ESRF websi e (89.98◦) ha exploi di e en Si(n, n, n) e lec ion o de s. Each o hese e lec ions selec s an ini ial ene gy Eiand a lineshape wid h ∆E( esolu ion), o a X- ays lux ha dec ease by inc easing he e lec ion o de n(see able below). ID28 monoch oma o s no de Ei(keV) ange ∆E(meV) lux (pho ons/s) Si(777) 13.84 5.3 10.5×1010 Si(888) 15.817 4.4 9 ×1010 Si(999) 17.794 2.2 2.7×1010 Si(111111) 21.747 0.83 6.6×109 Si(121212) 23.725 0.73 5.8×109 Si(131313) 25.704 0.5 1.47 ×109 A e he monoch oma ion, he beam is ocused on he sample, posi ioned on a o- a ing goniome e wi h o a ional, il ing and ansla ional deg ees o eedom, back in o he expe imen al cabin. He e lays he second ”axis” o he spec ome e , a 7 me e s a m, equipped wi h nine analyze , each o hem cons i u ed by 1200 single c ys als ben in a sphe ical shape, o ocusing he beam on he de ec o s o e an angula accep ance o 40 m ad2. The nine analyze s a e placed in he ho izon al sca e ing plane, wi h an angula displacemen o 0.75◦, and selec nine di e en momen um ans e  Qin he ange 1-80 nm−1wi h a momen um esolu ions o 0.03 nm−1. The indi idual Si de ec o s cons i u e he hi d ”axis” o he spec ome e and selec he inal sca e ing ene gy E . They a e in eg a ed on a monoli hic Si chip, oge he wi h hei p eampli ie and he Pel ie cooling modulus ( o elec onic noise educ ion). The leng h o he spec ome e a ms is jus i ied by he backsca - e ing geome y and he need o o se he incoming whi e beam wi h espec o he 48 Simone Chicco Chap e 2 monoch oma ed and ocused one Ei. INS and IXS, di e ences and simila i ies Since in his hesis bo h INS and IXS echniques a e used, i is wo h men ioning some o he main di e ences be ween he wo p obes. In pa icula , we will ocus on he aspec s ha makes X- ays a o able wi h espec o neu ons o he in es iga- ion o phonon dispe sions in molecula c ys als. The i s aspec ela es o he c oss sec ion ha o X- ays is highly cohe en , while in neu on sca e ing some elemen s like 1H, abundan ly p esen in molecula complexes, display an highly incohe en c oss sec ion ha causes he blu ing o he expe imen al phonon dispe sions due o incohe en sca e ing. Con e sely in IXS, he incohe en sca e ing in ol es ene gies way highe han he ypical phonon ene gies o molecula magne s we a e in e es ed in. Thus, i is no necessa y o, e.g., deu e a e he c ys als employed. Toge he wi h he absence o incohe en sca e ing, mul iple sca e ing e en s can also be neglec ed, making IXS a basically backg ound ee echnique. Thus, wi h espec o INS he da a educ ion p ocedu e esul s smoo he . Con e sely o INS, IXS do no su e o s ong dynamical limi a ion wi hin he (Q,E) space and he o de o magni ude di e ence be ween he inciden and he exchanged ene gies leads o he decoupling o ene gy and momen um ans e (which depends only on he sca e - ing angle) and causes he esolu ion o he expe imen o be independen om he ene gy. Mo eo e , he almos absence o X- ay magne ic c oss sec ion makes his p obe a o able o he s udy o phonon dispe sions and pola iza ion ec o s, since he magne ic exci a ions wi hin he ene gy ange o in e es a e na u ally emo ed. Fo INS ins ead, whe e he magne ic c oss sec ion is compa a ble a low Q o he nuclea one, he sub ac ion mus be done manually by measu ing a diamagne ic analogue o he s udied c ys al. Howe e , he wo echniques ea u es also some simila i ies, such as he coupling s eng h, which is compa able be ween X- ay-elec on and neu on-nucleus. Fi- nally, he easie ocusing o he X- ay beam and i s in insic highe lux makes IXS bes sui ed o he s udy o c ys als o small dimensions in easonable imes, wi h espec o INS ha , in absence o la ge c ys als, equi es long coun ing imes. 49 Simone Chicco Chap e 3 Ex ensi e b oadband NMR s udy o Vanadium-based molecula qudi s In his chap e we ocus on he cha ac e iza ion o Vanadium-based molecula qubi s. By exploi ing he b oadband NMR echnique, we access a comp ehensi e desc ip ion o he coupled elec o-nuclea spin Hamil onian and o he spin cohe ence imes o hese sys ems. By u he combining nuclea and elec onic esonance echniques, we u he inc ease he ex en and he accu acy o he desc ip ion o his complexes, achie ing complemen a y and non edundan esul s. Mo eo e , we demons a e ha a ge ed adio- equency ( ) and mic owa e (mw) pulses can be exploi ed o im- plemen monoch oma ic cohe en manipula ions o he elec onic and nuclea s a es o hese sys ems. Finally, he abili y o manipula e nuclea s a e wi h pulses is pushed u he by implemen ing he elemen a y encoding ope a ions o a Quan um E o Co ec ion algo i hm, unde he guide o ealis ic simula ions. This capabili y demons a es ha hese complexes a e p omising uni s o quan um in o ma ion p ocesses applica ions. 3.1 Vanadium-based qudi s Molecula complexes embedding a single ansi ion me al ion ha e been in ensi ely s udied in he las decade o quan um in o ma ion p ocessing applica ions. In- deed, he chemical abili y o ailo he ligands en i onmen and hus o enginee he quan um p ope ies o he sys em makes hese complexes an ideal es bed o disce n he di e en con ibu ion o elaxa ion and decohe ence. Among plen y o possibili ies, Vanadium and Oxido anadium(IV) based complexes a e eme ging as p omising candida es, because o an inna e obus quan um cohe ence. [5,6,35,67] Mo eo e , hese Vanadium-based sys ems ea u e a c ucial ad an age. The Vana- dium nuclea spin I=7 2can in ac be exploi o expand he a ailable compu a ional space wi hin a single quan um objec . Indeed, he hype ine coupling be ween he nuclea spin deg ees o eedom and he elec onic spin double gene a es a mul- ile el s uc u e which is called Qudi , i.e. a quan um digi ea u ing mo e han wo le els. [42,51] Encoding in o ma ion on his mul ile el s uc u e expands he dimensions o he compu a ional Hilbe space om a wo-dimensional qubi o a 50 Chap e 3 d-dimensional (d > 2) qudi . In his ame, esonan echniques ha exploi elec omagne ic pulses (as NMR and EPR, espec i ely in sec . 2.1.1 and 2.1.2) play a c ucial ole, since hey allow bo h he cha ac e iza ion and he cohe en manipula ion o hese mul ile el s uc u es. [55,100] In pa icula , in he ollowing, we demons a e why b oadband NMR is an ideal echnique o cohe en ly manipula ing he nuclea s a es o p o o ypical molecula qudi s and how his echnique can be used o cha ac e ize hem. [4,12] 3.2 NMR in es iga ion o he [VO(TPP)] molec- ula qudi The i s p o o ypical Vanadium-based molecula qudi s udied he e is Vanadyl e aphenylpo phy ina e [VO(TPP)]. The complex is composed by a single Vana- dium ion bound o an oxygen o o m a Vanadyl g oup which is inse ed a he cen e o a po phy ine ing. The molecule c ys allizes as a squa e py amid wi h a e agonal I4 space g oup. In he c ys al cell, he po phy ine g oup lies in he c ys- allog aphic ab-plane, wi h he O-V bond de eloping along he c-axis (see ig. 3.1). [4,14] Mo eo e , he squa e py amid basis ma ches he c ys allog aphic ab-plane, while he py amid heigh co espond o he c ys allog aphic c-axis. Figu e 3.1: [VO(TPP)] molecula s uc u e. The p incipal o hogonal symme y di ec ions o he molecula c ys al a e shown independen ly: he po phy in lying in he ab plane and he oxido anadium bond di ec ion along he caxis The NMR spec a o 51Vnuclei a e measu ed by he home-buil b oadband spec- ome e ”HyReSpec ” desc ibed in sec ion 2.1.1 and in Re .[11]. The da a we e collec ed a T= 1.4 K, in o de o supp ess as much as possible decohe ence e - ec s induced by empe a u e, on a single c ys al o [VO(TPP)] dilu ed a 2% in o i s isos uc u al diamagne ic analogue [TiO(TPP)]. Dilu ion ep esen s he mos e icien way o supp ess he dipola in e -molecula in e ac ions as a sou ce o deco- he ence. Howe e , dilu ion mus also be uned in o de o pe mi he de ec ion o he NMR signal om he a ge ed nuclei. The dilu ion pe cen age used o [VO(TPP)] measu emen s he e o e ep esen s a good ade-o be ween hese wo ac o s. [100] The single dilu ed c ys al is placed on a cus om 3D p in ed plas ic sample-holde and hold in place by acuum g ease. This sample holde pe mi s o easily o ien he 51 Simone Chicco Chap e 3 c ys al in o de o apply he ex e nal s a ic ield along speci ic symme y di ec ions. The i s s ep o he analysis was he cha ac e iza ion o he model spin Hamil onian pa ame e s by an ex ensi e b oadband NMR s udy as a unc ion o he applied ield. We measu ed 51V spec a by collec ing Hanh-echoes o e a wide equency ange a a ious applied s a ic ields in he ange B0= 0.05−0.3 T. The pulse sequence used is he uncon en ional 2π 3−τ−2π 3sequence which p o ides an op imized esonance signal,[100,110] wi h pulses du a ion o Tpulse = 0.202 µs and delay τ≈6µs. The disc e e equency s eps can be hickened in da a pos -p ocessing by me ging he o e lapping equency domains o spin-echoes as Fou ie ans o m o subsequen equency shi ed s eps. [111] In Figu e 3.2 we show wo examples o 51Vspec a, measu ed by applying he s a ic ield along he wo nonequi alen and o hogonal symme y di ec ions o he [VO(TPP)] molecule, he po phy in plane ab and he oxido anadium bond di ec ion c. Figu e 3.2: Spec a collec ed a di e en applied ield B0along he molecule sym- me y di ec ions: he ab plane (a), and he caxis (b). In inse , a spu ious peak is iden i ied om he absence o any ime-domain spin-echo signals. Indeed black line shows he echo signal in quad a u e o a eal nuclea exci a ion in co espondence o τ(dashed line), whe eas a spu ious appea s as he pe sis en g ay oscilla ions. Rep oduced om Re . [4] wi h pe mission om he Royal Socie y o Chemis y. The [VO(TPP)] molecule can be desc ibed by a simple coupled elec o-nuclea spin Hamil onian composed as ollows: H0=ˆ I·A·ˆ S+pˆ I2 z+µBˆ S·gS·B0+µNgNˆ I·B0.(3.1) The i s e m in eq. 3.2 accoun s o he Hype ine coupling be ween he elec- onic ˆ Sand he nuclea ˆ IVanadium spins, while he second one exp esses a small axial nuclea quad upola con ibu ion (see sec . 1.3). Finally, he las wo e ms ep esen he nuclea and elec onic Zeeman in e ac ions. As seen in sec ion 1.3, he pa ame e s o he spin Hamil onian de e mine he ield e olu ion o he nuclea esonances. The e o e, by compa ing he measu ed ield dependence o he spec a peaks wi h he simula ed spec a, we can i he spin Hamil onian pa ame e . By conside ing he c ys allog aphic axes as e e ence ame abc =xyz, we assume he Hamil onian enso s o be diagonal, axial and collinea . Mo eo e , his choice o axes combined wi h he c ys al shape simpli y he de ini ion o he di ec ion o he 52 Simone Chicco Chap e 3 applied ields  B0, B1, because he xy plane ma ches he squa e py amid basis while he z axis coincides wi h he e ex. Figu e 3.3: Red do s co espond o he measu ed ansi ion equencies in he b oad- band NMR spec a o 51V. Black lines ep esen he calcula ed e olu ion o he ansi ion equencies as a unc ion o he ield B0applied (a) in he ab-plane, and (b) along c-axis. Wi h capi al le e s AB# and C# we label he ∆mI=±1 ansi ions o each di ec ion. Shaded a eas we e no expe imen ally explo ed. Re- p oduced om Re . [4] wi h pe mission om he Royal Socie y o Chemis y. The bes ag eemen be ween he calcula ions and he expe imen al spec a is ound wi h an axial Hype ine enso and a small axial quad upola coupling (see able 3.1). [4] The elec onic spec oscopic spli ing enso gsis kep ixed o alues ob ained om a p e ious EPR s udy [14], since ha echnique ea u e an highe sensi i i y on he de e mina ion o he elec onic g- alues. Finally, he nuclea g- ac o is ixed o he cons an iso opic alue o he 51Vnucleus µNgN=−11.213 MHzT−1. [112] The bes i pa ame e o [VO(TPP)] spin Hamil onian a e lis ed in able 3.1. The ag eemen achie ed be ween he expe imen al esonance equencies and he model Hamil onian cu es is excellen o bo h he o hogonal ield di ec ions (plane ab and caxis), as shown in igu e 3.3. By diagonalizing he spin Hamil onian 3.2 53 Simone Chicco Chap e 3 [VO(TPP)] spin Hamil onian A(MHz) p(MHz) gS(MHz) x−170 ±1 0 1.9865 y−170 ±1 0 1.9865 z−480 ±1−0.35 ±0.07 1.963 Table 3.1: Bes i pa ame e o he [VO(TPP)] spin Hamil onian 3.2. [4] wi h he ob ained pa ame e s we can access he eigen alues o he sys em and plo i s ene gy le els diag am as a unc ion o he applied ield ( ig. 3.4). The calcula ed ene gy le els a e labeled as |mS, mI⟩, hus wi h he componen o he elec onic Sand nuclea Ispins pa allel o he applied s a ic ield B0. This a ibu ion is accu a e abo e B0= 0.25 T, whe e gS,xµBB0>|Az|wi h he ield ap- plied along he xaxis o he molecule ame. In his egime, he sys em eigens a es a e ac o ized o mo e han 98%. Figu e 3.4: Ene gy le els as a unc ion o he applied ield om he diagonaliza ion o he spin Hamil onian wi h s a ic ield B0applied (a) in plane ab and (b) along c-axis. The ed and blue shades highligh he mS=±1/2 mul iple s, espec i ely. On he le o each plo , wo inse s show a zoom in o he nuclea le el spli ing o he lowe and uppe mul iple s, wi h he le els ma ked by he nuclea spin componen s mIalong he ield. Ve ical ma ks highligh he nuclea ansi ions iden i ied in he spec a o igu e 3.2. Rep oduced om Re . [4] wi h pe mission om he Royal Socie y o Chemis y. Fo he pu pose o exploi ing he nuclea s a es o his sys em as he mul i-le els o a qudi , hey mus ul ill wo undamen al equi emen s. Fi s ly, he nuclea spin dynamics mus be condi ioned by he elec onic spin s a e, which means ha he ene gies o he nuclea ansi ions belonging o an elec onic spin s a e a e di e - en om he ene gies o he same nuclea ansi ions o he o he elec onic spin mul iple . As a consequence o his, any manipula ion o he nuclea s a e is condi- ioned by he elec onic s a e. The second equi emen conce ns he add essabili y o indi idual nuclea ansi ions. Indeed, any quan um ope a ion on he nuclea spin s a es can be exp essed as a combina ion o ansi ions be ween single pai s o nuclea consecu i e le els mI. To ensu e his, he di e ence be ween subsequen nuclea gaps δmI= (EmI+1 −EmI)−(EmI−EmI−1) mus be bigge hen he i a- 54 Simone Chicco Chap e 3 dia ion bandwid h used o he nuclea manipula ion. Indeed, his ea u e enables one o add ess single nuclea ansi ions, wi hou pe u bing nea by s a es. The [VO(TPP)] complex ul ills bo h he equi emen s epo ed abo e. In pa ic- ula , he pseudo quad upola coupling desc ibed in sec ion 1.3.4 plays a key ole, since he di e ences in he ansi ions ene gies a e mos ly impu able o his e ec . Indeed, he di e ences 2ζbe ween he ansi ion ene gies ob ained by diagonalizing he e ec i e Hamil onian o sec ion 1.3.4 (composed by he de ined pseudo-Zeeman and pseudo-quad upola con ibu ions), ma ch hose calcula ed by diagonalizing he eal spin Hamil onian o he sys em δ(mI) (see ig. 3.5). I is he e o e demons a ed ha his pseudo-con ibu ion is pa icula ly e ec i e in sys ems wi h s ong Hype - ine couplings. Figu e 3.5: Di e ence in subsequen nuclea ansi ions ene gies δ(mI) = (EmI+1 − EmI)−(EmI−EmI−1) in he mS= 1/2 mul iple (c osses), as a unc ion o he applied ield Bˆ x. These alues a e in sound ag eemen wi h wha expec ed om he diagonaliza ion o he pseudo-quad upola Hamil onian (black line). Rep oduced om Re . [4] wi h pe mission om he Royal Socie y o Chemis y. The dephasing a es o he nuclea spin 1/TmI,mI′ 2a e also in es iga ed by means o he 2π 3−τ−2π 3Hahn pulse sequence wi h inc easing τ, as a unc ion o he ap- plied s a ic ield B0along molecule c-axis and ab-plane. The echo in ensi y decays ollowing a single exponen ial law as a unc ion o τ, o all he measu ed ansi- ions ( ig. 3.6). The ex ac ed ime cons an TmI,mI′ 2is he spin phase memo y ime o he sys em, co esponding o he li e ime o he induced s a es supe posi ion. The alues ob ained o [VO(TPP)] a e epo ed in igu e 3.7 as a unc ion o B0 and show an inc emen om 10 µs o 60 µs wi h inc easing ield. The esul s a e ema kable, since [VO(TPP)] T2 alues a e among he highes in his ca ego y o complexes. Mo eo e , he phase memo y imes esul much longe han he ime expec ed o implemen ing elemen a y ope a ions on he qudi sys em. Thus, hese equisi es make his complex e y p omising o Quan um In o ma ion P ocessing (QIP) applica ions. 55 Simone Chicco Chap e 3 Figu e 3.6: Echo in ensi y decay as a unc ion o he delay be ween he Hahn echo exci ing and e ocusing pulses. The expe imen al da a o a selec ed nuclea an- si ion ((a) AB2, (b) C1) a e plo ed oge he wi h he elaxa ion a e T2single exponen ial i o di e en applied ields. Rep oduced om Re . [4] wi h pe mis- sion om he Royal Socie y o Chemis y. Figu e 3.7: Nuclea phase memo y ime o di e en nuclea ansi ions, measu ed wi h a e ocusing Hahn-echo sequence, as a unc ion o he applied ield B0(a) in he ab-plane and (b) along c-axis. Rep oduced om Re . [4] wi h pe mission om he Royal Socie y o Chemis y. In a dilu ed sample he dipola coupling be ween elec onic spins and he in e - ac ion o he nuclei wi h su ounding elec on spins a e bo h s ongly supp essed. The e o e, he nuclea spin decohe ence is go e ned by he in e ac ion wi h nea by magne ic nuclei media ed by he i ual exci a ion o he elec onic spins. In ou [VO(TPP)] molecule, his mechanism domina es o e he di ec nucleus-nucleus mechanism as epo ed o o he simila compounds. [45] The elec o-nuclea mix- ing o he sys em wa e unc ions p omo es his elaxa ion mechanism. Thus, he inc emen o he measu ed phase memo y imes wi h he applied ield, is jus i ied by an inc ease in he ac o iza ion o he elec onic and nuclea componen s o he wa e- unc ion. Mo eo e , he inc emen in he elec onic pola iza ion by inc easing B0supp esses he elec onic spin luc ua ions and u he educe hei in e ac ion wi h he p obed nuclei. [113,114] Nuclea spin s a e popula ions can be manipula ed by means o pulses. Indeed, 56 Simone Chicco Chap e 3 (M) (see ig. 3.13). Howe e , because o he c ys allog aphic inequi alence, each o he p e iously de ined di ec ions comp ises wo di e en con igu a ions o he ield wi h espec o he wo molecules ( ig. 3.14). Figu e 3.13: Pic u e o he c ys al collec ed om mic oscopy (le ) and a ske ch o he c ys al shape ( igh ). The wo a e compa ed o show he edge leng h hie a chy and he ela i e indexing o c ys al aces wi h he co esponding c ys allog aphic planes. Rep in ed wi h pe mission om Re . [12]. Copy igh 2021 Ame ican Chemical Socie y. Figu e 3.14: The c ys allog aphic planes (111), (010) and (10-1), ha co espond o he c ys al aces, a e highligh ed in he uni cell. This ske ch assis in he isual- iza ion o he applied ield di ec ions wi h espec o he molecula e e ence ame. Rep in ed wi h pe mission om Re . [12]. Copy igh 2021 Ame ican Chemical So- cie y. To de e mine he spin Hamil onian pa ame e o his sys ems a hyb id NMR-EPR app oach was used o he i s ime. Indeed, he wo echniques (see sec . 2.1.2 and 2.1.1) ea u es di e en esolu ion and sensibili y o each o he spin Hamil onian e ms. The s a ing poin is he elec o-nuclea spin Hamil onian, w i en as: H0=µBˆ S·gS·B0+ˆ I·A·ˆ S+µNgNˆ I·B0+ˆ I·P·ˆ I.(3.3) He e, he i s wo e ms co espond o he elec onic Zeeman e m and o he Hy- pe ine coupling be ween he elec onic and he nuclea spins. The o he wo e ms cons i u e he ine-s uc u e Hamil onian o sec ion 1.3 and co espond o he nu- clea quad upola and he nuclea Zeeman couplings. 63 Simone Chicco Chap e 3 Con inuous wa e (CW) X-band and Q-band EPR spec a (9.405 GHz and 33.7 GHz espec i ely) we e measu ed by ou collabo a o s a he Uni e si y o Flo ence and a he Labo a oi e Na ional des Champs Magn´e iques In enses (LNCMI) in G eno- ble, (see he Lis o Collabo a o s) on he dilu ed powde s (shown in ig. 3.15-(a,b) o sample 3c). The spec a clea ly show he eigh spec al lines induced by he Hype ine coupling wi h he 51V nuclea spin, co esponding o he eigh elec onic ansi ions mS=±1/2 a ixed nuclea componen s ∆mI= 0. Figu e 3.15: (a) Con inuous Wa e (ν= 9.405 GHz, black) and Echo De ec ed Field Sweep (ν= 9.70 GHz, blue) X-band EPR powde spec a o sample 3c measu ed a T= 20 K. The spec a a e bo h simula ed ( ed) using he Hamil onian pa ame e o able 3.2. EDFS is ansla ed in B o be e compa ison. (b) CW-EPR Q band spec um (ν= 33.7 GHz, black). (c,d) 2D colo map (blue nega i e, ed posi i e) o 3c single c ys al X-band EPR spec a as a unc ion o θAand θB o a ions. Black lines co espond o he simula ed spec a o he wo inequi alen molecules. I is e iden he o e lap o he wo spec a along he e e ence ame axis (θA,B = 0◦,90◦) wi hin he EPR esolu ion. Rep in ed wi h pe mission om Re . [12]. Copy igh 2021 Ame ican Chemical Socie y. The X and Q band EPR spec a a e i ed only wi h he i s wo componen s o he spin Hamil onian 3.3, i.e. he elec onic Zeeman and he Hype ine couplings, e ining he spec oscopic gand Hype ine A enso s. This because i was no pos- sible o deduce he small nuclea in e ac ion e ms om he expe imen al esolu ion o his echnique. To ep oduce he da a he wo enso s a e conside ed hombic and collinea . The inal alues a e shown in able 3.2 a e he subsequen NMR e- 64 Simone Chicco Chap e 3 inemen s. In e es ingly, i was no necessa y o simula ing he expe imen s esul s o assume di e en pa ame e s o he wo molecules in he uni cell. This p o es ha he wo molecules di e s only in he o ien a ions wi hin he c ys al, which a e a e aged ou by measu ing powde samples. The single c ys al EPR in es iga ion enables he de e mina ion o he ela i e molecules o ien a ions. S a ing wi h he ex e nal ield applied along he (111) c ys al ace, he sample has been o a ed s ep by s ep inside he ca i y and he ield dispe sion o he esonances has been eco ded in he anglua ange θ= 0◦ o θ= 180◦( ig. 3.15). The o a ional axes we e he edge be ween (111) and (010) planes (θA) and he edge be ween (111) and (¯ 101) planes (θB). The o a ional axes de ined abo e co espond wi h he molecula ame de ined in igu e 3.12, wi h θAco esponding o he o a ion om x o zand θBco esponding o he o a ion om y o z. Fo bo h he o a ional axis, a 0◦and 90◦ he spec a o he wo molecules o e lap in a single eigh -line Hype ine s uc u e. This con i ms ha , wi hin he expe imen al esolu ion o CW-EPR, hei magne ic enso s appea o sha e he same o ien a ion wi h espec o he applied ield (see ig. 3.12). As we will see in he ollowing, his aspec will be e ined by he NMR analysis which ea u es a ine sensi i i y on he hype ine componen s and enables o obse e a sligh misalignmen o he magne ic enso s, no app eciable in CW-EPR. Figu e 3.16: B oadband NMR spec a measu ed a T= 4 K a ixed applied s a ic ield B0= 0.2 T, o he ield di ec ions BS,BM,BLschema ized in inse wi h espec o he c ys al aces. In black, he Gaussian spec al lines i ing. Rep in ed wi h pe mission om Re . [12]. Copy igh 2021 Ame ican Chemical Socie y. Fo in es iga ing he nuclea ine-s uc u e componen s o he spin Hamil onian 3.3, 51V (I= 7/2, γ/2π= 11.21 MHz) b oadband NMR spec a we e measu ed a T=4 K in he ield ange B0=0 o 0.4 T, exploi ing he ”HyReSpec ” spec ome e [11] a he Uni e si y o Pa ma. The [V(Cp)2Cl2] single c ys al 3b was ins alled on 65 Simone Chicco Chap e 3 a speci ically p in ed sample holde , in o de o apply he s a ic magne ic ield as much as possible along he c ys al aces labeled as L, M and S. The ield o ien a ion was subsequen ly u he e ined as a i ing pa ame e o he spec a. Figu e 3.16 shows an example o spec a collec ed in each o he ield con igu a ions (BS,BM and BL). Despi e he esonance o e c owding (14 nuclea ansi ions mul iplied o he wo molecules), hanks o he ine expe imen al esolu ion, all he ansi ions a e dis inguishable and can be i ed by a ibu ing a Gaussian line-shape o each esonan equency. The e olu ion o he NMR peak equencies as a unc ion o he applied ield has been i ed wi h he model spin Hamil onian in o de o e ine he pa ame e ob ained by CW-EPR and o e alua e also he nuclea quad upola and Hype ine con ibu ions. Only he elec onic spec oscopic g enso was kep ixed om he EPR analysis, which is mo e sensi i e o i s de e mina ion. The i ing o he spec al componen s a di e en applied ield is shown in igu e 3.17. The achie ed ag eemen be ween he da a and he modeled elec o-nuclea spin Hamil onian is excellen e en when conside ing bo h he molecules in he uni cell ( ed and black lines). All he spin Hamil onian pa ame e s ob ained om his dou- ble echnique e ining app oach a e shown in able 3.2 below. [V(Cp2)Cl2] spin Hamil onian A(MHz) p(MHz) gS(MHz) x−55 ±2 0.09 ±0.01 2.0010 ±5×10−4 y−216 ±2 0.09 ±0.01) 1.9834 ±5×10−4 z−315 ±2−0.18 ±0.01 1.9721 ±5×10−4 Table 3.2: Bes i pa ame e o he [V(Cp)2Cl2] spin Hamil onian 3.3. [12] Bo h he spec oscopic gand Hype ine A enso s show a hombic symme y, while a modeled nuclea quad upola con ibu ion, axial along he Cp-V-Cp double-decke axis, was essen ial o he ep oduc ion o NMR spec a. Apa om he nuclea ine s uc u e componen s o 3.3, b oadband NMR also enables an accu a e e inemen o he Hype ine enso alues and absolu e sign. The la e is in ac no accessible in CW-EPR. By diagonalizing he spin Hamil onian 3.3, we calcula e he sys em ene gy le els o he h ee ex e nal ield con igu a ions, shown in igu e 3.18. While he elec onic mul iple s spli ing is no widely a ec ed by he o ien a ion o he ield, he nuclea mul i-le el s uc u e is s ongly modi ied. Indeed, he ema kable di e ences in he hype ine componen s causes he o e all nuclea spli ing o a y signi ican ly be- ween he h ee con igu a ions. Mo eo e , a he han he small nuclea quad upo- la coupling, i is he hype ine e ms ans e se o he applied ield esponsible o he second-o de pseudo-quad upola e ec ha play a key ole he e in he in a- mul iple gaps aniso opy. The calcula ed ene gy le els ul ill he equi emen s o a p omising qudi candida e. Indeed, he nuclea dynamics is condi ioned by he elec onic spin s a e, since he wo mSmul iple s a e well dis inguished in ene gy. Mo eo e , he aniso opy in he nuclea ansi ions exci a ion ene gies is su icien o enable hei selec i e exci a ion wi h s a e o he a esona o s. The ene gy le els o he wo inequi alen molecules a e iden ical, wi h he excep ion o small di e ences due o he di ec ion o he applied ield in he e e ence ame o he wo molecules. These small di e ences a e howe e su icien o dis inguish he exci a ion spec al equencies o each molecule, in mos o he applied ield 66 Simone Chicco Chap e 3 Figu e 3.17: The measu ed NMR equencies o a single c ys al o 3b (yellow and cyan do s) a e compa ed wi h he spin Hamil onian i ing model (black and ed do s o each molecule, espec i ely). The applied ield di ec ions BS,BM,BLa e schema ized in inse . Cyan expe imen al poin s label he ansi ions whose nT2 was measu ed. Shaded a eas we e no explo ed expe imen ally. Rep in ed wi h pe mission om Re . [12]. Copy igh 2021 Ame ican Chemical Socie y. 67 Simone Chicco Chap e 3 Figu e 3.18: Calcula ed ene gy le els o he expe imen al ield con igu a ions BS, BM,BLwi h espec o he c ys al aces. Black and ed lines ela es o he wo inequi alen molecules. Fo he nuclea ansi ions labeled by e ical colo ed lines i was also measu ed he nuclea phase memo y ime nT2( ig. 3.19). Rep in ed wi h pe mission om Re . [12]. Copy igh 2021 Ame ican Chemical Socie y. 68 Simone Chicco Chap e 3 condi ions explo ed expe imen ally (see ig. 3.17). The mul i echnique app oach exploi ed he e has p o en o be also essen ial in he s udy o he elaxa ion and cohe ence imes o he elec o-nuclea coupled sys em. An echo de ec ed ield sweep (EDFS) expe imen pe o med by ou collabo a o a he Uni e si y o Tu in (see he Lis o Collabo a o s) wi h he pulsed EPR ech- nique in he X band ( ig. 3.15-(a)) con i ms he p esence o elec onic cohe ence also a empe a u es abo e 20 K. The elec onic spin-la ice elaxa ion ime eT1 was measu ed as a unc ion o empe a u e on powde s, wi h an in e sion- eco e y sequence. The applied ield was ixed a 0.334 T in o de o ma ch he highe in- ensi y ansi ion in he spec a, co esponding o ∆mS=±1 and ∆mI= 0 om mI=−1/2. The eco e y a es a ixed empe a u e we e i ed wi h a s e ched single exponen ial sa u a ion model I=I0+C1exp(−(τ/eT1)β). The ex ac ed elaxa ion imes eT1a e shown in igu e 3.19. We obse e a maximum and hen la elaxa ion a es below 10 K, wi h alues inc easing wi h dilu ion. A highe em- pe a u es he elaxa ion ime dec eases ab up ly and becomes independen om he dilu ion pe cen age. This anomalous empe a u e dependence o eT1has been mod- eled by he sum o a di ec and a Raman-like con ibu ions o elaxa ion, domina ing a lowe and highe empe a u e, espec i ely: eT−1 1=aT +bTn.(3.4) The e iden di e ence wi h espec o o he Vanadium based molecula sys em ha display spin cohe ence app oaching oom empe a u e,[5,6] can be linked o an unexpec edly e icien Raman-like pa h o elaxa ion, whose exponen app oach n= 5 in he phenomenological model used he e. The dilu ion-dependen di e ences a e ins ead ela ed o he enhanced dipola in e ac ion. A ine model o elaxa ion in he high empe a u e egime can be se up by making explici he coupling o he spin wi h local ib a ional modes:[116] T−1 1=cT +dexp(¯hω/kT) (exp(¯hω/kT)−1)2.(3.5) This mo e de ailed model indeed e ined he i in igu e 3.19 and sugges s ha a po en ial impo an ole in he ab up dec ease o he elaxa ion ime a 30-40 K is played by an op ical phonon wi h ene gy ¯hω ≈14 meV, oge he wi h low ene gy phonons (¯hω ≈2−6 meV) commonly in ol ed in hese p ocesses. Spaces o a ine and in-dep h analysis o he elaxa ion p ocesses we e le o a u u e dedica ed wo k. Elec onic quan um cohe ence eTmas a unc ion o empe a u e was also in es i- ga ed by ou collabo a o s a he Uni e si y o Tu in (see he Lis o Collabo a o s) by pulsed X-band EPR on he same spec al line (0.334 T), wi h a Hahn-echo de- cay expe imen . The echo decays we e i ed wi h a s e ched exponen ial decay I=I0+C2exp[−(2τ/eTm)β] and he ex apola ed cohe ence imes a e shown in igu e 3.19-(a). The a es a e empe a u e independen below 40 K and in he o de o magni ude o he µs (2.3 µs o he highe dilu ion). Thus, he low empe a u e cohe ence imes a e compa able wi h o he pa en s compounds. [6,35] The con- cen a ion dependence also below 1% sugges s ha in his empe a u e ange he elaxa ion is s ill due o he elec onic spin-spin in e ac ion, ha domina es o e he less e ec i e elec on-nuclea in e ac ion, despi e he abundance o spin ac i e nuclei (1H). A highe empe a u e he spin cohe ence ime eTmsha es wi h he 69 Simone Chicco Chap e 3 spin la ice eT1a simila ab up d op. E en i eTmis no limi ed by eT1in his empe a u e ange, i is clea ha he wo phenomena a e co ela ed. A possible explana ion is ha he as e spin-la ice elaxa ion eT1causes as e p onounced elec onic luc ua ions ha di ec ly impac he spin cohe ence eTm. Thus, in he de ini ion o he elec onic cohe ence ime 1/eTm= 1/eT2+ 1/2eT1 he spin-la ice eT1canno be neglec ed, e en i he spin-spin elaxa ion eT2is s ill he dominan con ibu ion. Figu e 3.19: (a) Tempe a u e e olu ion o he elec onic spin-la ice eT1and spin cohe ence eTm elaxa ion imes, o each dilu ion pe cen age o [V(Cp)2Cl2]. Solid and dashed lines ep esen espec i ely he models 3.4 and 3.5 i ing esul s. (b,c,d) Nuclea phase memo y ime nT2measu ed on dilu ion 3b o he ansi ions labeled in igu e 3.18 a T= 4 K as a unc ion o he applied ield, o BS,BM,BL con igu a ion, espec i ely (ske ch in inse ). Rep in ed wi h pe mission om Re . [12]. Copy igh 2021 Ame ican Chemical Socie y. Nuclea phase memo y imes we e also in es iga ed o a single c ys al o sample 3b (1 % dilu ion). The measu emen s ha e been pe o med a he Uni e si y o Pa ma on he ”HyReSpec ” [11] b oadband NMR spec ome e (see 2.1.1). Hahn- echo decays we e measu ed as a unc ion o he applied ield, o all he ansi ions labeled in he ene gy le els scheme o igu e 3.18. The echo decays as a unc ion o he delay τbe ween he exci ing and e ocusing 2π/3 -pulses, was modeled wi h a single exponen ial M(τ) = M0exp(−2τ/nT2) ( ig. 3.20). The ex ac ed phase memo y imes nT2≈20 −30 µs a e compe i i e wi h espec o pa en compounds as he [VO(TPP)] epo ed in sec ion 3.2 and in Re .[4]. 70 Simone Chicco Chap e 3 Figu e 3.20: T ans e se magne iza ion decay om NMR echo elaxa ion expe i- men s, as a unc ion o he delay τbe ween exci ing and e ocusing pulses, a T= 4 K and ixed s a ic ield B0, along BS(a), BL(b), BM(c) o ansi ions S1,L2, M2. Lines shows he single exponen ial decay i ing. Rep in ed wi h pe mission om Re . [12]. Copy igh 2021 Ame ican Chemical Socie y. The nuclea cohe ence inc eases wi h he applied ield, because o he simul aneous educ ion o he elec o-nuclea mixing o he wa e unc ions and inc emen o he elec onic pola iza ion. Indeed, he dominan phenomenon ha imes he nuclea decohe ence is he indi ec dipola in e ac ion wi h nea by nuclea and elec onic spins. The la e is se e ely educed by dilu ion in diamagne ic analogues ( i a- nium(IV) in his case). Conside ing he abo e esul s o he elec onic and nuclea spin cohe ence imes, his sys ems can be unequi ocally conside ed a p omising coupled qubi -qudi sys- em. The u he s ep o con i m his claim and p o e ha he cohe ence imes allows a bi a y and selec i e manipula ions o he elec onic and nuclea mul ile el s uc- u e, is o ealize expe imen ally elec onic and nuclea spin nu a ion expe imen s. Low empe a u e elec onic Rabi oscilla ions we e ealized on c ys al powde s a X-band equency by ou collabo a o s a he Uni e si y o Tu in (see he Lis o Collabo a o s). The sys em is a ge ed by a i s esonan pulse o a iable leng h θ( ) ha exci es he chosen ansi ion, ollowed by he canonical π−τ−π/2 Hahn- echo e ocusing sequence o s imula e he spin-echo. The Rabi equency eωR( ig. 3.21) scales linea ly wi h he manipula ion pulse ampli ude. To u he suppo he 71 Simone Chicco Chap e 3 selec i i y o he exci a ion, he calcula ed equency domain spec a ( ig. 3.21-(b)) a e basically monoch oma ic, showing only he genuine Rabi eωRcon ibu ion and a small con amina ion in co espondence o he p o on (1H) nuclea La mo equency eω H, which a e ine i ably coupled wi h 51V. Figu e 3.21: (a) Elec onic Rabi nu a ion expe imen s (EPR X-band) on 3c a T= 20 K a di e en exci a ion powe s. (b) F equency domain pic u e o highligh he monoch oma ici y o he oscilla ions. (c) Linea dependence o Rabi equency (ΩR=ωR/2π) on he exci ing mic owa e pulse ampli ude B1. Rep in ed wi h pe mission om Re . [12]. Copy igh 2021 Ame ican Chemical Socie y. Nuclea cohe en manipula ions we e also ealized a he Uni e si y o Pa ma o a ious ield di ec ions wi h espec o he 3b single c ys al (see ig. 3.22). A i s exci ing pulse o a iable leng h θ( ) esonan wi h a speci ic nuclea ansi ion and wi h a na ow bandwid h is used o selec i ely induce he ansi ion o in e es . Then, he s imula ed spin exci a ion is e ocused by a subsequen πpulse o he de ec ion o he echo in ensi y. The induced Rabi oscilla ions a e damped by a ac- o λ ha combines he e ec s o igina ing om he sys em nuclea cohe ence imes and he inomogenei ies o he manipula ing - ield. The la e gene ally domina es o e he long phase memo y imes o hese sys ems. The oscilla ions a e hus mod- eled wi h an exponen ially damped oscilla o unc ion I∝exp(−λ ) sin(nωR ). Fo all he ield di ec ions and pulse powe s p obed, he damping a e is conside ably longe han he ime equi ed o induce a single in e sion (πpulse) o he nuclea spin sys em. Combining his esul s wi h he ac ha he damping is mainly due o 72 Simone Chicco Chap e 4 Un eiling phonons in a p o o ypical molecula qubi by IXS In o de o push u he he pe o mances o molecula qubi s owa d hei use in quan um de ices, i is undamen al o each a ho ough unde s anding o hei e- laxa ion dynamics and decohe ence. In his wo k we exploi ed o he i s ime high- esolu ion Inelas ic X- ays Sca e ing (IXS) oge he wi h cu ing-edge ab ini io calcula ions o s udy phonon dispe sions in he p o o ypical molecula qudi [VO(TPP)] [4,14] (see sec . 3.2 o i s cha ac- e iza ion wi h b oadband NMR). Wi h his syne ge ic expe imen al and heo e ical app oach, we alida ed he ab ini io phonon calcula ions by compa ing he measu ed spec a wi h he simula ed sca e ing c oss sec ion. IXS da a pe mi o in es iga e phonon dispe sions o e a wide sec ion o he ecip ocal space and o un a el he ex- is ence o ex emely low ene gy op ical phonons. Finally, we de e mined he c i ical ole o hose ex emely low ene gy modes in he spin elaxa ion, by calcula ing hei spin-phonon coupling cons an s. Iden i ying he ib a ions which mos ly con ibu es o elaxa ion indeed p o ides a iable op ion o syn he ically imp o ing he pe o - mances o molecula magne s. This new and powe ul syne gis ic app oach ga e us an unp eceden ed insigh s on he spin dynamics o he [VO(TPP)] molecule, pa ing he way o he applica ion o his echnique o he cha ac e iza ion o phonons in MNMs. 4.1 S a e o a Reaching a p o ound unde s anding o Molecula qubi s elaxa ion dynamics and decohe ence mechanisms is an aspec o pa amoun impo ance in o de o unleash he po en iali ies o hese sys ems o quan um in o ma ion applica ions and com- pu a ion. The cohe ence imes o hese sys ems mus be long and empe a u e esilien in o de o implemen quan um algo i hms.[70] A key ole in de e mining hose cohe ence imes and hei empe a u e dependence is played by he in e ac ion o molecula spin wi h in e - and in a-molecula ib a ions.[29,86] Recen ly, Inelas ic Neu on Sca e ing expe imen s (INS) and ab ini io spin dynam- ics simula ion ha e been used o he in es iga ion o phonons in molecula qubi s 79 Chap e 4 and single-molecule magne s. Fo ins ance, phonons ha e been ecen ly in es iga ed in he VO-based compound [VO(acac)2] wi h he 4-dimensional inelas ic neu on sca e ing echnique. [30,118]. This s udy un a eled he p esence low ene gy op i- cal phonons, in ol ed in an i-c ossing wi h he acous ic b anches, causing a mu ual spin-phonon coupling s eng h ans e . [29,30] This pic u e was in clea con as wi h he canonical ea men o phonons wi h he Debye model, in which acous ic phonons a e p edic o be well sepa a ed in ene gy om he op ical modes, being also he main playe s in he spin elaxa ion o MNMs. An expe imen al echnique allowing us o access bo h phonons ene gies and pola iza- ion ec o s is undamen al in o de o cons uc a sound model o phonon-induced elaxa ion mechanisms. Indeed, only wi h such a solid s a ing poin i is possible o de e mine he ole o speci ic phonon b anches in di e en elaxa ion p ocesses. [28]. In ac , be o e he ecen INS expe imen s on phonons, we only had in o ma- ion on phonon ene gy spec a a he Γ poin ob ained wi h Raman, IR o THZ spec oscopies, which can only be in e p e ed in he con ex o a Debye-like pic u e, which is iola ed in ou molecula complexes. Howe e , INS echnique poses s ong limi a ions (be e de ailed in he ollowing sec ion) in e ms, e.g., o samples di- mensions and chemical composi ions. The pu pose o his wo k is hen o exploi o he i s ime inelas ic X- ays sca e ing o a molecula qubi , o expe imen ally in es iga e he ene gies and he pola iza ion ec o s o phonons in a p o o ypical molecula qubi [VO(TPP)]. A he same ime he implemen a ion o ab ini io s a e- o -a spin dynamic simula ions enables o unde s and he c i ical ole o di e en phonon b anches in he elaxa ion dynamics. The [VO(TPP)] complex is chosen no only as a che ypes o he new gene a ion o molecula qubi s wi h p omising pe o mances (long cohe ence imes e en a high empe a u es), [4] bu also because, among he amily o anadium-based sys ems, i is conside ed o be one o he mos obus o high ene gy i adia ion. The expe - imen al obse a ion o ul a-low ene gy op ical modes makes his sys em an ideal es -bed o s udy he ole o hese non-Debye modes in he spin dynamics. 4.2 Inelas ic X- ay sca e ing expe imen The expe imen al s udy o phonon dispe sion and pola iza ion ec o s on a molecula c ys al is ealized o he i s ime by exploi ing high- esolu ion Inelas ic X- ay Sca - e ing (IXS). The ad an ages o his echnique wi h espec o he complemen a y Inelas ic Neu on Sca e ing (INS) a e se e al. Fi s o all, mode n high- esolu ion IXS spec ome e s as ID28 [119] a he Eu opean Sinch o on Radia ion Facili y (ESRF) allows o measu e acous ics and op ical phonons also o e y small samples (o de o 1 mm3), while he small dimensions o molecula c ys als s ill ep esen an hu dle e en o mode n high- lux neu on spec ome e , ypically p e en ing he use o INS o s udying phonon dispe sion. Mo eo e , ano he downside o neu ons is ep esen ed by he la ge incohe en c oss sec ion o ligh elemen s, like 1H, which a e abundan in MNMs, causing he blu ing o he phonons cohe en sca e ing signal and hus o phonon dispe sions. This d awback can be o e come by subs i u ing he elemen s wi h la ge incohe en c oss sec ion wi h chemical analogues, such as deu e ium in place o Hyd ogen. Howe e , deu e a ion has signi ican e ec s on he INS signal only i high pe cen age can be ob ained, which, in he case o MNMs, 80 Simone Chicco Chap e 4 i is e y di icul . On he con a y, he la ge ene gy ans e in ol ed in he X- ay incohe en sca e ing p ocesses (∼eV ) is o de o magni ude la ge han he ene gy scale o in e es when measu ing phonons in MNMs (∼meV ), hus i does no con- ibu e o he measu ed c oss sec ion. This, combined wi h he nea ly absence o mul iple sca e ing p ocesses, makes IXS basically a backg ound- ee echnique in he meV ange. Fu he mo e, he employmen o ha d X- ays wi h high inciden ene gies (∼keV ), se e al o de o magni ude la ge han he ene gy scale o MMNs molecula and la ice ib a ions, causes he spec al esolu ion o be independen om he ene gy and momen um ans e o he sca e ing e en . This is an im- po an ea u e o he cha ac e iza ion o op ical phonons. Indeed, in p eceden s udies pe o med wi h INS, acous ic modes dispe sions we e success ully obse ed, while he ene gy-dependen esolu ion was no su icien o esol e op ical modes wi h ene gies abo e 10-12 meV. [30] The ene gy and momen um ans e also esul o be comple ely decoupled. All he lis ed IXS ad an ages and ID28 [119] unique capabili ies allowed us o suc- ceed a measu ing o he i s ime acous ic and op ical phononic exci a ion in a molecula c ys al o a p o o ypical qubi . A [VO(TPP)] single c ys al (c ys allog a ic cell in ig 4.1) o dimensions 1 ×1×0.5 mm3was syn hesized by ou collabo a o s a he Uni e si y o Flo ence (see he Lis o Collabo a o s). The c ys al was o i- en ed by means o di use sca e ing on he ID28 beamline side s a ion. We ocused on he (h0l) sca e ing plane, and in pa icula on B agg e lec ions 006 and 600, which o e ed he bes comp omise in e ms o in ensi y o he inelas ic signal and supp ession o he elas ic one. Wi h his choice o B illouin zone (BZ) we could explo e he Γ −Kz, Γ −Kxand Γ −Npa h o he [VO(TPP)] ecip ocal space, co esponding o di ec ions (00l), (h00) and (h0h) espec i ely, in longi udinal and ans e se con igu a ion. Figu e 4.1: (a-b) iew o he [VO(TPP)] c ys al s uc u e along he a and c axis espec i ely. (c) B illouin zone pic u e, in which he symme y di ec ion p obed in ou expe imen a e highligh ed ( ed a ows). 81 Simone Chicco Chap e 4 IXS measu emen we e collec ed a 300 K, wi h he sample glued on a s anda d sample holde . The ID28 spec ome e was used in backsca e ing con igu a ion, exploi ing wo di e en e lec ions o he Si monoch oma o s: he Si(9 9 9), which selec s an incoming ene gy E= 17.794 KeV and ea u es a esolu ion o δE = 3.0 meV, and he Si(12 12 12), whose selec ed ene gy is E= 23.725 KeV wi h a esolu- ion δE = 1.5 meV. We e e o his wo con igu a ions as Low and High- esolu ion con igu a ions. By exploi ing he iple-axis geome y o he spec ome e in low- esolu ion con igu a ion we pe o med an explo a ion scan o he phonon modes. The p incipal symme y di ec ions in he ecip ocal space Γ−N, Γ−Kxand Γ−Kz (see he i s B illouin Zone o he c ys al in ig. 4.1) we e sampled spanning he ene gy om -25 meV o 25 meV a cons an sca e ing ec o Q alues. In igu e 4.2 examples o low esolu ion spec a a e shown, o longi udinal and ans e se scans a di e en poin s in he ecip ocal space, o all he explo ed symme y di ec ions1. Figu e 4.2: Longi udinal (a,b,c) and ans e se (d,e, ) IXS scans along he di ec ions Γ −N, Γ −Kxand Γ −Kz, collec ed in low- esolu ion con igu a ion. F om hese scans is al eady e iden he p esence o low ene gy phonons, oge he wi h highe ene gy op ical ones a a ound 8 and 15 meV ( ig. 4.2-(a-c)). To be - e in es iga e low-ene gy phonon modes, we measu ed he di ec ion in which hese exci a ions a e expec ed wi h he high- esolu ion con igu a ion o he spec ome e (see ig. 4.3). High esolu ion measu emen s indeed con i med he ul a-low ene gy o [VO(TPP)] op ical phonons (≤2 meV) in mos o he p obed BZ di ec ions. The su p isingly- low ene gy o hese modes is he lowe e e measu ed in a molecula c ys al. The ID28 beamline has a se o nine analyze s, each o hem co esponding o a de ec o in he de ec o desk. This ea u e enables he simul aneous collec ion o phonon spec a o nine di e en Q alues. This is an impo an esou ce, since i 1I wo h no icing ha he de ini ion o longi udinal and ans e se scans along a diagonal ecip ocal space di ec ion as Γ −Nis essen ially meaningless. The eason o which i is done is simply a no a ion con en ion o uni o m he language. 82 Simone Chicco Chap e 4 Figu e 4.3: Longi udinal (a) and ans e se (c) IXS scans along he di ec ions Γ−Kx, Γ−N, and Γ −Kz, collec ed in high- esolu ion con igu a ion, o esol e he low ene gy phonons con ibu ion close o he elas ic line. allowed us o de ec a seconda y se o exci a ions, e y close o he p obed sym- me y di ec ions and wi h sligh ly di e en |q|. Combining he wo se o da a we inc ease se e ely he sampling o he BZ. 4.2.1 Resul s The measu ed spec a ha e been i ed using he beamline so wa e @ i 28, by in- cluding an elas ic line cen e ed a ze o exchanged ene gy and a easonable numbe o inelas ic peaks, depending on he exci a ion clea ly isible in each scan. The spec al unc ion used was a damped ha monic oscilla o (DHO) dho =I0(cδE)2/(E2+cδE), ep oducing he lineshape o he ins umen , wi h a FWHM co esponding o he measu emen esolu ion (δE = 3 meV o δE = 1.5). The ene gies o he esul ing phonon exci a ions as a unc ion o he BZ sampling ep esen he phonon dis- pe sions along he p obed symme y di ec ions. The e o e, hey can be di ec ly compa ed wi h hose ob ained wi h pe iodic densi y unc ional heo y (DFT) cal- cula ions. Howe e , no e e y phonon b anch has he same IXS c oss-sec ion. The expe imen al da a mus he e o e be compa ed wi h he simula ed IXS c oss sec ion ∂2σ ∂E∂Ω. The sca e ing in ensi y o each phonon b anch can be simula ed s a ing om calcula ed phonon eigen alues and eigen ec o s, as desc ibed in sec ion 2.2.3. These DFT calcula ions on [VO(TPP)] phonons we e ealized by ou collabo a o s om he T ini y College Dublin and om he Uni e si y o Flo ence (see he Lis o Collabo a o s). The mos isible exci a ions in he measu ed spec a a e hus expec ed o be he one wi h he s onge sca e ing c oss sec ion. The compa ison be ween da a and DFT calcula ions is done by supe imposing he phonon ene gies ex ac ed om he da a i ing o he 2D colo map o he simula ed c oss sec ion in he ene gy/phonon wa e ec o plane. The esul is shown in igu e 4.4 o longi udinal and ans e se modes along he h ee sampled symme y di ec ions Γ−N, Γ−Kxand Γ−Kz. The ag eemen is excellen and he majo i y o he op ical o acous ics phonons wi h a signi ican c oss sec ion a e de ec ed expe imen ally. F om igu e 4.4 he p esence o he ex emely low ene gy op ical phonons obse ed in high- esolu ion con igu a ion is also con i med by DFT esul s. These modes a e pe sis en in he whole BZ and cha ac e ized by a ema kable sca e ing in ensi y. A comple e desc ip ion o phonon modes is undamen al o calcula ing he spin- 83 Simone Chicco Chap e 4 Figu e 4.4: The 2D colo map shows he inelas ic c oss-sec ion calcula ed s a ing om phonon ene gies and pola isazion ec o s ex ac ed om ab ini io calcula ions, along he h ee sampled symme y di ec ions in he eci pocal space in bo h lon- gi udinal and ans e se con igu a ion: (a,e) Γ −N, (b, ) Γ −Kx, (c,d) Γ −Kz. The line-wid h associa ed o each phonon modes is chosen small enough o enable he dis inc ion o he con ibu ion a ising om any b anch. Cyan do s and squa es (δE = 3 / 1.5 meV espec i ely) ep esen he exci a ion ene gies ex ac ed om he spec a in igu es 4.2 and 4.3. Whi e do s/squa es ep esen he ene gies ex ac ed om he inspec ion o seconda y analyze s. Shaded a eas highligh he egions non accessible expe imen ally due o he dominan elas ic con ibu ion. phonon coupling coe icien s. The e o e, he alida ion o DFT calcula ions mus be suppo ed by he analysis o bo h phonon ene gies ωj(q) and pola iza ion ec o s σd j(q). Bo h hese quan i ies, as seen in sec ion 2.2.3, con ibu e o he inelas ic sca - e ing c oss sec ion. Thus, a compa ison be ween he expe imen al spec a and he IXS c oss sec ion simula ed wi h he DFT calcula ed ωj(q) and σd j(q) gi es a di ec eedback on he accu acy o calcula ions. In o de o pu o he es he pola iza ion ec o o each phonon mode ob ained wi h DFT, he mos e ec i e app oach is o implemen a con olu ion o he c oss sec ion o each mode wi h he expe imen al esolu ion and di ec ly compa e i wi h he expe imen al spec a as a unc ion o he ene gy. The compa ison is shown in igu e 4.5 o some ep esen a i e Q alues and is excellen o he whole ene gy ange o bo h low and high- esolu ion con ig- u a ions. This es u he con i ms he eliabili y o he calcula ed phonon model. A ypical small escaling o ∼10% is applied [30] o accoun o an de Waals in- e ac ion and empe a u e e ec s. Mo eo e , small disc epancies in he low ene gy egime exci a ion in ensi y a e mainly due o he Bose ac o nj(q)=(exp(βωj(q))− 1)−1which exponen ially depends on he exac phonon ene gy and hus enhance small di e ences. 84 Simone Chicco Chap e 4 Figu e 4.5: Expe imen al IXS spec a (black do s) a e compa ed wi h he simula ed c oss-sec ions ( ed line). The la e a e con olu ed wi h he expe imen al esolu ion ((a-e,g): δE = 3 meV; ( ,h,i): δE = 1.5 meV). A ep esen a i e spec a is shown o each p obed symme y di ec ion. Blue lines ep esen he inelas ic c oss-sec ion, calcula ed wi h a line-wid h ha enables o disc imina e he con ibu ion o each phonon mode. The elas ic line is omi ed o cla i y. 85 Simone Chicco Chap e 4 4.3 The c i ical ole o ul a-low ene gy ib a- ions This ex ensi e expe imen al IXS cha ac e iza ion o he [VO(TPP)] phonon dis- pe sions enabled he alida ion o he ab ini io phonon calcula ions. F om his alida ed esul s i has been possible o mo e a s ep u he and simula e also he sys em spin elaxa ion. The ab ini io calcula ion o spin-phonon coupling pa ame- e s equi es o simula e he e ec o phonons on he pa ame e s o he sys em spin Hamil onian. As in sec ion 3.2, he spin Hamil onian o [VO(TPP)] is w i en as H0=ˆ I·A·ˆ S+pˆ I2 z+µBˆ S·gS·B0+µNgNˆ I·B0and he pa ame e s a e de i ed bo h expe imen ally in [4] and con i med by DFT calcula ions. The modula ions induced on he Hype ine Aand spec oscopic spli ing g en- so s a e calcula ed om he i s and second pa ial-de i a i e o hose enso s wi h espec o he phonon wa e ec o o each phonon b anch, Vα= (∂A/∂qα) and Vαβ = (∂2A/∂qα∂qβ), espec i ely. E en o a mode n compu a ional ha dwa e, his huge calcula ion is e y demanding in e ms o ime and compu a ional e- sou ces. This hu dle is o e come, by ou collabo a o s om Dublin T ini y College and om Flo ence Uni e si y (see he Lis o Collabo a o s), exploi ing a ained neu al ne wo k o in e pola e ab ini io esul s and compu e he di e en ia ion e i- cien ly. [120] The calcula ed spin-phonon coupling coe icien s, ep esen ed by he modulus |Vα|, a e shown in igu e 4.6 oge he wi h he calcula ed phonon densi y o s a e. Figu e 4.6: Phonon densi y o s a e (black line) and modulus |Vα|o he spin-phonon coupling coe icien ( ed line, nega i e axis) compu ed ab ini io, as a unc ion o ene gy. In inse he compa ison be ween he low ene gy pDOS o [VO(TPP)] (black) and he pa en molecula qubi [VO(acac)2] (g een). [13] Two impo an conclusions can be ex ac ed he e: he i s is ha he ex emely low ene gy op ical modes possess a ema kable coupling wi h he spin and hus a e ex- 86 Simone Chicco Chap e 4 pec ed o play a key ole in spin elaxa ion p ocesses. Secondly, om he calcula ed low ene gy phonon densi y o s a es (pDOS), we obse e a comple e de ia ion om he Debye model. Indeed, he [VO(TPP)] pDOS g ows linea ly wi h he phonons ene gy below 5 meV, while a ∼ω2beha io would be expec ed o he canonical ep esen a ion o he Debye model, in which op ical and acous ics phonons a e well sepa a ed in ene gy. This second poin is u he suppo ed in he inse o igu e 4.6 in which he low ene gy pDOS p o ile o [VO(TPP)] is compa ed o he one o he pa en compound [VO(acac)2], ecen ly s udied by 4D-INS [30] which does no ea u e op ical phonons a ene gy lowe han ∼6 meV a Γ-poin . [13] I has been demons a ed ha , o molecula sys ems, he spin li e ime a empe - a u e highe han ∼10 K is s ongly in luenced by he coupling wi h phonons. [13] In molecula qubi s like [VO(TPP)] elaxa ion dynamics is mos ly igge ed by non- esonan wo-phonon Raman p ocesses. [116] The spin elaxa ion induced by hese phonon abso p ion and e-emission phenomena can be simula ed by including, in he secula Red ield equa ion, he spin-phonon coupling coe icien s |Vα|and |Vαβ|. A mode a e applied ield (0.3 T) he leading e m o he elaxa ion a e he modula ion o he hype ine Aand spec oscopic spli ing g enso s. The la e con ibu ion is he e neglec ed in he calcula ion wi hou loss o gene ali y, since he con ibu ion o he wo o he spin-phonon coupling is analogous and ollows a simila end o he conside ed ields. The ex apola ed spin la ice elaxa ion ime T1as a unc ion o empe a u e is compa ed in igu e 4.7 wi h he da a measu ed in [14] by X-band EPR wi h an in e sion and eco e y expe imen (see sec . 2.1.2). The ag eemen is ema kable abo e ∼10 K and bo h he simula ion and he expe imen display a T−2dependence, ypical o phonon-induced Raman elaxa ion p ocesses. Figu e 4.7: Calcula ed spin elaxa ion ime T1 o [VO(TPP)] ( ed squa es) com- pa ed o he T1measu ed in [14] wi h pulsed X-band EPR (black do s). These da a a e hen compa ed o he simula ed T1a e he emo al o phonons wi h ene gy lowe hen 6 meV (blue iangles). 87 Simone Chicco Chap e 4 We no e ha he low empe a u e expe imen al T1de ia es om he end isible a highe empe a u e. This can be ela ed o he e ec o spin-spin dipola in e ac ion, which is only pa ially emo ed by dilu ion a %2 in he isos uc u al [TiO(TPP)] diamagne ic analogue. The Raman-like end in he spin la ice elaxa ion a es down o low empe a u es suppo s he leading ole o he ul a-low ene gy op ical modes which a e demons a ed o be s ongly coupled wi h he molecula spin ( ig. 4.6). Indeed, by emo ing low-lying phonons (¯hω < 6 meV) s ongly coupled wi h he spin, he simula ed T1inc eases by wo o de o magni ude in he whole p obed empe a u e ange ( ig. 4.7). This es u he s ess he pa amoun impo ance o low-ene gy op ical phonons in supp essing spin li e ime. The ib a ions associa ed wi h low ene gy op ical phonons (∼2meV ) co espond, a he molecula si e, o dis o ion o he molecula s uc u e. In his pa icula case i has been obse ed ha he ib a ions in ol ed cause dis o ion o he phenyl g oups o he molecule, and he bending o he po phi in plana ing (see ig. 4.8). Being able o associa e he c i ically coupled phonon modes wi h a speci ic molecula dis- o ion ep esen s an e ec i e app oach o imp o ing he spin li e ime pe o mances by chemically ailo ing he molecule s uc u e and hus he ib a ional spec a. Figu e 4.8: Ske ch o he dis o ion o he molecula s uc u e, om he equilib ium s uc u e (yellow) o an ex ensi ely dis o ed one ( ed). These dis o ions consis in a bending o he po phi in plana ing and a wis o he phenyl g oups, and a e associa ed wi h he lowes ene gy op ical mode a Γ-poin . I is wo h men ioning ha e en hough he expe imen s a e ypically pe o med on single molecula c ys als, he applica ion o hese molecules in quan um de ices e- qui es he exploi a ion o single molecules. Thus, he in es iga ion o single molecule ib a ions is also an impo an piece o in o ma ion, when compa ed wi h he sin- gle c ys al esul s. The pe sis ence o he low ene gy in a-molecula dis o ions in [VO(TPP)] was con i med by gas phase calcula ions on a single molecule. I was also e i ied ha by emo ing he phenyl g oups om he molecule, he lowes ene gy op ical phonons a e shi ed up o 6.5 meV. The e o e, he s a egy o chemically enginee ing he molecule by emo ing he phenyl ings as in Vanadyl po phi in can po en ially imp o e he elaxa ion imes by emo ing low ene gy ib a ions c i ically coupled wi h he spin. 88 Simone Chicco Chap e 5 elaxa ion p ocesses, wi h and an ex ac ed e ec i e ba ie Ue = 1760 K. A lowe empe a u e ins ead, elaxa ion in 1 esul s o be domina ed by powe law Raman-like mechanisms, wi h τ−1=CTnand a exponen n= 1.1, hal o he one epo ed o a. Indeed, o e all, 1displays a elaxa ion 10 o 100 imes as e wi h espec o a, wi h a 100 seconds blocking empe a u e o 23 K, compa ed o 56 K o a. Since a hese empe a u es he spin dynamics is con ined o he lowes K ame s double , he elaxa ion is d i en by he coupling wi h low ene gy dispe - si e phonons ha igge non- esonan Raman mechanisms. The e o e, he as e elaxa ion can be a ibu ed o he di e ences in he low ene gy phonon spec a. Insigh s on he low-ene gy phonons spec a a e hen c ucial o unde s and he p o- cesses go e ning he elaxa ion and o un a el he key ing edien s o each a high blocking empe a u e.[31] Fo samples a,2and 3 o example, simula ions o he pDOS e eal wo e y peculia ends in he low and high ene gy spec al egions. The high ene gy DOS, domina ed by localized ib a ional modes esonan wi h he c ys al ield gaps and p omo e o O bach elaxa ion o e he e ec i e ene gy ba - ie Ue , display simila spec al ea u es, especially abo e 100 meV. Con e sely, he DOS o low-ene gy dispe si e modes, igge ing non- esonan Raman p ocesses a ies signi ican ly (see ig. 5.6). Figu e 5.6: Compa ison o he calcula ed pDOS o samples a,2,3. Panel (a) exhibi s he whole calcula ed ene gy ange, while panel (b) shows he es ic ed low-ene gy egime, highligh ing he signi ican di e ences in he low ene gy spec a, compa ed o he less p onounced ones in he in e media e-ene gy spec a. 95 Simone Chicco Chap e 5 By compa ing hese ab ini io pDOS calcula ions wi h hose measu ed by INS e- po ed in he ollowing sec ion, we can alida e he DFT esul s o be used as he s a ing poin o he calcula ion o he spin-phonon coupling coe icien s and he simula ion o he empe a u e dependence o he elaxa ion a e in igu e 5.4. I mus be ema ked ha in his app oach we a e neglec ing he e ec o he quan- um unneling o he magne iza ion (QTM), which is empe a u e-independen and hus is no igge ed by he in e ac ion wi h phonons. The e ec o QTM can be in ac quan i ied only by measu ing he empe a u e and ield dependence o he magne iza ion elaxa ion a es. Howe e , his mechanism was demons a ed o be non dominan a low empe a u e in his compounds class, due o he s ong axial ligands en i onmen ha supp ess he e iciency o his elaxa ion p ocess.[17] 5.1.1 Inelas ic neu on sca e ing expe imen The inelas ic neu on sca e ing (INS) expe imen was pe o med a he ins i u e Laue-Lange in (ILL) on he indi ec geome y spec ome e IN1-Lag ange (LA ge GRaphi e ANalyse o Genuine Exci a ions), desc ibed in sec ion 2.2.2.[109] This beamline, ins alled on he ho neu on sou ce o he eac o , ea u es a combina ion o wide ene gy- ange and high esolu ion wi h an high neu on lux. Fo his eason, i is well sui ed o he s udy o molecula ib a ions in complex sys ems, h ough he e alua ion o he pDOS. In pa icula , a pa om cons an ac o s (see sec . 2.2.2), he measu ed sca e ing unc ion co esponds o he neu on weigh ed pDOS o he s udied sys ems (once he elas ic and quasi elas ic con ibu ion a e sub ac ed). The expe imen was pe o med a he empe a u e o 5 K on ≈2 g o powde o each o he sample unde in es iga ion (1,2,3). The ene gy ans e was chosen in o de o co e he 0 o 200 meV ange. To co e his low and in e media e ene gy ange wi h a cons an ene gy s ep, h ee monoch oma o s we e used, each o hem ea u ing di e en esolu ions:  Si(1 1 1), a ailable in a 4.5 o 20 meV ange and ea u ing a 0.8 meV esolu ion;  Si(3 1 1), a ailable in a 16.5 o 35 meV ange and ea u ing he same 0.8 meV esolu ion;  Cu(220), in he wide ange 26 o 200 meV and a p opo ional esolu ion o 2-3 % o he inciden ene gy. Each da ase was hen no malized by he numbe o inciden neu on, be o e me g- ing hem oge he in a single cu e. Fu he e inemen s o he collec ed da a equi e he sub ac ion o he emp y sample holde signal and he spec al spike smoo hing by compa ison wi h a wa e spec a p e iously collec ed. Elas ic and quasi-elas ic sca e ing con ibu ions a e hen sub ac ed by i ing he sum o a Gaussian cu e cen e ed a E=0 and a Lo en zian unc ion cen e ed a non-ze o exchanged ene gy, keeping he wid h as a i ing pa ame e . The esul s o hese da a educ ion p o- cedu e (pe o med wi h he so wa e LAMP [121]) a e shown in igu e 5.7 o he h ee Dy complexes, compa ed wi h he da a published by Chiesa e al. o he benchma k Dysp osocenium a.[17] The measu ed pDOSs con i m wha was expec ed om ab ini io calcula ion. The high ene gy spec al componen s o he h ee samples ea u e analogous peaks, wi h 96 Simone Chicco Chap e 5 Figu e 5.7: pDOS measu ed on IN1 o sample 1(g een line), 2(ligh blue line) and 3(magen a line), compa ed wi h he same measu emen collec ed o he benchma k Dysp osocenium on he MERLIN spec ome e a ISIS (do ed black line).[17] The wo panels (a) and (b) show he whole collec ed ene gy ange and a close up o he low ene gy pDOS, espec i ely. 97 Simone Chicco Chap e 5 a compa able ela i e in ensi y. This end is con i med also by compa ison wi h sample a. Con e sely, a low ene gy he pDOSs display s ong di e ences in bo h he ene gy o he modes and hei in ensi y. The la ges di e ences a e ound in he e y low ene gy b anches, om 0 o 30 meV, whe e he sca e ing in ensi y inc ease se e ly wi h espec o sample a, eaching he maximum in sample 3. This sample in pa icula display an o e all highe sca e ed in ensi y, which is jus i ied by he highe numbe o 1H in he coun e ion, yielding a ele an incohe en con ibu ion. 5.1.2 Resul s The p elimina y conclusions ou lined om he compa ison be ween he expe imen al pDOS a e in sound ag eemen wi h hose d awn om he ab ini io calcula ed ones. Bo h expe imen al da a and DFT esul s in ac con i m ha he main di e ences in he pDOS o he h ee compunds a e in he low-ene gy ange and hus can be di ec ly linked o he di e en beha iou o he elaxa ion a es in igu e 5.4 in he Raman empe a u e egime. To con i m hese hesis, i is undamen al o de elop a sound model o he phonon-induced elaxa ion o he magne iza ion. This can be done exploi ing he DFT phonon esul s o de e mine he coupling o molecula and la ice ib a ion wi h he molecula spin. The compa ison be ween he mea- su ed pDOSs wi h he calcula ed ones allows us also o alida e he exploi ed DFT me hod. The pe iodic DFT calcula ions o he pDOSs epo ed in igu e 5.6 ha e been pe o med by ou collabo a o s a he Uni e si y o Manches e (see he Lis o Collabo a o s) wi h he code VASP [122], exploi ing he PBE exchange-co ela ion unc ional [123,124] and D2 an dee Waals co ec ions.[125] Phonon densi y o s a es a e compu ed using he ini e displacemen me hod as implemen ed in he PHONOPY package.[126,127] The compa ison ou come is shown in igu e 5.8 o sample a,2,3 espec i ely. Figu e 5.8: Compa ison o he expe imen al phonon densi y o s a e (5 K) wi h he neu on weigh ed pDOS calcula ed ab ini io o (a) sample a, (b) sample 2and (c) sample 3. The ab ini io calcula ions we e pe o med by ou collabo a o s a he Uni e si y o Manches e (see he Lis o Collabo a o s). The ag eemen be ween he expe imen al da a and he calcula ed pDOSs is excel- len . The mos ele an spec al ea u es a e indeed ep oduced by he calcula ions. As a esul o his success ul compa ison, he phonon model is assumed o be ac- cu a e and can be used o he de elopmen o a ealis ic model o he elaxa ion 98 Simone Chicco Chap e 5 o hese compounds. This wo k in pa icula is s ill ongoing, since he ab ini io calcula ion o he spin phonon coupling coe icien s and he simula ion o he spin elaxa ion is e y demanding in e ms o compu a ional ime. 5.2 Measu ing phonon modes o a Dy-based SMM in applied p essu e Chemical and s uc u al modi ica ions a e no he only s a egy ha can be exploi ed in o de o igge a ia ions in he phonon spec a o molecula nanomagne s. In- deed, by applying an ex e nal hyd os a ic p essu e, i is possible o induce g adual dis o ion o he c ys al and molecula s uc u e o hese sys ems, and consequen ly o modi y he phonon spec a. By in es iga ing how phonons and hus he phonon- induced elaxa ion dynamics a y in SMMs as a unc ion o he applied p essu e, we can in p inciple disen angle he ole played by speci ic la ice and molecula i- b a ions in he domina ing elaxa ion mechanisms. In pa icula , his in o ma ion can gi e undamen al insigh in o de o de elop u u e syn he ic s a egies o im- p o ing he pe o mances o SMMs. To apply his inno a i e app oach, we decided o cha ac e ize a mononuclea Dys- p osium based SMMs [Dy(H2O)5(HMPA)2]I3·2HMPA (wi h HMPA = hexame hylphos- pho amide, [(CH3)2N]3PO), syn hesized by ou collabo a o s a he Uni e si y o Glasgow (see he Lis o Collabo a o s). The syn hesis o his sys em, oge he wi h magne ome y da a ha e al eady been published in 2018 by A. Canaj e al. [18]. The compound p esen s an almos ideal pen agonal bipy amidal geome y, wi h a p ominen axial c ys al ield p omo ed by he wo HMPA ligands (see ig. 5.9), which o m a ∼182◦axial angle. The sys em shows magne ic hys e esis up Figu e 5.9: (a) Molecula s uc u e o [Dy(H2O)5(HMPA)2] and i s coun e ion I3·2HMPA a ambien p essu e. The Dy ion (seag een) and he O ( ed) en i onmen a e highligh ed o cla i y. (b) Magne iza ion da a om Re .[18] as a unc ion o he applied ield showing hys e esis loops a di e en empe a u e in ambien p essu e (Y-dilu ed sample). (c) Magne iza ion elaxa ion a es om Re .[18] as a unc ion o empe a u e (black do s), ex ac ed om ac suscep ibili y measu emen s. The O bach egime is highligh ed by he A henius i ( ed line) o an ene gy ba ie Ue = 600 K and τ0= 1.2×10−11 s, as epo ed in [18] o T=9 K (10 K i he molecule is dilu ed in o i s Y diamagne ic analogue). The quan um unneling mechanism is no supp essed comple ely by he axial symme y; 99 Simone Chicco Chap e 5 he e o e, a a ound ze o applied ield he hys e esis loop becomes na owe , as a signa u e o he ac i a ion o his elaxa ion p ocess (see ig. 5.9). F om ac suscep- ibili y measu emen s a elaxa ion a e was also ex ac ed and i ed wi h a single exponen ial A henius cu e abo e 30 K ( ig. 5.9). This sugges s ha he dominan elaxa ion p ocess abo e 30 K is an O bach-like mechanisms wi h an ene gy ba ie Ue = 600 K, induced by he coupling o he spin wi h phonons wi h ene gy compa- able wi h he c ys al ield gaps. A empe a u e below 30 K ins ead, we do expec he de ia ion om he O bach end o ep esen he insu gence o non- esonan wo phonons Raman p ocesses as he leading mechanisms o he elaxa ion dynamics ( ig 5.9).[18] When an hyd os a ic p essu e is applied o he sys em, c ys allog aphic in es iga- ions o he molecule wi h X- ay di ac ion ha e e ealed a g adual il ing o he ligands axial angle, ha in an applied p essu e o 1.1 GPa becomes 190◦deg ee (see ig. 5.10). This loss o axiali y causes se e al di e ences in he magne iza ion cu es measu ed unde p essu e. A an applied p essu e o 1.1 GPa he magne ic hys e esis Figu e 5.10: (a) Molecula s uc u e o [Dy(H2O)5(HMPA)2] and i s coun e ions I3·2HMPA unde an applied p essu e o 1.1 GPa. Dysp osium ion (seag een) and he Oxygen ( ed) en i onmen a e highligh ed. (b) Magne iza ion da a as a unc ion o he applied ield showing hys e esis loops a di e en empe a u e in an applied hyd os a ic p essu e o 1.1 GPa. These measu emen s we e pe o med by ou col- labo a o s a he Uni e si y o Glasgow (see he Lis o Collabo a o s) loop is in ac comple ely closed in ze o applied ield, also a 2 K, due o he elax- a ion induced by he magne iza ion unneling unde he ba ie . When he ield is applied he hys e esis cycle opens, bu wi h a educed ela i e a ea i compa ed o he da a in ambien p essu e and equi alen empe a u e (see ig. 5.10). In o de o unde s and he modula ions induced in he phonon spec um by he applied p essu e and he e ec s on he elaxa ion dynamics, we combine he magne- iza ion measu emen s wi h unde p essu e INS expe imen s, ha allows he mea- su emen s o phonon exci a ions as a unc ion o he applied p essu e. These esul s can hen be compa ed wi h DFT calcula ions o he phonon modes. These DFT esul s, alida ed by compa ison wi h he expe imen al da a, will be exploi ed o simula e he sys em spin dynamic unde di e en p essu e egimes. 100 Simone Chicco Chap e 5 5.2.1 Inelas ic neu on sca e ing expe imen The inelas ic neu on sca e ing expe imen unde applied p essu e was pe o med a he Ins i u e Laue-Lange in (ILL) on he he mal neu on iple-axis spec ome e IN8 (mo e de ails on IN8 see sec . 2.2.2).[108] This high inciden lux spec ome e is op imized o he cha ac e iza ion o la ice ib a ions and magne ic exci a ions in he ange om ze o o ∼100 meV, in sample o small olume and weak signals. Mo eo e , IN8 can be equipped wi h a high p essu e Pa is-Edimbu g cell ha can emula e he ange o p essu e applied p e iously o he magne ome y expe imen s (0 o 1.1 GPa). Due o he chemical incompa ibili y o he [Dy(H2O)5(HMPA)2] sin- gle c ys al wi h he liquid ec o s used o apply he hyd os a ic p essu e, we used lead ilings as a ec o o dis ibu e uni o mly he ex e nal p essu e. The c ys al has been placed inside he CuBe gaske o he Pa is-Edimbu g cell, he emp y spaces ha e been closed wi h he lead ilings and hen p essed o a gapless pocke . A e he alignemen o he single c ys al sample, we in es iga ed he phonon exci- a ions a Γ-poin in he ecip ocal space o wo di e en  Q ec o alues:  Q1= (0 3 −12) and  Q2= (0 8 −16). The ene gy scans a e collec ed by ixing he inal wa e ec o k and a ying he inciden ki, o keep cons an he sca e ing  Q alue. We exploi ed wo di e en monoch oma o s, Si (111) and PG (002), ea u ing wo di e en esolu ions (0.88 meV and 3 meV, espec i ely). The low esolu ion con ig- u a ion ha e been used o an exhaus i e scan o he ene gy ange 0 −70 meV wi h a inal wa e ec o k = 4.11 ˚ A−1. Then he low ene gy ange 0 −34 meV has been e ined in he high esolu ion con igu a ion wi h k = 2.66 ˚ A−1. In his case, he ac ual accessible ene gy was limi ed below 34 meV due o he p esence o a spu ious signal om an analyze a monics o ki= 2k . The Γ-poin ene gy scans shown in igu e 5.11 we e collec ed a he empe a u e o 77 K o wo di e en applied p essu es 0 GPa and 1 GPa. Each da ase has been no malized by he numbe o sca e ed neu ons a he p e-sample moni o . Then, he emp y p essu e cell signal, collec ed wi h he same CuBe gaske bu wi hou he c ys al inside he lead iling, was sub ac ed, oge he wi h a quasi-elas ic lo en ian- shape signal a e y low ene gies. These da a a e also compa ed wi h he inelas ic double di e en ial c oss-sec ion o he Dy-complex (see sec.2.2.2) simula ed wi h he eigen alues and eigen ec o s o pe iodic DFT-calcula ed phonon modes. F om he compa ison i is e iden ha he majo di e ence induced by he applied p essu e lies in he low ene gy egion, be ween 0 and 25 meV. In his ange, phonon modes shi o highe ene gy by inc easing he applied p essu e (wi h a sizeable shi o he o de o some meV) and change he sca e ed spec al in ensi y. Some changes in he phonon exci a ions a e also obse able a highe ene gies; howe e , non sizeable as he e ec s on he low ene gy phonons. Da a we e also collec ed as a unc ion o empe a u e ( ig. 5.12). The di e en scans a e no malized o he empe a u e dependen Bose Fac o (sec . 2.2.2). This app oach can be use ul o de e mine he eliabili y o he spec al de ails, ha a e expec ed o g adually b oaden by inc easing he empe a u e. F om he compa ison o di e en scans we obse e ha mos o he spec al ea u es a low empe a u e su i e a highe empe a u es, bu b oaden signi ican ly and lose in ensi y. Va ious single c ys als ha e been hen milled in o ew g ams o powde and placed in a hin Aluminum cylinde , ha is sc ewed on he c yos a p obe. By measu - ing powde samples we a e age ou he  Qdependence o he sca e ing in ensi y o e a wide ange o o ien a ions. Thus, he sca e ing unc ion can be desc ibed 101 Simone Chicco Chap e 5 Figu e 5.11: INS c oss sec ion measu ed a 77 K o wo di e en applied p essu e on a [Dy(H2O)5(HMPA)2] single c ys al (do s), a he Γ-poin o  Q1= (0 3 −12). He e, he p essu e cell+gaske blank signal is al eady sub ac ed om he sca e ed in ensi y. The solid line ep esen he INS c oss sec ion calcula ed s a ing om DFT-calcula ed eigen alues and eigen ec o s and con olu ed wi h expe imen al es- olu ion o 3 meV (low esolu ion con igu a ion). The low-ene gy di e ences in he INS c oss sec ion induced by he applied p essu e a e e iden and eme ge conside - ably om he e o ba s. Figu e 5.12: Tempe a u e dependence o he high- esolu ion INS c oss sec ion mea- su ed a  Q1= (0 3 −12) in a [Dy(H2O)5(HMPA)2] single c ys al, o a cons an applied p essu e o 1 GPa. 102 Simone Chicco Chap e 5 in he so-called incohe en app oxima ion (see sec . 2.2.2), whe e i esul s di ec ly p opo ional o he neu on weigh ed phonon densi y o s a es (nw-pDOS). F om he powde spec a measu ed a 77 K, we can he e o e ob ain he nw-pDOS o he sample. In igu e 5.13 we compa e he measu ed nw-pDOS o wo di e en | Q| alues. Fo bo h hese alues, he phonon densi y o s a es p esen he same spec al de ails, con i ming ha he obse ed exci a ions a e pu ely phononics. The measu ed nw-pDOS can hen be compa ed wi h he one calcula ed wi h pe iodic DFT. Figu e 5.13: Phonon densi y o s a es om no malized inelas ic neu on sca e ing expe imen on [Dy(H2O)5(HMPA)2] powde s. The spec a was collec ed a 77 K o wo | Q| alues. The high and low esolu ion scans a e me ged in o de o maximize he ex en o he high esolu ion egion. 5.2.2 Resul s The pDOS as a unc ion o he applied p essu e can be calcula ed ab ini io wi h pe iodic DFT. These calcula ions we e pe o med by ou collabo a o s a he Uni- e si y o Pa ma, P. Bon `a and I. J. Onuo ah (see he Lis o Collabo a o s), using he Quan umESPRESSO package [128,129,130] which uses a plane wa e basis se and pseudopo en ials. The ene gy cu o o he wa e unc ions was se o 1225 eV while he ecip ocal space in eg a ion was pe o med wi h a 2 ×2×1 Monkhos -Pack [131] g id when conside ing he p imi i e uni cell. The ONCV no m conse ing pseudopo en ials [132] we e used o all elemen s and he exchange and co ela ion con ibu ion was desc ibed wi h he Pe dew, Bu ke, and E nze ho (PBE) unc ional.[123] The a omic posi ions we e op imized un il a omic o ces we e smalle han 2.57×10−4 103 Simone Chicco Chap e 5 eV˚ A−1and o al ene gy di e ences among op imiza ion s eps we e smalle han 2 meV while he expe imen al la ice pa ame e s we e kep ixed du ing he elax- a ion. Phonon equencies we e ob ained wi h he Phonopy package.[133] In his case a 2 ×2×1 supe cell is used o collec he in e -a omic o ce cons an s by pe - o ming 0.011 ˚ A posi ion displacemen s. The ecip ocal space is sampled using only he Γ poin in his la e case. The s a ing c ys allog aphic cell has been adap ed o each p essu e condi ion and e i ied by compa ison wi h X- ay di ac ion da a unde he same condi ions. The esul s o he simula ed pDOS ( ig. 5.14) con i ms he conclusions de i ed om he expe imen al da a, hus ha he la ges di e ence as a unc ion o he applied p essu e is obse ed o low ene gy phonon modes. Figu e 5.14: (a) Simula ed pDOS a wo di e en applied p essu es (a mosphe ic and 1 GPa). The pDOS is con olu ed wi h a sha p esolu ion σin o de o be e e alua e he con ibu ion o di e en phonon b anches ha would o e lap wi h he expe imen al one. (b) Zoom o he lowe ene gy egime, whe e he la ge p essu e- induced di e ences a e obse ed. In o de o alida e ou DFT esul s, we ha e compa ed he nw-pDOS measu ed a a mosphe ic p essu e wi h he one calcula ed ab ini io ( ig. 5.15). The spec al ea u es a e ep oduced by he calcula ion in bo h he low and high ene gy egime. The only e iden di e ence esides in he high-ene gy expe imen al backg ound ha esul s enhanced i compa ed o he simula ed pDOS. This e ec is no comple ely unde s ood up o da e, bu can be a ibu ed o a non negligible cohe en con ibu- ion o he sca e ed in ensi y, no included in he incohe en app oxima ion, and o mul i-phonon con ibu ions a high ene gy, excluded by ocusing only on he i s o de phonon expansion o he sca e ing unc ion (see sec . 2.2.2). Indeed, he sha p aspec o all he o he spec al ea u es pe mi o exclude a p io i ha his e ec is ela ed o he c ys al phonon s uc u e. Once alida ed, he DFT-calcual ed phonon modes will be used o simula e he elaxa ion dynamics and hus unde s and he e ec s o p essu e-induced phonons modi ica ions in p omo ing di e en elaxa ion mechanisms. 104 Simone Chicco Chap e 6 The echnique used o measu ing nuclea nT1is no a simple in e sion- eco e y. Con e sely, o measu ing he eco e y o he ans e se nuclea magne iza ion we add essed he sys em wi h a sa u a ing pulses ain, ollowed by a s anda d Hahn- echo e ocusing sequence π/2−τ−π/2. Wi h his p ocedu e we a e con iden ha he nuclea abso p ion line is ully i adia ed (maximum s a is ical empe a u e con- igu a ion). The sa u a ion eco e y pa h ollows a bi-exponen ial law I( ) = I1(1−exp(− /nT1,1))+ I2(1 −exp(− /nT1,2)), wi h a ime scales di e ence o se e al o de o magni ude; hus, he as es elaxa ion we a e in e es ed in is easily dis inguishable ( ig. 6.6). The ex ac ed spin-la ice elaxa ion a es 1/nT1 o he as es elaxing componen a e shown in igu e 6.7 as a unc ion o empe a u e o each applied ield. Figu e 6.6: (a) Fas es 1H-NMR no malized sa u a ion eco e y a 10 K o di e - en applied s a ic ield B0. (b) No malized single exponen ial 1H-NMR echo decay measu ed a 10 K o wo applied ields. The 1/nT1 empe a u e dependence shows a peak a 11 K in 0.33 T applied ield ha , by inc easing ield, lowe s in in ensi y, b oadens and shi s a highe empe a u es. A mo e de ailed analysis o his elaxa ion dynamics is ongoing and he i ing o he da a wi h a de ailed heo e ical amewo k, will gi e us in o ma ion on he phonon coupling s eng h and, on he numbe o domina ing elaxa ion a es/ equencies and on hei empe a u e dependence. In his analysis, an impo an e ec ha mus be aken in o accoun is he wipe ou e ec .[141] I consis s in a g adual NMR signal loss a low empe a u e, due o an inc emen in he elaxa ion a es nT−1 2o ce ain nuclei. The as e elaxa ion causes he signal o exi om he de ec able window o ou expe imen al se up ( he so-called ”dead- ime”, which is o he o de o ≈10 µs). Thus, o simula ing he spin-la ice elaxa ion a e we will e ain only he 1Hnuclei wi h a ans e se e- laxa ion a e nT−1 2smalle hen he iden i ied wipe ou h eshold. To e alua e his e ec we measu ed he nuclea spin-spin elaxa ion a es 1/nT2o e a wide empe - a u e ange, wi h a s anda d Hahn-echo e ocusing sequence. The elaxa ion a es ( ig. 6.6) a e i ed wi h a sligh ly s e ched mono-exponen ial unc ion (wi h al- mos empe a u e-independend s e ching pa ame e ) and he alues ex ac ed a e shown in igu e 6.8. The ex ac ed elaxa ion a es show a peak a 10 K in 0.33 T applied ield ha b oadens, lowe s in in ensi y, and shi s a highe empe a u es by 111 Simone Chicco Chap e 6 Figu e 6.7: P o on nuclea spin-la ice elaxa ion a es 1/nT1ex ac ed om i ing he sa u a ion eco e y cu es in igu e 6.6 as a unc ion o he sys em empe a u e, a h ee di e en applied s a ic ield B0. Figu e 6.8: (a) P o on nuclea spin-spin elaxa ion a es 1/nT2a wo s a ic applied ields, ex ac ed om i ing he echo ampli ude decay wi h a s e ched single ex- ponen ial unc ion a ixed s e ching pa ame e β. (b) T ans e se magne iza ion MH x,y(0)T om he echo ampli ude ex apola ed a ze o delay in he echo decays. Mul iplied by he empe a u e, i gi es a hin on he numbe o nuclei wi h a s ill measu able decohe ence. 112 Simone Chicco Chap e 6 inc easing he applied ield. A isualiza ion o he wipe ou e ec can be ob ained by plo ing he s a ing ans e se magne iza ion alues MH x,y(0) ex apola ed om he τ= 0 poin in he echo decay cu es, mul iplied by he measu ing empe a u e ( ig. 6.8). This quan i y MH x,y(0)Tco espond o he numbe o esona ing nuclei. In igu e 6.8, a he applied ield o 0.33 T, he numbe o esona ing nuclei is sa u a ed abo e 20 K, and dec eases ab up ly below 15 K, owa ds a low empe a u e ( ew Kel in deg ees) cons an minimum alue. 6.3 Elec onic spin Manipula ions To be e in es iga e he hype ine spli ing in he Cu gzcomponen , we milled se e al polyc ys als (5 % dilu ed in he diamagne ic analogue {Ti7Ga}[Zn(h ac)2]) in o ine powde s and measu ed he echo de ec ed ield sweep in he Q-band EPR spec ome e ( ig. 6.9). The signal o noise a io is ex emely high, hanks o a ine uning o he esonan ca i y, and consequen ly all he spec al ea u es a e well esol ed. Indeed, in his con igu a ion also he ou old Hype ine spli ing o he gzcomponen is isible and enables he obse a ion o he u he spli ing ( i e e iden peaks a e ma ked in ig. 6.9) induced by he Jcoupling. Taking ad an ages o his ine uning o he expe imen al se up, we pe o med a Rabi nu a ion expe imen in o de o demons a e he abili y o manipula e bo h he ing and Cu elec onic spins cohe en ly and independen ly om each o he . The sys em was a ge ed by a manipula ing pulse o a iable leng h θ( ) ollowed by a Hahn- e ocusing sequence π/2−τ−π. The Rabi oscilla ions o he gx,y componen s o he {C 7Ni} ing and [Cu(h ac)2] a e shown in igu e 6.10 o di e en powe a - enua ion. Figu e 6.9: Q-band EPR-EDFS a 3K on a 5% dilu ed ine powde o {C 7Ni}[Cu (h ac)2]. Red and black icks ma k he in plane gx,y and axial gz ing and coppe signal espec i ely. 113 Simone Chicco Chap e 6 Figu e 6.10: Rabi nu a ion expe imen s a 3 K on ine powde o {C 7Ni}- [Cu(h ac)2]. The ield was ixed o ma ch espec i ely he wo highe spec al com- ponen s (a) B ing = 13.597 kG and (b) BCu = 11.704 kG (as ske ched in inse ) co esponding o he gx,y plane. The pulse powe -a enua ion is a ied o highligh he Rabi equency eωRand damping λRdependencies. Fo each pulse-powe , he pulse leng h in he Rabi-sequence was modi ied in o de o main ain he e ec i eness o he π/2−π e ocusing. The ob ained Rabi oscilla ions esul s cohe en and monoch oma ic. Rabi equency eωRscales linea ly wi h he pulse powe a enua ion, oge he wi h he oscilla ion damping, which is hampe ed in highe powe con igu a ions by he pulse inhomo- genei ies. 6.4 Conclusions The elec onic and nuclea spins elaxa ion dynamics o a {C 7Ni}[Cu(h ac)2] com- plex has been ho oughly in es iga ed by exploi ing a combined pulsed EPR-NMR app oach. The da a analysis and he heo e ical simula ions on he elaxa ion dy- namics a e s ill ongoing. The heo e ical model desc ibing he sys em dynamics will be con i med by i ing he elaxa ion a es o bo h he {C 7Ni}and he Cu elec- onic spins and Cu nuclea one. In e p e ing hese elaxa ion measu emen s will gi e undamen al insigh s o he de e mina ions o he leading elaxa ion p ocesses, such as hei dependence on he empe a u e egime and he s eng h o spin-phonon coupling.[138,139] Pu ing oge he all his in o ma ion, we will be able o deduc he key ing edi- en s go e ning he sys em elaxa ion o his p omising sup amolecula assembly. Mo eo e , in iew o he ema kable esul s achie ed in he elec onic spin manip- ula ions h ough he Rabi nu a ions expe imen s on powde samples, s udies on single c ys als a e ongoing. By exploi ing a combined EPR-NMR app oach, we ex- pec o ealize cohe en manipula ions o he qubi -qudi sys em, demons a ing he possibili y o implemen o dina y single qubi quan um algo i hms and o ealize e icien ly he in o ma ion swapping be ween he elec onic spin p ocesso and i s nuclea memo y. [49] 114 Simone Chicco Gene al Conclusions In conclusion, in his hesis we ha e p esen ed he esul s o di e en expe imen- al cha ac e iza ions o molecula nanomagne s. The spin Hamil onian and he spin dynamics o hese sys ems ha e been in es iga ed by means o se e al di e en cu ing-edge expe imen al echniques. We ha e demons a ed ha b oadband NMR is an elec ion expe imen al ool o ou s udies, since i enabled a p ecise e alua ion o he pa ame e s o he elec o-nuclea spin Hamil onian o wo V-based molecu- la qubi s, oge he wi h hei nuclea phase memo y imes (sec . 3). P o en ha hese sys ems possess he ene gy le els s uc u e and he cohe ence p e equisi es o an ideal qudi , by means o NMR -pulses we ealized cohe en manipula ions o hese elec o-nuclea sys ems, inducing selec i ely he desi ed ansi ions be ween hei nuclea s a es. In he case o he [VO(TPP)] complex, we also succeeded in simula ing he encoding o a Quan um e o co ec ion algo i hm and we add essed expe imen ally he same ansi ions in ol ed in he encoding sequence, wi h eally p omising esul s, such as as manipula ions, negligible cohe ence losses and encod- ing eliabili y. By exploi ing his echnique and his, o simila complexes, we expec o succeed in ealizing he i s s p oo -o -p inciples expe imen s on he implemen a- ion o quan um e o co ec ion algo i hms in molecula qudi s. In pa icula , an in e es ing pe spec i e-wo k would be he cha ac e iza ion o complexes om he class o Lan hanide-based monome s, dime s o ime s wi h he NMR expe imen al se up desc ibed abo e. [47,44,45] Indeed, also hese sys ems ep esen ideal es - beds o he implemen a ion o quan um algo i hm on he expanded compu a ional space ha esul s om he coupling o di e en magne ic ions and o he elec onic and nuclea spins wi hin each ion. The phonon-induced elaxa ion p ocesses play a c ucial ole in he spin dynamics o molecula qudi s such as he [VO(TPP)] molecule. The phonon dispe sions o his complex we e ho oughly in es iga ed by exploi ing, o he i s ime in a molecula c ys al, inelas ic X- ay sca e ing (sec . 4). The echnique ea u es se e al ad an- ages wi h espec o he pa en inelas ic neu on sca e ing and he ou s anding quali y o he esul s achie ed will pa e he way o a sys ema ic usage o his echnique in pa en compounds cha ac e iza ions. In pa icula , in he [VO(TPP)] compound we obse ed ex emely low-ene gy op ical phonons and we alida ed he pe iodic DFT phonon calcula ions by compa ing he measu ed IXS c oss sec ion wi h he simula ed one. The alida ed phonon model was hen used o simula e he spin elaxa ion o [VO(TPP)], iden i ying he c ucial ole o he low ene gy modes, which esul o be c i ically coupled wi h he elec onic spins. The esul s achie ed in his s udy will be c ucial o he de elopmen o new classes o molecula magne s wi h op imized cohe ence by ailo ing o phonon modes. In his hesis we also ocused on he s udy o single molecule magne s wi h high en- e gy ba ie s. In pa icula , ou scien i ic e o was de o ed o he de e mina ion o 115 Chap e 6 he key ac o s go e ning he phonon-induced elaxa ion p ocesses. Focusing on Dy- based SMMs, we s udied by inelas ic neu on sca e ing he e ec s on he phonon densi y o s a es o chemical/s uc u al subs i u ions in he molecule o o an ex e - nal applied p essu e (sec . 5). By exploi ing INS, we accessed he phonon densi y o s a es and, in bo h sys ems, we demons a ed ha he main di e ences a e in he ene gy ange o phonons in ol ed in Raman elaxa ion p ocesses. These elaxa ion p ocesses a e indeed he c ucial ones o he de e mina ion o SMMs pe o mances. The compa ison o he expe imen al phonon densi y o s a es wi h he ones simu- la ed wi h pe iodic DFT was exploi ed o alida e he phonon models, o be used o simula ing he spin dynamics o hese complexes. This app oach ep esen a aluable es -bed o he phonon model alida ion, ha needs o be implemen ed sys ema ically in u u e wo ks. Finally, we p esen ed a combined EPR-NMR cha ac e iza ion o he spin dynamics o a sup amolecula complex in which a {C 7Ni} ing qubi is linked o a Cu-based nuclea qudi wi h long cohe ence imes (sec . 6). By exploi ing pulsed Q-band EPR we measu ed he elec onic spin-la ice and spin-cohe ence elaxa ion imes, and we ealized cohe en indi idual manipula ions o bo h he ing and he Cu elec onic spins. Mo eo e , wi h 1H-NMR we accessed he p o on nuclea spin-la ice and spin-spin a es. These esul s will be used o elabo a e a model o he elaxa ion dynamics o he sys em. To summa ize, in his hesis we exploi ed se e al ad anced expe imen al echniques which a e essen ial in he s udy o MNMs and enabled o pinpoin he key in- g edien s o he op imiza ion o hei pe o mances o applica ions in quan um echnologies. 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Ca e a, ”Cohe en manipula ion o molecula Qudi s by b oadband NMR”, Il Nuo o Cimen o C, 45 (6). 1-4 2022. Publica ions abou o be published, a e mino e ision  E. Ga la i, A. Albino, S. Chicco, V.H.A. Nguyen, F. San anni, L. Paolasini, C. Mazzoli, R. Caciu o, F. To i, P. San ini, A. Lunghi, R. Sessoli and S. Ca e a, ”The c i ical ole o ul a-low ene gy ib a ions in he elaxa ion dynamics o molecula qubi s”, Na u e Communica ion, 2023. 129 Acknowledgmen s To conclude his hesis, I would like o since ely hank all he scien is and collabo- a o s ha made i possible. Fi s and o emos , I would like o hank my supe iso P o . S e ano Ca e a, o gi ing me he oppo uni y o s a my esea ch ac i - i y in his an as ic g oup, and o his ex ao dina y kindness. He in oduced me, du ing my mas e deg ee, o he ascina ing ield o molecula magne ism and his en husiasm has ne e ceased o inspi e me. E e y discussion we had o ad ice I ecei ed was p iceless and I am ex emely g a e ul o e e y hing ha he augh me. I would conside ed mysel sa is ied i I succeeded in lea ning e en a en h o his boundless knowledge. I am hono ed and p oud o ha e had he oppo uni y o wo k wi h him and I also hope ha hese h ee yea s we e only he beginning o a long un. I will be o e e hank ul o D . Elena Ga la i, who co-supe ised my PhD, o he ex ao dina y e o she pu in helping me, wi h he kindness o a sis e and an ex ao dina y compe ence. Wi hou he suppo and supe ision I would ha e been los . I hope I ha e cap u ed he passion o esea ch and he expe ise. Many special hanks o P o . Giuseppe Allodi, who supe ised me du ing he whole expe - imen al NMR ac i i y. Du ing he ime we spen oge he in he labo a o y I ied my bes o lea n as much as possible om his immense expe imen al a i ude and compe ence. I would also like o hank D . Alessand o Chiesa o he collabo a ions we had du ing hese h ee yea s and o being so en husias ic in eaching me. I is also impo an o me o hank my colleague D . Emilio Macaluso o he p ecious ime spen discussing all he i ial ques ions I was no able o answe mysel and I was ashamed o ask o my supe iso s. All he people in he g oup ha e been, and always will be, an example o me, and I am immensely g a e ul o each o hem. I would like o hank ou collabo a o s a he Uni e si y o Manches e P o . Richa d Winpenny and P o . E ic McInnes o gi ing me he oppo uni y o spend h ee mon hs in hei an as ic labo a o ies, welcoming me in hei esea ch g oup. I am e y g a e ul o D . Selena F.J. Lockye o he p ecious suppo du ing my s ay in Manches e , i has been a g ea lea ning and wo king oppo uni y. I hope o collabo a e again in he nea u u e. I would also like o hank all he ins umen scien is s o he X- ays, neu ons and EPR acili ies whose help and assis ance was p iceless: D . Luigi Paolasini, D . Claudio Mazzoli, D . Cai Yong, P o . Ta iana Guidi, D . And ea Pio ano, D . Alexand e I ano , D . Monica Jimenez Ruiz, M . Adam B ook ield. Thanks o all o hem I am e y p oud o he expe imen al esul s achie ed in his hesis. Las bu no he leas , I would like o wa mly hank my high school Physics eache Gianni Melega i, who plan ed he seed many yea s ago and ook ca e o i in he ea ly s ages. Finally I would like o hank all he collabo a o s ci ed a he beginning o his hesis. Simone Chicco, Janua y 2023 130 FUNDING PROJECTS The wo k p esen ed in his hesis has ecei ed unding om he Eu opean Union’s Ho izon 2020 Resea ch and Inno a ion P og amme, FET-OPEN p ojec FATMOLS (”FAul Tole an MOLecula Spin p ocesso ”) unde g an ag eemen No. 862893, including a h ee mon hs s ay a he Uni e si y o Manches e o he esea ch p ojec : “Elec on Spin Resonance measu emen s on molecula complexes o Quan- um In o ma ion P ocessing”. O he p ojec s in ol ed in his wo k we e he Eu o- pean P ojec “Scaling Up quan um compu a ion wi h MOlecula spins” (SUMO) o he call Quan ERA, and he PRIN P ojec 2017CR5WCH Q-chiSS “Quan um de ec ion o chi al induced spin selec i i y a he molecula le el” o he I alian Minis y o Uni e si y and Resea ch (MUR).