MANIFOLD FIELD EFFECTS
AT A COMPLEX OXIDE INTERFACE
MANIFOLD FIELD EFFECTS
AT A COMPLEX OXIDE INTERFACE
Manifold field effects
at a complex oxide interface
D I S S E R TAT I O N
to obtain
the degree of doctor at the University of Twente, on the authority of the Rector Magnificus,
prof. dr. T. T. M. Palstra,
on account of the decision of the Doctorate Board, to be publicly defended
on Friday the 22ndof February, 2019, at 14:45 hours by
Alexander Everardus Maria Smink
born on the 16th of May, 1990 in Amersfoort, the Netherlands
prof. dr. ir. W. G. van der Wiel
The research described in this thesis was performed at the Faculty of Science and Technology, at the Faculty of Electrical Engineering, Mathematics and Computer Science, and at the MESA+ Institute for Nanotechnology of the University of Twente. It was funded through the DESCO program of the Foundation for Fundamental Research on Matter.
Manifold field effects at a complex oxide interface
PhD thesis, University of Twente
Printed by: GildePrint Drukkerijen, Enschede, the Netherlands ISBN: 978-94-632-3513-6
© A. E. M. Smink, 2019.
All rights reserved. No parts of this thesis may be reproduced, stored in a retrieval system or transmitted in any form or by any means without permission of the author. Alle rechten voorbehouden. Niets uit deze uitgave mag worden vermenigvuldigd, in enige vorm of op enige wijze, zonder voorafgaande schriftelijke toestemming van de auteur.
Graduation committee: Chair:p
prof. dr. J. L. Herek University of Twente Secretary:
prof. dr. J. L. Herek University of Twente Supervisors:
prof. dr. ir. J. W. M. Hilgenkamp University of Twente prof. dr. ir. W. G. van der Wiel University of Twente Members:p
dr. A. D. Caviglia Delft University of Technology prof. dr. J. Lesueur ESPCI, Paris
prof. dr. R. Pentcheva Universität Duisburg-Essen prof. dr. J. Schmitz University of Twente prof. dr. ir. H. J. W. Zandvliet University of Twente
B r i e f s u m m a r y
The fundamental physical process underlying all consumer electron-ics on the market today is the (electric-)field effect. In the broadest sense of the word, it encompasses tuning the properties of a material by an electric field, which is usually done by externally applying a voltage. The elementary electronic building block, the field-effect tran-sistor, nowadays comes by the billions on the size of a postage stamp, and is conventionally based on the material silicon. Replacing silicon by a different material may lead to electronics with novel function-ality, such as devices that compute more efficiently or in a different way: for example, future devices may closely resemble the way nature ’computes’ interactions between atoms and molecules; the realization of such devices is projected to enable breakthroughs in medical and material sciences. In this light, the conducting interface between lan-thanum aluminate (LaAlO3) and strontium titanate (SrTiO3) is espe-cially appealing: it exhibits exotic phenomena like superconductivity and magnetism, which are highly tunable by an externally applied voltage. These unique characteristics make this interface a promising platform for better understanding these phenomena on a fundamen-tal level, as well as for creating new electronic devices for innovative computing methods.
The main result of this thesis is that the field effect at the LaAlO3 -SrTiO3 interface goes beyond accumulating and depleting charges in a channel. Correlated-electron phenomena such as superconductivity and magnetism are deeply connected to the electronic band structure; the magnetotransport studies described in this thesis show that an ex-ternally applied gate voltage can change the band structure of this interface. A direct manifestation of the Schrödinger equation, this pre-mier result establishes electrostatic control over the fundamental elec-tronic properties of a material. The following chapters show that this evolution of the band structure with gate voltage is closely related to tuning superconductivity in this system, and that the electrostatic confinement is the main factor distinguishing bulk and surface states. Finally, the limits of complex-oxide field-effect devices are explored in terms of electric field strength, dielectric layer thickness, and lateral dimensions. In general, this thesis contributes both to the fundamen-tal understanding of the field effect in advanced materials, and to the boundary conditions of using such materials in functional devices.
C o n t e n t s
1 Introduction
1
1.1 Field-effect devices 21.2 Interfaces 3
1.3 Complex oxides 4
1.4 Field effects at the LaAlO3-SrTiO3interface 6 1.5 This thesis 7
2 SrTiO3 s u r f a c e s tat e s
9
2.1 Introduction 102.2 Properties of bulk SrTiO3 11 2.2.1 Electronic properties 12
2.2.2 Dielectric properties 14
2.3 Creation of conducting surface states in SrTiO3 16 2.3.1 Tunable interface properties by overlayer growth 18
2.4 Electronic structure of SrTiO3surface states 20
2.4.1 The self-consistent Schrödinger-Poisson framework 21
2.4.2 Electrostatic tuning of the potential well 25
2.5 Conclusion 29
3 Minimization of gate currents in metal-oxide-i n t e r f a c e d e v metal-oxide-i c e s
31
3.1 Introduction 32
3.2 Device fabrication 35
3.2.1 Process steps 36
3.2.2 Pulsed Laser Deposition of LaAlO3and overlayers 39 3.2.3 Pulsed Laser Deposition of Au in high Ar pressure 41
3.3 Device characterization 44
3.3.1 Gate current 46
3.4 Conclusion 51
4 Gate-tunable band structure of the LaAlO3-SrTiO3 i n t e r f a c e
53
4.1 Introduction 54
4.2 Experimental aspects 55
4.3 Topgate tuning of magnetotransport 56
4.4 Schrödinger-Poisson calculations with interband interactions 60
4.5 Conclusion 63
5 Correlation between superconductivity, band fill-i n g, and electron confinement at the LaAlO3 -SrTiO3 i n t e r f a c e
65
5.1 Introduction 66
5.2 Experimental aspects 67
5.3 Dual-gate tuning of superconductivity 68
5.4 Dual-gate tuning of band filling 71
5.5 Schrödinger-Poisson calculations 73
5.6 Discussion 77
5.7 Conclusion 78
6 Field-effect modulation of quantum oscillations at h i g h-mobility LaAlO3-SrTiO3 i n t e r f a c e s
79
6.1 Introduction 806.2 Experimental aspects 83
6.3 Backgate tuning of magnetotransport 84
6.4 Subband properties 87
6.5 Discussion 91
6.6 Conclusion 93
7 Size limits for LaAlO3-SrTiO3 f i e l d-effect tran-s i tran-s t o r tran-s
95
7.1 Introduction 96 7.2 Experimental aspects 97 7.3 Room-temperature operation 99 7.3.1 Transfer characteristics 99 7.3.2 Device stability 102 7.3.3 Capacitance-voltage characteristics 103 7.4 Low-temperature operation 106 7.4.1 Transfer characteristics 108 7.4.2 Short-channel effects 108 7.5 Conclusion 1118 Conclusions and perspectives
113
8.1 Synopsis 1148.2 Parallel developments 115
8.3 Perspectives 116
8.4 Conclusion 120
a Device fabrication details
121
a.1 Epitaxial growth of complex oxide thin films 121
a.1.1 Pulsed Laser Deposition 121
xiii
a.2 Extended process flow 125
a.2.1 SrTiO3substrate preparation 125 a.2.2 Ultraviolet lithography 127
a.2.3 AlOxhard mask 127
a.2.4 Au/Ti interface contacts 128
a.2.5 Au top electrode etching 129
b Additional analysis of magnetotransport
131
b.1 Tuning magnetotransport with top- and backgate voltage 131b.2 Low-field magnetotransport 134
Acknowledgments
137
Summary143
Samenvatting
147
Bibliography153
1
I n t r o d u c t i o n
The field-effect transistor – the backbone of modern electronics – is operated by using a voltage to control the electronic
proper-ties of a channel material. Engineering of such devices has led to great improvement in their performance and functionality, which went hand in hand with revealing new physical phenom-ena. Conventional field-effect devices are made from semicon-ducting materials; using new materials may expand the func-tionality of field-effect devices even more, and enable the ob-servation of additional physical effects. In the material class of complex oxides, the field effect may be used to control the inter-actions between individual electrons. This allows to explore the field effect to a whole new level, of great interest to fundamental physics and for advanced electronic devices.
1.1 Field-effect devices
The electric-field effect is the fundamental physical process driving the technology of the information age we live in today. Its basis is to control the electronic properties of a material by varying an electric in-put, such as a voltage or a current. For example, an externally applied voltage may be used to control the current flowing through another part of the device, much like a mechanical switch used to turn on the light. If current can flow, the device is ON; if it cannot, the device is switched OFF. These two states define the well-known 0 and 1 in all (digital) electronic devices.
Nowadays, such field-effect devices are fabricated by the billions on the area of a postage stamp. This is usually done on a wafer, a disk of up to thirty centimeters in diameter, made from silicon (Si), quartz sand from which the oxygen was removed. This material, very abundant on Earth, belongs to the material class of semiconductors. These materials can be seen as the middle ground between conductors, which normally conduct electricity very well, and insulators, which do not. Pure semiconductors are weakly conducting, but small perturba-tions can change this drastically: minute amounts of chemical doping (typically one in a million atoms) or small electric fields, induced by a voltage, can already change the conducting properties of a semicon-ductor by orders of magnitude. The latter is known as the field effect, used in electronic devices like switches and (radio) signal amplifiers.
The first solid-state electric-field-effect device, or transistor, was in-vented in 1947 by Bardeen, Brattain, and Shockley (1). They attached three electrodes to a block of germanium (Ge), another semiconduc-tor material. By sending a current into one of these terminals, they could adjust the current that flows between the other two terminals. This control distinguishes such ‘active’ devices from ‘passive’ elements like resistors, capacitors and inductors, which have only two termi-nals. Combining such transistors into integrated circuits created chips able to perform a wide variety of tasks. This started with simple cal-culations, but the functionality of integrated circuits soon expanded. Together with improvements in material quality, the miniaturization of individual components allowed to “cram” more components onto a single chip without increasing its cost (2). This long-standing trend in miniaturizing transistors has resulted in the enormous variety in electronic devices on the market today.
The core chip in any computer – for example desktop, smartphone or operating system of a car – is the microprocessor. Within these microprocessors, the most abundant component is the metal-oxide-semiconductor field-effect transistor (MOSFET). Its operation
princi-1.2 Interfaces 3
Semiconductor bulk
Active channel (Metal) gate (Oxide) insulator Drain Source(a)
OFF(b)
ON V TG V DS Current Electric fieldFigure 1.1: Operation of a field-effect device.
(a) Building blocks of a metal-oxide-semiconductor field-effect transistor (MOSFET). In this OFF state, no current can flow between the source and drain terminals, because there are no free charges present in the active channel. The bulk is colored dark green to indicate the (bound) p-type charge. (b) The application of a positive gate voltage, indicated by Vtg>0, accumulates positive charges on the metal gate electrode (shaded green), which by the electric field are coupled to negative charges in the channel (shaded red). With a voltage between drain and source (Vds), a current can now flow through the channel, turning the device ON. As the shaded areas indicate, the field effect is bound to the interfaces with the insulator.
ple is as follows: as schematically presented in Figure1.1, a metal gate
electrode controls the conductivity of a channel between the drain and source terminals, which are highly-doped regions of a semiconductor. The channel and the gate electrode are separated by a thin layer of an insulating (oxide) material. Although MOSFETs can operate with positive charges (p-type, holes) just as well as with negative charges (n-type, electrons), Figure1.1depicts the n-type device as it is most
rel-evant to this thesis. In panel (b), a positive voltage on the gate electrode attracts electrons to the channel underneath. By applying a voltage be-tween drain and source (Vds), a current of these charges can now flow through the channel and the device is switched ON.
1.2 Interfaces
As Figure1.1 shows, the magic happens at the interface between the
semiconductor and the oxide. Or, as Herbert Kroemer stated upon receiving the 2000 Nobel Prize in Physics: “often, it may be said that the interface is the device.” (3). Besides in transistors, interfaces play a crucial role in devices like touchscreens, (light-emitting) diodes, and semiconductor lasers. Over the past decades, the methods for growing
the thin layers that make up these devices have progressed greatly. Advanced control over the formation of thin layers has been one of the crucial aspects for the miniaturization of components referred to above. It also enabled creating new components: for example, the white LED that now finds widespread use in energy-efficient lighting would not have been possible without careful interface engineering (4).
Engineering the structural quality of interfaces has also progressed fundamental science greatly. Optimizing the purity of semiconductor channels in field-effect devices enabled the experimental observation of phenomena like the Quantum Hall Effect (5). Boosting the mobility of electrons further by using other semiconductors like GaAs (6) re-sulted in the observations of states with fractional charge (7), and of self-organized anisotropy in ultrapure systems (8). These astounding discoveries were all made at interfaces of ‘conventional’ semiconduc-tors; what can we expect from interfaces of more ‘exotic’ materials? 1.3 Complex oxides
A prominent example of such materials is the class of complex oxides. The adjective ‘complex’ is used to describe materials that contain, be-sides oxygen, at least two other elements; often from the transition metal groups in the periodic table. Their structural, electronic, and magnetic properties are all closely related and thereby also highly tun-able, resulting in huge responses of these properties to external stimuli (9). Because the fundamental building block, the unit cell, of most com-plex oxide crystals is almost equal in size, they can be stacked as if they were Lego bricks to create interfaces and stacks of interfaces (superlat-tices) with unique functionality (10). This offers a tremendous number of possibilities to combine different characteristics of materials. More-over, the physical properties of these interfaces are not limited to com-binations of those characteristics, as new phenomena may emerge that are uniquely observed at interfaces (11).
The electronic properties of the complex oxides studied in this thesis are mainly determined by the d orbitals of the transition metal ion(s). Contrary to the s and p orbitals determining the physics of most semi-conductors, the d orbitals have more complex shapes, as Figure 1.2
illustrates. In a crystal, these orbitals can order themselves in many different ways; in other words, there is a large orbital degree of free-dom. Together with the possibility of all kinds of structural deforma-tions, this orbital degree of freedom enables the stabilization of many different electronic phases.
1.3 Complex oxides 5
(c)
(b)
(a)
z
y
x
px pz dz2 s dxz dyz dxy dx2-y2 py px
s p(c)
d pz d In cr eas in g c o m p lexityFigure 1.2: Families of atomic orbitals.
Representations of electronic orbitals surrounding atoms in a crystal lattice. The s orbital (a) is highly symmetric, due to its spherical shape. The p or-bitals (b) have less symmetry and can take three different shapes. In the d-orbital manifold (c), the symmetry is reduced even further and five pos-sible shapes emerge. In free space, different orbitals of the same family are degenerate, meaning that they have the same energy. Within a crystal, the interaction of an orbital with those on neighboring atoms shifts its energy. The electronic system will condense in the lowest energy configuration pos-sible, by ordering these electronic orbitals and the positions of the atoms that form the crystal structure. Tuning this order yields a great richness in physical phenomena, which may be harnessed in future electronic devices.
Another driving force for the formation of distinctive electronic phases are interactions between electrons, or electronic correlations. Because the kinetic energy of d-orbital electrons is typically relatively small, they are particularly susceptible to the effects of these interactions (12). Prominent examples of correlated-electron effects are supercon-ductivity, the Mott insulator state, and different types of magnetism. Field-effect control of such correlations may lead to the discovery of new physics, as well as to realizing new components for electronics (13). This applies in particular to the interfaces between complex ox-ides, where the mesoscopic physics of semiconductor interfaces can be combined with the electronic correlations found in oxides (14).
1.4 Field effects at the LaAlO3-SrTiO3 i n t e r f a c e
The exemplary complex oxide interface is the one between lanthanum aluminate (LaAlO3) and strontium titanate (SrTiO3). In the bulk, these two materials are highly insulating. Surprisingly, when growing a thin – typically 10 unit cells, or 4 nm – layer of LaAlO3on top of a SrTiO3 crystal, the interface between them becomes conducting (15). The free electrons at the interface occupy the normally empty d orbitals of the Ti atom in SrTiO3, and exhibit phenomena related to electronic cor-relations, such as superconductivity (16) and signatures of magnetic scattering (17). These effects are normally absent in the undoped bulk hosts, making them pure interface effects. The most outstanding as-pect about these correlated-electron effects is that they can be tuned by the field effect (18). In most other correlated-electron systems, this is impossible due to the high carrier density required for correlations, to which (doped) SrTiO3is a notable exception (13,19).
Considered a model system for many (conducting) interfaces be-tween complex oxides, the LaAlO3-SrTiO3 interface has been the sub-ject of intense research for more than a decade (20–23). Following the field-effect tuning of superconductivity (18), other characteristics like the carrier mobility (24) and Rashba spin-orbit coupling (25,26) were also found to be highly tunable by electric fields. In an engineered in-terface with a magnetic EuTiO3interlayer, even ferromagnetism could be tuned by the electric-field effect (27). This marked the realization that these SrTiO3-based interfaces possess all ingredients for creating exotic states like Majorana quasiparticles (28), and that all these prop-erties can be controlled by externally applied gate voltages.
Such properties can normally be described using the electronic struc-ture of a material, which for the LaAlO3-SrTiO3interface is formed by the d orbitals of the Ti atom. These orbitals form bands of quantum-mechanically allowed energies for the electrons within the material; these bands can be filled or emptied by an electric field. However, the factors that determine the structure of these bands at the interface are not firmly established, and experiments trying to relate it to tuning the conducting properties appear to contradict each other (29–32). A fundamental understanding of how the field effect affects the conduc-tive properties of the interface is highly desired, and would present an important step towards realizing functional devices that make use of the electronic correlations in these exciting materials.
1.5 This thesis 7
1.5 This thesis
This thesis aims to contribute to the understanding of the field effect at the LaAlO3-SrTiO3interface, which we consider to be a model for many correlated-electron systems. To this end, we enhance the struc-tural quality of devices to apply large electric fields across the LaAlO3 layer of previously impossible magnitude. We focus on the relation between the electronic (band) structure and the field effect, in which important aspects are the anisotropic shape of the d orbitals and the unusual response of SrTiO3to externally applied electric fields. Both have a unique effect on the conducting states at the interface, which might hold for all interfaces with electronic structures based on d (and probably f ) orbitals.
In the next chapter, Chapter 2, we start by exploring SrTiO3, the host material in which the conducting states at the interface reside. The relevant electronic and dielectric properties are used to derive a self-consistent model for the electronic structure of its surface states. This model is in principle independent of the material on top of the SrTiO3, and is used in later chapters to interpret experimental results. The fabrication of devices with a high structural quality required for field-effect studies is described in Chapter3. Of the geometries in which an electric field can be applied, we identify topgating – similar to the MOSFET in Figure 1.1– as the most promising for functional
electronic devices. Topgating is challenging to realize at complex oxide interfaces, mainly due to the large electric fields that are required to change the number of carriers at the interface significantly. The main technological advance that enables application of such electric fields is depositing the metal topgate electrode in a very gentle way.
Chapter4describes how a topgate voltage tunes the occupation of the d orbitals by accumulating and depleting charges in a LaAlO3 -SrTiO3 interface channel. The nontrivial results show that the field ef-fect in this material goes beyond changing the charge density alone: it affects the electronic (band) structure of the interface as well. We sup-port this conclusion by applying the self-consistent model developed in Chapter2to the data, which also reveals the influence of electronic
correlations on the band structure.
In Chapter5, the topgate voltage is complemented by a voltage on the backside of the substrate (backgating) to demonstrate that the effec-tive electronic structure can be electrostatically controlled with great accuracy. Moreover, measurements down to a very low temperature of 0.01 K allow to compare the field-effect tuning of the band occupation to that of the critical temperature for superconductivity. This compari-son reveals that the optimal conditions for superconductivity are met
exactly at a special point in tuning the band occupation, which calcula-tions point out to be related to a satellite state (subband) of one of the d orbitals, induced by the confinement of the electron system to the interface. Although this does not yet reveal the mechanism of (tuning) superconductivity at the interface, it does show that interface effects are essential elements to consider in a description of this mechanism.
In high magnetic fields up to 33 Tesla, the high-mobility samples de-scribed in Chapter6show oscillations in their resistance as a function of magnetic field. We use these quantum oscillations to describe how the occupation of several subbands depends on the backgate voltage. In line with the previous chapters, many confinement-induced sub-bands are occupied at the interface, the population of which evolves with the gate voltage in a very unusual way. This indicates a changing subband structure with gate voltage, much alike the lower-mobility samples discussed in the previous chapters. These results strongly sug-gest that the band structure of conducting interfaces on the (001) sur-face of SrTiO3, independent of their origin, is derived simply from the bulk electronic structure. Between all samples with different growth conditions, and with various overlayers, the only significant difference is the electric field confining the states to the interface or surface.
The structural quality of our devices allows to explore the funda-mental limits for the geometry of LaAlO3-SrTiO3 field-effect transis-tors. Chapter7shows that, despite the intrinsically very high density of carriers, reducing the LaAlO3 layer thickness to only 1.5 nm still yields devices with excellent switching characteristics; they also oper-ate stably in enormous electric fields exceeding 10 MV/cm. At liquid-helium temperatures, the very large polarizability of SrTiO3 hinders proper operation of these transistors. This poses fundamental lower limits to the lateral size of SrTiO3-based electronics. Accordingly, we propose that the realization of nanoscale electronics in complex oxides requires engineering of electrochemical and structural aspects, rather than optimizing multiple-gate structures for field-effect tuning.
In Chapter8, we summarize our newly found understanding of the field effect at the LaAlO3-SrTiO3interface. We discuss how our results compare to parallel developments in other research groups and iden-tify trends in the field to which this thesis can contribute in a meaning-ful way. Based on these trends and on our understanding, we provide perspectives for future research, formulating several open questions to be answered. Understanding the field effect at this interface even bet-ter may lead to a widespread application of electric fields in all kinds of quantum-matter heterostructures.
2
S r Ti O
3s u r f a c e s t a t e s
The unique electronic properties of the workhorse perovskite oxide, SrTiO3, have been the motivation for intense research in-terest for decades. When used in field-effect devices, these prop-erties could result in novel computing methods. In this chapter, we discuss the relevant material properties of SrTiO3to devise a model for understanding the field effect in its surface states. We start by a discussion of the main electronic and dielectric proper-ties, after which we focus on the creation of surface states by de-position of a polar crystalline overlayer. Armed with knowledge on the boundary conditions, we then implement a Schrödinger-Poisson framework to model the effects of a gate voltage on the surface states. This chapter aims to provide a solid basis for understanding the experimental results in the other chapters of this thesis.
2.1 Introduction
As silicon is the workhorse material for the semiconductor industry, so is strontium titanate (SrTiO3) for the material class of complex oxides. Its crystalline beauty is exploited in jewelry as fake diamonds, but to materials physicists, it is much more than that. Already in the 1960s, it was reported to become superconducting (19), starting the field of conductivity in complex oxides culminating in the discovery of high-temperature superconductivity in 1986 (33).
Following the semiconductor revolution, it was soon realized that utilizing correlated-electron materials, such as complex oxides, in sim-ilar field-effect devices as used in the semiconductor industry, could lead to a large variety of novel device functionalities (13). Such correla-tions usually require very high electronic carrier densities, which are above the limits for conventional electric-field-effect devices to manip-ulate significantly. In SrTiO3 however, the carrier density required for a correlated-electron effect such as superconductivity is the lowest re-ported to date (34), leading to the label "most dilute superconductor" (35). If field-effect devices with conventional geometries can be real-ized with SrTiO3 as the channel material, this could result in a new class of electronic devices, able to perform computations in alternative ways (13,14).
In this chapter, we start our exploration of SrTiO3-based electron-ics by discussing the properties of the bulk material and of its sur-face states, which may be utilized as channels in field-effect devices. Among the many interesting properties of the bulk crystal, we iden-tify the electronic structure and the dielectric permittivity to be cru-cial aspects to describe the surface states. Then, we introduce how surface states in SrTiO3 can be formed, in particular by epitaxial de-position of a polar crystalline overlayer such as LaAlO3. Using the knowledge on how surface states are realized, we can then construct a model to describe the electronic structure of these surface states. In a Schrödinger-Poisson framework, we calculate the electronic struc-ture self-consistently with the electrostatic confinement potential; this allows us to study how both are affected by externally applied gate voltages. In the following chapters, we apply this model to fit our ex-perimental data, and to acquire a deeper understanding of the mech-anisms underlying the gate tuning of electronic phases in SrTiO3 sur-face states.
2.2 Properties of bulk SrTiO3 11 a z/(001) x/(100) y/(010) Sr Ti O
Figure 2.1: Perovskite crystal structure.
Unit cell of the SrTiO3crystal. The Sr atoms are located on the corners and the Ti atom in the center of the unit cell. The Ti atom is surrounded by six O atoms on the faces of the cube, forming an octahedron shaded in blue. For LaAlO3, the Sr and Ti atoms are replaced by La and Al, respectively.
2.2 Properties of bulk SrTiO3
In any surface state, the characteristics are derived from the properties of the bulk crystal. To describe the surface states of SrTiO3, we there-fore start by discussing the basic structural properties of the bulk, and the resulting electronic and dielectric characteristics. For the interested reader, broader overviews of the physics of SrTiO3 and SrTiO3-based heterostructures are discussed in elaborate reviews (20–23).
SrTiO3 is a member of the perovskite crystal family, which has the general chemical formula ABX3, where A and B are generally (transi-tion) metals and X is usually a halogen or an oxygen atom. Perovskites based on halides instead of oxides are currently being widely explored for their use in solar cells (36). Figure2.1shows the unit cell of SrTiO3
at room temperature. The strontium atoms on the corners (’A-site’) enclose the blue-shaded octahedron of six oxygen (O) atoms. In the middle of the octahedron, the titanium atom resides on the ’B-site’. The difference between the A and B sites is the distance to the oxy-gen ions: the A-site atoms are at slightly larger distance, which is why in most perovskites, the larger of the two cations occupies this site (37). The lattice parameter, a, of about 3.905 Å is close to that of many other complex oxide perovskites, offering tremendous possibilities to combine the properties of different materials in superlattices or at in-terfaces (11).
Upon cooling down from room temperature, the cubic symmetry is lowered to a tetragonal phase around 105 K (38). The tetragonal unit cell is elongated along one of the axes, by rotation of the oxygen
octahedron (39,40). The B-site cation, Ti, can then lower its energy by sitting just above or below the unit cell center. As discussed in Section
2.2.2, this results in quite unusual dielectric properties.
Because this elongation is equally likely to occur along the x, y, or z axis, domains of differently oriented tetragonal unit cells emerge. These domains, and especially the walls forming between them, may have profound consequences for conductivity at SrTiO3surfaces (41–
47). In the work described in this thesis, we find that two-dimensional models for conductivity apply to the data very well: the effect of these domain walls on our experiments appears to be negligible.
2.2.1 Electronic properties
Bulk SrTiO3 is a band insulator in which the Fermi energy, Ef, is lo-cated within a wide band gap of ∼3.3 eV (48). By means of chemi-cal doping, an insulator can turn conducting: atoms of different va-lence are introduced to create free charge carriers in either the con-ductance (n-type) or the valence (p-type) band. In SrTiO3, substituting Sr2+ or Ti4+ by a higher-valence cation like La3+ or Nb5+, respec-tively, results in n-type doping (49,50). The same n-type doping can be achieved by reduction of the crystal through the formation of oxy-gen vacancies (19), where the cations remain bound in the crystal. Un-like in Mott insulators and high-Tc superconductors (51), successful p-type electronic doping of SrTiO3was not yet reported: substituting lower-valence cations usually results in formation of oxygen vacan-cies restoring charge neutrality without the need for free carriers (37). Unlike in some perovskite-like high-temperature superconductors, e.g. YBa2Cu3O6+y, the simple perovskite unit cell is not thermodynami-cally stable with the excess oxygen that would result in p-type doping (37).
The n-type doping of SrTiO3puts electrons into the normally empty 3d orbitals of the Ti atom. As depicted in Figure 2.2(a), two of the
five 3d orbitals, the egorbitals, are pushed upwards in energy because their orbital lobes point in the direction of the O atoms on the edge of the unit cell (11). The other three orbitals, labeled t2g, thus form the conduction band minimum into which electrons are doped. In the simplest approach, these are degenerate in energy: in the cubic sym-metry, the labels x, y, and z are interchangeable without affecting the physics of the system. Effects like spin-orbit coupling (32,52–57) and the low-temperature tetragonal symmetry (40,58) lift this degeneracy, but only to the extent of∼5 meV and∼3 meV, respectively.
2.2 Properties of bulk SrTiO3 13
(b)
(a)
Ti 3d eg t2g ~2 eVd
xyd
xzd
d
yzd
x2-y2 eVd
d
3z2-r2Figure 2.2: Electronic structure of bulk SrTiO3.
(a) 3d orbitals of the Ti atom. Crystal field splitting lifts the egorbitals, dx2-y2 and d3z2-r2, up in energy by∼2 eV. Hence, the t2gorbitals – dxy, dxz, dyz – form the conduction band bottom. (b) Band dispersion at the conduction band bottom; the left-hand (right-hand) side depicts dispersion along ky (kx). As described in the text, the tetragonal distortion of the unit cell shifts the energy of the dxyorbital up by∼3 meV.
Along the orbital lobes, the bands are highly dispersive with an ef-fective mass, m* ≈ 0.7 me (59), where me is the electron rest mass in vacuum. Perpendicular to the lobes, the overlap with the next or-bital on the nearest neighbor is much smaller, yielding a weakly dis-persive band with m* ≈ 14 me (59). This results in the conduction band structure of Figure 2.2(b), where we have neglected spin-orbit
coupling for simplicity, but where we have illustrated the effect of the low-temperature tetragonal distortion: an elongation along the z di-rection moves the dxy band to a slightly higher energy (40, 58). For a more detailed description of the bulk band structure of SrTiO3, we refer to Refs. (48,59). The population and dispersion of these bands determines the momentum and spin of the electrons around the Fermi level, and thereby the possibilities for them to condense into Cooper pairs. Consequently, correlation effects such as superconductivity de-pend critically on the population of bands (60).
Because the 3d orbitals in stoichiometric SrTiO3are completely empty and doping can be used to fill them, it has become a model system for studying the physics of electrons in these orbitals. Other supercon-ducting complex oxides such as YBa2Cu3O7-δor Sr2RuO4have partly filled bands and therefore, a much higher density of free electrons that
is harder to tune (61). This makes SrTiO3especially appealing for elec-trostatic control over superconductivity, but also to serve as a model system for tuning electronic correlations in general (13).
2.2.2 Dielectric properties
In perovskite oxides with a non-cubic unit cell, the displacement of the B-site cation often leads to ferroelectric order (62). In stoichiometric SrTiO3, this transition into a ferroelectric phase at low temperature is thwarted by quantum fluctuations (63,64). Despite the tetragonal crys-tal structure, it remains paraelectric down to very low temperatures. This "quantum paraelectric" phase is highly tunable by several fac-tors. For example, the ferroelectric phase can be recovered by doping (65), epitaxial strain (66), or by cation non-stoichiometry (67). On the other hand, the permittivity is strongly suppressed by defects, rough-ness and other inhomogeneities emerging during thin film growth (68). Therefore, studies on the electronic properties of SrTiO3 are usually done on single crystals and their surfaces.
An important result of the quantum paraelectricity was found long before that term was introduced: as first reported in 1959 by Weaver (69) and in more detail later by Sakudo et al. (70), the permittivity, er, of SrTiO3 is strongly dependent on temperature. Shortly after, it was found that it also varies with electric field below T≈65 K (71). There-fore, er of SrTiO3 cannot not be considered as a ’dielectric constant’, but it must be evaluated explicitly depending on the experimental con-ditions.
To date, quantitative descriptions of how the permittivity of SrTiO3 depends on temperature and electric field are empirical; we are not aware of a consensus on a complete description of er. We will address two of these models below. A first expression for the field dependence of erwas put forward by Neville et al. (71):
er(E) = 1
A(T) +B(T)|E| (2.1)
with A(T) the inverse zero-field permittivity as function of temper-ature, and B(T) the field-dependent part as function of temperature. At 4.2 K, A = 4.097×10−5 and B =4.907×10−10 m/V for the (001) direction. After a few updates to these values in later studies (68,72), Fête et al. put forward an alternative field dependence for
tempera-2.2 Properties of bulk SrTiO3 15
(a)
(b)
Figure 2.3: Permittivity of quantum paraelectric SrTiO3.
(a) Measured relative permittivity, er, of SrTiO3 versus temperature. Reprinted with permission from AIP Publishing, from Ref. (71). (b) Overview of empirical models (71–73) for the electric-field dependence of erat T=4.2 K.
tures below 10 K (74), which was fit to low-field measurements and first-principles calculations at high field (73–75):
er(E) = er,0 [1+ (|E|/E0)2] 1 3 (2.2) where er,0 is the zero-field permittivity, taken equal to 25462, and E0 is a typical field, with a value of 82213 V/m. Figure 2.3(b) plots the
outcome of these equations according to Gariglio et al. (73), Neville et al. (71), and Copie et al. (72). In all these models, we observe that er remains constant up to a (small) field of∼10 kV/m, and then drops linearly or with a power 2/3. In all practical low-temperature devices, the permittivity of SrTiO3is thus field-dependent, and very large.
In general, the permittivity of a material is an important parameter for electrostatics: it determines the energy of an electric field in matter. The higher the permittivity, the higher the energy cost to sustain an electric field in a material. Therefore, the screening length – the typical distance over which differences in electric potential decay – increases with permittivity. In case of a fixed potential difference over a set dis-tance, the opposite is true: a larger modulation of charge density can be achieved with higher permittivity. This is the driving force behind the use of high-permittivity dielectrics in the semiconductor industry (76). For the SrTiO3surface states studied of interest here, the peculiar permittivity has two important implications. Its magnitude puts fun-damental limits to the lateral size of field-effect devices (Chapter7of
this thesis), and together with the high carrier density, its nonlinearity enables strong confinement at low temperature (58,72).
2.3 Creation of conducting surface states in SrTiO3 Despite its wide band gap, stoichiometric SrTiO3 offers several possi-bilities for the creation of conducting states at its surface (21). Of these, the deposition of a polar crystalline overlayer is the most elegant, and unique to oxide interfaces (11). Insulating, polar crystalline overlayers are also the most appealing for electrostatic devices: crystalline layers usually have higher permittivity than amorphous layers (76), and are less prone to the formation of leakage current paths thanks to their high packing fraction and strong chemical bonds.
In their pioneering work, Ohtomo and Hwang reported formation of a conducting state at the interface between the band insulators LaAlO3 and SrTiO3 (15). Similar to SrTiO3, LaAlO3 is a perovskite insulator, with a band gap of ∼5.6 eV and a lattice parameter of 3.789 Å. Sur-prisingly, the emergence of a conducting state depends critically on the atomic termination of the SrTiO3 surface. Only with termination on the TiO2 planes, a conducting interface emerges, for which Hall measurements indicated n-type doping. The expected p-type equiva-lent for SrO termination remained elusive. For this extreme difference between systems that only differ by half a unit cell, Nakagawa et al. proposed an electronic reconstruction to occur in the n-type case, and an ionic reconstruction in the p-type case (77).
As depicted in Figure 2.4, the electronic reconstruction scenario is
based on a discontinuity of the polarization at the interface. Along the (001) crystal direction, perovskite oxides with composition ABO3 have a plane stacking of AO-BO2. For SrTiO3, this results in stacking of Sr2+O2−and Ti4+O4−2 planes, which are electrically neutral on their own: (SrO)0and (TiO
2)0. In the case of LaAlO3, the layers are not elec-trically neutral: the sub-unit cell planes La3+O2−and Al3+O4−2 have a net charge: (LaO)1+and (AlO2)1−. This results in a diverging potential for LaAlO3films grown on SrTiO3, which needs to be resolved to min-imize the total energy of the system. Nakagawa et al. suggested that this is done by transferring half an electron per u.c. from the LaAlO3 surface to the conduction band of SrTiO3(77). There, they are confined to the interface by the strong electric field. In this scenario, this charge transfer is triggered by raising the LaAlO3 valence band maximum above the SrTiO3conduction band minimum. The polar discontinuity predicts this to happen above a LaAlO3film thickness of 4 u.c, which was confirmed experimentally by Thiel et al. (78). Below the critical thickness, the diverging potential gives rise to a polar distortion of the LaAlO3film (75,79), which relaxes into a non-polar, antiferrodistortive structure for thicker films (80).
2.3 Creation of conducting surface states in SrTiO3 17
(a)
(b)
Al3+O 2 4-Al3+O 2 4-Al3+O 2 4-Al3+O 2 4-Ti4+O 2 4-Ti3.5+O 2 4-Ti4+O 2 4-Ti4+O 2 4-La3+O 2-La3+O 2-La3+O 2-La3+O 2-La3+O 2-La3+O 2-Sr2+O 2-Sr2+O 2-Sr2+O 2-Sr2+O 2-+1 +1 +1 +1 +1 +1 0 0 0 0 -1 -1 -1 -1 0 -0.5 0 0 ρ ρ E E φ φ (001) (100) From surface:1/
2e
-Figure 2.4: Electronic reconstruction at the LaAlO3-SrTiO3interface.
(a) Schematic of an uncompensated interface along the (001) direction. The SrO and TiO2 planes are charge neutral, indicated by the 0 next to the schematic blocks. The polar LaAlO3 on top has charged planes with alter-nating charge of+1 and−1. The columns on the right depict the charge density, ρ, the electric field E, and electrostatic potential φ, respectively. Be-cause of the polar discontinuity, a continued growth will result in a huge potential buildup, known as the polar catastrophe. (b) Compensated inter-face. The polar catastrophe is avoided by transferring a charge of -½e per u.c. from the LaAlO3surface to the interface. The emergent q-2des is indi-cated by the red shading. In reality, the transferred charge is usually less than -½e and is distributed across a region with a thickness of∼5 nm into the SrTiO3substrate.
Since its introduction, the electronic reconstruction scenario has been subject of much debate as to being the sole reason for interface con-ductivity (20). The doping mechanisms for bulk crystals, reduction by formation of oxygen vacancies and cation substitution, can also lead to interface conduction. Especially when the overlayer is grown at low pressure, many oxygen vacancies form (17,81,82). However, the fact that the substrate termination has such a dramatic effect on conduc-tivity (15,77), as well as the critical thickness phenomenon (78) rule out doping by oxygen vacancies as the singular mechanism to trigger interface conductivity. This notion is supported by the observation of highly conducting interfaces when the density of oxygen vacancies is strongly suppressed (83). Cation substitution could be triggered by in-termixing across the interface (84), but this possibility is also strongly challenged by the critical thickness phenomenon (85), as well as by the dependence of interface properties on cation stoichiometry of the LaAlO3 overlayer (86), and by analysis of Transmission Electron Mi-croscopy images (83,87).
By considering electrochemical formation of defects, Yu and Zunger (88) refined the mechanism of purely electronic reconstruction. In their calculations, the polar discontinuity at the interface affects the forma-tion enthalpy of oxygen vacancies at the LaAlO3surface (89,90). It is reduced with increasing thickness, and becomes negative at the criti-cal thickness of 4 u.c.: oxygen vacancies form spontaneously and the binding electrons they leave behind tunnel into the SrTiO3conduction band, as in the electronic reconstruction scenario sketched above. This process requires much less energy than the purely electronic recon-struction, which requires to overcome the band gap of LaAlO3. They also show that the incorporation of defect chemistry in modeling the interface can explain the phenomenology to a greater extent, making this polarity-induced defect formation scenario the most likely expla-nation for conductivity at polar-nonpolar oxide interfaces reported to date.
2.3.1 Tunable interface properties by overlayer growth
As readily observed by Ohtomo and Hwang, the interface properties can be tuned by varying the LaAlO3 thickness and the growth con-ditions for the overlayer (15). Especially the oxygen process pressure can change the interface conductivity by orders of magnitude (17,81,
82, 91). Later, also the substrate temperature (92,93) and the cation stoichiometry of the overlayer (86,94,95) proved to be useful methods of tuning the interface properties during overlayer growth. Complex
2.3 Creation of conducting surface states in SrTiO3 19
oxide overlayers (83) and surface adsorbates (96,97) also induce very large changes in interface properties, the latter being of interest for ap-plications in electrochemical sensors. Of special interest for field-effect devices is the influence of a metal layer on top of LaAlO3. The work function (98, 99) and chemical activity (100–103) of metal overlayers can sensitively tune the carrier density at the interface, and even drive metal-insulator transitions.
Since 2004, many more combinations of polar and non-polar com-pounds have shown to host conducting interface states, e.g. the ones discussed in (104–110). By using solid solutions like LaxSr2yAlx+yTayO3 or (LaAlO3)0.5(SrTiO3)0.5, the polar discontinuity can be weakened, leading to a higher critical thickness and an alteration of interface properties (111–113). All these systems may have different properties emerging from the interaction of the SrTiO3 surface with the (po-lar) overlayer, although their electronic structure is remarkably simi-lar (114). Therefore, studying the SrTiO3 surface states that form at the interface with LaAlO3 should yield results applicable to all such interfaces.
For our description of the electrostatics and the electronic structure of the surface states, this tuning of interface properties is relevant through their effect on the electrostatic boundary conditions, and on the distribution and number of (bound) charges induced by crystallo-graphic defects. For instance, it is well-established that depositing at very low oxygen pressure gives rise to a thick conducting layer (115), in which the conductivity has a three-dimensional character (81). High-mobility conduction was later also achieved in the two-dimensional counterpart, evidenced by characteristic quantum transport phenom-ena such as Shubnikov-de Haas oscillations (92,116,117) and Rashba spin-orbit coupling (25,26). These intrinsically two-dimensional effects evidence the strong confinement in the out-of-plane direction; further evidence for a two-dimensional system is provided by conducting-tip atomic force microscopy (CT-AFM) imaging (72), superconducting crit-ical field anisotropy (118), and hard x-ray photoelectron spectroscopy (HAXPES) (119). This confinement forms several subbands, whose en-ergy depends on the quantized values of the out-of-plane momentum (120). Hence, the SrTiO3surface states are not strictly two-dimensional: we will therefore refer to them as a quasi-two-dimensional electron sys-tem, or q-2des. In the following section, we will explore the electronic structure of this q-2des, and how it may be affected by out-of-plane electric fields.
2.4 Electronic structure of SrTiO3 s u r f a c e s tat e s
Compared to the bulk electronic structure discussed in Section 2.2.1,
inversion symmetry is broken in a surface state. The crystal axes are no longer interchangeable: the xy plane along the interface is funda-mentally different than the z direction perpendicular to it. This gives rise to a splitting in energy between the in-plane-oriented dxyand out-of-plane-oriented dxz,yz orbitals (121). This splitting, ∆E, is generally much larger than the splitting in energy due to tetragonal distortions or spin-orbit coupling discussed above. This was first measured by x-ray linear dichroism, which showed ∆E ≈ 50 meV (122). These first reports also brought forward that two distinct types of carriers should contribute to electronic transport in parallel: light carriers originat-ing from the dxyorbitals, and heavier carriers populating dxz,yzstates. This suggestion was confirmed by in-plane transport measurements on LaAlO3-SrTiO3(29) and LaTiO3-SrTiO3interfaces (30).
The first calculations of the electronic structure of the LaAlO3-SrTiO3 interface used Density Functional Theory (DFT), in which the carriers reside on the individual TiO2planes (121). Each of these planes has its own band structure, the offset in energy between them is determined by the inversion symmetry as well as by the strong electric field at the interface (75). However, in angle-resolved photoemission spectroscopy (ARPES) experiments on bare SrTiO3 surfaces, a maximum of only three subbands was observed (120, 123), challenging this picture. A continuous potential well, in which subbands are split by electrostatic confinement, fits those experimental observations much better (120).
Following this idea, modeling of the system shifted towards a tight-binding approach with the effective-mass approximation (29,32, 52). This approach is commonly referred to as the ‘three-band model’ and works reasonably well to explain magnetotransport experiments as function of gate voltage, tuning the Fermi level up and down (29,32). In the three-band model, the spin-orbit coupling is taken into account easily as off-axis matrix elements in the Hamiltonian (52). The most im-portant ingredient is the splitting in energy between the dxyand dxz,yz orbitals, ∆E. In that respect, it is problematic that experimentally, ∆E was found to vary by almost two orders of magnitude: from∼250 meV (120) down to∼2 meV (117). Van Heeringen et al. showed that depend-ing on a model confindepend-ing field strength, F, the energy splittdepend-ing changes, producing reasonable agreement to the low-splitting experimental re-sults (53). They also report that in that case, the subbands also strongly hybridize because∆E is reduced to the order of the atomic spin-orbit splitting energy. Chapter6describes transport experiments on such a
2.4 Electronic structure of SrTiO3 s u r f ac e s tat e s 21
All calculations and experiments referred to above describe how elec-tronic confinement affects the band structure of (001) surfaces. We note that surface states on (011) or (111) facets of SrTiO3have different or-bital orientations with respect to the interface, hence their energy off-sets differ (124, 125). Although exploration of these surface states re-cently gained more widespread attention (126–128), we limit ourselves to the archetypical (001) interface in this thesis.
2.4.1 The self-consistent Schrödinger-Poisson framework
In the tight-binding models discussed above, the electrostatic confine-ment is not taken into account explicitly. They either use a fixed value for∆E that fits experimental results best (29,32,52,129), or assume a triangular potential with a fixed electric field strength F (53,120). In reality, electrostatic confinement is described by a potential that varies with position, which can be calculated by the Poisson equation. For a given charge distribution and electrostatic boundary conditions, the potential is uniquely defined. In turn, this charge distribution is deter-mined by the population of bound states in this potential; this popu-lation is described by the Schrödinger equation for a given potential. So, the coupling of these two equations should enable a self-consistent calculation of the potential as well as the energy offsets, but no ana-lytic solution for the coupled equations exists (130). As early as 1972, Stern was the first to solve the equation system: an iterative numeric algorithm calculated the charge distribution and energy levels in an Si inversion layer (131).
For SrTiO3surface states, such Schrödinger-Poisson calculations were first reported by Biscaras et al. (30) and later by the Geneva group (73,
74). Compared to Si inversion layers, SrTiO3 surface states are more complex to model: the nearly degenerate 3d orbitals with anisotropic m*, and the field-dependent permittivity must be taken into account. As a result, the Hamiltonian that enters the Schrödinger equation is formulated to describe multi-orbital conduction; the implication for the Poisson equation is that the permittivity as function of position must be calculated consistently with the electric field.
Schrödinger-Poisson modeling has proven itself to provide qualita-tive insight into experimental results (30,132) as well as predict inter-esting new properties of SrTiO3surface states. For instance, Scopigno et al. reported a possible thermodynamic instability as a result of the nonlinear permittivity, in certain regions of the gate-voltage/band-occupation phase diagram (133). By using the temperature depen-dence of the permittivity, Raslan et al. showed that the band structure
Electric field E(z) Electric potential V(z) Potential Φ(z) = -V(z) Charge distribution ρ(z) = ρb(z) - eΣ|ψi(z)| 2 Distribution |ψi(z)| 2 = n i|φi(z)| 2 Band occupation ni = mi*/πħ2(E F - Ei) Eigenvalues Ei Solutions φi(z)
Trial wave function
ψi(z) Carrier density Σni = n0 + dnTG + dnBG Bound charge ρb(z) Bound charge ρb(z)
e-e interaction term U Trial permi"ivity εr,0 SrTiO3 permi"ivity εr(E)
Schrödinger
equation
Poisson
equation
Band
structure
Figure 2.5: Schematic for self-consistent Schrödinger-Poisson calculations. The Schrödinger and Poisson parts of the calculation are shaded red and yellow, respectively. Input parameters and initial guesses for solutions are depicted in green. Connecting arrows indicate a numeric coupling of quan-tities, which are evaluated by the formulas inside the blocks.
of the surface states must be equally dependent on temperature (134). In an attempt to unite superconductivity in bulk SrTiO3and at its in-terface, Li et al. recently employed Schrödinger-Poisson calculations to predict when the three-dimensional carrier density is high enough to trigger superconductivity (113).
For the work described in this thesis, we are interested in using such calculations to describe the effect of a gate voltage on the self-consistent band structure1
. As identified earlier in Ref. (133), a gate voltage can be simulated by varying the mobile charge density, n2d: the electrostatic boundary conditions follow automatically. Figure2.5
2.4 Electronic structure of SrTiO3 s u r f ac e s tat e s 23
presents a schematic of our model. In the red and yellow shaded areas, the Schrödinger and Poisson parts of the model are depicted. We use the over-relaxation method as described by Fête (74) to ensure proper convergence. During convergence, we include a step in which the old solution is completely discarded−in the notation of Fête, this means
ζ = 1. This ensures that the result is independent of the trial wave
function and permittivity.
Our calculations aim to reproduce experimentally measured occupa-tions of the dxyand dxz,yzbands. To this end, we implement two fitting parameters: a background charge density, nb (73), and a phenomeno-logical parameter for interband Hubbard-type repulsion between elec-trons in the same unit cell, U (129). In practice, the background charge originates from defect states that form during the LaAlO3deposition; its density is of the same order as the mobile carrier density (135–
137). In the calculations, the presence of this background charge, ρb, stabilizes and deepens the potential well (133). Without it, the solu-tions in our model always diverged. Similar calculasolu-tions by Chen et al. converged without a bound background charge (132), but in those calculations ∆E is much smaller than the experimentally measured values (120,122). Very recently, large∆E values resulted from a model without background charge, through including a flexoelectric instabil-ity in the calculations (138). Here, we use the simplest model for the background charge (73): a flat distribution over a region with thick-ness d = 50 or 100 nm. An exponentially decaying density closer to measured defect density profiles (135,136) can also be used (133). In our calculations, we tried both distribution profiles and linear com-binations of the two: this does not affect the qualitative picture, only the values of the fitting parameters. For all calculations in this chapter, nb=1.3×1019 cm−3and d=50 nm. This background charge density is in good agreement with defect chemistry experiments (135,136) and with previous Schrödinger-Poisson calculations (73).
The phenomenological parameter for Hubbard-like repulsion, U, is included in the calculations to take into account interband interactions (129). It models the strength of interactions between electrons in dif-ferent subbands, shifting the energy of subbands up with respect to the energy without interactions. Every bound state does not interact with itself, but only with the other states and its spin-degenerate coun-terpart at the same energy. By this reasoning, Maniv et al. (129) for-mulated the interaction energy leading to an additional band offset as:
Eint,i=
∑
n0 2 4 6 8 -2 0 0 -1 8 0 -1 6 0 -1 4 0 -1 2 0 -1 0 0 -2 -1 0 1 2 -2 0 0 -1 8 0 -1 6 0 -1 4 0 -1 2 0 -1 0 0 0 2 4 6 8 1 0 18 1 0 19 1 0 20 1 0 21 d xz,1 d yz,1 d xy,1 d xy,2 d xy,5 V ( m e V ) an d | | 2 z (nm) V E F (a) E ( m e V ) k x (nm -1 ) (b) n 3D ( c m -3 ) z (nm) Bound d xy d xz,yz T otal (c)
Figure 2.6: Results of a self-consistent Schrödinger-Poisson calculation. (a) Out-of-plane potential well, V(z), and bound states in the well|ψ|2with different orbital character as indicated in panel (b). The Fermi level is indi-cated by the dashed line. V(∞) ≡0 meV. (b) In-plane parabolic band disper-sions corresponding to the bound states in panel (a). (c) Three-dimensional charge distribution away from the interface, corresponding to the states depicted in (a). Calculation parameters are described in the text.
where Eint,iis the interaction energy for subband i, δijis the Kronecker delta, and Nj is the 2d carrier density per unit cell of band j. Here, we extend the three-band model of Maniv et al. to take into account higher-order subbands. For all calculations presented in this chapter, we take U =0 for simplicity. The effect of including U is investigated in detail in Chapters4and5.
Besides the fitting parameters, important input parameters are the permittivity and the effective masses of the orbitals. For the permittiv-ity, we use Equation2.2(73). It is calculated iteratively within the
Pois-son equation loop as depicted in Figure2.5. For the effective masses
of the t2g orbitals, we use a light mass of 0.7 me and a heavy mass of 14 me (30,59,73). This means that for the dxy orbital, m*x =m*y =0.7 me, and m*z=14 me; for the other orbitals, the axis labels are rotated. Note that, for the dxzand dyz orbitals, the effective mass entering the density of states to calculate the band occupation, equals√0.7×14 me (133). Because of the difference in electron affinity (139), the LaAlO3 conduction band is 2 eV higher in energy than that of SrTiO3: this is modeled as a 2-eV-high step in the potential at z=0−.
Figure2.6shows the result of a Schrödinger-Poisson calculation for
a mobile carrier density, n2d=2.0×1013cm−2. In accordance with pre-vious results (30,73,134), we observe several dxy subbands and only
2.4 Electronic structure of SrTiO3 s u r f ac e s tat e s 25
one dxz,yzstate. For the dxz,yzorbitals, the energy splitting between sub-bands is much larger due to the smaller out-of-plane effective mass.
The current model does not (yet) include the effects of spin orbit cou-pling on the band structure. Although it is an important phenomenon determining the band structure of SrTiO3surface states (29,52–54,140,
141), it does not have a very profound effect on the energy of the bound states in the potential well. The associated spin-orbit energy, ∆so, is usually much smaller than∆E. Except for very weak confining poten-tials (53), the band occupation will thus only be weakly affected by the orbital hybridization resulting from the spin-orbit coupling. Con-sequently, including spin-orbit coupling in these Schrödinger-Poisson calculations will only change the phenomenological fitting parame-ters required to reproduce the experimental results. Still, including spin-orbit coupling can lead to accurate modeling of a wider range of experimental results, such as low-field magnetoconductivity (25) and spin-to-charge conversion (142).
2.4.2 Electrostatic tuning of the potential well
We now use the model to study the effect of externally applied gate voltages on the potential well and the resulting electronic structure. The effect of a gate voltage is modeled by the accumulation of charge in the potential well, and a opposite charge of equal magnitude on the corresponding gate electrode. The charge accumulation as a result of a backgate voltage, dnbg, is calculated using a simple parallel plate capacitor model, taking into account the field-dependent permittivity. We used 0.5 mm, the thickness of a typical SrTiO3 substrate, for that calculation. For the topgate voltage, the parallel plate capacitor model is less trivial, because the LaAlO3layer is very thin. This enhances the possible influence of ‘dead layers’ (143) and quantum capacitance (144,
145). Therefore, we model the topgate voltage simply by a change in the mobile carrier density, dntg, and a corresponding countercharge on the LaAlO3surface (133).
Figure 2.7 shows how individual top- and backgate voltages tune
the confining potential, with n2dranging from 1.0 to 6.0×1013 cm−2, and Vbgfrom−200 to 200 V. These ranges correspond to realistic lim-its in experiments, see e.g. Ref. (18) and Chapter 4. Below −200 V,
the mobile carrier density became so low that the calculation did not converge anymore. Panel (a) shows that a topgate voltage primarily deepens the potential well slightly, but that its electric field is screened entirely within the first 5 nm of SrTiO3. The situation is quite differ-ent for the backgate voltage, where the potdiffer-ential well minimum shifts
0 1 2 3 4 5 -2 6 0 -2 4 0 -2 2 0 -2 0 0 -1 8 0 -1 6 0 -1 4 0 -1 2 0 0 2 4 6 8 1 0 -5 0 0 -4 0 0 -3 0 0 -2 0 0 -1 0 0 0 6 .0 × 1 0 13 cm - 2 V ( m e V ) z (nm) 1 .0 × 1 0 13 cm - 2 (a) + 2 0 0 V V ( m e V ) z (nm) -2 0 0 V (b)
Figure 2.7: Effect of a gate voltage on the potential well.
(a) Schrödinger-Poisson calculations on the effect of a topgate voltage, mod-eled by a variation of the total carrier density in the range of 1.0−6.0×1013 cm−2. (b) Same as (a) for a backgate voltage ranging from−200 to+200 V, corresponding to carrier densities from 0.2 to 4.0×1013cm−2. Note that the light blue line in (a) and the yellow line in (b) are identical: they represent the situation without an applied gate voltage as presented in Figure2.6.
much more, and the potential well slope – local electric field – becomes smaller with positive gate voltage.
In Figure2.8, we quantify the effects of these gate voltages on the
band occupation, and on relevant properties of the potential well: its depth with respect to the conduction band edge in the bulk, V0, the Fermi energy Ef, and the center-of-mass (COM) of the envelope wave function. Panels (a)-(b) show that both top- and backgating produce a transition from single-band to multi-band occupation, associated with a Lifshitz point in the band structure (29). Above this point, the extra carriers induced by the gate voltage have to be distributed over the two bands, which we observe to be almost one-to-one. This is a bit different for backgate voltage, where the dxz,yz carrier density increases much faster than nxy. This suggests that the band filling depends crucially on the electrostatic boundary conditions, which we will explore in more detail in Chapters4and5.
Panels (c) and (d) show important differences between the effects of top- and backgating on the center-of-mass of the envelope wave tion. We consider the COM as an indicator for the extent of wave func-tion spreading into the SrTiO3interior, which tunes the carrier mobil-ity (24,143). It might also have effects on the topgate capacitance (144)
2.4 Electronic structure of SrTiO3 s u r f ac e s tat e s 27 1 2 3 4 5 6 0 1 2 3 4 -2 0 0 -1 0 0 0 1 0 0 2 0 0 0 1 2 3 4 1 2 3 4 5 6 -3 0 0 -2 5 0 -2 0 0 -1 5 0 -1 0 0 -2 0 0 -1 0 0 0 1 0 0 2 0 0 -5 0 0 -4 0 0 -3 0 0 -2 0 0 -1 0 0 0 n i ( 10 13 c m -2 ) n 2D (10 13 cm -2 ) (a) T otal n xy n xz + n yz n i ( 10 13 c m -2 ) V BG (V) (b) D epth of well Fermi energy V 0 , E F ( m e V ) n 2D (10 13 cm -2 ) (c) 0 .0 0 .5 1 .0 1 .5 2 .0 Center of mass C e n t e r -o f -m a s s ( n m ) V 0 , E F ( m e V ) V BG (V) (d) 0 1 2 3 4 C e n t e r -o f -m a s s ( n m )
Figure 2.8: Gate dependence of q-2DESparameters obtained by Schrödinger-Poisson calculations.
(a)-(b) Effect of a topgate (a) and backgate (b) voltage on the band occupa-tion of the q-2des at the SrTiO3surface. The initial state was deliberately set up to be close to the point where the dxzand dyzbands start to be pop-ulated. (c)-(d) Calculations of the potential well depth, V0, Fermi energy, Ef, and center-of-mass of the envelope wave function, corresponding to the band occupations in (a) and (b). The energies are taken relative to the bulk conduction band bottom, Ec,bulk.
