Effect of oxygen vacancies on the structural and transport properties of SrTiO3 thin films: experiments and ab-initio calculations

TESIS DE DOCTORADO Effect of oxygen vacancies on the structural and transport properties of SrTiO3 thin films: experiments and ab-initio calculations Lucía Iglesias Bernardo ESCUELA DE DOCTORADO INTERNACIONAL PROGRAMA DE DOCTORADO EN CIENCIA DE MATERIALES SANTIAGO DE COMPOSTELA 2019 DECLARACIÓN DEL AUTOR DE LA TESIS Effect of oxygen vacancies on the structural and transport properties of SrTiO3 thin films: experiments and ab-initio calculations Dña. Lucía Iglesias Bernardo Presento mi tesis, siguiendo el procedimiento adecuado al Reglamento, y declaro que: 1) La tesis abarca los resultados de la elaboración de mi trabajo. 2) En su caso, en la tesis se hace referencia a las colaboraciones que tuvo este trabajo. 3) La tesis es la versión definitiva presentada para su defensa y coincide con la versión enviada en formato electrónico. 4) Confirmo que la tesis no incurre en ningún tipo de plagio de otros autores ni de trabajos presentados por mí para la obtención de otros títulos. En Santiago de Compostela, 10 de julio de 2019 Fdo AUTORIZACIÓN DEL DIRECTOR / TUTOR DE LA TESIS Effect of oxygen vacancies on the structural and transport properties of SrTiO3 thin films: experiments and ab-initio calculations D. José Francisco Rivadulla Fernández D. Víctor Pardo Castro INFORMA/N: Que la presente tesis, corresponde con el trabajo realizado por Dña. Lucía Iglesias, bajo mi dirección, y a utorizo su presentación , considerando que reúne l os r equisitos exigidos en el R eglamento de Estudios de Doctorado de la USC, y que como director de ésta no incurre en las causas de abstención establecidas en Ley 40/2015. En Santiago de Compostela, 10 de julio de 2019 Fdo: José Francisco Rivadulla Fernández Fdo: Víctor Pardo Castro Acknowledgements Una parte del viaje es el final y esta tesis significa el final de un viaje que emprend´ ı hace poco m´ as de 4 a˜ nos. Desde luego el viaje no ha sido f´ acil, plagado de retos, ilusiones y aspiraciones, y por qu´ e no decirlo, de algunos fracasos, desilusiones y mucho trabajo duro. Ha sido un viaje en el que he re´ ıdo y llorado, pero sobre todo en el que he aprendido y madurado. En estas pocas l´ ıneas pretendo expresar mi agradecimiento a todas las personas que de una forma u otra me han ayudado a finalizar este viaje. Indudablemente el primero de la lista es uno de mis directores de tesis, Francisco Rivadulla. No tengo palabras suficientes para expresar todo lo que agradezco su gu´ ıa, entusiasmo y perserverancia, sin los cuales estoy segura de que habr´ ıa sido muy dif´ ıcil completar esta tesis. En ´ el he encontrado un amigo m´ as que un jefe, que me ha apoyado y ayudado cuando los ´ animos flaqueaban. Espero haber estado a la altura y haber cumplido al menos algunas de sus expectativas. Quiero agradecer tambi´ en especialmente a mi otro director de tesis, V´ ıctor Pardo. Su gu´ ıa y paciencia para ense˜ nar a alguien marcadamente experimental a introducirse en el mundo te´ orico es para enmarcar. Agradezco tambi´ en enormemente el apoyo de mis compa˜ neros de laboratorio durante estos a˜ nos, empezando por Eric, quien ha sido uno de los mejores apoyos en toda esta etapa, transmiti´ endome siempre su tenacidad y pasi´ on por la ciencia. Jose, por ense˜ narme a decir ”pode ser” y ”depende”. Carolina, por haber escuchado mis quejas en la parte final de esta tesis y Daniela, con la que sin duda he compartido grandes momentos. Y a V´ ıctor, que aunque con los colores tiene un gusto du7 Luc´ ıa Iglesias Bernardo doso, siempre se ha ofrecido para echarme una mano con todos los problemitas inform´ aticos que he tenido. Tambi´ en quiero agradecer a ´ Alex, El´ ıas, David, Carlos, Bea y Tinh. Quiero hacer tambi´ en especial menci´ on a Irene y Araceli, con las que he tenido el placer de colaborar. Ha sido un privilegio compartir estos a˜ nos con todos ellos, sin duda han sido mi segunda familia durante este tiempo y sin su apoyo habr´ ıa sido un viaje mucho m´ as dif´ ıcil de completar. De aqu´ ı me llevo un pu˜ nado de amigos, que espero conservar por mucho tiempo. I would also like to thank Silvia Picozzi, who was responsible for my stay in her group in Chieti, Italy. I have been fortunate to share three months with her, and she is undoubtedly one of those people who marks you due to her enthusiasm, patience and closeness. I also want to extend my thanks to my office mates, Hrishit and Huimin, who greatly helped me during my days in Chieti. E anche ai miei coinquilini (e amici) durante il soggiorno, Federica, Angelina e Ana, mi sono divertito molto con loro. A mi padre, sin la humildad y la perserverancia que me inculc´ o desde peque˜ na estoy segura que no podr´ ıa haber llegado hasta aqu´ ı. All´ a donde est´ es espero que est´ es orgulloso de m´ ı... A mi madre y a mi hermano, gracias por estar ah´ ı para apoyarme siempre, sin los valores que me habeis transmitido no podr´ ıa haber llegado a ser quien soy. Tambi´ en quer´ ıa agradecer a mis amigas, Laura, Sheila y Gala, por compartir momentos y risas desde siempre, sois de lo mejor. Tambi´ en a Nuria que ha compartido esta aventura conmigo en Santiago, y a Fernando, ´ el tambi´ en ha compartido una parte muy importante de este viaje y ha sabido escucharme y aconsejarme siempre acertadamente. Muchas gracias a todos, pero sobre todo al que no est´ a... Quiero dedicarle esta tesis a mi padre. 8 Contents Acknowledgements 7 Aim of the Thesis 13 1 Introduction 17 1.1 LaAlO3/SrTiO3interface ................ 18 1.1.1 The polar catastrophe scenario . . . . . . . . . 19 1.1.2 The oxygen vacancies scenario . . . . . . . . . 23 1.1.3 Cation intermixing at the LAO/STO interface . 26 1.2 Strain and defect engineering of transition metal oxides 28 1.3 Structure and properties of SrTiO3........... 33 2 Experimental techniques 39 2.1 Pulsed Laser Deposition . . . . . . . . . . . . . . . . 39 2.2 X-Ray Diffraction and Reflectivity . . . . . . . . . . . 42 2.3 Atomic Force Microscopy . . . . . . . . . . . . . . . 48 2.3.1 Electrostatic Force Microscopy and Kelvin Probe Microscopy................... 51 2.4 Electrical transport measurements . . . . . . . . . . . 54 2.4.1 Electrical resistivity . . . . . . . . . . . . . . 55 2.4.2 Hall effect measurements . . . . . . . . . . . . 56 2.4.3 Seebeck effect measurements . . . . . . . . . 58 3 Computational Modelling 61 3.1 Density Functional Theory (DFT) . . . . . . . . . . . 62 9 1. Introduction Life need not be easy, provided only that it is not empty. Lise Meitner, Austrian-Swedish physicist who worked on radioactivity and Nuclear Physics. Herbert Kroemer started his Nobel lecture in 2000 claiming that, ”Often, it may be said that the interface is the device” [1]. With this precept, he was referring to the surprising success of devices based on thin semiconductor films for photonic and electronic applications, which started more than 40 years ago. Many of them, such as transistors, solar cells or lasers take advantage of the interfacial phenomena. Thus, the introduction of interfaces into semiconductor structures gave rise to numerous devices of immense utility and interesting physical properties. Analogously, the introduction of active interfaces into oxide structures is expected to generate a new technological revolution in the coming years; the interfaces between complex oxides are currently in the spotlight of intense research in Condensed Matter Physics. A wide range of phenomena is exhibited by these interfaces such as magnetism, superconductivity, ionic conduction or ferroelectricity, which can find direct application for instance, in batteries, fuel cells, diverse information storage technologies, etc [2]. 17 Luc´ ıa Iglesias Bernardo In the past decades, the development of powerful deposition techniques such as pulsed laser deposition, sputtering or molecular-beam epitaxy have allowed the growth of well-defined interfaces of these complex oxides. This atomic-scale engineering opens a world of possibilities to explore and make use of the fundamental properties of such interfaces, with new and more fascinating properties awaiting to be discovered and studied. 1.1 LaAlO3/SrTiO3interface The electrical conductivity at the (001) interface between lanthanum aluminate (LaAlO3, LAO) and strontium titanate (SrTiO3, STO) was first discovered by Ohtomo and Hwang in 2004 [3]. Despite both materials being electrical insulators with wide band gaps (5.6 and 3.2 eV, respectively), a two-dimensional electron gas (2DEG) of a few unit cells was found at their interface. In addition, the 2DEG exhibits a high carrier mobility at low temperatures, exceeding 10.000 cm2/Vs [3]. This unexpected finding received a lot of attention, stimulating an intense research to understand the fundamental mechanism underlying and led to the emergence of a completely new field called oxide interfaces [4,5]. This field explores the possibility of oxide-based electronics as an alternative to conventional semiconductor technology based on silicon [6]. As a consequence of this extensive research, new exciting interfacial properties were revealed at the LAO/STO (001) interface, such as magnetoresistance [7], superconductivity [8] and ferromagnetism [9], most of them absent in their bulk constituents. Nevertheless, very specific conditions are required to make the 2DEG emerge at the interface between these two insulators. For instance, the LAO/STO interface presents a critical thickness of the LAO layer, above which an abrupt transition from insulator to metallic-like character is produced [10]. It was also found that such 2DEG is very sensitive to STO surface termination, only being observed when LAO is grown on TiO2-terminated (n-type interface) but not for SrO-terminated (p-type interface) STO 18 1. Introduction substrates, suggesting that the details of the interface do matter [3]. Also the growth conditions of the LAO layer are crucial; the conduction appears only when the thin film is grown at low oxygen pressures (P≤10−2mbar) [11–13]. The properties of this interface are well characterized and studied, but the exact mechanism to explain the phenomena is still debated and researchers continue to investigate this system. Three main scenarios have been proposed for describing the metallic conduction at the LAO /STO interface: i) An electronic reconstruction to cancel the electric field built up at a polar TiO2-LaO interface (the so-called polar catastrophe scenario); ii) the presence of charged oxygen vacancies at either STO or/and LAO; and iii) an ionic reconstruction in the form of cation intermixing. Only the first scenario is related with intrinsic properties of the system, whereas the other two are linked with extrinsic effects (vacancies). The consequences of each of these three hypotheses are explained in more detail in the Sections below. 1.1.1 The polar catastrophe scenario Two years after the discovery of the 2DEG at the LAO/STO interface, Nakagawa et al. [14] proposed the polar catastrophe model to explain this behaviour. This model establishes that an electronic reconstruction occurs in response to the polar discontinuity formed at the atomically abrupt interface between the polar (La3+O2−)+1/(Al3+O2− 2)−1ionic layers of (001) LaAlO3and the neutral (Sr2+O2−)0/(Ti4+O2− 2)0ionic layers of (001) SrTiO3. This results in a valence discontinuity across the interface as shown in Fig. 1.1 a), and builds up an electric potential (V), which diverges as the LAO thickness increases. Consequently, an spontaneous electronic reconstruction is produced in response to this instability, transferring half an electron per two-dimensional unit cell from the LaO+1 layer to the TiO0 2sublayer [15]. The overall structure remains neutral, with the Ti3.5+ ion at the interface layer and the potential no longer diverges (see Fig. 1.1 b). As a result of free electrons occupying Ti 3dxy orbitals in the STO layer, the interface becomes conducting and a high mobility 19 Luc´ ıa Iglesias Bernardo 2DEG is established at the unit cells closer to the interface. (-1) Al3+O24- … (+1) La3+O2- … (0) Ti4+O24- … (0) Sr2+O2- … (0) Sr2+O2- … (0) Ti4+O24- … (+1) La3+O2- … (-1) Al+3O24- … r (-1) Al3+O24- … (+1) La3+O2- … (0) Ti4+O24- … (0) Sr2+O2- … (0) Sr2+O2- … (-1/2) Ti3.5+O24- … (+1) La3+O2- … (-1) Al+3O24- … r V e-/2 e-/2 e-/2 e-/2 e-/2 a) b) E E V 2DEG Figure 1.1: Schematic illustration of the polar catastrophe scenario in an AlO2/LaO/TiO2interface. a) Before the electronic reconstruction. The polar layers of LAO grown on non-polar STO substrate produce a negative electric field (E), leading to an electric potential (V), which diverges with sufficient LAO thickness. b) After the reconstruction. The voltage divergence is avoided through the transfer of half an electron per unit cell to the TiO2layer. Thus, Eoscillates around 0 and the electric potential remains finite. ρrepresents the net charges at each layer. Grey, blue, orange, green and red spheres refer to the Al3+, La3+, Ti3.5+/4+, Sr2+ and O2−atoms, respectively. The excess of charge incorporated to the interface corresponds to an expected carrier density around 3.5×1014 cm−2, which is one order of magnitude above the experimental values reported by various groups (∼1013 cm−2) [16, 17]. Several explanations have been proposed for 20 1. Introduction this “missing charge” problem, among which the localization of part of the conduction electrons could be the most plausible. In this sense, DFT calculations by Popovic et al. [16] revealed that some electrons are confined to a single interfacial layer and therefore, they are susceptible to Anderson localization due to disorder, whereas other electrons with small masses and extended over several layers are expected to contribute to electronic transport. Such disorder could result from structural distortions at the interface due to the strain that arises from the lattice mismatch between LAO and STO (≈+2.9%) [18]. In 2006, Thiel et al. [17] reported that a minimum thickness of 4 unit cells for the LAO layer is required for the formation of the 2DEG, and also first-principles calculations based on the polar catastrophe scenario confirms this critical thickness [19]. In fact, the existence of a critical LAO thickness as a pre-requisite for interfacial conductivity to emerge is the strongest argument in favour of the electronic reconstruction picture. However, the observation of the 2DEG in amorphous layers [20] as well as in other polar interfaces [21] cannot be explained correctly by this model. Another important problem of the model is the extension of the 2DEG. Ab-initio calculations by Janicka et al. [22] predicted that the 2DEG is confined in STO within ≈1 nm away from the interface. However, Basletic et al. [23] mapped the spatial distribution of charge carriers in LAO/STO heterostructure using a conductive tip of atomic force microscopy. They found that the 2DEG is extended 500 µm to the STO side in samples grown by Pulsed Laser Deposition (PLD) at 10−5mbar (Fig. 1.2 a), meanwhile the conducting depth decreases to 7 nm in interfaces annealed at 300 mbar of oxygen (Fig. 1.2 b). Herranz et al. [24] also deduced from the analysis of ShubnikovHaas oscillations a thickness of ≈500 µm in LAO/STO samples grown by PLD at an oxygen pressure of 10−6mbar. Meanwhile, Reyren et al. [8] estimated the conducting thickness around 10 nm at low temperatures on samples deposited at an oxygen pressure of ≈10−5mbar and post-annealed in 400 mbar. All these results point towards the importance of the growth and annealing conditions in LAO/STO interfaces, which suggest an important role of oxygen vacancies. 21 Luc´ ıa Iglesias Bernardo a) b) c) SrTiO3 LaAlO3 SrTiO3 LaAlO3 Figure 1.2: a) Map of electrical resistance across the LAO/STO interface for a sample deposited at an oxygen pressure of 1×10−5mbar, and b) sample grown at 1×10−6mbar with a post-annealed procedure at 300 mbar. Reprinted by permission from [23]. c) Temperature dependence of the sheet resistance for 6 samples grown under different oxygen atmosphere and no-annealed. Reprinted by permission from [7]. The LAO/STO interfaces in which the 2DEG is reported so far, have been grown on STO substrates under reducing conditions [8, 25, 26]. In this regard, Brinkman et al. [7] revealed a pronounced increase of the interface conductivity as the oxygen pressure of the growth of the LAO thin film decreases, as can be seen in Fig. 1.2 c). On the other hand, G. Herranz et al. [27] demonstrated the formation of a 2DEG at interfaces involving (110) (non-polar) and (111) (polar) surfaces of STO interfaced with epitaxial layers of LAO and above the critical thickness. In the same paper, they also reported the conductivity at the interface between (110) STO substrate and amorphous LAO, STO and yttria-stabilized zirconia (YSZ), showing that epitaxial interfaces are not a pre-requisite for the formation of a 2DEG. This 22 1. Introduction along with the lack of polar discontinuity at the (110) interface, calls into question the polar catastrophe model and claims for an alternative explanation of the origin of this phenomenon. Therefore, although some experimental and theoretical studies support the idea of the polar reconstruction scenario, other important issues are still unresolved. 1.1.2 The oxygen vacancies scenario Soon after the polar catastrophe mechanism was proposed, several groups realized that the electronic properties of the interface are significantly affected by the presence of oxygen vacancies. Many experimental evidences suggest that the spontaneous formation of oxygen vacancies in STO during the typical conditions used to grow the LAO thin film play a fundamental role in the appearance and properties of the 2DEG. Even though the current deposition methods provide an excellent control in the fabrication of the thin films and interfaces, there is always the possibility that defects, such as anionic or cationic vacancies, will be added to the crystal structure. Their role can be specially important in depositions that involve perovskites, in which the cation valence can change depending on the oxygen content of the sample. In general, the growth of LAO layers is carried out at high temperatures and in a reducing oxygen pressure (i.e. PO2=10−4– 10−2mbar), conditions that are very suitable for creating oxygen vacancies in STO. For each ionized oxygen vacancy, two free electrons can be incorporated to the 3dconduction band of the STO, modifying the Ti4+/Ti3+ valence, as is illustrated in Fig. 1.3 a). Since STO undergoes an insulatorto-metal transition at remarkably low charge carrier densities (1×1016 cm−3), a slight deviation from stoichiometry of just one oxygen vacancy every 150 unit cells, is enough to make the oxide an electrical conductor. Therefore, a contribution from oxygen vacancies to the total carrier density cannot be excluded a priori. 23 Luc´ ıa Iglesias Bernardo EF E STO LAO 2DEG CB VB In-gap states E O: 2p Ti: 3d Ti4+/Ti3+ a) b) 2eEF CB VB eFigure 1.3: a) Schematic illustration of the ionization process of an oxygen vacancy. Every oxygen vacancy yields two electrons in the Ti-3dband of STO, which contributes to the conduction of the material. b) The creation of vacancies in STO produces the appearance of in-gap states close to the conduction band of STO, favouring the conduction at the interface. In addition, post–annealing processes at high oxygen pressure could result insufficient to recover the oxygen stoichiometry. Numerous studies have examined the conditions of growth and post-annealing in an effort to understand the role of oxygen vacancies. Chen et al. [20] reported that a post-annealing procedure removed the oxygen vacancies in the case of amorphous LAO overlayer, but some residual conductivity is preserved in the interfaces with crystalline LAO overlayer [11]. This reveals that oxygen vacancies are the dominant source of mobile carriers when LAO film is amorphous, but it seems that both oxygen vacancies and electronic reconstruction could contribute to the interface electron gas at the unannealed crystalline LAO/STO structure. Dittman et al. [13] also discussed about the role of the oxygen pressure used in the post-annealing process over the electronic properties of LAO/STO interface. They concluded that equilibrium defect formation is the dominant process for establishing the 2DEG properties, meanwhile growth dynamics plays a minor role in the typical growth regime of LAO/STO interfaces. Cavillieri et al. [28] observed a conductive interface for 2-3 unit cells of LAO when the deposition is performed at low oxygen pressures. However, a post–annealing procedure at high 24 1. Introduction oxygen pressure completely suppressed the conductivity, confirming the oxygen vacancies as responsible for the metallic conduction. Many theoretical studies also addressed this issue of oxygen vacancies at the LAO / STO interface, concluding that the presence of vacancies is essential for the formation of a 2DEG [29,30]. All these evidences raised doubts about whether charge transfer across the interface is really responsible of the conducting interface and supports the possibility that the 2DEG resides solely in STO due to the presence of vacancies. In addition, the 2DEG observed at LAO/STO interface shows much similarity to those reported in other STO-based heterostructures (e.g. LaTiO3/STO [31], GdTiO3/STO [32] or LaGaO3/STO [33]) and field–effect–transistor structures involving STO [34,35], which suggests that different forms of electronic confinement at the surface of STO would lead to essentially the same 2DEG. In this context, Santander-Syro et al. [36] reported Angle-Resolved Photoemission Spectroscopy (ARPES) of the surface of STO crystals cleaved under high vacuum. They observed the emergence of a 2DEG at the bare surface of STO, suggesting the critical role of oxygen vacancies on this phenomenology. In this sense, there are also studies that suggest that vacancies formed on the surface of the LAO could influence in the properties of LAO/STO interfaces. The electronic reconstruction scenario proposed by Nakawaka et al. [14] for n-type LAO/STO interface, demands the ionization of some of the oxygen atoms at the upper layer of AlO2. DFT calculations performed by Li et al. [29] revealed that these oxygen vacancies may reduce the electrostatic energy in LAO, being energetically the most stable. In addition, the charge carriers release by the oxygen vacancies on the LAO surface would be transferred to the interface, creating in-gap states located just below the LAO conduction band, as is depicted in Fig. 1.3 b). The charge carriers donated by these in-gap states would be confined to the interface, forming a 2DEG. Therefore, in view of the results of extensive research about the role of oxygen vacancies in either the SrTiO3substrate or the LaAlO3 surface, it seems unavoidable to consider its effect for a proper understanding of the properties of the 2DEG at the (001) LAO/STO interface. 25 Luc´ ıa Iglesias Bernardo through spin–orbit coupling [56,57]. In this regard, either the selection of the substrate or the presence of defects can lead to a determined cooperative rotation of the BO6octahedra. The substrate, apart from imposing stress on the thin film due to the connectivity of the BO6between film and substrate, can transfer its own octahedral configuration across the heterointerface, imprinting it into the film [58]. a) a+b0c0 b) a-b0c0 a c b Figure 1.7: Schematic illustration of the in-phase (a) and out-of-phase (b) cooperative TiO6rotations along the a-axis in a perovskite structure. Glazer introduced a notation to describe all the possible octahedral tilting distortions in perovskites [59]. This notation specifies the BO6 tilts or rotations along each of the three pseudocubic axes (a(100), b (010) or c(001)). The letters also show whether the magnitude of the tilt is equal or different along the spatial directions. For example, a+a+a+ indicates the same magnitude of rotation along the three axes whereas a+b+c+denotes different angle of rotation about each axis. Moreover, the superscript +indicates that the rotations of neighbouring octahedra along the given axis occur in-phase (Fig. 1.7 a), whereas the out-ofphase rotation is denoted as −(Fig. 1.7 b). The absence of tilting is expressed as 0. Using this notation, 23 different tilt systems describe all 32 1. Introduction the possible octahedral configurations in perovskites. Understanding the effect of epitaxial growth of oxoperovskites inevitably implies the correct determination of the octahedral rotations within the thin films. This is especially important, since it could strongly affect the formation of ionic defects and their diffusion through the lattice, as well as the electronic transport properties of the films. For these reasons, this aspect will be studied in detail in this Thesis. 1.3 Structure and properties of SrTiO3 Strontium titanate (SrTiO3) is a quantum paraelectric insulator [60], whose great chemical stability, diamagnetism and high dielectric constant makes it one of the preferred templates for the growth of other oxoperovskite thin films, and it is the basic ingredient of the aforementioned (001) LAO/STO interface. At room temperature, SrTiO3presents a cubic perovskite structure (space group Pm3m). Below 105 K, it undergoes a structural transition to a tetragonal structure (space group I4/mcm), which presents an outof-phase rotation of the TiO6octahedra along the (001) axis (a0a0c−). The undistorted STO structure is sketched in Fig. 1.8. The Ti4+ ions are sixfold coordinated by oxygen O2−, whereas each Sr2+ is surrounded by four TiO6octahedra (ninefold coordination to O2−ions). Within the TiO6octahedra, the hybridization of the O-2pstates with the Ti-3dstates leads to a pronounced covalent bonding, meanwhile Sr2+ and O2−ions present ionic bonding character. It is precisely this mixed covalent-ionic bonding properties that makes it a model electronic material. STO is a good example of the richness in properties of the oxoperovskites [61–64]. These can be further enhanced through doping, presence of vacancies or the change in bond distances and angles produced by epitaxial strain. Thus, n-doping through replacement of Ti4+ by Nb5+ or La3+ by Sr2+, transforms STO in a conducting system and gives rise to practical applications such as gas sensors [65], electrodes in solar cells [66] or dielectric in capacitors [67,68]. 33 Luc´ ıa Iglesias Bernardo Figure 1.8: Schematic representation of the cubic undistorted SrTiO3structure (space group Pm3m). Each Ti atom (orange spheres) is bonded to six O atoms (red spheres), forming an corner-sharing octahedral structure. Each Sr atom (green sphere) is surrounded by four octahedra. Nevertheless, oxygen vacancies (V ¨ O) are probably the most likely to add new functionalities to STO. They are easily formed by processing STO in an oxygen-reducing atmosphere without any further cation substitution. Moreover, due to their donor character and the very large electron mobilities characteristic of δ-doped STO, even the slightest concentration of vacancies produces a measurable electrical conductivity. Actually, reduced STO was the first oxide reported to show superconductivity, with a maximum Tc≈0.3 K for electron densities of n≈1020cm−3[69]. However, incorporation of oxygen defects may be detrimental for some applications of this oxide; for instance, those with high-k requirements of dielectric insulating layers for carriers density modulation can be compromised. Among the properties induced by oxygen vacancies, resistive switching is one of the most prominent [70–74]. The high mobility of oxygen vacancies in comparison with cation vacancies in STO can lead to an important oxide-ion conductivity at a not too high temperature. In addition, each V ¨ Ois positively charged, which makes them sensitive to 34 1. Introduction being moved within the material by an electrical field. Thus, the movement of vacancies along filaments based on dislocations has already been reported [75] in single crystals as well as in thin films [76]. On the other hand, each vacancy donates its electrons to a band that is anti-bonding in character with respect to the Ti-O bond. This effect produces a chemical expansion around it, and therefore the distribution of vacancies is sensitive to local strain fields. In this regard, Sharma et al. reported the movement of oxygen vacancies by means of the mechanical force exerted by an AFM tip in LAO/STO interfaces, taking advantage of the Vegard’s effect [77]. Another property that attracts extensive interest for its potential applications is ferroelectricity. As an incipient ferroelectric, STO can undergo a ferroelectric transition by diverse methods such as doping [78, 79], electric field [80] or isotope substitution [81]. But even more important, Haeni et al. [82] first demonstrated room-temperature ferroelectricity in STO by means of the application of strain. They found that STO thin films deposited under tensile strain present in-plane ferroelectric polarization at temperatures close to room temperature, whereas compressive strain only generates ferroelectricity at very low temperatures, and of a much smaller magnitude. The different ferroelectric phases stabilized on STO thin films by epitaxial strain are shown in the phase diagram of Fig. 1.9. Later, Jang et al. [83] examined the ferroelectricity in strain-free STO films and bulk crystals, determining that they are relaxor ferroelectrics, and the role of strain is to stabilize long-range correlation of pre-existing nanopolar regions within STO. These authors also point to the minimal presence of unintentional Sr deficiency in the samples as the origin of the nanopolar regions, emphasizing the delicate sensitivity of the STO to deviations in its stoichiometry. 35 Luc´ ıa Iglesias Bernardo Figure 1.9: Phase diagram of ferroelectricity in STO thin films as a function of temperature and in-plane epitaxial stress. The arrows indicate the direction of the polarization for strained STO. Reprinted by permission from [82]. Thermoelectricity is another important property that was recently reported at doped SrTiO3thin films and interfaces. Although the thermoelectric conversion performance of STO still remains low because of its high thermal conductivity, the large Seebeck coefficient and electrical conductivity reported in lightly doped thin films makes it a firm candidate to be used in thermoelectric devices. Several strategies were explored to improve its thermoelectric properties, such as strain [84] or cationic doping [85]. The most promising approach involved the 2DEG formed at the STO surface and the possibility to be modulated by means of diverse techniques, improving considerably the thermoelectric efficiency of the material. In this regard, Ohta et al. [86] proposed the fabrication of STO superlattices, in which e-doped SrTi0.8Nb0.2O3layers are inserted between insulating STO thin films, as is illustrated in Fig. 1.10 a). Thus, a 2DEG with high carrier density (≈1021cm−3) is formed at the TiO2/STO heterointerface. They reported an unusually large Seebeck coefficient (S), when the thickness of the conductive layers was less than 4 unit cells. The maximum value (≈490 µVK−1) is reached for 1 36 1. Introduction unit cell of the SrTi0.8Nb0.2O3, which is approximately 4.4 times larger than the Seebeck coefficient reported in bulk for similar carrier densities. a) b) c) Figure 1.10: a) Seebeck coefficient at room temperature as a function of the thickness of the SrTi0.8Nb0.2O3layer for superlattices composed by STO/SrTi0.8Nb0.2O3/STO. A sudden increase is produced when the thickness is smaller than 1.56 nm. Reprinted by permission from [86]. b) Sketch of the CAN-gated STO-based field effect transistor. c) Seebeck coefficient values of the 2DEG as a function of the charge carrier concentration (nsheet), which is modulated by the gate voltage. Reprinted by permission from [87]. Years later, they also achieved a large enhancement of the Seebeck coefficient modulating the 2DEG formed in a field effect transistor device fabricated on an STO substrate and capped by nanoporous calcium aluminate (CAN), which served as a gate insulator (see Fig. 1.10 b). An electric field application provided an extremely thin 2DEG (≈2nm), which exhibits a large S≈950 µVK−1for carrier densities of ≈1015 cm−2(Fig. 1.10 c). 37 Luc´ ıa Iglesias Bernardo These results open the way to obtain efficient and environmentallyfriendly high performance thermoelectric materials, which take advantage of the 2DEG formed at an oxide interface. In the following pages, we will present the main results obtained during the development of this PhD Thesis to contribute to the current knowledge of the properties of STO. In particular, we will focus on the influence of strain as well as oxygen vacancies in some of the interesting functional properties exhibited by this material in the form of thin films. At the same time, we will aim to contribute to the discussion about the role of oxygen vacancies in all the phenomenology observed in the famous LAO/STO interface. Thus, we will discuss the effect of epitaxial strain on the energy of formation of oxygen vacancies, as well as on its distribution and mobility at room temperature. Then, we will present a detailed discussion about the controlled movement of oxygen vacancies by means of large local electric fields induced by an AFM tip. Finally, the effect of vacancies on the ferroelectric and magnetotransport properties of STO thin films will be discussed in detail, from an experimental and theoretical point of view, through ab-initio calculations. 38 2. Experimental techniques Nothing in life is to be feared, it is only to be understood. Now is the time to understand more, so that we may fear less. Marie Curie was the first woman to win a Nobel Prize and the first person to win the award twice. Nobel Prize in Physics in 1903 and Nobel Prize in Chemistry in 1911. This Chapter presents the deposition technique used to grow the Nb:STO thin films as well as the characterization techniques used to verify the structural quality of the films. Finally, we also provide a description of the methods used to measure the Seebeck effect and the electric transport properties of the samples. 2.1 Pulsed Laser Deposition Since the late 1980s, Pulsed Laser Deposition is an extensively used technique for the growth of complex oxide thin films. The technique’s principle is very simple as is illustrated schematically in Fig. 2.1 b). A high-energy pulsed laser beam is focused on top of a stoichiometric solid target of the material to be deposited, creating an energetic plasma plume which is ejected from the target towards the substrate. The substrate is typically placed over the target and maintained at high 39 Luc´ ıa Iglesias Bernardo temperature (600-1000◦C). Once the ejected material reaches the substrate, the atoms rearrange to adapt to the crystalline structure of the substrate underneath. The entire process is carried out in the presence of a background gas, normally oxygen, nitrogen or argon. Due to the plasma plume is highly focused towards the substrate, PLD is not well suited for large-scale film growth and one of its principal drawbacks is the uneven coverage over large substrates [38]. Heater Substrate Laser beam Plasma plume Target a) b) Figure 2.1: Pulsed Laser Deposition system. a) Picture of our PLD chamber in the precise moment when the laser beam hits the target, producing a high energetic plasma plume. b) Sketch describing the different elements of the PLD working process. The growth of oxide thin films is normally performed under oxygen atmosphere at controlled pressure for two purposes: first, oxygen acts as a scattering center in the route of the atomic species ejected towards the substrate, and second, it provides the oxygen source necessary to achieve the film stoichiometry. Even so, oxygen content can also be modified after the deposition process inside the vacuum cham40 2. Experimental techniques ber itself without extracting the sample, by using oxygen pressure and temperature different from the deposition conditions. Another option is to change the oxygen stoichiometry ex-situ by a subsequent postannealing treatment at high temperature and a given oxygen pressure (this is the approach followed in this Thesis; more details are provided in Chapter 4). One of the most important advantages of this technique, which makes it attractive for the growth of complex oxides, is the possibility of a stoichiometric transfer from the multi-cation targets to the substrate, and the capability to control the stoichiometry of the films modifying the oxygen pressure of the vacuum chamber. For this purpose, an accurate optimization of the deposition conditions of both, laser (pulse time, wavelength, repetition rate and laser fluence) and growth parameters (target-substrate distance, substrate temperature and background oxygen pressure) is required. Among these, laser energy and oxygen pressure are probably the key parameters to control the degree of scattering inside the PLD chamber and the ablation of the materials with different atomic weight. As we assume stoichiometric transfer and negligible evaporation from the film surface, only the cation stoichiometry (not the composition of the target) should be identical to that of the films. For many materials, this implies a very narrow window of parameters in which the coherent growth and correct stoichiometry of thin films are achieved. For the growth of high quality epitaxial thin films with sharp interfaces, an atomically smooth and chemically homogeneous surface of the substrates are also crucial. If the termination layer is not uniquely defined, two types of interfaces can be formed, degrading the interfacial properties due to chemical and electronic inhomogeneities on a unit-cell scale. Typically, the as-received commercial substrates have a chemically mixed surface, so that a previous chemical treatment is required to achieve atomically flat and chemically single-terminated surfaces. Although the procedure is different depending on the substrate, it is usually composed by a chemical etching and/or a thermal treatment [88]. In the case of this Thesis, all the substrates were treated to obtain a single41 Luc´ ıa Iglesias Bernardo 0.0 0.5 1.0 1.5 2.0 Experimental Fitting Intensity (a.u.) /2() Figure 2.5: The experimental data correspond to a Nb:SrTiO3thin film deposited on a (001) LSAT substrate. The curve fitting using the X’Pert Reflectivity software provides the value of the density (ρ), thickness (t) and the surface roughness (R) of the film. For this particular sample, the adjusted values are ρ=5.1 g/cm3,t=18.2 nm and R=0.3 nm. 2.3 Atomic Force Microscopy Since the discovery in 1986 by Binning and Rohrer of the Atomic Force Microscopy (AFM) [94], this technique has become a powerful tool to characterize the thin film surfaces at the atomic level. It consists in a small tip attached to a cantilever, which is focused by a laser (see Fig. 2.6). When the cantilever is displaced over the sample surface using a piezo-electric crystal, it experiences an interacting force between the tip and the surface atoms, which bends the cantilever. The deflection of the cantilever is registered by the laser, and its reflection is amplified and recorded using an array of photodiodes. The processed signal is transformed into a height profile. Nowadays, the best cantilevers are able to sense forces down to 1pN with a tip radius less than 1nm. As a consequence, the resolution is typically around 1-10 nm in-plane and 0.1-100 nm in the out-of-plane direction [95]. 48 2. Experimental techniques a) b) Photodetector Laser Cantilever Tip Sample surface XYZ Scanner 0 1 2 3 4 0 1 2 3 4 5 6 Height (nm) Distance (m) c) 1.2 1.4 1.6 1.8 0.8 1.0 1.2 Distance (m) Height (nm) 0.35nm Figure 2.6: Atomic Force Microscopy. a) Scheme of AFM working operation, detailing the different elements of the system. b) An AFM topography image of the surface of SrTiO3thin film grown on (110) DyScO3substrate and c) the corresponding profile which shows the smooth terraces. The inset shows an enlargement of the height profile, in which the height difference between terraces is indicated. Although other enhanced modes of AFM operation have been developed for specific purposes, all of them are based on three basic modes: non-contact, contact and tapping mode. •Contact mode. It is the most common; the tip is in “close” contact with the sample surface and is dragged over it, keeping the deflection constant. The tip experiences a repulsive force, and both longand short-range forces add to the image signal. In this static mode, the control parameters are the displacement and the constant force with which the cantilever pushes the sample surface (typically around tens of nN). Therefore, the choice of cantilever is crucial, it must be softer enough to prevent its breaking 49 Luc´ ıa Iglesias Bernardo and also the damage of the surface of the sample. Under ambient conditions, most samples are covered by an adsorbed layer of water and nitrogen, which is 10-30 monolayers thick. When the tip scans the surface, a meniscus layer is formed. Additionally, many types of materials, including semiconductors and insulators, can trap electrostatic charge, which is dissipated and screened in liquid. This charge can increase the attractive forces between the tip and the sample surface, which can alter the sample surfaces and lead to an erroneous interpretation of the height profile. •Non-contact mode (NC-AFM). In order to avoid the problems associated to the presence of a meniscus, the scan of the surface can be performed in a non-contact mode. In this case, the tip scans the sample above the surface, without touching it, and uses the attractive inter-atomic forces (Van der Waals) between the tip and the sample to measure the surface topography. Due to the attractive forces from the sample being weaker compared to those in contact mode, the tip must vibrate close to its resonance frequency when it passes over a surface, and associate the changes in the cantilever’s vibration to variations in height. •Tapping mode. The tapping mode appears to overcome the operating problems of NC-AFM mode and other difficulties such as friction, adhesion or electrostatic forces. The operation of this mode is as follows: the tip slightly hits and lifts off from the sample surface on each oscillation, using high vibration frequency (50-500 kHz) and amplitude (∼20 nm). Normally the selection of the optimal oscillation frequency is software-assisted and maintained constant at the lowest possible level. When the tip moves over a hill in the sample surface, the cantilever has less space to vibrate and the amplitude of oscillation decreases. In contrast, when the tip passes over a cavity, the cantilever amplitude increases. 50 2. Experimental techniques 2.3.1 Electrostatic Force Microscopy and Kelvin Probe Microscopy The study of materials for electronic applications required the development of new AFM techniques, capable of mapping the electronic properties of materials. These AFM-based electronic techniques are mainly three: Electrostatic Force Microscopy (EFM), Kelvin Probe Force Microscopy (KPFM), and Piezoresponse Force Microscopy (PFM). Since only EFM and KPFM techniques have been used in this Thesis, this Section focuses on briefly explaining its main characteristics and differences. •EFM is based on NC-AFM mode. A bias voltage is applied between the conductive tip and sample surface while the cantilever scans the surface of the sample (Fig. 2.7). The amplitude of the cantilever oscillation depends on the electrostatic force interaction between the AFM tip and the sample surface. The electrostatic forces are proportional to 1/r2, while Van der Waals forces are proportional to 1/r6. Therefore, Van der Waals forces are dominant when the tip-sample distance is short, but as the distance increases electrostatic forces become dominant. Normally, the image is taken using the two-pass method. In the first scan, the tip scans close to the surface where the Van der Waals forces are dominant for topography imaging. After that, the tip is lifted to reach the distance in which the electrostatic force is dominant, it is biased and scanned following the topography line acquired from the first scan, generating the EFM image. EFM collects the information about the electrostatic force between the dc biased conductive tip and the sample surface. A variation in this force results in a change of the cantilever resonant frequency, which is proportional to the force gradient: ∆ω=ω0 2k dF(z) dz (2.12) 51 Luc´ ıa Iglesias Bernardo where kis the spring constant and w0is the resonance frequency of the cantilever. By adjusting the driving frequency wp, the resonance is maintained. Then, the frequency shift ∆w=wp-w0is acquired as the EFM image. Vdc XYZ Scanner a) b) c) Figure 2.7: a) Scheme of the EFM method; the bias voltage is applied between the tip and the sample holder. b), c) Example of EFM images corresponding to a Nb:STO thin film grown on a (110) GdScO3substrate after recording the central square region marked with dashed lines with +10 V and -10 V, respectively. •KPFM. The principal difference compared with EFM mode is that KPFM uses a compensation technique to allow quantitative measurements of the local surface potential (SP) (see Fig. 2.8), which can be related with the work function (WF) difference between the tip and sample. In this Thesis, EFM has been used in combination with the KPFM to avoid topographic artifacts when the spatial variation of the charge density in the films is determined. 52 2. Experimental techniques Vdc DC bias control = SP XYZ Scanner a) b) Figure 2.8: KPFM technique. DC bias is controlled by a feedback loop, which tries to nullify the measured force. The result is the surface potential of the sample with respect to that of the tip. b) Real KPFM image corresponding to a Nb:STO thin film grown on a (110) DyScO3, in which a previous scan with -10 V and +10 V inside the marked zones was performed. In the KPFM mode the cantilever is not driven mechanically when the image is acquired. Instead, the tip is biased directly using a sinusoidal signal with a dc offset. ∆V is the potential difference between VSP and the voltage applied to the AFM tip (Vtip) as follows: ∆V=Vtip ±VSP = (VDC ±VSP ) + VACsin(ωt)(2.13) where VAC is referred to the driving voltage. The electrostatic force between the tip and sample surface at potential Vis given by: F(z) = −1 2 ∂C(z) ∂z ∆V2(2.14) where C’(z) is the gradient of the capacitance between the tip and the sample, dependent on tip geometry, surface topography and tip-surface separation z. 53 Luc´ ıa Iglesias Bernardo The first harmonic of the force is expressed as follows: F1w=−∂C(z) ∂z (VDC ±VSP )VACsin(ωt)(2.15) This term is nullified by supplying a bias feedback loop which continually adjusts the constant component of the tip bias (VDC). The tip bias that cancels the electrostatic force is equal to the VSP . Therefore, the final KPFM image is a 2D map of spatial variation of the SP, which is related to local differences in charge density distribution. This compensation technique attenuates all other responses from the measurement, losing all information outside the modulation frequency (e.g. the information provided by the second harmonic) [96–98]. 2.4 Electrical transport measurements The transport properties of the samples were measured using either the Van der Pauw method, or by patterning a conducting Hall bar on top of the thin films by optical lithography (see Fig. 2.9). Hall bar configuration allows the measurement of multiple electric properties (resistivity, conventional and planar Hall effect, anisotropic magnetoresistance, etc), minimizing the leakage current and the undesired misalignment. For the results presented in this Thesis, the length of the Hall bar channel is l= 650 µm and the width is ω= 100 µm. The fabrication process consists in 6 steps, combining standard photolithography and physical etching as is schematized in Fig. 2.9. At the beginning, the thin film covers the entire surface of the substrate. The first step is doing a lithography with a positive mask in order to protect the thin film under the desired Hall bar configuration (2.9 a) and b)). Once the Hall bar is covered by the photoresist, a wet acid or dry ion-etching is carried out, which removes the unwanted part of the thin film (2.9 c)). After that, a litt-off process is performed to eliminate the protective photoresist layer (2.9 d)). In the second step of lithography, a negative 54 2. Experimental techniques mask is used, which covers only the channel region of the Hall bar, leaving the six pads (see 2.9 e)). Finally, 5nm of Cr and 50 nm of Pt are evaporated to create the ohmic contacts at the six contact pads. The electrical contacts are made by wire bonding with 30 µm Al wire (2.9 f). Lithography positive mask Lithography negative mask Thin film Substrate Photoresist Etching process Cr+Pt deposition Lit-off a) b) c) d) e) f) Final Hall bar Substrate Film + photoresist Figure 2.9: Scheme of the Hall bar fabrication. a) Thin film as-grown, b) lithography with positive mask, c) physical etching process, d) litt-off to remove the protective photoresist, e) second lithography step using a negative mask, and f) metal deposition on top of the six ends of the bar. Final Hall bar has 650 µm length and 100 µm width. 2.4.1 Electrical resistivity Van der Pauw technique is a convenient method used to measure the electrical resistivity of uniform samples, such as thin films, because it does not require sample preparation [99]. The samples should have a flat shape and uniform thickness, without inhomogeneities and isotropic with respect to the electrical transport. For the measurements, four small ohmic contacts are placed at the corners of the film, as is depicted in Fig. 2.10 a). The area of the electrical contacts should be much smaller than the area of the surface of the sample and additionally, these must have symmetry at the boundaries of the sample to minimize errors. 55 Luc´ ıa Iglesias Bernardo While Hall bar configuration requires only one voltage reading, Van der Pauw geometry needs two. In practice, an electrical current is injected from one side of the sample (for instance, I12) and the voltage is read at the opposite side (V34). Consequently, the resistance can be determined using the Ohm’s law. In our particular case, more accurate measurements were performed by repeating the same process through all four sides of the thin films, and then inverting the polarity of each measurement. Thus, an average resistance (R) is determined. In the case of square geometry using Van der Pauw measurement, the resistivity (ρ[(Ω.cm]) is given by: ρ=πRt ln(2) (2.16) where tis the thickness of thin film. Similarly, for the measurement using the Hall bar configuration illustrated in Fig. 2.10 b), the current is injected between I+and I−contacts and the voltage is read between D and E ones. Thus, using this configuration, ρwill be determined by the relation: ρ=Rtω L(2.17) where ωand lare the width and length of the Hall bar, respectively. 2.4.2 Hall effect measurements Hall effect is based on the physical principle of Lorentz force. When an electrical current is applied along a metal or semiconductor and perpendicularly to a magnetic field, the charge carriers inside the material experience a force acting normal to both directions, which changes their trajectories and produces a transverse voltage, the so-called Hall voltage (VH) [100]. From this voltage, the charge carrier concentration (n) can be obtained as follows: n=IB qtVH (2.18) 56 2. Experimental techniques Combining resistivity and charge carrier concentration, the mobility of charge carriers (µ) can be calculated using: µ=1 ρne (2.19) where eis the electron charge. I12 V34 1 3 4 2 I+ IC D E a) b) Figure 2.10: a) Sketch of Van der Pauw configuration for resistivity measurement in our samples. Indicating the four ohmic contacts placed at the corners of the sample. The current is injected between contacts 1 and 2 contacts, while the voltage is read in the opposite side from 3 and 4. b) Scheme of Hall bar configuration, in which is marked the current injection (I+-I−). The voltage is measured between D and E for electrical resistivity and between C and D for Hall measurements. For the measurement of Hall effect using the Van der Pauw method, the electrical current is injected through contacts 1 and 4 and the voltage is measured between contacts 2 and 3. For each measurement, the voltage is measured for different values of current, maintaining the magnetic field constant. Additionally, two measurement are performed, either with positive and negative magnetic field. The same procedure is repeated for different values of magnetic field, in our case up to 1.2 T, with a step of 0.1 T (see Fig. 2.11). Two measurements are required, so that we repeat the same protocol by injecting the current (I23) and reading the voltage (V14) (see Fig. 2.10 a). Moreover, in order to cancel 57 Luc´ ıa Iglesias Bernardo neous gas with density equal to the local density at any given point, and neglecting exchange and correlation among electrons. This approach illustrates the way density functional theory works today, although it is a very crude approximation that cannot even predict the existence of chemical bond. Another historically important strategy was the Hartree-Fock method [112], which is commonly used in Quantum Chemistry but it is not scaling appropriately for multi-electronic systems, as all the methods based on wavefunction solutions. It assumes that the multi-electron wavefunction can be approximated by a single Slater determinant, which is an antisymmetric combination of one wavefunction per electron (or spinorbitals) taking into account the Pauli exclusion principle. Although the accuracy required at present in the description of nuclei, atoms and molecules cannot be reached by the Hartree-Fock method, it is still widely used as the starting point of several more elaborate methods, even for periodic systems. 3.1.2 Khon-Sham approach In 1964, Hohenberg and Kohn [107] presented the statements which make the current DFT method possible. Basically, the idea is to substitute the initial many-body system by a non-interacting fictitious system where each electron is separately subjected to an effective potential but still sharing the electronic density of the global many-body problem. The two theorems were postulated as follows [113]: Theorem 1:For any system of interacting particles in an external potential Vext(r), the potential Vext(r) is determined uniquely, except for a constant, by the ground state particle density n0(r). Theorem 2:A universal functional for the energy E[n] in terms of the density n(r) can be defined, valid for any external potential Vext(r). For any particular Vext(r), the exact ground state energy of the system is the global minimum value of this functional, and the density n(r) that minimizes the functional is the exact n0(r) ground state density. 64 3. Computational Modelling Therefore, these two theorems demonstrate that the ground state density can be determined from the minimization of an energy functional for any particular Vext(r), being able to extract all the properties of the system. While Hohenberg-Kohn theorems proved that it is possible to calculate the properties of the system by using the ground state density, they did not provide a way of finding it. A route for that was the formulated by Kohn and Sham [108] almost exactly a year after the Hohenberg-Kohn theorems were published. The approach is to replace all electrons in the system with a system of non-interacting electrons, with the same electron density as the real system of interacting electrons, assuming that the ground state density of the original interacting system is equal to that of some chosen non-interacting system. To recover the many-body phenomena an exchange-correlation (XC) term is introduced, which contains all the missing many-particle effects. The auxiliary independent-particle system is defined by the auxiliary Hamiltonian (HKS): ˆ HKS =−1 2∇2+Veff (~r)(3.3) where Veff is the effective potential. The ground state is calculated by solving the Schr¨ odinger equation for a system with N independent electrons given by −1 2∇2+Veff (~r)ϕi(~r) = εiϕi(~r)(3.4) where εiis the orbital energy of the corresponding Kohn-Sham orbital ϕi(~r). These are the so-called Kohn-Sham equations, and from the N orbitals ϕi(~r)the density matrix of the auxiliary system can be constructed as follows ρ(~r) = N X i=1 |ϕi(~r)|2(3.5) 65 Luc´ ıa Iglesias Bernardo This approach leads to a self-consistent solution of the orbitals and electron density. In this way, at the beginning a trial electron density is introduced into the equations in order to obtain the wave functions of a single electron, which is then reintroduced into the equation to calculate the electron density. If the electron density is the same as the density we started with, then the ground state electron density is found, and the process finishes. If not, the electron density must be updated, continuing the process until a self-consistent solution that satisfies the convergence criterion is reached, as is illustrated in Fig. 3.1. Initial guess density ( r (r)) Calculate effective potential (Veff(r)) Solve Kohn-Sham equations Evaluate electron density and total energy: 𝜌 𝑟 = ∑𝑖𝜓𝑖(𝑟) 2⇒ 𝐸𝑇𝑂𝑇 𝜌(𝑟) Converged? No Output quantities: Forces, eigenvalues, etc… Yes Figure 3.1: Flow chart of a typical DFT loop. First, an initial electron density is assumed, which is used to calculate the effective potential Veff (r). Then, Khom-Sham equations are solved, and subsequently it is evaluated ρ(r) and ETOT . If the convergence criterion is not satisfied, the loop starts over with the last ρ(r). Once the convergence criterion is satisfied, the loop ends and all sorts of properties can be calculated based on the ground state electron density. Scheme adapted from ref. [113]. 66 3. Computational Modelling Kohn-Sham equations cannot be solved exactly unless the exchangecorrelation functional is determined, for which unfortunately there is no established way to proceed. Therefore, except for the homogeneous free electron gas, it should be approximated. 3.2 The Exchange-Correlation Functional The choice of an exchange-correlation functional is crucial but not a trivial issue, since there is not any universal DFT approach, and it depends strongly on the particular system and the property to be sought. The most commonly used functional approximations are Local Density Approximation (LDA) [108], Generalized Gradient Approximation (GGA) [114], Hubbard-corrected approximation (LDA+U or GGA+U) [115] and Hybrid Functional (HF) [116]. Focusing on this Thesis, we chose the Generalized Gradient Approximation (GGA) functional with the Perdew-Burke-Ernzerhof (PBE) [117] and PBEsol [118] approximations to perform DFT calculations, which will be briefly discussed below. 3.2.1 Generalized Gradient Approximation (GGA) This exchange-correlation energy functional was proposed as a natural improvement beyond the LDA, which is only based on the local density at a given point. As its own name indicates, the GGA energy functional can be expanded in terms of the density and its gradients: EGGA XC [ρ] = ZGGA XC (ρ, ∇ρ)ρ(~r)d~r (3.6) Compared to LDA, GGA significantly improves the lattice parameter determination and other properties related to crystal lattice constants of most transition metals systems and consequently, it is normally used to calculate their electronic structure. However, GGA tends to overcorrect LDA results in ionic crystals, in which LDA fits better to the experimental results. 67 Luc´ ıa Iglesias Bernardo A well-known problem is that both GGA and LDA usually underestimates the electronic band gap. One attempt to solve this issue is by introducing the Hubbard term (U) in a strongly correlated system, which adds the effect of Coulomb interaction between electrons into the localized orbitals. Another possibility is by using Hybrid functionals, in which fraction of the LDA or GGA exchange is replaced by a fraction of exact exchange energy functional. Although this approach considerably improves the band gap determination of the most semiconductor and insulator materials, its implementation is computationally expensive and not satisfactory in all cases. Within the GGA approximation, there are several ways to incorporate the density gradient. One of those methods is the one proposed by Perdew-Burke-Ernzerhof [117], which is the most commonly used in Solid State calculations today due to its computational efficiency, numerical accuracy, and reliability. This is the main functional we have utilized in this Thesis, although in certain parts (especially in the calculations related to ferroelectricity and strain) the PBEsol functional [118] has also been used. The latter is a functional specifically proposed for solids, improving some disadvantages of the GGA-PBE but without requiring a lot of additional calculation time. 3.3 Wien2k and VASP Packages There are several software packages which allow to perform electronic structure calculations of solids by DFT method. In this Thesis, we used Wien2k [119] and VASP (Vienna Ab initio Simulation Package) [120]. These are packages normally used to calculate quantum mechanical properties on periodic solids, and are very accurate for performing electronic calculations in crystals. The two packages compute an approximate solution to the manybody Schr¨ odinger equation, solving the Kohn-Sham equations within the DFT method. They are based on the augmented plane-wave method plus local orbitals (APW+lo) and projector augmented-wave (PAW), respectively. They include different exchange-correlation potentials such 68 3. Computational Modelling as LDA and various GGA approximations (PBE, Perdew-Wang, PBEsol, Wu-Cohen,..). •APW+lo: Augmented plane-wave plus local orbitals. APW method is a variational expansion approach which solves the equations of DFT by approximating solutions as a finite linear combination of basis functions. For such solutions, it is used the called muffin-tin approximation, which separates the unit cell into two parts: the muffin-tin spheres centered at the atomic sites and the remaining interstitial region (Fig. 3.2). Muffin-tin radii (Rmt) must be chosen carefully depending on the atomic size and the spheres can not overlap each other. Different basis are used in the two regions: linear combinations of atomic-like functions inside the spheres, and a plane wave expansion in the interstitial part. The main problem with the APW method is that the basis set is energy-dependent (within the muffin-tin spheres) and in addition, it turns out to be a slow and unpractical method. Rmt1 Rmt2 Interstitial region Rmt1 Rmt2 Figure 3.2: Representation of how the unit cell is divided in the muffin-tin approximation for a case with two atoms and different muffin-tin radii. It is also indicated the remaining interstitial region between the non-overlapping spheres. To improve it, the APW+lo method (based on APW method) was 69 Luc´ ıa Iglesias Bernardo recently proposed by Sj¨ ostedt [121]. By adding a local orbital in the basis set, this method makes the calculations faster and more efficient and above all, solves the energy-dependent problem of the APW method. The parameter RmtKmax=6-9 controls the size of the plane-wave basis, and is defined as the product of the planewave cut-off and the smallest muffin-tin radii [122,123]. •PAW: Projector augmented-wave. Another method to model the core electrons makes use of pseudopotentials. It is the so-called Projector Augmented-Wave (PAW), which expresses the single particle all-electron Kohn-Sham wave functions. PAW does this by writing the all-electron wave function as a sum of a few other functions, each of which can be expressed in a natural way in a basis. The pseudo-potential can be constructed to be weak and smooth, solving the Kohn-Sham equations in solids directly by using the Fourier space [124]. 3.4 Electrical polarization A workhorse throughout the history of electrical polarization has been how to define it in a crystalline solid. The problem was not solved until 25 years ago with the introduction of the so-called Modern Theory of Polarization by Vanderbilt and King-Smith [125, 126]. That was when researchers realized that one should work with changes in polarization instead of absolute values. The changes in polarization are well-defined and can also be compared with the experimental measurements. There are two principal ways to estimate the spontaneous polarization in a crystalline solid: by using the Born Effective Charges, (Z∗) or by performing Berry Phase calculation, which entails more difficulty but provides more accurate results. •The Born effective charge (BEC) is a tensorial quantity which evidences the coupling between lattice displacements and electrostatic fields. It differs from the nominal charge because it takes into account that when an ion of lattice is moved, the electrons 70 3. Computational Modelling redistribute as well. In most materials, nominal and BEC have the same sign but different magnitudes and the difference can become large. Therefore, BEC is useful because it indicates how the ions respond to external electric fields and it can be determined theoretically using perturbation theory. BEC is defined through: Z∗ ij =V e δPi δdj (3.7) where Piis the change in the polarization induced by the periodic displacement djin the direction i, that in addition can also cause a polarization change in the direction j. The unit cell volume is denoted by V. By summing the polarization over the contributions of the displacements of all sub-lattices, the total polarization of the system can be obtained: δPi=e VZ∗ ijδdj(3.8) •Berry phase method. Another way to calculate the spontaneous polarization is by using the Berry phase formalism [127]. Basically, the Berry phase concept describes how if you add all the curvatures of an object and then deform it, the total curvature will remain the same. In the modern magnetism, this concept plays an important role to understand a broad range of phenomena such as the spin-orbit coupling, the quantum Hall effect or the anomalous Hall effect, and it is also useful to calculate the electrical polarization. The Berry phase approach consists of determining the change in polarization between a paraelectric initial structure (high symmetry) of reference, and a final ferroelectric structure (low symmetry), both with the same volume. An adiabatic path should be built between them, moving carefully the ionic positions along the polarization branch of the lattice. Various calculations are necessary following the deformation path between the high and low symmetry structures to ensure unequivocally that it remains on the same 71 Luc´ ıa Iglesias Bernardo branch during the adiabatic path (Fig. 3.3). To perform that, the system should be insulating, and with the same number of bands occupied at every point in k-space. The relaxed ionic positions and self-consistent charge density will be used as an input of the calculation [128]. 0 0.2 0.4 0.6 0.8 1 Adiabatic distortion parameter -50 -40 -30 -20 -10 0 10 20 30 40 50 Pz (microC/cm2) Figure 3.3: Calculated out-of-plane polarization as a function of the adiabatic distortion parameter for STO system starting from the high symmetry paraelectric structure, I4mmm, to the polar structure, I4mm. Unfilled dots are calculated points and solid lines are a guide to the eye, showing the continuous path along the branches of the polarization lattice. 72 4. Thermodynamics of oxygen vacancy formation I hadn’t been aware that there were doors closed to me until I started knocking on them. Gertrude B. Elion, American biochemist and pharmacologist, who won the Nobel prize in Physiology and Medicine in 1988. The mechanisms to accommodate the lattice mismatch between an epitaxial film and a substrate has drawn attention for more than 50 years [129–133] as it strongly influences the morphology and properties of the films [45, 134]. A common way to adapt the mismatch in ABO3perovskite-based heterostructures is primarily through changes in the lattice constants (volume effects or octahedral tilts) [135]. These mechanisms will be studied in depth in Chapter 6. However, point defect formation is also a likely strain-relaxation mechanism to be considered [50]. In this regard, the oxygen pressure during the deposition of metal oxides (for instance, by PLD technique) plays a major role in the mechanism that controls epitaxial relaxation [136]. For instance, to achieve fully oxygenated thin films, a high oxygen pressure is required. Nevertheless, some oxides need a low oxygen pressure during deposition to guarantee a low valence state of the cations. Thus, the competition between crystalline perfection and stoichiometry 73 Luc´ ıa Iglesias Bernardo Substrate Acronym a ( ˚ A) b ( ˚ A) c ( ˚ A) apc (˚ A) s (%) aexp (˚ A) cexp (˚ A) sexp (%) LaAlO3LAO 3.790 3.790 3.790 3.790 -2.91 3.860 3.934 -1.15 (NdAlO3)0.39– (SrAl0.5Ta0.5O3)0.61 NSAT 3.840 3.840 3.840 3.840 -1.66 3.870 3.928 -0.90 (LaAlO3)0.29– (SrAl0.5Ta0.5O3)0.71 LSAT 3.890 3.890 3.890 3.890 -0.95 3.868 3.940 -0.95 LaGaO3LGO 5.523 5.491 7.773 3.885 -0.51 3.885 3.927 -0.51 SrTiO3STO 3.905 3.905 3.905 3.905 0 3.905 3.914 0 DyScO3DSO 5.540 5.710 7.890 3.950 +1.15 3.950 3.906 +1.15 GdScO3GSO 5.450 5.750 7.930 3.970 +1.66 3.970 3.884 +1.66 KTaO3KTO 3.989 9.989 3.989 3.989 +2.14 3.989 3.890 +2.15 Table 4.1: Summary of the principal structural parameters of all the substrates used in this Thesis. Bulk lattice parameters (a,b,c) and in-plane apc pseudocubic lattice parameters for each substrate. The nominal strain (s) imposed by each substrate in the STO thin films calculated according to Eq. 2.3 (af= 3.905 ˚ A) is also indicated. Experimental data extracted from XRD and RSM are also shown (aexp,cexp,sexp). 80 4. Thermodynamics of oxygen vacancy formation to the Poisson’s ratio, meaning an elastic deformation of the material. To remove a cationic vacancy is favoured in this case due to volume arguments, as an attempt to contract the structure and to accommodate the large in-plane compressive strain. Meanwhile, on the tensile side, the volume of the unit cell is slightly higher than expected from the Poisson’s ratio (about ≈1%), suggesting the formation of oxygen vacancies as the predominant mechanism to relax the stress. This forces the reduction of Ti4+ to Ti3+ and leads to changes in ionic radius, which manifests macroscopically in the extra expansion of unit cell volume. These relaxation mechanisms based on a simple chemical expansion model are common in perovskite-type structures [45,51,147]. -3 -2 -1 0 1 2 3 57 58 59 60 61 62 KTO GSO DSO STO LGO LSAT NSAT LAO Poisson ratio =0.23 Experimental Volume (Å3) Strain (%) Figure 4.5: a) Evolution of the unit cell volume with experimental strain for all samples grown in this work (orange unfilled dots). The solid black line indicates the theoretical volume of the stoichiometric unit cell considering a Poisson’s ratio of ν= 0.23. Orange solid line is a guide to the eye. Further insight into the microstructure and crystalline quality of the films can be provided by High Angle Annular Dark Field-Scanning Transmission Electron Microscopy (HAAD-STEM). This analysis was carried out in an FEI Titan 60-300 operated at 300kV and equipped with a high brightness Schottky field emission gun, a CETCOR probe aberration corrector form CEOS to achieve a spatial resolution better 81 Luc´ ıa Iglesias Bernardo than 1 ˚ Ain STEM mode, and a Gatan Imaging Filter 866 ERS for spectroscopic analysis. The specimens analysed correspond to lamellae extracted from the samples by focused ion beam (FIB) milling in a FEI Helios Nanolab 600 a 5 kV ion beam. The measurements were performed for two samples, one subjected to compressive (on LAO) and another to tensile stress (on DSO). The images are shown in Fig. 4.6 and corroborate the results obtained in RSM measurements. The sample grown on DSO presents an excellent crystalline quality and a sharp interface between the film and the substrate (Fig. 4.6 a). Nevertheless, the image of the film grown on LAO (Fig. 4.6 b), although it also shows a good crystalline quality along the film thickness, reveals the formation of defects close to the interface, which could be produced to accommodate the large epitaxial stress (s=-2.91%) imposed by the substrate. a) Nb:STO DSO b) Nb:STO LAO Figure 4.6: HAADF-STEM image of the as-grown Nb:STO thin films deposited on DSO (a) and LAO (e) substrates. White arrow indicates the presence of defects close to the interface in image b). Furthermore, Geometrical Phase Analysis (GPA) can provide the evolution of the in-plane and out-of-plane lattice parameter along the film thickness, unlike X-ray diffraction which gives an average of the lattice parameter for the whole film. GPA analysis confirms that the film on DSO presents a homogeneous in-plane distortion throughout the films, whereas the film on LAO shows different in-plane distortion in the film than in the substrate (see Fig. 4.7). 82 4. Thermodynamics of oxygen vacancy formation In the case of the sample grown on a DSO substrate (Fig. 4.7 a, b), the profile of GPA analysis confirms that the out-of-plane negative distortion (zz) in response to the tensile in-plane strain (xx) is homogeneous throughout the films and in complete agreement with the coherent growth observed in RSM data . For example, for the STO film deposited on DSO, the caxis parameter of the film contracts ≈1.1% with respect to the substrate, while xx does not exhibit any variation from the substrate to the film (Fig. 4.7 c). However, in the film deposited on LAO, both zz and xx show variation between the film and the substrate, demonstrating that the film was already partially relaxed (Fig. 4.7 d, e, f)). -10 -5 0 5 10 0 5 10 15 z (nm) xx zz % Deformation a) xx zz b) d) e) xx zz -1.5 -1.0 -0.5 0.0 0.5 0 5 10 15 20 z (nm) xx zz % Deformation c) f) Nb:STO DSO Nb:STO LAO Figure 4.7: GPA analysis corresponds to Fig. 4.7, showing the in-plane (xx) and out-of-plane (zz) contraction/elongation along with the corresponding line profiles with respect to the substrate for the films grown on DSO (a,b, c) and on LAO (d, e, f), respectively. 83 Luc´ ıa Iglesias Bernardo As mentioned before, the formation of oxygen vacancies produces an increase of the unit cell volume due to the higher atomic radius of Ti3+ compared to Ti4+, a higher vacancy concentration in reduced samples could affect the lattice parameters of the thin films. However, XRD measurements of the samples as-grown at 100 mTorr and after an annealing process at 1 x 10−6Torr show the negligible effect on the lattice parameters, as can be seen for instance in one of the films grown on a LAO substrate (Fig. 4.8 c). Therefore, we conclude that the main responsible for structural relaxation are the presence of strontium vacancies (VSr). This is consistent with the observations reported by other authors [106,148,149]. 42 44 46 48 50 52 As grown Anneal. 10-6 Torr 2 Intensity (a.u.) Nb:STO film LAO (002) Figure 4.8: c) XRD θ-2θscan of the sample grown on LAO as-grown at 100 mTorr and after the annealing process at 1 ×10−6Torr, showing the negligible effect on the lattice parameters. 4.2 Creation and annihilation of oxygen vacancies As discussed before, nominally undoped SrTiO3single crystals show a small concentration of acceptor impurities (Sr2+ vacancies) [144], which can trap the electrons donated by V ¨ Oto the conduction band 84 4. Thermodynamics of oxygen vacancy formation of the oxide. To avoid this effect, we have doped the films with a 2% of Nb5+. The doping introduces a small density of free electrons to compensate the unintentional presence of cationic vacancies. So that, even small changes in the V ¨ Oconcentration due to the different postannealing processes will produce a measurable change in the conductivity and carrier concentration of the thin films. During the PLD deposition at low oxygen pressures, the formation of oxygen vacancies in STO thin films occurs as Schottky defects. A VSr is formed at the same time as an oxygen vacancy to maintain the neutrality of charge: Srx Sr +Ox OVSr +V¨ O+SrO (4.1) Alternatively, V ¨ Ocan be introduced into a stoichiometric film by a post-annealing process in a reduced environment, in where the predominant mechanism is the loss of oxygen from the lattice to the atmosphere and a change in the oxidation of Ti: SrTiO3(s)SrTiO3−x(s) + x 2O2(g)↑(4.2) Although V ¨ Oin e-doped STO are not stoichiometric vacancies, their concentration is not too large and the crystal can be still considered nearly stoichiometric. Assuming this, a simple mass action law can be applied to the reaction 4.2 and formulate the following equilibrium for the material at low oxygen pressures: O2−1 2O2+V¨ O+ 2e−(4.3) where O2−represents the concentration of oxygen ions at their STO lattice sites. Denoting the concentration of defects by angular brackets (’[]’), the equilibrium constant can be defined as: K=[V¨ O]n2 eP1/2 O2 hOx ¨ Oi(4.4) 85 Luc´ ıa Iglesias Bernardo As the effect of acceptor vacancies (VSr) is compensated by Nb5+ donors originally present in the sample, the reduction reaction (Eq. 4.3) will produce a change in the carrier density that can be measured by Hall effect experiments. Assuming that [Ox ¨ O]is constant and that each oxygen vacancy contributes two electrons to the conduction band of the material (ne=2[V ¨ O]), then the eq. 4.4 can be expressed as: K=n3 e 2P1/2 O2(4.5) Solving the Eq. 4.5, we obtain [150,151]: log ne∝ −1 6log PO2(4.6) According to this equation, the logarithmic representation of the charge carrier concentration measured by Hall effect (after each postannealing) versus the oxygen pressure should give a slope of -1/6 as long as oxygen vacancies are doubly ionized. A controlled amount of oxygen vacancies was introduced in the films by post-annealing process at different temperatures (either 800◦C or 600◦C) and successive lower oxygen pressures in order to study the thermodynamics of an oxygen vacancy formation. In each postannealing, the samples were left with the background oxygen pressure during 2 hours to reach equilibrium, and then rapidly quenched to room temperature at a cooling rate of ≈100◦C/min approximately, to maintain the concentration of vacancies constant during the cooling. The results are shown in Fig. 4.9 a) for samples annealed at either 800◦C and 600◦C. At high temperature, there is a very good agreement with the expectations for doubly ionized vacancies (solid orange line). Note that if there was an extrinsic source of oxygen vacancies, for example from acceptor impurities or an excess of TiO2, Eq. 4.3 will not be the main source determining the concentration of vacancies, changing the slope of this representation. For instance, for single-ionized vacancies, the slope of the logarithmic representation of nversus PO2 86 4. Thermodynamics of oxygen vacancy formation will change from 1/6 to 1/4. This behavior has been frequently seen by other authors at intermediate oxygen pressures [150]. In Fig. 4.9 a), the expected slope of -1/4 for singly ionized vacancies is also indicated, which clearly does not fit our experimental data independently of the temperature. 10-6 10-4 10-2 1020 1021 -1/6 800C 600C n (cm-3) PO2 (Torr) -1/4 10-6 10-4 10-2 1020 1021 800C 600C n (cm-3) PO2 (Torr) b) a) Figure 4.9: a) Variation of the charge carrier concentration as a function of the post-annealing oxygen pressure for the Nb:STO thin film on LSAT substrate. The same experiments were performed at 800◦C (orange squares) and 600◦C (open blue diamonds). Solid orange and blue lines represent the behaviour for the slope of -1/6 corresponding with doubly ionized oxygen vacancies (eq. 4.6), while black dashed line indicates the slope of -1/4 for single ionized vacancies. b) Re-oxygenation process of the previously reduced samples at higher oxygen pressure. Solid lines are guides to the eye. At very low oxygen pressures (<10 −6Torr) the charge density saturates for both temperatures, indicating a limit in the V ¨ Oformation by this procedure. This can occur for two principal reasons: the time to reach the thermodynamic equilibrium is insufficient or the combination between temperature and very low oxygen pressure is not enough to activate the formation of vacancies. The first option does not seem to be the cause, as two hours should be sufficient to reach thermal stability. Therefore, higher temperature or lower oxygen pressure are required to increase further the density of V ¨ O. The films were re-annealed at higher oxygen pressures, following 87 Luc´ ıa Iglesias Bernardo the same protocol as before. While the samples completely recover the initial state performing the re-oxygenation at 800◦C and higher oxygen pressures (orange dots in Fig. 4.9 b)), the same does not occur when the samples are re-annealed at 600◦C or lower temperatures. Actually, the charge carrier density remains constant at 600◦C irrespectively of the oxygen pressure used. This result is of vital importance to understand the reduction/oxidation mechanism of STO, both in the form of thin film and single crystal (substrate). It demonstrates that the temperature is a critical factor in the creation/annihilation of V ¨ Oin STO thin films and STO-based heterostructures in general. For instance, in the literature, LAO/STO interfaces are usually grow at low oxygen pressure and then, a subsequent re-annealing at high oxygen pressure and a temperature around 500◦C is performed with the aim to fully reoxygenate the film [8,26]. In view of our results, this temperature would not be enough to reoxygenate the samples completely. Thus, the temperature is a fundamental factor in the synthesis and post-annealing protocols, and therefore it should be considered carefully when understanding their effects in terms of oxygen stoichiometry in complex oxides in general. 10-6 10-5 10-4 10-3 10-2 10-1 1020 1021 -1/6 LAO STO GSO n (cm-3) PO2 (torr) Figure 4.10: a) Variation of the charge carrier density for samples grown on different substrates (LAO, STO and DSO) and annealed at 800◦C under low oxygen pressures. 88 4. Thermodynamics of oxygen vacancy formation Although the previous thermodynamic analysis was shown using the Nb:STO thin film grown on LSAT substrate, similar experiments were carried out in the other samples with different degrees of strain. As it is illustrated in the Fig. 4.10 a), a slope with value -1/6 was determined irrespectively of the substrate, indicating that V ¨ Oare doubly ionized in Nb:STO thin films independently of the applied strain. 4.3 Enthalpy of formation of oxygen vacancies Another important parameter to understand the formation of oxygen vacancies in complex oxides is the energetic cost of creating a vacancy, i.e. the enthalpy of formation (∆H). Given that STO is one of the most used substrates in the growth of oxides and the strain could be a key parameter for controlling the formation and annihilation of vacancies, it is surprising that the dependence of ∆Hhas not already been determined experimentally. Theoretical calculations of ∆Hdependence with strain have been reported, but there are not many experimental studies performed in this regard. To gain insight about it, the same mass action law used above can be taken as starting point (Eq. 4.3). This thermodynamic equilibrium constant is related to the Gibbs free energy (∆G) of the crystal, which in turn depends on ∆Hand entropy (∆S) of formation of oxygen vacancies as follows: ∆G= ∆H−T∆S(4.7) Then, the equilibrium constant for Eq. 4.3 can be expressed as: K=K0e−∆H/kBT(4.8) where kBrepresents the Boltzmann constant and K0is a constant. Here we assumed that ∆Sis only related to the increasing possibilities 89 Luc´ ıa Iglesias Bernardo out of resonance KPFM was employed, speeding up our data acquisition and avoiding crosstalk between first and second resonance mode. Moreover, to minimize artifacts and offsets from the measurement conditions, all measurements were performed with the exact same type of probe, AC bias magnitude, frequency, tip-sample distance, and scanning/recording rate for KPFM/EFM and topography acquisition. 5.2 Mobility of vacancies under an external electric field The thin films used in this Chapter are the ones characterized at the beginning of Chapter 4. Before the experiments, the surface topography of the samples must be carefully examined to verify the correct growth of the films, as a rough surface could determine very much the oxygen exchange with the atmosphere and the diffusion of the vacancies. Figure 5.1 shows the results for two of the films grown on STO (0% strain) (a) and GSO (≈1.66%) (b) and the corresponding profiles (c,d). The line scan of the topography exhibits smooth surfaces and regular terraces with step height of ≈0.2 nm, consistent with a layerby-layer growth, which reproduces the single-terminated surface of the substrates. These results demonstrate that the films present similar surface morphology independently of the sign of strain. The electric-field-dependent vacancy mobility was carefully investigated as a function of strain in this Section. This important parameter gives an idea of how difficult it is to move an oxygen vacancy at room temperature in SrTiO3. The mobility is also an important parameter in many applications, like resistive switching devices, in which it is crucial to determine the threshold voltage necessary to drag V ¨ Oacross the thin film. 96 5. Controlled movement of oxygen vacancies a) b) 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 0.2nm Height(nm) Distance(m) 0.0 0.5 1.0 1.5 -1.0 -0.5 0.0 0.5 1.0 0.2nm Height(nm) Distance(m) c) d) STO DSO STO DSO Figure 5.1: AFM images of the surface topography for Nb:STO thin films grown on STO (a) and DSO (b) substrates. The respective profiles along the white lines are represented below (c,d). In order to quantify the effect of epitaxial strain on the V ¨ Omobility, a region of the surface of each sample was first poled with negative/positive AFM tip bias in contact mode, and then the same area was mapped by KPFM scanning at zero DC bias. An scheme of this approach is depicted in Fig. 5.2 a, b). Given the nanoscale dimensions of the tip radius (<25 nm), the electric field generated by the AFM tip bias (∼6 MV/m) is highly localized and it is sufficiently strong to modify the initial homogeneous distribution of positively charged V ¨ Oof the thin films. A negative (positive) tip bias attracts (repeals) the positively charged oxygen vacancies, increasing (decreasing) the local work function. As a consequence the SP is lower (higher) compared with the average background potential of the sample. Then, depending on the electric field direction, the accumulation (depletion) of V ¨ Owill produce a localized decrease (increase) of the local SP [168]. A schematic illustration of the electric field direction depending on the tip bias, and the corresponding 97 Luc´ ıa Iglesias Bernardo movement of the oxygen vacancies away/near of the sample surface can be seen in Fig. 5.2 c, d). Figure 5.2: Scheme of the approach followed in this Section to study the room temperature diffusion of V ¨ Oin STO. a) A negative/positive voltage is applied to the sample surface, b) generating depletion/accumulation regions of oxygen vacancies. c, d) Detailed sketches specifying the direction of the electric field and the movement of V ¨ Odepending on the sign of the applied voltage. Following the procedure explained above, four samples with different degrees of strain were recorded with positive and negative voltages (±10 V). The enriched or depleted zones produce a clear contrast in the KPFM signal independently of the sample (Fig. 5.3). From these results, we are able to compare the relative change of the surface potential between the two well-differentiated areas. This difference expressed in percentage with respect to the film grown on STO is denoted as ∆SP, and its strain-dependence can be seen in Fig. 5.4 a). ∆SP varies substantially with strain, increasing up to ≈40%for the most stressed films, either compressive (on LSAT) or tensile (GSO), with respect to the unstrained film. Therefore, epitaxial strain influences very much the V ¨ O 98 5. Controlled movement of oxygen vacancies mobility, being 40-50%easier to drag or repel the vacancies in the most stressed films. It is important to remark that these experiments provide information about the drift mobility, i.e. the movement of oxygen vacancies across the film exclusively under the influence of an electric field. This is not the free diffusion of V ¨ O, which must be obtained from measurements in absence of an applied tip bias, as will be done in the next Section. LSAT (-0.95%) STO (0%) DSO (1.15%) GSO (1.66%) +10 V -10 V a) b) c) d) Figure 5.3: SP characterization for the thin films grown on LSAT a), DSO b), STO c) and GSO d). In image d), the voltages applied on the surfaces of the samples are indicated. The degree of strain in each film is also shown. It is worth mentioning that significant differences in the SP background of the unpolarized regions are evident in Fig. 5.3, which depend on the magnitude of the strain. This can be related to a different initial concentration of V ¨ Oin the as-grown films, also reflected in the different initial electrical conductivity of the samples [169]. All the as-grown films present very large electrical resistance, being only possible to determine the carrier density for the films on LSAT, LGO and STO. In order to compare the charge density with respect to the degree of strain, an annealing process at 10−6Torr was necessary to achieve a measurable carrier density produced by the formation of oxygen vacancies. The charge carrier concentration, n, determined by Hall measurements, versus the experimental strain is plotted in Fig. 5.4 b). Although npresents a value in the same order for all the samples (≈1020 cm−3), it reaches the maximum value for less stressed samples, i.e. those grown on STO 99 Luc´ ıa Iglesias Bernardo and LSAT. Considering the nominal charge contributed by the 2%Nbdoping, n≈3.27x1020 cm−3(indicated on Fig. 5.4 b)), this shows that samples with more defects (more degree of either positive or negative strain) exhibit a carrier density similar to the expected nominal value. This indicates that more stressed samples exhibit a higher concentration of defects (VSr) which trap the extra charge donated by the oxygen vacancies created in the post-annealing process. -1 0 1 2 0 25 50 GSO DSO STO Strainexp(%) SP(%) LSAT -1 0 1 2 2 4 6 8 KTO LAO NSAT GSO DSO LGO LSAT STO n (1020cm-3) Strainexp (%) a) b) 2% Nb doping Figure 5.4: a) Evolution of ∆SP as a function of epitaxial strain. ∆SP is defined as the difference between the minimum/maximum in the SP signal of the V ¨ Oaccumulation/depletion regions. b) Charge carrier density as a function of experimental strain for some of the samples studied in this Thesis. The dashed line is a guide to the eye. 5.3 Room temperature diffusion coefficient of oxygen vacancies As mentioned before, the understanding of the V ¨ Odiffusion can be particularly interesting for many applications, including solid oxidefuel cells, catalysts, resistive switching memory devices and photoelectrochemistry. For many of these applications, the knowledge of the room temperature diffusion coefficient is crucial. In order to determine the diffusion coefficient of the strained thin 100 5. Controlled movement of oxygen vacancies films, the experimental approach is similar to the one used in the previous Section: First, a rectangular area (3×8µm) of the sample surface is poled with a positive voltage (+10V) and on the right side, another rectangle with the same dimensions is recorded with a negative voltage (-10V). The positive/negative tip bias disturbs V ¨ Odistribution of the whole STO thickness under the poled area. Then, the electric field is set at 0 V, allowing the free relaxation of vacancies at room temperature across the thin film. The time-dependence of the EFM signal is registered during one day, taking an image every 30 minutes. Essentially, the EFM signal will decrease over time, describing how the concentration of vacancies approaches the equilibrium state. As the sample is recorded with positive and negative voltages, the diffusion is studied for both the accumulation and the depletion of V ¨ O(positive/negative concentration gradient). Theoretically, oxygen vacancies could diffuse along and perpendicular to the film surface; hence the effect of surface mobility should also be considered apart from diffusion into the film bulk. However, a careful analysis of the profiles extracted from the time-dependence of the EFM images, shows that the border of the poled regions remains well defined over time (Fig. 5.5 a)). For quantifying the broadening, the EFM profiles were fitted to a Gaussian function, extracting the full width at half maximum (FWHM) at different times and comparing it with the voltage amplitude in the central part of the recorded part; Figures 5.5 b) and c), respectively. From these results, we can conclude that the lateral diffusion would not perturb the EFM signal and the FWMH variation over time is negligible compared with the rapid decrease in amplitude even at short times, justifying the assumption that on average, the diffusion occurs mainly along the thickness of the films. 101 Luc´ ıa Iglesias Bernardo 0 2 4 6 8 10 12 0 25 50 75 Amplitude FWHM % variation Time (h) 0 2 4 6 8 0 200 400 600 800 t=12 h Experimental Fitting Amplitude (mV) Distance (m) t=0 h 1h 2h 3h 4h 8h a) b) c) Figure 5.5: a) 3D plot of the evolution of EFM signal during 8 hours. The size of the area poled is 3 ×8µm. b) Time dependence of EFM profiles over 12 hours, extracted from the images. Orange solid lines are Gaussian fittings in each case. c) Amplitude and FWHM variation versus time, obtained from the fittings. Under these conditions, the second Fick’s law in its one dimensional form can be used to follow the V ¨ Oredistribution and to determine their diffusion coefficient: ∂[V¨ O] ∂t =D∂2[V¨ O] ∂2z2(5.1) where Dis the diffusion coefficient. This law describes how the oxygen vacancies spread over time from a region of higher concentration towards a region of lower concentration, in order to recover the initial homogenous distribution. 102 5. Controlled movement of oxygen vacancies Solving the above equation by the method of separation of variables or by the Laplace transformation, the solution becomes [170,171]: [V¨ O](t) = 1 −8 π2 ∞ X n=0 1 (2n+ 1)2exp −D(2n+ 1)2π2t l2(5.2) The corresponding solution for Eq. 5.2 at very long times (t→∞) is: ∆V¨ O(t) = [V¨ O](t)−[V¨ O(∞)] [V¨ O](0) −[V¨ O(∞)] =8 π2e −π2Dt l2(5.3) where [V ¨ O](t) denotes the average vacancy concentration in the sample surface at any given time t, [V ¨ O](∞) refers to the time required for the vacancy concentration within the enriched/depleted areas to recover equilibrium with the average concentration of the sample surface and [V ¨ O](0) represents the initial V ¨ Oconcentration immediately after recording with positive/negative tip bias. Assuming that the EFM amplitude is proportional to the oxygen vacancy concentration, we can consider Eq. 5.3 to determine the slow diffusion coefficient at room temperature. The results for the timedependent EFM amplitude of a thin film of STO grown on GSO are shown in Fig. 5.6 e-h). In the same figure (a-d), the whole process for the enriched V ¨ Oregion recorded with positive voltage is schematically represented. 103 Luc´ ıa Iglesias Bernardo Figure 5.6: Sketch of the V ¨ Orelaxation (a-d) and the corresponding evolution of the EFM amplitude over time at zero DC bias (e-f). a, e) Immediately after removing the AFM tip bias, b, f) after 2 hours, c, g) after 4 hours and d, h) after 12 hours. The results correspond to a film of STO grown on GSO. By extracting the profiles of the EFM amplitude over the time (Fig. 5.7) and taking the average value of the positive (negative) poled region, we can obtain the decay of the EFM signal for the depletion (accumulation) of V ¨ O, see Fig. 5.7 b). 0 5 10 15 0.00 0.25 0.50 -10V +10V EFM amplitude (mV) t (h) 0 5 10 15 -0.6 -0.3 0.0 0.3 0.6 0.9 0h 2h 4h 8h 12h EFM Amplitude (mV) Distance (m) a) b) +10V -10V DSO Figure 5.7: a) Profile of the EFM amplitude at zero dc bias over time extracted from Fig. 5.6. The results correspond to a film on GSO. b) Decay of the EFM signal for both accumulation (-10 V) and repulsion (+10 V) of V ¨ O. 104 5. Controlled movement of oxygen vacancies By repeating the same process for all the samples with different degrees of strain (both positive and negative), the strain dependence of diffusion coefficient of V ¨ Ocan be extracted using Eq. 5.3. Fittings of the experimental data for two representative cases (films grown on LSAT and GSO) are shown in Fig. 5.8 a). The values of the diffusion coefficient are the averaged values from the initially depleted/accumulated surface regions (±V). The two values show similar results for each film, meaning that the incorporation of atmospheric oxygen to the surface is not an important contribution to the equilibration of the EFM signal (see Fig. 5.7 a). Oxidizing the initially reduced sample surface will require adsorption of oxygen from the atmosphere, dissociation, and diffusion through the lattice. On the other hand, reducing an initially oxidized region will require diffusion of the vacancies towards the surface, recombination (probably forming peroxide-ions) and formation of molecular oxygen. These processes are expected to show quite different activation energies. Therefore, the observation of a similar Dafter applying a positive or negative field suggests that oxygen exchange through the surface is not an important contribution to time-dependent surface voltage in our case, and that oxygen redistribution in the material is the mechanism governing this process. -1 0 1 2 0 2 4 6 8 KTO GSO DSO STO LSAT LAO D (10-17cm2/s) Strain (%) 010 20 30 40 -3 -2 -1 0 GSO D=(5 2) x10-17cm2/s LSAT D=(2 2) x10-17cm2/s ln(VÖ(t)) t (103s) b) a) Figure 5.8: a) Fittings according to Eq. 5.3 of the EFM signal for samples grown on LSAT and GSO. b) Strain dependence of the diffusion coefficient of V¨ Oat room temperature. The dashed line is a guide to the eye. 105 6. TiO6-Octahedral rotations and ferroelectric-like response in strained SrTiO3 thin films People who have it too easy in early life have a disadvantage for later on, because they get to thinking that everything is going to be easy. Mildred Dresselhaus. She was the first woman Institute Professor and Professor Emerita of Physics and Electrical Engineering at the Massachusetts Institute of Technology. Most ABO3perovskites undergo structural distortions associated with the tilt or rotation of the BO6octahedra about one or more of the crystal axes in order to increase the bonding strength and coordination number. As the metal-oxygen-metal (or in general, cation-anion-cation) bond angles can influence the electronic structure of the materials, small changes in the oxygen octahedra tilting have important implications on the electronic and magnetic properties. The A-site substitution with cations of different radii is an effective strategy to tune the degree of the BO6octahedra rotation, which can profoundly affect the spin in113 Luc´ ıa Iglesias Bernardo teraction along the crystal. Thus, the ability to control these rotations in the perovskite structure is very important to understand structureproperty relationships and also to create multifunctional materials with new properties [47,56]. Apart from the change in the cation size, epitaxial strain is an alternative tool for tuning the connectivity of the BO6octahedra [57, 176]. Nonetheless, it is becoming clear that the octahedral rotation mismatch between substrate and film could potentially play a more dominant role in controlling film distortions [177]. In addition to imposing epitaxial strain on the film, the substrate force the BO6octahedra to stay connected, transferring octahedral distortions and tiltings across the interface and imprinting them into the film. This is particularly important considering that most of the commercially available substrates used for the growth of perovskite thin films present symmetries with robust octahedral distortions, such as a−b+c−in orthorhombic scandates or a−a−a−in LAO. Thus, Liao et al. [178] reported the control of the magnetic and electronic properties in manganite heterostructures by transferring the octahedral rotations present in a NdGaO3substrate to the La2/3Sr1/3MnO3thin film. Therefore, a deterministic control over the film distortions necessarily implies the understanding of how the octahedral units in perovskites respond to the strain and to the substrate octahedral rotations [58]. In this regard, one of the emerging properties that can be induced by means of the octahedral rotations is the ferroelectricity. Octahedral rotations by themselves cannot induce ferroelectric polarizations in the perovskite structures, but combined with defects or even strain, they can do so [48, 179]. This relationship has not been extensively studied too much from the theoretical standpoint, due to the complexity of implementing the simulations [180–182] and experimentally, due to the problem to isolate the effect of the different parameters involved in the phenomena. Recently, A. Gruverman et al. [183] reported a ferroelectric-like response in LAO/STO heterostructures mediated by the electric field induced by an AFM tip bias, which produces oxygen vacancy migration, similar to a previous report performed by C. B. Eom et al. [184]. 114 6. Octahedral rotations and ferroelectric-like response Furthermore, although SrTiO3is an incipient ferroelectric that remains paraelectric down to 0 K, epitaxial tensile strain can modify its delicate state, turning it into a ferroelectric [82]. Given these reports, it is therefore reasonable to expect that octahedral rotations as well as strain, cationic and oxygen vacancies could play a major role in controlling the ferroelectricity in STO thin films and more generally, in perovskites. In this Chapter, we will present the results of experiments and abinitio simulations, focusing on the structural properties of the STO thin films. The different octahedral rotation patterns of the films will be determined experimentally as a function of strain, and by means of abinitio calculations we will address their origin. Afterwards, the ferroelectricity in STO thin films under specific conditions will be studied, examining which parameters are the most relevant to induce it, such as strain, octahedral rotations or oxygen vacancies. 6.1 Strain-dependence of octahedral rotation pattern In perovskites, a progressive reduction of the ionic radius at the A site is expected to induce a cooperative rotation of the BO6octahedra, which could reduce the symmetry from cubic (t= 1) to tetragonal (I4/mcm), rhombohedral (R¯ 33c, rotation along [111] axis), and orthorhombic (Pbnm or Pnma, rotation along [110] axis) as tdecreases (see Chapter 1, Eq. 1.1 for more details). STO is cubic at room temperature having no octahedral rotations (a0a0a0) and t≈1. Below 105K, a phase transition occurs in STO, which becomes tetragonal with an out-of-phase rotation along the c-axis (a0a0c−) [185]; this is also true for 2%-Nb:STO. Nevertheless, the presence of VSr could introduce a distortion similar to an average reduction of the ionic radius of the A cation, which coupled with the effect of strain (and its cooperative nature) can drastically change the configuration of octahedral rotations of the material. In addition to epitaxial strain and defects, an alternative interfacial coupling effect can also take place when an octahedral rotation mismatch 115 Luc´ ıa Iglesias Bernardo between substrate and film exists. The mismatch of atomic displacement at the heterointerface is generally accommodated by either deforming or rotating the oxygen octahedra in the film. By determining the octahedral rotation pattern in the films, we can investigate whether such rotations are controlled by the strain, cationic vacancies or the rotation pattern of the substrate underneath. The different structures and octahedral distortions of the substrates are summarized in Fig. 6.1. Both GSO and DSO present orthorhombic structure with a distortion compatible with the space group Pnma, whereas LAO is rhombohedral with rotations described by the a−a−a−(R3cspace group) tilting pattern in Glazer notation [59]. LSAT, STO and KTO possess cubic structure without any rotations (a0b0c0). LAO Rhombohedral a-a-a- (-2.91%) LSAT Cubic a0a0a0 (-0.95%) STO Cubic a0a0a0 (0%) DSO Orthorhombic a-b+c- (+1.15%) KTO Cubic a0a0a0 (+2.14%) Figure 6.1: Summary of the phase and octahedral distortions of the substrates studied in this Chapter, indicating the precise octahedral rotation pattern in each case. The nominal degree of strain induced in STO thin films grown on these substrates is also indicated in each case. The different octahedral rotation patterns can be extracted by carefully investigating the presence or absence of X-Ray half-order reflections in the thin films, as explained in Chapter 2. On the compressive side, the films exhibit in-phase rotation along the aand baxes and lack of rotation along caxis, a+b+c0(Immm space group), independently of the BO6rotation pattern of the substrate (Fig. 6.2). Consequently, we can conclude that under compressive strain the substrate does not impose its own octahedral pattern on the films. 116 6. Octahedral rotations and ferroelectric-like response LAO (a-b-c-) STO thin film a+b+c0 LSAT (a0a0a0) STO thin film a+b+c0 a) b) c) d) e) f) LAO LAO a+ (0 1/2 3/2) b+ (1/2 0 3/2) a+ (0 1/2 3/2) b+ (1/2 0 3/2) Figure 6.2: High resolution reciprocal space maps around half-order reflections for compressive Nb:STO thin films. The reflections correspond to the rotations pattern characteristic of a+b+c0found for samples grown on LAO (a,b) and LSAT (d,e). c,f) Sketches showing the distortions of the films and substrates. However, the film grown under tensile strain on an orthorhombic substrate with robust distortions (DSO, a−b+c−) exhibits the same octahedral configuration than the substrate underneath (see Fig. 6.3). Therefore, the substrate has imprinted its characteristic octahedral rotation pattern into the film. However, the film grown on cubic KTO does not show hints of any octahedral rotations, despite of the large tensile strain applied. This indicates that the interfacial coupling could play a dominant role in controlling the octahedral rotations pattern of the film under tensile stress. Surprisingly, the film deposited on an STO substrate, in which the lattice mismatch can be considered negligible, presents again the a+b+c0configuration compatible with a tetragonal distortion and described by Immm space group (Fig. 6.4 a,b). Because from our experiments it is not possible to determine the precise amplitude of the octahedral rotation, which could be the same along the aand baxes, 117 Luc´ ıa Iglesias Bernardo the tilt system can also be described as a+a+c0, corresponding to the I4/mmm space group. STO thin film a-b+cDSO (a-b+c-) a) b) c) DSO DSO DSO a- (1/2 1/2 3/2) b+ (1/2 0 3/2) c- (3/2 1/2 3/2) d) Figure 6.3: a-c) High resolution reciprocal space maps around half-order reflections for the sample grown on DSO under tensile strain. The reflections correspond to the rotation pattern a−b+c−characteristic of the space group Pnma. d) Sketch showing the distortions of the film and the substrate. Given the negligible lattice mismatch between the Nb:STO film and the STO substrate, the distortion could be induced by the presence of VSr throughout the lattice of the film, which are introduced unintentionally during the PLD deposition process. Note also that the clear observation of the x-ray half order reflections in all cases implies that the tilting of the TiO6octahedra is propagated along the whole sample, not just around a local vacancy. In addition, as can be seen in Fig. 6.4 c,d), the rotation pattern does not change in films grown at very low PO2, i.e. with a higher concentration of oxygen vacancies. Therefore, oxygen vacancies can be ruled out as the main source of the distortion. The presence of VSr introduces a distortion similar to an average reduction of the ionic radius of the A cation. This decreases the tolerance factor below 1 and subjects the Sr-O (Ti-O) bonds to tensile (compressive) strain, resulting in a rotation of the TiO6octahedra to accommodate this stress. The compressibility of Sr-O bonds is often larger than the one of Ti-O bonds [53], so that compressive stress goes in the 118 6. Octahedral rotations and ferroelectric-like response same direction as the reduction of the ionic radius. Consistent with this hypothesis, the same rotation pattern was observed for all the films grown under compressive stress, either on cubic LSAT or orthorhombic LAO, and also for the unstrained film on STO substrate due to the presence of Sr2+ vacancies. STO substrate (a0a0a0) STO thin film a+b+c0 STO 100mtorr STO 100mtorr STO 1x10-6 torr STO 1x10-6 torr a+ (0 1/2 3/2) b+ (1/2 0 3/2) a+ (0 1/2 3/2) b+ (1/2 0 3/2) a) b) c) d) e) Figure 6.4: High resolution reciprocal space maps around half-order reflections for unstrained Nb:STO thin film on STO substrate. a, b) As-grown film at 100 mTorr of oxygen pressure and c,d) film after a post-annealing procedure at 1×10−6Torr. The rotation pattern observed corresponds to the space group Immm (a+b+c0) or I4/mmm (a+a+c0). e) Sketch showing the distortions of the film and the substrate. Nevertheless, positive stress dominates over the effect of VSr, imposing on the films the rotation pattern of the crystal underneath. These results demonstrate that different octahedral rotation patterns are induced in thin films prepared under similar conditions (identical composition) but subject to different degrees (and sign) of epitaxial stress (i.e., deposited on different substrates). This particular tuning of the octahe119 Luc´ ıa Iglesias Bernardo dral configuration by means of strain, substrate rotations and/or cationic vacancies has not been previously reported neither theoretically nor experimentally for Nb:STO thin films. The next step will be to understand the mechanism that produces it. Once the mechanism is understood, experiments can be designed in order to control it. 6.1.1 Origin of the distortions: ab-initio calculations To determine the influence of the strain, cationic vacancies or substrate distortions on the different octahedral configuration of the STO film, we performed DFT calculations and coupled the results with the experiments. Computational details: The calculations were carried out using the Wien2k code and the generalized gradient approximation (GGA) in the PBE scheme. The values of the atomic sphere radii (Rmt) were chosen as 1.87 a.u. for Ti, 2.5 a.u. for Sr and 1.69 a.u. for O. We used a plane wave cut-off described by RmtKmax= 6, and the values for the kmesh were 4×4×4 sampling of the full Brillouin zone. A pseudocubic 2×2×2 supercell consisting of 40 atoms (8 Ti, 8 Sr and 24 O) was used. Removing an oxygen in the structure to simulate the effect of an oxygen vacancy represents a concentration close to 4%, which is comparable to the experimental values (ranging from 2 to 4%) [106]. In order to find the appropriate computational parameters to describe our system in the best possible manner, we decided to relax the initial structure without rotations. Thus, we carried out a simple calculations for the supercell without any rotations or oxygen vacancies, optimizing the volume, and the results can be seen in Fig. 6.5 for GGA-PBE functional. The same calculations extracting an oxygen atom were also performed. For this case, the optimized lattice constant is also overestimated compared to the bulk parameter of the STO (a=3.905 ˚ A). However, the disagreement between lattice constant and experimental values is less than 1% (less than 3% in volume), which is in good agreement 120 6. Octahedral rotations and ferroelectric-like response with the reported in the literature [186,187]. The increase in volume for the system with the oxygen vacancy is fully consistent with the unit cell expansion due to the larger ionic radius of the Ti+3 (0.81 ˚ A) compared to Ti+4 (0.75 ˚ A). 58 59 60 61 62 63 0 50 100 257100 257150 257200 Stoichiometric Oxygen vacancy E (meV/Ti) Volume (Å3) Figure 6.5: Total energy as a function of the cell volume for a 2×2×2 supercell of stoichiometric STO (solid squares) and with an oxygen vacancy (open squares). The energies were determined taking as a reference the ground state of stoichiometric STO. As a second step, we introduced the three possible octahedral rotation patterns observed experimentally (a+a+c0, a−b+c−and a0b0c0) in the structure with 40 atoms and strain between -6% to +6%. The simulations were carried out using lattice constants very similar to those obtained experimentally in STO thin films (see Chapter 4). Consequently, a=bwere set according to the degree of strain chosen in each case, taking as a reference the experimental lattice parameter of STO in bulk (a=3.905 ˚ A), and cwas determined assuming that the Poisson coefficient is conserved, i.e. ν= 0.23 (more details in Chapter 4 and Fig. 4.5). Regarding the octahedra rotation angles in the structures, given that there are no experimental data about it, a similar value (4◦) to those reported in the literature has been taken [188,189]. During the simulations, both in-plane and out-of-plane lattice constants were locked and only the internal coordinates of the ions were relaxed until the conver121 Luc´ ıa Iglesias Bernardo exhibit a clear distinguishable minimum independently of the sign and magnitude of strain. Nevertheless, the energy minimum is uncertain for PBE functional, for positive and negative strain values. For example, in Fig. 6.10 a) the energy curve corresponding to -3% of strain presents a flattening for high values of c, being difficult to determine the minimum unambiguously. -5 0 5 10 15 20 -39.7 -39.6 -39.5 -39.4 -39.3 -39.2 s = -3% s = -2% s = -1% E (eV) c (%) -6 -4 -2 024 -43.8 -43.7 -43.6 s = -3% s = -2% s = -1% E (eV) c (%) -4 -2 0246810 -33.1 -33.0 -32.9 -32.8 -32.7 -32.6 s = -3% s = -2% s = -1.5% s = -1% s = -0.5% E (eV) c (%) -4 -2 0246 -39.7 -39.6 -39.5 -39.4 s = +3% s = +2% s = +1% E (eV) c (%) -6 -4 -2 0 2 4 -43.8 -43.7 -43.6 -43.5 -43.4 s = +3% s = +2% s = +1% E (eV) c (%) -4 -2 0 2 4 -33.1 -33.0 -32.9 -32.8 s = +3% s = +2% s = +1.5% s = 1% s = 0.5% E (eV) c (%) LDA PBEsol PBE a) b) c) d) e) f) Figure 6.10: Energy versus caxis value after the relaxation setting the polarization along the out-of-plane direction. Simulations were performed for PBE (a,d), LDA (b,e) and PBESol (c, f) functionals. Comparable calculations performed fixing the in-plane polarization do not exhibit this artifact independently of the XC functional used, as can be seen in Fig. 6.11. By plotting the optimized c-axes obtained for out-of-plane and in-plane polarizations as a function of the a-axis, a clear overestimation of cis evident by PBE functional and out-of-plane polarization, especially for compressive strain (Fig. 6.12 d). This artifact is called supertetragonality and it has been previously reported in the literature [199, 200]. The lattice distortion that we introduced is driven by the Ti off-center displacement along the respective polarization directions ([001] and [110]). This produces an imbalance between two nonequivalent Ti-O bonding interactions: shorter Ti-O has the more 128 6. Octahedral rotations and ferroelectric-like response covalent character, while longer Ti-O bond presents ionic character. LDA functional tends to homogenize the electron density, smearing out the differences between the shorter and longer Ti-O bonds. Therefore, the Ti off-center displacement and the distortion of the lattice parameters are underestimated. -6 -4 -2 0 2 4 -43.8 -43.7 -43.6 s = -3% s = -2% s = -1% E (eV) c (%) -4 -2 0 2 4 6 -39.6 -39.5 -39.4 -39.3 s = -3% s = -2% s = -1% E (eV) c (%) -6 -4 -2 0 2 4 -43.8 -43.7 -43.6 s = -3% s = -2% s = -1% E (eV) c (%) -2 0 2 4 6 -39.7 -39.6 s = +3% s = +2% s = +1% E (eV) c (%) -6 -4 -2 0 2 4 -43.8 -43.7 -43.6 -43.5 s = +3% s = +2% s = +1% E (eV) c (%) -4 -2 0 2 4 -33.1 -33.0 -32.9 -32.8 s = +3% s = +2% s = +1.5% s = +1% s = +0.5% E (eV) c (%) LDA PBEsol PBE a) b) c) d) e) f) Figure 6.11: Energy versus caxis value after the relaxation setting the polarization along the in-plane direction. Simulations were performed for PBE (a,d), LDA (b,e) and PBESol (c, f) functionals. However, PBE is designed to soften the overestimated bonding calculated by LDA, favouring stronger bondings (i.e. covalent and metallic bonds) and thus underestimates the relatively weak bonds (i.e. ionic bonds). Thus, the short Ti-O bond is energetically favoured by PBE, producing the supertetragonality artifact [199]. Similar phenomenology is also observed for PBESol, although much less pronounced and only in the case of -3% of compressive strain. It is worth noting that by comparing these results with the experimental values (Fig. 6.12 df), PBESol is clearly the functional that provides the best comparison with experimental values and therefore, this will be the XC functional chosen to perform the simulations from now on the rest of the Chapter. 129 Luc´ ıa Iglesias Bernardo -3 -2 -1 0 1 2 3 -39.7 -39.6 -39.5 -39.4 Polarization 001 Polarization 110 Ground state Fitting E (eV) s (%) -3 -2 -1 0 1 2 3 -43.8 -43.7 -43.6 Polarization 001 Polarization 110 Ground state Fitting E (eV) s (%) -3 -2 -1 0 1 2 3 -33.1 -33.0 -32.9 Polarization 001 Polarization 110 Ground state Fitting E (eV) s (%) 3.8 3.9 4.0 3.8 3.9 4.0 Polarization 001 Polarization 110 Experimental c (Å) a (Å) 3.8 3.9 4.0 3.8 3.9 4.0 4.1 Polarization 001 Polarization 110 Experimental c (Å) a (Å) LDA PBEsol PBE a) b) c) d) e) f) 3.8 3.9 4.0 3.8 4.0 4.2 4.4 4.6 Polarization 001 Polarization 110 Experimental c (Å) a (Å) Figure 6.12: Energy versus strain for optimized values of cat each value of strain using PBE (a), LDA (b) and PBESol (c). Optimized value of cas a function of the a-lattice parameter for PBE (d), LDA (e) and PBESol (f). All simulations were performed for out-of-plane (solid symbols) and in-plane polarization (open symbols). In d-f) the experimental values are also plotted for comparison with the theoretical predictions (solid orange circles). The atomic displacements between Ti-O, which indicates the offcentering of Ti cation, were determined in the relaxed structures using the PBESol and is shown in Fig. 6.13 a) as a function of strain. The out-of-plane polarization is favoured by compressive strain, whereas it vanishes under large tensile strain. On the contrary, in-plane polarization exhibits the opposite trend. Moreover, Fig. 6.13 b) shows the dependence of the polarization calculated by Berry phase method on epitaxial strain. The maximum values of the polarization are close to 40 µCcm−2for out-of-plane polarization under -3% of compressive strain and in-plane polarization for +3% of tensile strain. These polarization values are comparable with those estimated by Berry phase formalism for (111) epitaxially strained BaTiO3(Ps=25 µCcm−2) and lower than PbTiO3(Ps=80 µCcm−2). 130 6. Octahedral rotations and ferroelectric-like response -3 -2 -1 0 1 2 3 0 10 20 30 40 Polarization 001 Polarization 110 P (C/cm2) s (%) -3 -2 -1 0 1 2 3 0.00 0.05 0.10 0.15 0.20 0.25 Polarization 001 Polarization 110 Ti-O displacement (Å) s (%) a) b) Figure 6.13: Relative Ti-O displacement (a) and polarization (b) as a function of strain determined for the relaxed structures, for the out-of-plane and in-plane polarizations. The spontaneous polarization were calculated using the Berry phase method, taking the difference between a paraelectric structure (SrTiO3unit cell without any distortions) and the ferroelectric structures after the relaxation. The next step is considering space groups with distinct octahedral tilting and polarization direction, in search of the ground state and the complete phase diagram of the stoichiometric system. The space groups used are summarized in Table 6.2, following the notation of Stokes et al [201]. This notation expresses in a compact way both the tilt system and the distortion caused by the displacement of the cations. The superscripts denote the octahedra tilts, while the subscripts +and −mean ferroelectric B-cation (Ti in this case) displacements along a given axis. The space group with the rotations observed in the STO below 105 K (i.e. a0a0c−) is among the phases studied, considering the polarization in the [110] and [100] directions. Furthermore, the structure that represents the octahedral rotation pattern experimentally observed for compressive strain is chosen (I4mmm), considering in-plane polarization (I4mm) and out-of-plane polarization (Fmm2(II)). Space groups Fmmm and Ima2(II)with rotations out-of-phase around the in-plane axes are included for comparison. The total energies of the superlattices with different symmetries are calculated varying the strain from -2% to +2% (Fig. 6.14). The lattice parameters used are those previously optimized using PBESol 131 Luc´ ıa Iglesias Bernardo Space group Rotations Polarization System Noatoms I4/mcm a0a0c−000 a0 0a0 0c− 020 Ima2 a0a0c−pp0 a0 +a0 +c− 020 Fmm2 a0a0c−p00 a0 +a0 0c− 040 Fmmm a−a0c0000 a− 0a0 0c0 040 Ima2(II) a−a−c0pp0 a− +a− +c0 020 I4mmm a+a+c0000 a+ 0a+ 0c0 040 I4mm a+a+c000p a+ 0a+ 0c0 +40 Fmm2(II) a+a+c0pp0 a+ +a+ +c0 040 Table 6.2: Space groups considered in this Section. The principal characteristics of these phases are also indicated: octahedral rotation pattern around the pseudocubic axes, Cartesian coordinates of the polarization vector and the atoms that composed the supercell in each case. functional (see Fig. 6.12), which are closest to the experimental results. In absence of strain, the ground state is the non-polar I4/mcm phase, which represents the structure of STO at low temperatures, overlapping in energy with the polar structures with the same octahedral rotation pattern (Ima2and Fmm2) and the space groups Fmmm and Ima2(II) with out-of-phase and in-plane rotations. However, the structures observed experimentally are less stable, having higher energy. In addition, the variation of the rotation angles also shows a trend with strain. Space groups with out-of-plane tilts increase the degree of rotation for compressive stress and decrease for tensile strains. By contrast, space groups with a+a+c0present the opposite trend, meanwhile Fmmm and Ima2(II)phases do not show appreciable variations in the range of strains studied (see Fig. 6.14 b). For compressive strain, the phase diagram splits in three branches with clearly differentiated energies, each of them containing the systems with the same rotation pattern regardless of whether they are polar or non-polar. The space groups with a0a0c−configuration are the most stable. Then, the symmetries with out-of-phase and in-plane rotations 132 6. Octahedral rotations and ferroelectric-like response (Fmmm and Ima2(II)) are higher in energy (about ≈30 meV) with respect to the first ones. And lastly, the group of the phases with a+a+c0 are the least favourable configuration among those studied, although this is the configuration experimentally observed. Therefore, the octahedral configuration seems to play a major role, being quite difficult to stabilize a tilt system different from a0a0c−under compressive strain. The situation for tensile strains is completely different. Ima2(II) is the most stable phase for all the range of positive strain studied. However, the other phases are very close in energy, despite of them being less stable, and well-defined branches cannot be observed as in the case of compressive stress. This remarkable result indicates the existence of phases with very similar energy, suggesting that metastable phases could exist in the phase diagram for tensile strain. Therefore, the properties of the samples under positive stress could be very sensitive to growth conditions and also to the octahedral rotation pattern imposed by the substrate, which coincides with experimental observations. -2 -1 0 1 2 0 25 50 75 100 I4mcm Ima2 Fmm2 Fmmm Ima2II I4mmm I4mm Fmm2II E (meV/Ti) s (%) -2 -1 0 1 2 0 3 6 9 () s (%) a) b) Figure 6.14: a) Free energy of the different phases considered in this Thesis and detailed in Table 6.2 as a function of strain. The energy of high-symmetry P4/mmm phase is taken as reference. b) Average rotation angles determined after the relaxation for the space groups considered in this Section. The angles are determined for the axes around which octahedral rotation exists in each space group. The legend is common to both panels. After the atomic relaxation, the total polarization was estimated by 133 Luc´ ıa Iglesias Bernardo using either BEC and Berry phase method. The results obtained with BEC are shown on Fig. 6.15. For clarity, the space groups are separated in 3 groups, which possess the same rotations although different initial polarization directions. The first group characterized by the a0a0c−rotation pattern exhibit in-plane polarization for tensile strain (Fig. 6.15 a)), similar result is obtained for the second group (Fmm2and Ima2II, Fig. 6.15 b). Nevertheless, the third group, which is composed of the symmetries with the rotations observed experimentally (a+a+c0), shows polarization for both compressive and tensile strain. Under tensile stress, the symmetry denoted as Fmm2(II)preserves the in-plane polarization, whereas the I4mm (a+ 0a+ 0c0 +) symmetry is the only one that can stabilize the spontaneous polarization for compressive strain (out-ofplane polarization). The latter result is completely in agreement with the experimental evidences, which point out to the appearance of a FE response only for samples with the a+a+c0octahedral configuration. Moreover, these results are in accordance with other studies already reported in the literature for STO [202, 203], which consider some of the space groups studied in this Thesis. The ferroelectric instability of perovskite compounds is dependent on the volume and more generally strongly affected by strain. Since ferroelectricity is a collective phenomenon, when an atom is displaced from its position, it feels a force that tries to brings it back to its initial position. Moreover, when the volume is reduced, the ferroelectric instability is progressively suppressed. For this reason, it is very difficult to retain the polarization for compressive strain. It is worth noting that despite of the high distortion imposed to the different space groups due to the large negative or positive strain, the symmetry after the relaxation is conserved in all the cases studied. This was verified using the free software ISOCIF [204]. Since the BEC method often fails to provide numerical values in agreement with those reported experimentally, the spontaneous polarization has also been estimated by means the Berry phase method, which is based on a more sophisticated theoretical scheme and gives more accurate results compared to experiments [205–207]. 134 6. Octahedral rotations and ferroelectric-like response -2 -1 0 1 2 0 2 4 6 Fmmm Ima2II P (C/cm2) s (%) -2 -1 0 1 2 0 2 4 6 I4mmm I4mm Fmm2II P (C/cm2) s (%) -2 -1 0 1 2 0 2 4 6 I4mcm Ima2 Fmm2 P (C/cm2) s (%) a) b) c) -2 -1 0 1 2 0 10 20 30 40 Ima2 Fmm2 P (C/cm2) s (%) -2 -1 0 1 2 0 10 20 30 40 Ima2II P (C/cm2) s (%) -2 -1 0 1 2 0 10 20 30 40 I4mm Fmm2II P (C/cm2) s (%) d) e) f) Figure 6.15: Spontaneous polarization estimated by BEC (a-c) and Berry phase approach (d-e) for the symmetries studied in this Section. The results are grouped by space groups with similar rotations but different initial polarization, a, d) a0a0c−octahedral configuration, b, e) a−a0c0or a−a−c0and c, f) a+a+c0pattern. Despite of the problems of the BEC method to provide accurate numerical values, the trend predicted by this model is reliable, so that only symmetries and strains that already exhibited polarization by the BEC method were also studied through the Berry phase formalism. The results are shown on Fig. 6.15 d-f), the values for 2% tensile strain ranging from ≈20 to ≈33 µC/cm2and for compressive strain values of ≈17 µC/cm2for -2% of strain and ≈10 µC/cm2for -1% are obtained. These values are slightly higher than other experimentally reported, for example, in SrTiO3ultrathin films deposited on Si (001) (P∼9µC/cm2) [208] and comparable with the value of ∼20-30 µC/cm2for the BaTiO3 [200,209], which is the exemplary room temperature ferroelectric. However, the values of the spontaneous polarization are far from those of other typical ferroelectric materials, such as BiFeO3(P∼80-100 µC/cm2) [210–212] or PbTiO3(P∼70-80 µC/cm2) [200]. 135 Luc´ ıa Iglesias Bernardo The results can be summarized in the phase diagram shown in Fig. 6.16, from which two main conclusions can be extracted: i) tensile strain favours in-plane polarization along the direction [110] (or [100],[010]) for several symmetries with different octahedral rotation patterns. In-plane polarization is not observed in any case under compressive strain; ii) out-of-plane polarization can only be stabilized under compressive strain and for the symmetry with a+a+c0octahedral configuration, which is completely in agreement with the experimental observations of the previous Section. Therefore, we can conclude that octahedral rotations could be the key factor to stabilize out-of-plane polarization in STO under compressive strain. -2 -1 0 1 2 Polarization s (%) a+a+c0 a-a-c0 a0a0ca+a+c0 Pz Pxx Figure 6.16: Phase diagram summarizes the results of the simulations for the different space groups considered in this Thesis. Compressive strain only stabilizes out-of-plane polarization with the a+a+c0rotation pattern, however several octahedral configurations can exhibit in-plane polarization for tensile stress. Arrows indicate the direction of the polarization. 6.2.2 Oxygen vacancy influence on the polarization: abinitio calculations Once the effect of the different rotation patterns on the ferroelectric response was examined, we introduce an oxygen vacancy in the simula136 6. Octahedral rotations and ferroelectric-like response tions to study the interplay between rotations, strain, defects and FE. In ferroelectric materials, there is a delicate balance between the shortrange repulsion, which favours the non-polar structure and the longrange Coulomb interaction which favours the ferroelectric state. Free carriers, as those donated by the oxygen vacancies, can screen out the long-range Coulomb interaction, reducing the polar distortion. Unfortunately, the most oxygen-deficient systems present a metallic band structure where both the Born effective charge calculation and Berry phase method fail [213]. Nevertheless, in practical ferroelectrics, polarization still exists in spite of the inevitable presence of oxygen vacancies and the material being semiconductor or insulator. The idea of our approach here is to demonstrate that the combination of rotations and oxygen vacancies may give rise to a polar distortion. To demonstrate this premise, the simulations in this Section were performed for all the space groups studied previously. As starting structures we took those already relaxed in the previous Section, and a new relaxation was carried out introducing a vacancy in each symmetry. Furthermore, two scenarios were considered to explore the influence of the vacancy site on the energetics: one with an apical oxygen vacancy and other with an equatorial one. In addition, the unit cells were doubled with respect to those used in the previous Section to ensure a concentration of oxygen vacancies similar to experiments and test the long-range distortions caused by the presence of an oxygen vacancy. Thus, the cells with 20 and 40 atoms were doubled until reach 40 and 80 atoms, respectively (oxygen concentration of 2% and 4%). The k-mesh was adjusted for the supercells of 40 and 80 atoms to 4×4×4 and 3×3×4, respectively. In general, removing an oxygen atom from the lattice pushes the cations (Sr and Ti) away from the vacancy and attracts the anions (O), to redistribute the charge and minimize the forces. The results of the relative energies of all the space groups as a function of strain after the atomic relaxation are shown in Fig. 6.17. As a general trend, under tensile stress, the effects of an apical or an equatorial oxygen vacancy are not reflected in the total energy, presenting a very close energy for both phases or even being indistinguishable the effect of the vacancy 137 Luc´ ıa Iglesias Bernardo acteristic low temperature upturn in the electrical resistivity of the 2DEG at the (001) LAO/STO interface, which is commonly associated to Kondo effect [219,220]. In this regard, Rice et al. [221] observed the emergence of a net magnetic moment in oxygen-deficient STO crystals below T≈18 K, after irradiation of the crystal with polarized light (λ=405 nm). The optically induced magnetization persisted for hours below 10 K, and only occurs in crystals containing oxygen vacancies, demonstrating the coupling between magnetism and ionic defects in STO. On the other hand, a Kondo-like phenomenology was also reported in electrolyte-gated STO crystals, which suggests that this effect (whether it is Kondo effect or not [222]) may be related to the interaction between the conduction electrons and the local Ti+3 moments inevitably associated to the presence of oxygen vacancies in SrTiO3[223]. Therefore, a deeper understanding of the effect of oxygen vacancies in the magnetotransport properties of STO thin films becomes very important, in particular with the aim to elucidate the role of defects on the phenomenology observed in the LAO/STO interface [5,157,224,225]. In this Chapter we will describe the main results of our study of the magneto-transport properties of the STO thin films fabricated under different conditions. The resistivity and magnetoresistance of the films under different degree of epitaxial strain is discussed, as well as the possible existence of Kondo effect and other alternative possibilities to explain this interesting phenomenology. 7.1 Out-of-plane magnetoresistance In 2010, Caviglia et al. [226] reported the observation of Rashba effect in the 2DEG confined at the (001) LAO/STO interface. The lack of interfacial inversion symmetry of the heterostructure originates a large electric field (E) perpendicular to the plane in which the 2DEG is restricted to move. As a result, the conducting electrons will experience a magnetic field (the spin-orbit field, B=E×p/mc2) which couples to its spin (the spin-orbit coupling), and produces a splitting of the bands 144 7. Transport properties of strained SrTiO3thin films at the Fermi energy (EF) for opposite spin momentum. What is very interesting is that because the strength of the Rashba spin-orbit coupling depends on the electric field at the interface, it can be tuned through a voltage in a field-effect transistor configuration. Using this approach, Caviglia et al. [226] as well as Ben Shalom et al. [227] tuned the spinorbit interaction in the 2DEG until it condenses into a superconducting state below ≈0.3-0.4 K. This was a remarkable discovery, and suggested an unconventional order parameter related to some type of spin fluctuations to explain superconductivity in STO, like in heavy-fermion superconductors [228,229]. However, it must be remembered that superconductivity was already reported in oxygen deficient SrTiO3in 1964 by Schooley, Hosler, and Cohen [69]. Later on, Baratoff and Binning observed the simultaneous reduction of Tcand the contribution of the longitudinal-optical (LO) phonon modes to the resistivity of Nb:STO [230], and suggested the main role played by polar modes in the superconducting state of this oxide. Recent experiments in strained thin films of (La,Sm):SrTiO3 reinforced the conclusion of a proximity between a ferroelectric and a superconducting phase in e-doped STO [231]. But more importantly, irrespective of the nature of the superconducting state, it has been demonstrated that lightly e-doped (oxygen deficient) STO shows a Tc≈0.25 K-0.6 K, very similar to the superconducting critical temperatures reported for E-gated LAO/STO interfaces. In Chapter 5 we showed how a local electric field may modulate the local concentration of oxygen vacancies and therefore, the local electrical conductivity. The effect of an electric field applied to the interface in a field-effect transistor configuration, may therefore modify the local concentration of oxygen vacancies, which accumulate at the interface to the concentration required to form the superconducting state. The effect of the electric field in this scenario will be to change the local electrondensity, moving Tcacross the superconducting dome reported by other authors in oxygen deficient STO [230]. In view of this possibility, the origin of the superconducting state and the conclusions about the role of Rashba spin-orbit coupling in the transport properties of LAO/STO interfaces should be revised. 145 Luc´ ıa Iglesias Bernardo To probe this hypothesis, we measured the electrical resistance and charge density (n) as a function of temperature for a series of samples with different thicknesses deposited on LAO and LSAT (s=−1.15% and s=−0.95%, respectively), as can be seen in Fig. 7.1. -9 -6 -3 0 3 6 9 -6 -3 0 3 6 5K 10K 15K 25K 50K Rxy () H (T) 110 100 3 4 5 6 110 100 1 15 20 LAO 35nm LAO 19nm n (1020cm-3) T (K) LAO 35nm LAO 19nm LSAT 19nm LSAT 19nm (mcm) T (K) a) b) Hop Hip q c) d) Figure 7.1: a) Sketch of the set-up used to measure the resistivity and the magnetoresistance of the samples, with the magnetic field applied in-plane (Hip) or out-of-plane (Hop). θindicates the angles between the in-plane magnetic field and the current that flows along the Hall bar. b) Hall resistance for the 19 nm film grown on LAO, at different temperatures. c) Temperature dependence of resistivity and d) carrier density, for thin films on LSAT and LAO. The thickness of each film is indicated in c) and d). The arrows in c) indicate the temperature of the resistivity minimum (Tmin). The temperature dependence of the resistivity exhibits a metalliclike behavior at high temperature (T>Tmin), with an increasing number of carriers as temperature increases, which suggests a thermally activated contribution. At low temperatures (T<Tmin), the resistivity shows an upturn similar to a Kondo effect, irrespectively of the substrate and/or thickness of the sample. Since the samples were deposited side-by-side, a similar density of oxygen vacancies is expected a priori. However, the larger carrier 146 7. Transport properties of strained SrTiO3thin films density and smaller rise of the resistivity in the film deposited on LSAT suggests a smaller density of defects in this sample, with respect to those grown on LAO. In this regard, Bergmann [232, 233] showed that the resistance of two-dimensional electronic conductor shows deviations from classical conductivity (Drude formula) caused by quantum corrections. This effect due to interference of self-crossing trajectories between random scattering events, resulting in an increased resistivity, the so-called Weak Localization (WL) effect [234]. Given that the probability of these selfcrossing trajectories decreases as dimensionality increases, the WL contribution is more relevant in the conductivity of thin films. Also, the disorder created by the presence of oxygen vacancies would act as random scattering sites, resulting in a diffusive rather than ballistic propagation of the electrons in the system. Inelastic electron-electron (e−e) or electron-phonon (e−ph) scattering also reduce the number of coherent scattering events, and therefore, the number of loops contributing to WL. The total change in the conductance due to WL, taking into account the inelastic scattering time τiis therefore [234]: ∆σWL σ∝ln τe τi(7.1) where τeis the elastic scattering time. What is important for our discussion is that the time-reversed paths of WL are a consequence of preservation of the time reversal symmetry. Therefore, the application of a perpendicular magnetic field Bthat breaks this symmetry also reduces the probability of WL, and restores partially the conductance (positive magnetoconductance). On the other hand, the presence of a strong spin-orbit coupling modifies completely this picture: the spin of the diffusive electron rotates and randomizes as it travels around the self-intersecting trajectories, which leads to destructive interferences. This is the so-called Weak Anti-localization (WAL) effect. In this case, a perpendicular B suppresses this positive contribution, and therefore decreases the conductance (negative magnetoconductance). Thus, taking all these contributions to σinto account, the shape of 147 Luc´ ıa Iglesias Bernardo the magnetoconductance curves at a given temperature depends on the relative magnitude of their characteristic relaxation times, τi,τeand the spin-orbit scattering time (τso) (see Fig.7.2). 𝜏 𝑠𝑜 𝜏 𝑒𝑙 < 𝜏 𝑖< 𝜏 𝑠𝑜 𝜏 𝑒𝑙 < 𝜏 𝑠𝑜 < 𝜏 𝑖 𝜏 𝑠𝑜 ≃ 𝜏 𝑒𝑙 < 𝜏 𝑖 B sxx WAL WAL/WL WL Figure 7.2: The electrical conductance for a ≈2D conductor as a function of the applied magnetic field. Three regimes can be observed depending on the relative value of the different scattering times: i) for τe<τi<τso, WL dominates showing positive conductance (green curve); ii) for τe<τso<τi, crossover between WL and WAL regimes, i.e. a negative conductance appears at low field followed by a positive one at higher fields (red line); and iii) for τso≃τe<τi, dominating WAL results in a negative conductance in the whole range of B (blue line). We measured the electrical resistance (1/σ) at different temperatures under the influence of an increasing perpendicular magnetic field for a series of 19 nm-thick films deposited on LSAT and LAO. The experimental results are shown in Fig. 7.3. For better comparison with the theory, we present the results of the magnetoconductance (MC), which is defined as follows: MC =δσ(B) σ0 =δσ(B) e2 πh (7.2) where σ0is the quantum of conductance and δσ(B)=σ(B)-σ(0). 148 7. Transport properties of strained SrTiO3thin films The sample grown on LSAT shows a completely positive MC; only small hints of negative values can be seen at the very lowest fields, which are within the error of the measurement (see Fig. 7.3 a). This is consistent with WL effect with a weak spin-orbit coupling (see also Fig. 7.2). On the other hand, the sample deposited on LAO exhibits a negative cusp at low magnetic fields (<3 T), which is a characteristic feature of WAL effect. This cusp is more pronounced at low temperatures and disappears as the temperature increases (see Fig. 7.3 b). Hence, this sample shows a crossover between WAL and WL regimes, suggesting an increasing spin-orbit coupling with respect to the film deposited on LSAT. Rashba spin-orbit coupling produces a magnetic field perpendicular to the direction of the motion of the electrons. If the spin of the electron is not aligned with this internal magnetic field, it precesses at a given frequency, ωL. This is the origin of the Dyakonov–Perel spinrelaxation, in which the spin relaxation time varies inversely with the elastic scattering time (τSO ∝1/(ωLτe) [235]. On the other hand, impurities and/or vacancies with an associated net magnetic moment can also act as spin-orbit scattering centers, leading to a spin-flip which is characterized by a relaxation time proportional to the elastic scattering (τSO ∝τe) [236]. This is the so-called Elliot-Yafet spin-orbit relaxation. We obtained the different scattering times in our samples from the fitting of MC curves to the equation proposed by Hikami-LarkinNagaoka (HLN) for quasi 2D systems [237]: ∆σ=−e2 2πh ΨBi B+1 2−ln Bi B −e2 πh ΨBso+Be B+ 1/2−ln Bso+Be B +3e2 2πh hΨ(4/3)Bso+Bi B+1 2−ln (4/3)Bso+Bi Bi (7.3) where Bi, Beand Bso are the inelastic, elastic and spin-orbit effec149 Luc´ ıa Iglesias Bernardo tive fields, respectively. They are directly related to τi,τeand τso by the expressions: Bn=} 4eDτn (7.4) where Dis the diffusion constant and the indices n=e, i, so have their usual meaning. -4 -2 024 -0.005 0.000 0.005 MC B (T) LAO (s = -1.15%) a) -4 -2 0 2 4 -0.005 0.000 0.005 MC B(T) LSAT (s = -0.95%) b) c) d) -9 -6 -3 0 3 6 9 0.00 0.02 0.04 2 K 5 K 10 K 15 K 25 K (H)/(e2/h) B (T) -9 -6 -3 0 3 6 9 0.0 0.5 1.0 2K 5K 10K 15K 25K (H)/(e2/h) H(T) Figure 7.3: Magnetoconductance of Nb:STO epitaxial thin films on top of LSAT a) and LAO b). The magnetic field is applied perpendicular to the film plane. Figs. c) and d) show an enlargement of the curves at low magnetic fields. The thickness of the films is 19 nm in both cases. The experimental epitaxial strain is indicated in each case. Red lines represent the fittings according to Eq. 7.3. The fitting of the experimental data to Eq. 7.3 is very satisfactory in the entire range of fields up to 9 T; see Fig. 7.3. The values of the different scattering times obtained from these fits are shown in Fig. 7.4. The values of the scattering times confirm the conclusions extracted from the shape of the MC curves. For the film grown on LSAT (with a 150 7. Transport properties of strained SrTiO3thin films lower concentration of defects), τe<τi<τso, indicating the existence of WL effect in all the range of magnetic fields and temperatures (see Fig. 7.4 a). In this case, τso is at least one order of magnitude higher than τe and τi, which points out to a weak spin-orbit coupling in the system. Nevertheless, the value of the different scattering rates is more similar in the case of the sample grown on LAO (see Fig. 7.4 b). In this case, τso shows the same order of magnitude as τi, which evidences the existence of a stronger spin-orbit interaction. These values are compatible with the observed crossover between WAL and WL regimes in the low-field magnetoconductance. 0 5 10 15 20 25 1 10 100 1000 i e so  (ps) T (K) 0 5 10 15 20 25 10 100 1000 i e so (ps) T (K) a) LSAT (s = -0.95%) b) LAO (s = -1.15%) Figure 7.4: Different scattering times as a function of temperature extracted from the fitting using Eq.7.3 for the films deposited on LSAT a) and LAO b). Moreover, plotting τso versus τeshows a direct proportionality relationship (Fig. 7.5), which is a strong evidence of the existence of Elliot-Yafet mechanism for spin-relaxation in the STO thin film deposited on LAO. This points out to the local magnetic moments at the Ti3+ as the source of the spin-orbit coupling in this system. Therefore, the proposed tunable Rashba effect and superconductivity observed by other authors [226, 238, 239], could be a misinterpretation of the effect of the electric field over the local concentration of (highly mobile) oxygen vacancies (and the associated spin-orbit coupling effect discussed in this Section). 151 Luc´ ıa Iglesias Bernardo 015 30 50 100 150 so (ps) e (ps) Figure 7.5: Spin relaxation time vs elastic scattering time showing linear dependence consistent with Elliot-Yafet mechanism. The grey solid line represents the linear fitting. In the next Section, we will investigate the origin of the low temperature upturn in the resistivity. Looking at ρ(T)in Fig. 7.1 c), Kondo-like behavior is more evident in the films with more vacancies and increases as the thickness decreases. Given the discussion before, the effect of vacancy scattering (precursor of Anderson localization) close to the interface or surface, must be investigated. For that we have measured the ρ(T)and in-plane anisotropic magnetoresistance for different amounts of strain/disorder, and the discussion is presented below. 7.2 Temperature dependence of the electrical resistivity: effect of strain and defects Defects have a profound effect in the transport properties of a solid, particularly on the electrical conductivity of a metal or semiconductor. According to the Bloch theory, an electron moving in the periodic potential created by a perfect ionic lattice will not experience any collision. But the presence of defects (in the form of impurities, substitutions or vacancies) introduces imperfections in this potential, leading to collisions and scattering events which contribute to the magnitude and tempera152 7. Transport properties of strained SrTiO3thin films ture dependence of the electrical resistivity ρ(T) of the conductor. Of course, there exit other sources of deviation from the periodic potential of the lattice, such as thermal vibrations of the ions, as well as other inelastic scattering events like e−einteractions [240]. The different terms that contribute to ρ(T) will be considered independently of each other (Matthiessen’s rule [241]), so that in the relaxation time approximation [242]: ρ=m∗ ne2 1 τ=m∗ ne2 n X i=1 1 τn(7.5) where m∗is the electron mass, nis the charge carrier density, eis the electron charge and τrepresents the average time between scattering events, in which all the possible sources of scattering are included. At low temperatures, few phonons are excited, so that the e−e interaction and localized impurities are the main source of scattering, contributing with the characteristic Fermi-liquid ∼T2term [243] and a temperature independent term (ρ0), respectively. The latter term is the so-called residual resistivity due to the scattering by impurities, defects or vacancies within the crystal lattice [234]. When temperature increases, there is an increase in the density of phonons, and, therefore, in the probability of e−ph scattering. Thus, the resistivity is mainly governed by the scattering of electrons by the lattice vibrations. At high temperatures, well above the Debye temperature (ΘD), the Bose-Einstein distribution tends to kBT/hω(q), so that the number of phonons (and therefore the e−ph scattering probability) increases linearly with temperature. This leads to a ρ(T)∝T, at T>> ΘD. However, at lower temperatures, T<< ΘD, the phonons participating in the scattering events suffer certain restrictions about the ratio between their wavelength and the Fermi wavevector (only phonons with energies around kBTof the Fermi energy can be absorbed or emitted by electrons), resulting in a faster variation of the resistivity with temperature. So in this regime, ρ(T)∝T5[244–246]. Given that ΘD(STO)∼690 K [247], and the considerations above, the temperature dependence of electrical resistivity is expected to vary 153