Introduction
Quantum confinement involves the use of spatial modulations of chemical composition and electric fields to localize electrons to regions that are sufficiently small (at least in one direction) that their quantum mechanical properties are affected. Quantum wells, sketched in Figure 1 , are basic to semiconductor science and technology. Reference Weisbuch and Vinter1 In its simplest form, a quantum well involves a thin layer of one material, such as InGaN, sandwiched between thick layers of another (often a wider bandgap) material, such as GaN. This creates a potential well in which carriers can move in two directions but are confined in the third.
Figure 1. Schematic of the wave functions in an infinite square quantum well. Three states or sub-bands (n) are shown.
The basic ideas of quantum confinement—using a gradient of the electrochemical potential, such as band offsets, band bending, and applied fields, to localize the electrons in particular spatial regions—apply to transition metal oxides, but the physics is much richer. First and foremost, electrons in the narrow d-bands of transition metal oxides are subject to strong electron-electron interactions (thus the electrons are referred to as being “correlated,” and materials containing such electrons are referred to as “correlated electron materials”). The electron-electron interactions (“correlations”) lead to a rich variety of physical phenomena that can be accessed, modified, and controlled in quantum wells. Examples include magnetism with high Curie temperature, “Mott” (electron correlation-driven) metal-to-insulator transitions, high transition temperature (T c) superconductivity, and unique charge and magnetic order.
The electron densities required to obtain the Mott insulator and many of the related correlation-driven effects are typically very high, on the order of one electron per 1022 cm–3, or a sheet carrier density of several 1014 cm–2 in a single atomic plane. Reference Ahn, Bhattacharya, Di Ventra, Eckstein, Frisbie, Gershenson, Goldman, Inoue, Mannhart, Millis, Morpurgo, Natelson and Triscone2 Such sheet carrier densities are an order of magnitude higher than the highest density two-dimensional electron gases (2DEGs) achievable in conventional semiconductors (the III-nitrides). One of the unique aspects of oxide quantum wells is that these densities can be achieved. Reference Kim, Seo, Chisholm, Kremer, Habermeier, Keimer and Lee3–Reference Nakagawa, Hwang and Muller5 The article by Hilgenkamp in this issue discusses one example of such an interface, namely that between the band insulators LaAlO3 and SrTiO3. In this article, we discuss the issues arising in the theoretical description of new high density oxide quantum wells, describe some aspects of the materials, fabrication techniques and diagnostics used, and discuss a few prototypical examples at the forefront of current research interests.
Theoretical challenges
Theoretical and practical challenges arise for high electron density quantum wells. The high carrier densities imply that the physics can be much more local than in conventional semiconductors, because the interelectron distances become comparable to the distances between atoms in the crystal lattice. Materials must be controlled on subnanometer length scales, and the specifics of the interface are important. The richer physics means that information beyond just the spatial extent of the electron wave function is needed.
In conventional semiconductors, electron-electron interactions may, to a good approximation, be neglected (except for the electrostatic potential variations produced by spatially varying charge distributions). The situation may be discussed in terms of the electron wave function, which obeys the single-particle Schrödinger equation appropriate for a particle moving in a spatially varying potential, V. Further, the electron densities are typically low enough that an effective mass or k⋅p approximation may be made for the electronic band structure (this has been done in Equation 1). For the simple quantum well sketched in Figure 1, the physics may thus be encapsulated in a Schrödinger equation of the form:
$$- {{{\hbar ^2}} \over {2{m^ * }}}{{{d^2}{{\rm{\psi }}_n}\left( {z,{k_ \bot }} \right)} \over {d{z^2}}} + V\left( z \right){{\rm{\psi }}_n}\left( {z,{k_ \bot }} \right) = \left( {{E_n} + {{{\hbar ^2}k_ \bot ^2} \over {2{m^ * }}}} \right){{\rm{\psi }}_n}\left( {z,{k_ \bot }} \right)$$
. Here
${k_ \bot }$
is the wave vector corresponding to motion transverse to the interfaces separating the two materials, z is the coordinate perpendicular to the interface, and the index n labels discrete states of motion transverse to the interfaces. Solutions of different n are referred to as sub-bands. If for some n the wave functions vanish as z goes to ±∞ (so E
n
is less than the value of V far from the well), it can be stated that sub-band n is confined in the direction transverse to the interface. Quantum confinement refers to the situation when only a few sub-bands are occupied.
In transition metal oxides, the relevant electronic states are the transition metal d orbitals, which are highly spatially compact (typically 0.5 Å compared to the ∼4 Å spacing between transition metal ions) and are delocalized only via their overlap with ligand orbitals (typically oxygen 2p); thus the k⋅p approximation is not generally relevant. The high density and small size of the d orbital means that the interesting physics is to a large degree driven by a short range (effectively onsite) Coulomb interaction. In transition metal ions in free space, the d-levels occur in groups of five degenerate (equal energy) states. The degeneracy may be partially or fully lifted in crystals, so that different d-states may have different energies and therefore occupancies. The orbital occupancies are controlled by symmetry-breaking electric fields, transition metal-oxygen bond lengths, and octahedral rotations and strongly affect the physics. Bond angles play a very important role in correlated electron physics and may be controlled by judicious choice of interface materials.
To model the correlated situation, one must go beyond Equation 1 and consider a many-body Green function, which depends on orbital and spin as well as position transverse to the interface and momentum along it, and contains information about many-body effects such as magnetic, superconducting, and Mott insulating states. Even on a very basic level, the calculation of confining potentials is on a much less firm theoretical footing for correlated oxides, because the band offsets are determined by the energies of the frontier orbitals, which are set, in part, by beyond-band theory correlations. Reference Zaanen, Sawatzky and Allen6 Systematic investigations of these and related issues remain for the future.
Materials
Transition metal oxides crystallize in many structures, but to date, the experimental community has largely concentrated on quantum wells made from materials such as RTiO3, SrTiO3, SrVO3, and RNiO3 (R is a trivalent rare-earth ion, including Y). These materials crystallize in perovskite-derived (chemical formula: ABO3) structures. The B site ion in ABO3 is a transition metal with a partially filled (empty in case of SrTiO3) d-shell and is octahedrally coordinated with six oxygen ions, while the A site ion is typically an alkali earth (Ca, Sr, or Ba) or a trivalent rare-earth ion, including Y.
The A site ion affects the electronic properties in two ways. First, via filling of the d-shell: Sr or Ba donates two electrons, while Y, La, or other lanthanides (R) donate three so that the nominal configuration of SrVO3 is d
Reference Weisbuch and Vinter1
, while that of LaVO3 or YVO3 is d
Reference Ahn, Bhattacharya, Di Ventra, Eckstein, Frisbie, Gershenson, Goldman, Inoue, Mannhart, Millis, Morpurgo, Natelson and Triscone2
. Second, via internal strains, if the chemically preferred A-O distance is smaller (larger) than
$\sqrt 2$
times the B-O distance, the B-O complex will be subject to compressive (tensile) strain.
Reference Woodward7
Compressive internal strains are common and lead to rotational distortions, relative to the ideal cubic perovskite structure.
Reference Woodward7–Reference Glazer9
A prominent example, the orthorhombic GdFeO3 structure (space group Pbnm), is shown in
Figure 2
. BO6 rotations and distortions couple strongly to electronic properties both by changing bandwidths and by lifting degeneracies in the d-shell.
Figure 2. Schematic of the orthorhombic GdFeO3 structure adopted by many perovskite oxides (Pbnm notation). Shown are the BO6 octahedra, the A-site cations (large blue spheres), and the oxygen ions (small orange spheres). The B-site cations are located at the center of the octahedra.
The rare-earth nickelates (RNiO3) and the rare-earth titanates (RTiO3) are two prototypical perovskite oxides illustrating the close relationship between properties, oxygen octahedral tilts/distortions, and orbital ordering or polarizations. In the RNiO3 series, the temperature of the metal-to-insulator transition systematically increases with increasing deviation of the Ni–O–Ni bond angle from the ideal 180° angle in the cubic perovskite structure. Reference Torrance, Lacorre, Nazzal, Ansaldo and Niedermayer10 The RTiO3 series are prototype Mott insulators, with a single electron occupying the Ti t 2g orbitals. Magnetic ordering in the RTiO3 is closely coupled with Ti-O octahedral tilts and distortions, which remove the orbital degeneracy, Reference Goodenough and Zhou11–Reference Mochizuki and Imada13 often via long-range ferro-orbital or antiferro-orbital ordering. Reference Itoh, Tsuchiya, Tanaka and Motoya14–Reference Pavarini, Biermann, Poteryaev, Lichtenstein, Georges and Andersen18
In thin films and heterostructures, octahedral tilts are modified by epitaxial coherency strains and by interfacial coupling (connectivity) to adjacent layers or the substrate. The close coupling of tilts to orbital polarization and properties influences the phenomena observed in ultrathin oxide quantum wells, as will be further discussed later. In recent years, methods have been developed to establish octahedral tilt patterns as a function of epitaxial lattice mismatch strain even in few atomic layer thick films and superlattices; Reference He, Wells, Ban, Alpay, Grenier, Shapiro, Si, Clark and Xi19–Reference Hwang, Zhang, Son and Stemmer28 an excellent review of this subject has recently appeared in MRS Bulletin. Reference Rondinelli, May and Freeland29
Quantum wells are created by a spatial variation of the electrochemical potential. This may be accomplished in several ways. One is by varying the A-site ion, for example by sandwiching a few layers of LaMnO3 between layers of SrMnO3. The difference in charge between La (3+) and Sr (2+) ions creates a spatially varying electric potential, which can cause a spatially varying electron density. A second choice is to vary the B-site ion, for example by sandwiching a few layers of SrVO3 between layers of SrTiO3. In this example, the different electronegativities of V and Ti define the quantum well.
Most of the quantum wells studied to date have been grown on (001) planes (using [pseudo] cubic notation), although (111) structures have been theoretically discussed Reference Yang, Zhu, Xiao, Okamoto, Wang and Ran30,Reference Ruegg and Fiete31 as giving rise to interesting topological properties.
Fabrication and diagnostics
A number of different deposition methods are currently being used to fabricate oxide quantum structures (for a review, see Reference Reference Posadas, Lippmaa, Walker, Dawber, Ahn, Triscone, Rabe, Ahn and Triscone32). Addressing challenges such as stoichiometry control Reference Ohnishi, Shibuya, Yamamoto and Lippmaa33,Reference Jalan, Moetakef and Stemmer34 and defects created by energetic deposition (pulsed laser deposition [PLD], sputtering Reference Cuomo, Doyle, Bruley and Liu35 ) remains critical and is non-trivial. Advanced transmission electron microscopy techniques, in particular those based on scanning transmission electron microscopy (STEM), provide quantitative information about interfacial chemical abruptness. Reference Kourkoutis, Xin, Higuchi, Hotta, Lee, Hikita, Schlom, Hwang and Muller36,Reference Pennycook, Chisholm, Lupini, Varela, Borisevich, Oxley, Luo, van Benthem, Oh and Sales37 An example is shown in Figure 3 . Unit-cell resolved studies of BO6 rotations and bond-length changes are more challenging, even with the most advanced aberration-corrected transmission electron microscopes. A recently developed technique is position averaged convergent beam electron diffraction (PACBED) Reference LeBeau, Findlay, Allen and Stemmer38 in STEM, which is sensitive to picometer-scale structural distortions, Reference LeBeau, D’Alfonso, Wright, Allen and Stemmer39 has unit cell spatial resolution, and has been used to quantify oxygen octahedral tilts at interfaces and in superlattices. Reference Hwang, Son, Zhang, Janotti, Van de Walle and Stemmer27,Reference Hwang, Zhang, Son and Stemmer28
Figure 3. Atomic resolution, aberration-corrected STEM energy-dispersive x-ray spectrometry elemental maps of a thin GdTiO3 region (∼3 GdO layers wide) embedded in SrTiO3, using Ti-K, Sr-K, and Gd-L edges, respectively, and the corresponding STEM high-angular annular dark-field image. Figure courtesy of D. Klenov (FEI). Adapted from Reference 4. In the upper left panel, the light area is the GdTiO3 region, while the darker areas are SrTiO3. In the other panels, element-specific images of the same area highlight variations of particular elements.
Transport and device characteristics, including carrier mobilities, magnetic ordering temperatures, unintentional charge carriers, or, conversely, charge carrier densities that are lower than the expected concentrations, are the most sensitive measure of material quality. Disorder can give rise to carrier localization unrelated to true correlation physics. Reference Lee and Ramakrishnan40 For example, in SrTiO3, unintentional carriers can be due to oxygen vacancies, while unintentional acceptors (such as Sr vacancies) act as charge traps. Doping studies and measurements of carrier mobilities have demonstrated the excellent materials quality that can be achieved by oxide molecular beam epitaxy. Reference Son, Moetakef, Jalan, Bierwagen, Wright, Engel-Herbert and Stemmer41 Careful control of growth parameters in PLD allowed for the synthesis of superconducting Sr2RuO4 films. Reference Krockenberger, Uchida, Takahashi, Nakamura, Kawasaki and Tokura42 The particular properties of strongly correlated materials, such as low carrier mobilities and high electrical resistances, add extra challenges to characterizing material quality and electronic structure using transport properties.
Examples of oxide quantum well studies
SrTiO3/SrVO3/SrTiO3
Yoshimatsu et al. Reference Yoshimatsu, Okabe, Kumigashira, Okamoto, Aizaki, Fujimori and Oshima43–Reference Yoshimatsu, Sakai, Kobayashi, Horiba, Yoshida, Fujimori, Oshima and Kumigashira45 used photoemission (a technique based on the photoelectric effect in which the energies and momenta of electrons emitted in response to incident light are analyzed to obtain information about the occupied electronic states in a material) to study quantum wells consisting of n layers of SrVO3 (a moderately correlated metal crystallizing in the ideal cubic perovskite structure) embedded in SrTiO3 (a cubic perovskite band insulator), see schematic in Figure 4 a. The vanadium ion is nominally in the d Reference Weisbuch and Vinter1 configuration, and in the cubic bulk material, the one d-electron is shared between the three t 2g symmetry d orbitals, d xy, d xz, d yx. To a first approximation, an electron in the d xy orbital can move easily in the x–y plane but has only a very weak dispersion in the z direction.
Figure 4. SrVO3 quantum wells. (a) Schematic of a quantum well with 5 SrVO3 monolayers (MLs) with the V-atoms indicated in orange embedded in SrTiO3 (Ti = green). (b) Angle-integrated photoemission spectrum, symmetrized with respect to the Fermi energy, for quantum wells consisting of a different number of layers of SrVO3. (c) Energy of the bottom of a xz band plotted against a number of SrVO3 layers in a quantum well, showing progressive depopulation of xz-derived orbitals as well as thickness decreases. Adapted from References 43 and 45.
Figure 4b shows the photoemission spectra (many-body density of states) for quantum wells of a varying number of SVO layers. Photoemission measures the occupied portion of the electronic density of states; this figure has been symmetrized in energy. One sees that down to approximately six unit cells, the spectrum remains quite similar to that of the bulk material. Below this thickness, changes become apparent—first a suppression of the density of states peak at the Fermi level, then the appearance of a suppression (pseudogap), then the appearance of a full gap for two unit cells, which becomes much larger in magnitude for n = 1. Many oxide quantum wells and ultrathin films exhibit a metal-insulator transition when the film thickness is decreased below a few unit cells. Reference Yoshimatsu, Sakai, Kobayashi, Horiba, Yoshida, Fujimori, Oshima and Kumigashira45–Reference Boris, Matiks, Benckiser, Frano, Popovich, Hinkov, Wochner, Castro-Colin, Detemple, Malik, Bernhard, Prokscha, Suter, Salman and Morenzoni50 The spectroscopic evidence for gap formation suggests an intrinsic origin (as opposed to disorder causing localization).
Figure 4c shows that these effects occur in parallel with a thickness dependent orbital polarization effect, whereby the xz and yz bands (shown in the figure), which involve motion transverse to the interface, get progressively pushed upward in energy. As n approaches 1, what remains is a small number of xy bands (not shown) that become insulating. Also interestingly, this experiment reports no evidence for any leakage of electrons in SrVO3 into the SrTiO3 region. The quantum well evidently provides essentially complete confinement.
The SrVO3 quantum well, along with many other experiments, shows that quantum confinement can qualitatively alter the electronic properties, but confinement effects become detectable only at very short length scales, in this case, films with less than six unit cell thickness.
LaMnO3/SrMnO3 superlattices
Figure 5 shows a schematic of an electrostatically (A-site) defined set of quantum wells of alternating layers of LaMnO3 and SrMnO3. The different ionic charges of La and Sr mean that in bulk LaMnO3, the nominal Mn configuration is d Reference Moetakef, Cain, Ouellette, Zhang, Klenov, Janotti, Van de Walle, Rajan, Allen and Stemmer4 , while in bulk SrMnO3, it is d Reference Kim, Seo, Chisholm, Kremer, Habermeier, Keimer and Lee3 . SrMnO3 crystallizes in the cubic perovskite structure and is a simple Neel (two sublattice) antiferromagnet, while LaMnO3 exhibits a strong Jahn–Teller distortion that organizes the size Mn–O bonds into three inequivalent pairs with bond lengths ranging from 2.07 to 1.95 Å. This leads to a layered antiferromagnetic structure consisting of ferromagnetic planes with alternating spin orientations.
Figure 5. Neutron reflectometry of a manganite quantum well. (a) Schematic of the structure, showing one extra LaMnO3 layer (pink) introduced into every 9th LaMnO3/SrMnO3 pair in a multilayer structure. (b) Neutron reflectometry data in two different reflectance configurations R++ and R– – obtained on a sample with Sr density x = 0.47 (i.e., one extra La layer per 9 La/Sr bilayers) and fit; (c) depth dependence of magnetization inferred from neutron data, shown with layer-by-layer schematic of quantum well composition (color coding as on left). Adapted from Reference 51.
Santos and co-workers Reference Santos, Kirby, Kumar, May, Borchers, Maranville, Zarestky, Velthuis, van den Brink and Bhattacharya51 fabricated quantum well superlattices of this type and used neutron reflectometry to extract the space-dependent magnetic moment. Neutron reflectometry is a technique in which neutrons are scattered off of a solid at a very shallow angle of incidence. The dependence of the reflected neutron beam on angle of incidence can be analyzed to provide detailed information about the spatial variation of the magnetism in the direction perpendicular to the surface. They found that units of two LaMnO3 planes (pink unit in Figure 5) are ferromagnetic (all spins aligned in the same direction) with a high moment per Mn; this is different both from bulk LaMnO3 (where the magnetic state is antiferromagnetic, meaning that the moment directions alternate from layer to layer, so there is no net moment) and from the intervening regions (alternating layers of LaMnO3 and SrMnO3), where the magnetic moment is considerably smaller, roughly consistent with what is found in bulk La0.5Sr0.5MnO3 (this compound exhibits a complicated interplay of ferromagnetism and anti-ferromagnetism). Closer inspection of the figure reveals that the length scale over which the magnetization changes from a high to a low value is about two unit cells. Thus, we see that (1) length scales in electrostatically defined quantum wells are on the order of 2–3 lattice constants, consistent with earlier theoretical predictions, Reference Okamoto and Millis52,Reference Okamoto, Millis and Spaldin53 (2) electronic properties other than the charge can be controlled in quantum wells, and (3) experimental technique has advanced to the point where layer-dependent electronic and magnetic properties can be determined.
RNiO3 quantum wells
The use of quantum wells to control the electronic properties of complex oxides received a significant impetus from the proposal of Chaloupka and Khaliullin
Reference Chaloupka and Khaliullin54
to convert LaNiO3 into a high-Tc superconductor. LaNiO3 is rhombohedral with a nominally orbitally degenerate d
Reference Woodward7
configuration, involving one electron shared between the two e
g
orbitals
${d_{{x^2} - {y^2}}}$
and
${d_{3{z^2} - {r^2}}}$
(so two bands cross the Fermi level), whereas an essential characteristic of the high-Tc copper-oxide superconductors is a one-band Fermi surface involving the transition metal
${d_{{x^2} - {y^2}}}$
orbital. It was suggested that a sufficiently narrow quantum well (perhaps one unit cell thick) could, by lowering the orbital symmetry, split the d-levels sufficiently that only one band remained at the Fermi level, thereby replicating the electronic structure of a copper oxide superconductor.
Reference Chaloupka and Khaliullin54,Reference Hansmann, Yang, Toschi, Khaliullin, Andersen and Held55
It is fair to say that the original goal has not been realized and probably cannot be realized (at least in these materials). However, attempts to test this hypothesis have produced considerable insight into materials and have helped hone experimental techniques. Orbital reflectometry (polarization-dependent resonant x-ray scattering in a surface reflection geometry) has enabled estimates of the occupancy of the relevant d-orbitals. Very recent results
Reference Wu, Benckiser, Haverkort, Frano, Lu, Nwanko, Brück, Audehm, Goering, Mache, Hinkov, Wochner, Christiani, Heinze, Logvenov, Habermeier and Keimer56,Reference Frano, Schierle, Haverkort, Lu, Wu, Blanco-Canosa, Nwankwo, Boris, Wochner, Cristiani, Habermeier, Logvenov, Hinkov, Benckiser, Weschke and Keimer57
indicate that even very narrow quantum wells can produce some degree of orbital polarization, in semiquantitative agreement with theoretical predictions,
Reference Han, Wang, Marianetti and Millis58
and show that the relative occupancies of the different orbitals is substantially affected by lattice strain.
Reference Wu, Benckiser, Haverkort, Frano, Lu, Nwanko, Brück, Audehm, Goering, Mache, Hinkov, Wochner, Christiani, Heinze, Logvenov, Habermeier and Keimer56
The results underscore the importance of lattice strain in the physics of LaNiO3 quantum wells, as already demonstrated in a number of studies.
Reference May, Kim, Rondinelli, Karapetrova, Spaldin, Bhattacharya and Ryan21,Reference Son, Moetakef, LeBeau, Ouellette, Balents, Allen and Stemmer47,Reference Ouellette, Lee, Son, Stemmer, Balents, Millis and Allen59,Reference Stewart, Brownstead, Liu, Kareev, Chakhalian and Basov60
In addition to the question of orbital polarization, a major ongoing issue is the nature of the insulating state observed in the thinnest films. Reference Gray, Janotti, Son, LeBeau, Ueda, Yamashita, Kobayashi, Kaiser, Sutarto, Wadati, Sawatzky, Van de Walle, Stemmer and Fadley46–Reference Boris, Matiks, Benckiser, Frano, Popovich, Hinkov, Wochner, Castro-Colin, Detemple, Malik, Bernhard, Prokscha, Suter, Salman and Morenzoni50 Complex spin and charge ordering appear to be integral to the insulator in bulk, Reference Medarde61 but may not be present in the two-dimensional limit. Reference Lee, Chen and Balents62–Reference Lau and Millis64 Ongoing studies that reveal what order Reference Frano, Schierle, Haverkort, Lu, Wu, Blanco-Canosa, Nwankwo, Boris, Wochner, Cristiani, Habermeier, Logvenov, Hinkov, Benckiser, Weschke and Keimer57 and modifications of octahedral tilt patterns are present Reference Hwang, Son, Zhang, Janotti, Van de Walle and Stemmer27 and how they influence electronic states may shed light on the mechanisms of interaction-driven metal-insulator transitions more generally.
A comparison Reference Hwang, Son, Zhang, Janotti, Van de Walle and Stemmer27 of epitaxially strained LaNiO3 films with LaNiO3/SrTiO3 superlattices has also revealed how competing structural influences determine BO6 octahedral tilt patterns in ultrathin films and quantum wells, and how these in turn can influence the orbital polarization. For LaNiO3 layers that are connected to the substrate, strongly modified tilt patterns have been observed, Reference May, Kim, Rondinelli, Karapetrova, Spaldin, Bhattacharya and Ryan21,Reference Hwang, Son, Zhang, Janotti, Van de Walle and Stemmer27,Reference Hwang, Zhang, Son and Stemmer28,Reference Son, Moetakef, LeBeau, Ouellette, Balents, Allen and Stemmer47 which appear to be correlated with insulating behavior below a critical thickness (bulk LaNiO3 is metallic at all temperatures) that has been reported in several independent studies. Reference Son, Moetakef, LeBeau, Ouellette, Balents, Allen and Stemmer47,Reference Scherwitzl, Gariglio, Gabay, Zubko, Gibert and Triscone48,Reference Boris, Matiks, Benckiser, Frano, Popovich, Hinkov, Wochner, Castro-Colin, Detemple, Malik, Bernhard, Prokscha, Suter, Salman and Morenzoni50,Reference Liu, Okamoto, van Veenendaal, Kareev, Gray, Ryan, Freeland and Chakhalian65 In contrast, LaNiO3 layers in superlattice geometry exhibit octahedral tilts that relax toward bulk values, even when they are under the same epitaxial coherency strain as individual films. Reference Hwang, Son, Zhang, Janotti, Van de Walle and Stemmer27 This relaxation is facilitated by correlated tilts in the SrTiO3 spacers and allows for metallic behavior of even very thin LaNiO3 layers. Reference Hwang, Zhang, Son and Stemmer28 An important lesson is that octahedral tilt patterns, often substantially modified in ultrathin films, should be quantified along with reporting transport and other properties, as they may be at the core of many observed phenomena.
SrTiO3 quantum wells with extreme electron densities
Two-dimensional electron gases at interfaces between SrTiO3 and RTiO3 exhibit mobile carrier densities of several 1014 cm–2. Reference Kim, Seo, Chisholm, Kremer, Habermeier, Keimer and Lee3,Reference Moetakef, Cain, Ouellette, Zhang, Klenov, Janotti, Van de Walle, Rajan, Allen and Stemmer4 These extreme electron densities are introduced into the d-bands of SrTiO3, a material that is a band insulator in the bulk (i.e., not a correlated material). The ideal RTiO3/SrTiO3 interface contains ½ electron per cubic interface unit cell (or 3.5 × 1014 cm–2), which, even if these carriers are confined to a single interfacial TiO2 plane, is half the density required for a Mott insulator, namely one electron per 3D unit cell (or 7 × 1014 cm–2 per TiO2 plane). Fortunately, RTiO3/SrTiO3 interfaces are symmetric, that is, both types of interfaces, RTiO3 on SrTiO3 and SrTiO3 on RTiO3, provide ½ electron per cubic interface unit cell. Reference Moetakef, Cain, Ouellette, Zhang, Klenov, Janotti, Van de Walle, Rajan, Allen and Stemmer4,Reference Moetakef, Jackson, Hwang, Balents, Allen and Stemmer66 Quantum wells that are bound by two such interfaces contain electron densities of 7 × 1014 cm–2, spread out over the width of the quantum well, which has to be made very narrow to achieve high 3D electron densities ( Figure 6 ).
Figure 6. Schematic showing an extreme-electron density quantum well, enabled by sandwiching a thin SrTiO3 film between two RTiO3 layers. In RTiO3, the RO and TiO2 planes carry –1 and +1 formal ionic charges. Each RO layer donates ½ electron to the upper/lower lying TiO2 planes, including the interfacial TiO2 plane, as indicated by the arrows. The TiO2 and SrO layers in the SrTiO3 are charge neutral. At the interface, a two-dimensional electron gas (2DEG) with mobile carrier densities on the order of 3 × 1014 cm–2 is formed, which also compensates for the net fixed positive interface charge of the terminating RO planes. Transport studies have established that the 2DEG is located in the SrTiO3, Reference Moetakef, Cain, Ouellette, Zhang, Klenov, Janotti, Van de Walle, Rajan, Allen and Stemmer4,Reference Cain, Lee, Moetakef, Balents, Stemmer and Allen74 consistent with experimentally and theoretically determined band offsets. Reference Conti, Kaiser, Gray, Nemsak, Palsson, Son, Moetakef, Janotti, Bjaalie, Conlon, Eiteneer, Greer, Perona, Rattanachata, Saw, Bostwick, Stolte, Hloskovsky, Drube, Ueda, Kobayashi, Van de Walle, Stemmer, Schneider and Fadley75 We note that the RTiO3 surfaces are polar but are expected to solve the polar problem by one or more of the several routes that are available to such surfaces. Reference Noguera76 To achieve high 3D carrier densities, the SrTiO3 quantum well has to be made very thin (no more than a few SrO layers wide), as otherwise the 2DEG will spread out over many TiO2 layers. Reference Khalsa and MacDonald77,Reference Park and Millis78
Signatures of mass enhancement were observed in the DC transport of extreme-density quantum wells of SrTiO3 sandwiched between the Mott insulator GdTiO3 that was on the verge to a transition to an insulator. Reference Moetakef, Jackson, Hwang, Balents, Allen and Stemmer66 Such mass enhancement is similar to what is observed in doped (metallic) bulk RTiO3 near the Mott transition. Reference Tokura, Taguchi, Okada, Fujishima, Arima, Kumagai and Iye67 At a critical thickness (2 SrO layers), or 3D electron density, a correlated insulating state emerges ( Figure 7 a). Reference Moetakef, Jackson, Hwang, Balents, Allen and Stemmer66 Using atomic resolution STEM, it was shown that orthorhombic-like Sr-site displacements (which are closely coupled with oxygen octahedral tilts) are observed only for SrTiO3 quantum wells that were 1 and 2 SrO layers wide, Reference Zhang, Hwang, Raghavan and Stemmer68 in precise agreement with the observed metal-to-insulator transition (Figure 7b). Metallic quantum wells (those thicker than 2 SrO layers) showed no Sr displacements and octahedral tilts. Reference Zhang, Hwang, Raghavan and Stemmer68 Therefore, similar to bulk Mott insulators, the transition to the insulating state is accompanied by a reduction in symmetry. These results support true “Mott” physics, controlled by the combination of on-site repulsive interaction at large electron densities and structural distortions in these quantum wells.
Figure 7. (a) Electrical transport (resistance) data of SrTiO3 quantum wells embedded in GdTiO3 layers as a function of SrTiO3 quantum well thickness (expressed here in terms of the number of SrO layers). The metallic SrTiO3 quantum wells contain mobile 7 × 1014 cm–2 carriers. A transition to an insulating state is observed at a thickness of 2 SrO layers. (b) High-angle annular dark-field image of a sample containing multiple SrTiO3 quantum wells with different thicknesses used for structural characterization. The GdTiO3 layers appear bright, and the SrTiO3 quantum wells are darker. (c) Magnified image of the SrTiO3 quantum well that is 2 SrO layers wide. The red line is a guide to the eye showing buckling of the Sr columns, which only appears in the insulating quantum wells. Adapted from References 66 and 68.
Recent theoretical studies of quantum wells of 1 SrO layer in GdTiO3 suggest the formation of a dimer Mott insulator state, in which electrons are localized to bonding orbitals on molecular dimers formed across a bilayer of two TiO2 planes. Reference Chen, Lee and Balents69 These calculations were carried out for a different crystallographic orientation relationship than the experiments, and thus a complete theoretical understanding of the experimentally observed insulating state is still lacking. Alternative routes to an insulating state, such as charge ordering, may exist.
In addition to the polar/nonpolar interfaces described previously, modulation doping can be used to introduce large densities of carriers into such quantum wells. Reference Son, Jalan, Kajdos, Balents, Allen and Stemmer70,Reference Kajdos, Ouellette, Cain and Stemmer71 Moreover, by judicious use of the materials that interface the quantum well, octahedral tilts and other properties may be controlled. For example, in the experiments described previously, the quantum well was bound by GdTiO3, which is among the more strongly distorted RTiO3s, and is also an insulating ferrimagnet (magnetic moment directions alternate from layer to layer but net magnetic moment remains, unlike in antiferromagnetic materials). Not only does this give rise to fairly large distortions in the insulating SrTiO3 quantum wells, but also to ferromagnetism in the metallic quantum wells below a critical thickness. Reference Moetakef, Williams, Ouellette, Kajdos, Goldhaber-Gordon, Allen and Stemmer72
Conclusions and future outlook
Combining spatially resolved studies of structural distortions and orbital polarization with measurements of properties (such as magnetism and electrical transport) should allow for addressing some of the most interesting and challenging scientific questions in the field of strongly correlated materials physics and electronics. In particular, it will be important to determine if the orbital/structural/bandwidth control afforded by proximity effects in heterostructures can be used to (1) rationally engineer emergent states (beyond metal-insulator transitions); Reference Rondinelli, May and Freeland29 (2) bring strongly correlated oxides closer to quantum phase transitions to allow for electric field control of correlated phenomena; and (3) conduct systematic studies that distinguish the relative roles of orbital coupling to the lattice and electron-electron interactions in correlated materials physics and phenomena.
Figure 8 illustrates the different routes to control strongly correlated materials using narrow quantum wells and interfacial proximity effects. Specifically, structural distortions, high electron (or hole) densities introduced by electrostatic doping, and a host of properties, such as ferromagnetism and superconductivity, can be induced in high-quality epitaxial structures. Such hybrid and proximity effects are likely to be of great interest for applications in spintronics, quantum computing, as a route to Majorana fermions (novel quantum states, which, unlike electrons, are their own antiparticles and that have been proposed as a substrate for quantum computing applications), and, more generally, for artificially designing novel states of matter that do not exist in bulk. Modulating extreme carrier densities by field effect could also be of great interest for RF transistors Reference Boucherit, Shoron, Cain, Jackson, Stemmer and Rajan73 or tunable plasmonic devices. Ultrathin layers are necessary, requiring exquisite control not only over atomic layers but also of defects and doping, to avoid artifacts from localization due to disorder. Advances in theoretical understanding are also needed. The theoretical and experimental results achieved to date demonstrate that the exciting possibility of using oxide quantum wells to design correlation physics from the bottom up is now becoming a reality.
Figure 8. Summary of possible proximity effects in complex oxide quantum wells. Ultrathin quantum wells are required for all effects.
Acknowledgments
S.S. would like to acknowledge discussions and collaborations with Jim Allen, Leon Balents, Pouya Moetakef, Clayton Jackson, Dmitri Klenov, Jack Zhang, Jinwoo Hwang, Junwoo Son, Chris Van de Walle, Anderson Janotti, Siddharth Rajan, David Goldhaber-Gordon, and Jimmy Williams. She also thanks Adam Kajdos, Jack Zhang, Jinwoo Hwang, and Clayton Jackson for help with some of the figures for this article. Work at UCSB was supported by the Army Research Office (Grant No. W911-NF-09–1-0398), the US National Science Foundation (Grant Nos. DMR-1006640 and DMR-1121053), the US Department of Energy (Award No. DE-FG02–02ER45994), and DARPA (Award No. W911NF-12–1-0574). A.J.M. thanks Hanghui Chen, Hung Dang, M.J. Han, Chungwei Lin, Peter Littlewood, Chris Marianetti, Satoshi Okamoto, Hyowon Park, Seyoung Park, Darrell Schlom, and Jean-Marc Triscone for collaborations and discussions and the US Department of Energy (Award No. DOE-FG-BES2–04ER46169) and the US Army Research Office (Award W911NF-09–1-0345) for support.
