General considerations of the electrostatic boundary conditions in oxide heterostructures
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y General considerations of the electrostatic boundaryconditions in oxide heterostructures
Takuya Higuchi and Harold Y. Hwang , Department of Applied Physics, University of Tokyo, Hongo, Tokyo 113-8656, Japan Department of Applied Physics and Stanford Institute for Materials and Energy Science,Stanford University, Stanford, California 94305, USA Correlated Electron Research Group (CERG), RIKEN-ASI, Saitama 351-0198, Japan
When the size of materials is comparable to the characteristic length scale of their physicalproperties, novel functionalities can emerge. For semiconductors, this is exemplified bythe “superlattice” concept of Esaki and Tsu, where the width of the repeated stacking ofdifferent semiconductors is comparable to the “size” of the electrons, resulting in novelconfined states now routinely used in opto-electronics [1]. For metals, a good example ismagnetic/non-magnetic multilayer films that are thinner than the spin-scattering length,from which giant magnetoresistance (GMR) emerged [2, 3], used in the read heads ofhard disk drives. For transition metal oxides, a similar research program is currentlyunderway, broadly motivated by the vast array of physical properties that they host. Thislong-standing notion has been recently invigorated by the development of atomic-scalegrowth and probe techniques, which enables the study of complex oxide heterostructuresapproaching the precision idealized in Fig. 1(a). Taking the subset of oxides derived fromthe perovskite crystal structure, the close lattice match across many transition metaloxides presents the opportunity, in principle, to develop a “universal” heteroepitaxial1igure 1: (a) Schematic illustration of ideal heterointerfaces of two perovskites ABO andA ′ B ′ O stacked in the [001] direction. (b) Charge sequences of the AO and BO planesof perovskites plotted together with their pseudocubic lattice parameters. (c) Scanningtransmission electron microscopy image of a LaTiO /SrTiO (001) superlattice (Ohtomo et al. [4]).materials system.Hand-in-hand with the continual improvements in materials control, an increasinglyrelevant challenge is to understand the consequences of the electrostatic boundary con-ditions which arise in these structures. The essence of this issue can be seen in Fig. 1(b),where the charge sequence of the sublayer “stacks” for various representative perovskitesis shown in the ionic limit, in the (001) direction. To truly “universally” incorporatedifferent properties using different materials components, be it magnetism, ferroelectric-ity, superconductivity, etc., it is necessary to access and join different charge sequences,2abelled here in analogy to the designations “group IV, III-V, II-VI” for semiconductors.As we will review, interfaces between different families creates a host of electrostaticissues. They can be somewhat avoided if, as in many semiconductor heterostructures,only one family is used, with small perturbations (such as n-type or p-type doping)around them . However, for most transition metal oxides, this is greatly restrictive. Forexample, LaMnO and SrMnO are both insulators in part due to strong electron corre-lations, and only in their solid solution does “colossal magnetoresistance” emerge in bulk[6]. Similarly, the metallic superlattice shown in Fig. 1(c) can be considered a nanoscaledeconstruction of (La,Sr)TiO to the insulating parent compounds. Therefore the aspira-tion to arbitrarily mix and match perovskite components requires a basic understandingof, and ultimately control over, these issues.In this context, here we present basic electrostatic features that arise in oxide het-erostructures which vary the ionic charge stacking sequence. In close relation to theanalysis of the stability of polar surfaces and semiconductor heterointerfaces, the vari-ation of the dipole moment across a heterointerface plays a key role in determining itsstability. Different self-consistent assignments of the unit cell are presented, allowingthe polar discontinuity picture to be recast in terms of an equivalent local charge neu-trality picture. The latter is helpful in providing a common framework with which todiscuss electronic reconstructions, local-bonding considerations, crystalline defects, andlattice polarization on an equal footing, all of which are the subject of extensive currentinvestigation. polar discontinuity picture The surface of crystals determines many of their physical, mechanical and chemical prop-erties. Due to the lack of translational symmetry in the perpendicular direction, the These effects can in principle also be reduced by choosing a (110) growth orientation, but otheraspects of stability may be limiting [5]. ρ , the dipole µ in the unit cellstarting from the top-most layer, the electric field E induced by the dipoles, and theelectrostatic potential φ at the surface of an ionic crystal with dipole moment in eachunit cell.stable charge distribution at the surface can be completely different from that of thebulk, and the surface may reconstruct in a manner different from the bulk states. Imag-ine an ideal ionic crystal which consists of charged ions bound together by their attractiveinteractions, and all the ions are taken as fixed point charges. Since the charges are lo-cally preassigned to the ions in this model, the ideal ionic surface apparently requires noreassignment of the charges from that of the bulk. However, the electrostatic potentialin an ionic crystal diverges when there is a dipole moment in the unit cell perpendicularto the surface. The potential φ should be constant in vacuum in the absence of externalfields, and the potential can be obtained by integrating the electric field caused by thecharged sheets, as shown in Fig. 2. A finite shift in the potential emerges due to thedipole moment of each unit cell, and as the unit cells are stacked, so the potential grows,and diverges into the crystal. Due to this effectively infinite surface energy, such surfacescannot exist without reconstructions, and the stability of an ionic surface randomly cutfrom the bulk is not trivial without knowing the stacking sequence of the charged sheetsprecisely. This surface instability and the associated reconstructions have been indeedobserved by means of low-energy electron diffraction (LEED) and ion scattering, whereabsorption of foreign atoms, surface roughening, or changes in surface stoichiometry werefound [7–9].In order to survey the stability of such surfaces, Tasker introduced a classification of Note that literature on this topic uses both the electrostatic potential (for a positive test charge)and the electron potential energy (as in band diagrams) — we use the former here. starts from the top-most layer [10]. Note he discussed the stability of bulkfrozen surfaces , where the top-most layer is one of the constituent atomic sheets of thebulk crystal and has no reconstruction. Tasker described three types of the surfaces, asshown in Fig. 3: • Type 1 has equal numbers of anions and cations on each plane, and therefore theunit cell has no dipole moment. For example, the (001) and (110) surfaces of therocksalt structure
M X (e.g. NaCl, MgO, NiO) are classified as this type. • Type 2 has charged planes, but no net dipole moment perpendicular to the surface.The anion X terminated (001) surface of the fluorite structure M X (e.g. UO ,ThO ) is an example. • Type 3 has charged planes and a net dipole moment normal to the surface. Exam-ples include the (111) surface of the rocksalt structure, and (001) or (111) surfacesof the zincblende structure
M X (e.g. GaAs, ZnS).In Tasker’s classification, type 1 and type 2 surfaces are stable while type 3 is not,since the instability of the surface comes from the stacking of the dipole in each unitcell. Macroscopically, the instability of the type 3 surface arises from the change inthe potential slope when crossing the surface. The term “polar surface” can be definedfollowing this classification, namely we call a surface “non-polar” when it is type 1 ortype 2, and “polar” when it is type 3.These definitions are analogous to the definition of polar crystals, although polar sur-faces and surfaces of polar crystals are not equivalent. Dielectric polarization is observedwhen an electric field is applied to a material, but even in the absence of the field, somecrystals retain a “spontaneous” polarization [12]. Only 10 out of 32 point groups showthis behavior, and their members are called polar crystals, while the others are non-polar.Twenty-one out of 32 point groups do not have inversion centers, and the polar crystals This definition of “ bulk frozen ” follows description of Goniakowski et al. [11]. ρ on planes for three stacking sequences parallel to thesurface. (a) Type 1, (b) type 2, and (c) type 3 (Tasker [10]).are included among them. When a material shows macroscopic spontaneous polariza-tion, the electrostatic potential of one end is different from the other end as a result ofthe stacking of the dipole in each unit cell, and the surface usually has “compensating”charge to reconcile this potential difference. Since Tasker took unit cells from the top-most layer, even crystals with inversion symmetry can show dipoles in the unit cells inhis model. For example, although NaCl is cubic and has inversion symmetry, its (111)surface is classified into type 3. Therefore, the word “polar” should be used with somecare since it has different meanings in different contexts. At the surface of covalent crystals, lacking full coordination, the top-most atoms havevalence electrons which are not used for bond formation. These non-bonding electronsare called dangling bonds, and have higher energy than the bonding electrons, whichcauses the movement of the atom positions to decrease their number [13].In a covalent crystal, since the bonds are formed as a hybridization of the valenceelectrons of charge-neutral atoms, one can describe the charge distribution starting from abulk unit cell which is charge neutral and dipole-free. However, when the electronegativityof the atoms are different e.g., Ga and As in GaAs, charge transfer between anions andcations occurs, similar to the ionic case. This charge transfer is realized by the differenceof the contribution of each bond, namely 1 + α electrons to the anions and 1 − α electronsto the cations. Here α is a parameter to describe the ionicity of the bond, determined by6he electronegativity of the two bonding atoms. As a consequence, the unit cell can havea dipole moment normal to the surface, which causes the same instability as that in theionic crystal case. Therefore, even in a covalent crystal, a surface instability emerges fromthe dipole in the unit cell, independent of the surface instability naturally arising fromdangling bond formation. Even though Tasker’s classification was introduced to describethe stability of ionic surfaces, it is also relevant for covalent surfaces in the presence offinite ionicity.Both dangling bond formation and the instability of polar surfaces are at play, andthey are reconciled simultaneously at the surface of covalent crystals. Therefore, directobservation of the instability of polar surfaces in covalent systems has been difficult.When we consider an epitaxial interface which has similar charge structure as the polarsurface, the instability from the dangling bonds disappears, and we can solely discuss thestability in the same manner as that for the ideal ionic surfaces, as discussed in the nextsection. Similar to the instability of polar surfaces, dipoles in the unit cells stacking from theinterface can cause potential divergence and instability, and require reconstruction. Thispoint was first proposed at the heteroepitaxial interface of GaAs and Ge in the [001]direction by three groups from different starting points for treating ionicity. Based ontheir considerations, we define polar and non-polar interfaces, in analogy to polar andnon-polar surfaces.
Frensley and Kroemer calculated the energy band diagram at abrupt semiconductor het-erojunctions [14]. Their starting point was to describe the alignment of atoms around theinterface without considering the ionicity, and then calculate the charge transfer based onthe ionicity using the electronegativity of the atoms, under assumption that the chargetransfer only occurs between the nearest neighbors. The Phillips electronegativity values7igure 4: (a) Model for a (001) Ge/GaAs heterojunction considering the ionic characterof the bonds. The atomic positions and effective ionic charges are shown above. Belowis a diagram of the plane-averaged potential (Frensley and Kroemer [14]). (b) Schematicdiagram of the charge transfer from the neutral atoms with respect to the electronega-tivity. X Ph were used ( X Ph (Ga) = 1 . X Ph (Ge) = 1 .
35, and X Ph (As) = 1 .
57 [15]). In the bulkzincblende structure AB , the A site is tetrahedrally coordinated by four B atoms and vice versa , and based on their calculation [16], the ionic charges e ∗ of the atoms are givenby e ∗ ( A ) = − e ∗ ( B ) = 0 . q × [ X Ph ( B ) − X Ph ( A )] , (1)where q is the elementary charge. This is equivalent to assuming a charge transfer of0 . q × [ X Ph ( B ) − X Ph ( A )] between any pair of the nearest neighbors.Consider the case of Ge/GaAs interfaces as shown in Fig. 4(a), where the charge e ∗ (Ga int ) on the Ga ions adjacent to the interface is e ∗ (Ga int ) = 0 . q × (cid:20) X Ph (Ge) + 12 X Ph (As) − X Ph (Ga) (cid:21) = 0 . q . (2)Similarly, the charges e ∗ (Ge int ) on the Ge ions at the interface and e ∗ (As) at the As sitesare e ∗ (Ge int ) = − . q and e ∗ (As) = − . q .Without ionicity, the electrostatic potential is constant, and even with ionicity, thepotential does not diverge. This can be understood easily by tracking the virtual chargetransfer processes from the starting alignment of the charge neutral atoms. The charge8igure 5: (a) Calculated contour plot of charge density for the partially occupied interfaceband around Ga-Ge bonds (Baraff et al. [17]). (b) Schematic model for counting electronsfrom the local-bond point of view.transfer is equivalent to the situation that each neutral atom loses 0 . q × X Ph charges,and half of them are transferred to the left atoms and the other half to the other side,as shown in Fig. 4(b). Therefore, charges are always transferred symmetrically, and eachmodulation creates no dipole, resulting in no potential shift. Here, a change of the numberof electrons in the ions compared to the bulk state is assumed, which is equivalent tochanging the valence assignments. Baraff et al. performed a self-consistent calculation of the potential, charge density, andinterface states for the abrupt interface between Ge and GaAs, terminated on a (001)Ga plane [17]. As shown in Fig. 5(a), their calculation showed fractional occupancy ofelectronic states only at the interface, which cannot exist in bulk.These interface states can be discussed from a local-bond counting point of view aswell. The number of the valence electrons is 4 for Ge, 3 for Ga, and 5 for As. When weassume all the valence electrons of an atom are equally distributed to the four covalentbonds around it, Ge, Ga, and As atoms supply 1, 0.75, and 1.25 electrons to each bond,respectively. As shown in Fig. 5(b), the Ge-Ga bonds at the interface have 1.75 electronsper bond, while other bonds have 2 electrons in each. These partially occupied bondsare considered to form the interface states.When the number of electrons are smaller than that in the bulk, the attractive inter-action between the bonded atoms should be weaker. According to their calculation, the9nergy is minimized when the Ge-Ga bond length is 4 % larger than that of the bulk.Without the change of the bonding length, the system requires long-range disturbanceof the lattice, which is unlikely to be realized.
Both the charge transfer model by Frensley and Kroemer and the electron counting modelby Baraff et al. predict (indeed require) that there are interface states at a Ge/GaAs (001)interface even if it is perfectly abrupt, with no crystalline defects. The central point raisedby these studies is that despite having the same crystal structure, and having very closelymatching lattice constants, this interface must accomodate charge arising from interfaceboundary conditions. However, experimentally no considerable density of interface stateswas observed [18], and a model to treat this interface without changing the number ofcharges at the interface was required.Harrison et al. constructed a model for the Ge/GaAs (001) heterojunctions by stack-ing the fully ionized atoms, and calculated the electrostatic potential based on the fixedassignment of the charges [19]. As shown in Fig. 6(a), the potential is very similar to thecase of polar surfaces since the unit cells which start from the interface have dipoles inGaAs, while the unit cells in Ge are always charge neutral. Therefore, the stacking ofthe dipoles causes potential divergence in this case as well.Since the charge of each ion was fixed, the solution to the instability of this interfacerequires compensation by changing the stoichiometry at the interface. They proposed asimple model where 1/4 of the Ge atoms are replaced by As atoms at the interface while1/4 of the Ga atoms adjacent to the interface are replaced by Ge. In this reconstructedmodel with two transition layers, the electrostatic potential does not diverge, as shownin Fig. 6(b).It might be surprising that from two completely different starting points, namely onefrom covalent (Frensley and Kroemer) and the other from ionic (Harrison et al. ) pictures,exactly the same potential diagrams were obtained. However, rearranging the numberof charges at the perfectly abrupt interface or changing the interface composition while10igure 6: Schematic crystal structure and electrostatic potential φ in the heterojunctionsof Ge and GaAs in the [001] direction. (a) An atomically abrupt interface. (b) A Ge/GaAsheterojunction with two off-stoichiometric transition layers (Harrison et al. [19]).maintaining the ionic charges of the atoms can give the same net charge distribution. Theexperimental absence of localized states suggests the atomic reconstruction based on theionic picture. This is equivalent of saying that the electronic state at this semiconductorheterointerface cannot deviate so strongly from that of the bulk constituents — it is en-ergetically inaccessible. This is the fundamental aspect which can be quite different incomplex oxide heterointerfaces, and is the subject of much current excitement. Namely,there is a possibility that the charge transfer picture (Frensley and Kroemer) and theelectron counting picture (Baraff et al. ), which require large deviations of electron num-bers from the bulk values, can be energetically accessed in transition metal oxides withmulti-valency, as described in Section 3. Following the model by Harrison et al. , we can define the polar nature of a bulk frozen interface between two materials, where the interface consists of two bulk frozen surfacesconnected together . First let us classify the interfaces by the polar nature of the twoconstituent surfaces, as shown in Fig. 7. Here, we consider only interfaces between two semi-infinite materials — we ignore the coupling toother interfaces or surfaces, which is discussed in Sections 4.3 and 4.4. ρ on planes, the dipole moment µ in unitcells starting from the interface, and the electrostatic potential for the four stackingsequences parallel to the interface, (a) type I, (b) type II, (c) type III, and (d) type IV.In order to treat a bulk frozen interface as a set of two bulk frozen surfaces, vacuum isinserted between them (dashed lines). The electrostatic potential φ was calculated takingthe vacuum as the potential reference except for the right plot in (c), where a constantelectric field was added to show the absence of macroscopic band bending at the interface. • Type I is formed between two non-polar surfaces, and the potential is flat. • Type II is formed between polar and non-polar surfaces, and the potential divergesfrom the interface in the material with the polar surface. • Type III is formed between two polar surfaces, and the direction and the size of thedipoles in the unit cells which start from the interface is the same. Although φ looksto diverge from the interface in both materials, there is no macroscopic differencein the potential slope across the interface. We can cancel the potential slope onboth sides by adding a constant electric field, as shown in Fig. 7(c). Therefore, thisinterface is stable as constructed. • Type IV is formed between two polar surfaces, where the dipoles in the two differentunit cells are not identical, which results in a macroscopic difference in the potentialslope at the interface. It is impossible to find any constant electric field to cancelout the potential divergence in both media, due to this difference . Note type IV is the most common and general case in reality, since the electronegativity can neverperfectly match between different materials. bulk frozen interface is non-polar, when the dipoles in the unit cells starting fromthe interface are identical across the interface, and thus no change of the macroscopicpotential slope exists. On the other hand, it is polar if a bulk frozen interface has adiscontinuity of the dipole moment in each unit cell. For example, from a purely ionicviewpoint (Harrison et al. ) the abrupt Ge/GaAs (001) interface is classified as type II,and thus polar.This definition of the polar nature of interfaces is consistent with that of surfaces.When the vacuum is treated as a charge neutral medium, the non-polar surface is typeI, and the polar surface is type II in this classification of the interfaces, and the polarnature is maintained following the definitions for the interface. We can treat surfacesand interfaces in one framework, which is the polar nature of discontinuities at materi-als boundaries. In summary, polar discontinuities, which consist of polar surfaces andinterfaces, are unstable due to the macroscopic potential folding arising from the discon-tinuity of the stacking of dipoles in the unit cells, and require reconstructions to stabilizethem. Interfaces with continuous polarity, on the other hand, are stable without anyreconstructions.
As noted in the introduction, the modern ability to approach atomic control in complexoxide heterostructures has newly enabled the experimental investigation of their polardiscontinuities. As illustrated in Fig. 1(a), an immediate question arises regarding thechoice of the termination layer at the interface, and the consequences of this degree of free-dom. This issue has been most explicitly raised, and hotly debated, for the electron gasobserved at the interface of two perovskite insulators, LaAlO and SrTiO [20]. Specif-ically, the (001) heterointerface was found to be insulating when grown using a SrTiO -terminated. Given therapid evolution of the field, and the numerous reviews of this heterostructure in the lit-erature, we do not attempt a comprehensive review here. Instead the LaAlO /SrTiO interface will be used to illustrate the various mechanisms suggested to explain the in-terface electronic structure, and the electrostatic boundary conditions which arise. Itshould be stressed, however, that all oxide heterointerfaces should be considered type IVto varying degrees, and thus these issues are quite general – even underlying the interfacecharge in the superlattice shown in Fig. 1(c). polar discontinuity scenario Assuming pure ionicity, the charge sequence of the (001) perovskite planes are differentin these two materials, namely the planes of LaAlO are (La O − ) + and (Al O − ) − ,while those of SrTiO are (Sr O − ) and (Ti O − ) . Therefore, the abrupt interfacebetween LaAlO and SrTiO is type II polar and requires reconstruction as shown inFig. 8, just as for the GaAs/Ge (001) interface.Unlike polar semiconductor interfaces where only atomic reconstructions are avail-able due to the fixed ionic charge of each element , we have another possibility to rec-oncile the instability of polar interfaces, through electronic reconstructions [23]. At theLaAlO /SrTiO interface, when the interface termination is LaO/TiO , it requires a nethalf negative charge per 2D unit cell to reconcile the potential divergence (n-type). Ac-cessing Ti can source this negative charge by accommodating electrons at the Ti 3 d level, as was spectroscopically observed [24]. On the other hand, when the interface isterminated by AlO /SrO, the sign of the required charges is opposite (p-type). Due tothe difficulty of accommodating holes in this structure (such as Ti ), the positive chargesare realized by the formation of oxygen vacancies close to the interface, as inferred frommeasurements of the O-K edge fine structure. Oxygen vacancies are known as electron Allowing for covalency, SrO and TiO planes in SrTiO are no longer charge neutral, and thus the(001) surface is weakly polar, but still the dipole size of the unit cells which start from the interface isdifferent from that of LaAlO . While here we discuss large scale charge modifications, small polar discontinuities can induce freecarriers in semiconductors, such as in AlGaN/GaN heterostructures [21, 22]. /SrTiO interfaces. The unreconstructed(a) LaO/TiO terminated interface, (b) AlO -SrO terminated interface, (c) and (d) thecorresponding reconstructed interfaces, respectively (Nakagawa et al. [24]).donors, but in this case they are formed to provide positive charges, and thus no electronsare supplied from these vacancies compensating the polar discontinuity. Therefore thesystem does not have itinerant electrons and remains insulating [20, 25].One of the key corollaries of this scenario is the LaAlO thickness dependence. Thisis because the size of the potential difference arising from the stacking of the dipoles inthe unreconstructed structure is finite in thin films, and if it is small enough, the systemmay be stable without any reconstruction. Indeed, a critical thickness of LaAlO wasobserved [26], where the n-type LaAlO /SrTiO interface is insulating if the thickness ofLaAlO is up to 3 unit cells, and metallic above that thickness. A similar tendency wasalso observed in SrTiO /LaAlO /SrTiO trilayer structures where the distance betweenthe two polar interfaces was varied [27]. SrTiO is known to be a material which readily accommodates oxygen vacancies thatact as donors to provide itinerant electrons [28, 29]. Either kinetic bombardment of theSrTiO substrate by the ablated species (early studies of this interface all used pulsedlaser deposition), or gettering by a reduced film, can induce oxygen vacancies, and theywere suggested to be the dominant origin for the observed conductivity by several groups15igure 9: Conducting tip atomic force microscopy mapping around the LaAlO /SrTiO interface of (a) the as grown sample ( P O = 10 − Torr) and (b) the postannealed sample(Basletic et al. [34]).[30–32]. Indeed, the first report found a strong dependence of the Hall density for n-typeinterfaces on the oxygen partial pressure ( P O ) during growth, while the p-type interfacewas robustly insulating [20]. For n-type samples with similar variations in the transportproperties, Kalabukhov et al. found when grown at P O = 10 − Torr, the samplesexhibited blue cathode- and photo-luminescence at room temperature [30], similar tothat of reduced SrTiO by Ar bombardment [33]. In addition to transport studies,Siemons et al. demonstrated that the photoemission spectra from these interfaces showeda larger amount of Ti in samples grown at low pressures without oxygen annealing[31]. Herranz et al. observed Shubnikov-de Haas oscillations in reduced LaAlO /SrTiO samples which were quite similar to bulk doped SrTiO , and rotation studies indicated athree-dimensional Fermi surface [32].The strong P O dependence of the conducting channels in LaAlO /SrTiO was ob-served and mapped by means of conducting tip atomic force microscopy on cross-sectionsof the interface, which revealed a conducting region extending > µ m into the substratefor samples grown at low pressure [34], as shown in Fig. 9. After annealing, the widthof the conductive layer decreased to ∼ P O were dominated by oxygen vacancies, since the density far exceeded that neededto stabilize the polar discontinuity. For high P O , or after post-annealing, the origin wasless clear. 16igure 10: (a) Occupancies and (b) cumulative displacements ∆ z of the atoms at theLaAlO /SrTiO interface. (c) Concentration of Ti determined by a minimization ofthe electrostatic potential. (d) Predicted cumulative unit cell displacements from bulkpositions, based on the component ionic radii (Willmott et al. [35]). Willmott et al. studied a five unit cell film of LaAlO on SrTiO (001) by means ofsurface x-ray scattering [35], using coherent Bragg rod analysis (COBRA) [36] and furtherstructural refinement. Their analysis revealed intermixing of the cations (La, Sr, Al,and Ti) at the interface [Fig. 10(a)], as well as significant local displacement of theatomic position both in the film, and in SrTiO close to the interface [Fig. 10(b)]. Thedistribution of Ti valence was also inferred by minimizing the electrostatic potential[Fig. 10(c)] following the obtained atomic positions, and the atomic displacements wereexplained by the larger ionic radii of Ti compared to Ti [Fig. 10(d)]. Based on theseobservations, the origin of the interface conductivity was suggested to be the formation ofthe bulk-like solid solution La − x Sr x TiO in a region of approximately 3 unit cells. Thisexplanation can be considered a diffused version of local bonding arguments — i.e., thateven in the abrupt limit, the Ti at the interface has La on one side, and Sr on the other. At present, it can be fairly stated (we believe) that no one scenario can completely ex-plain the vast and growing body of experimental work on this system. Even theoretical17alculations show an extreme sensitivity to the choice of boundary conditions and as-sumptions of site-occupancy [37–41]. To give examples for each perspective: The polardiscontinuity picture should lead to a significant internal field in ultrathin LaAlO , whileexperiments [42] put an upper bound far below that expected theoretically, even allowingfor significant lattice polarization [43]. Oxygen vacancies induced by growth are difficultto reconcile with the notion that a single layer of SrO can prevent their formation. Localbonding and interdiffusion considerations do not address the constraints of global chargeneutrality. Furthermore, discriminating between these mechanisms is often difficult, sincethe change of Ti valence shows similar transport, spectroscopic, and optical properties,independent of its origin.It is extremely likely that multiple contributions exist to varying degrees, dependenton the growth details of a given sample. While this is a matter for further experimentalinvestigation and refinement, an equally difficult issue appears to be one of semantics.For example, one of the conceptual difficulties of the polar discontinuity picture hasbeen the question: Where do the electrons come from? In many presentations [24,26, 44, 45], the charges at the interface are depicted to originate from the surface ofthe LaAlO film, but it is not so obvious that they travel through the insulating filmsindependent of its thickness [26]. Fundamentally, the “non-locality” of these electrostaticdescriptions has sometimes been considered less intuitive and compelling than “localchemistry” mechanisms such as vacancies or interdiffusion [30, 31, 34, 35, 46].A related difficulty is how to treat the charge density to describe the macroscopicelectric field. Based on the polar discontinuity picture, the stacking of the dipoles inthe unit cells creates a macroscopic electric field, resulting in the change of the potentialslope at the discontinuity. In this picture, the unit cells start from the discontinuity, andthus the composition is always the same as that of the bulk, which is charge neutral. Itmight be strange that the potential starts to bend at the discontinuity although all theunit cells start out charge neutral, because the source of an electric field is charge. Infact, the source is the macroscopic bound charge density at the interface, as discussedin Section 5, but the existence of such implicit charge density has caused a fair bit of18onfusion. To address these concerns, it is useful to treat the electrostatics in a purelylocal description, as well as the boundary charges explicitly. Furthermore, the effects ofdefects and diffusion can be discussed more simply by re-framing the polar discontinuitypicture in local form. Therefore, this local charge neutrality picture based on dipole-freeunit cells is developed in the next section first for idealized models, followed by discussionof incorporation of chemical defects. local charge neutrality picture One of the origins of confusion regarding the stability and reconstructions of polar discon-tinuities arises because of the various choices of a unit cell in a crystal, which determinesthe size and direction of the dipole in it . For example, when we take the unit cell ofLaAlO , this stacking can be treated as a dipole of [(AlO ) − (LaO) + ], as shown in Fig.11(a). However, it is also possible to take [(LaO) + (AlO ) − ] as a unit cell, and the signof the dipole is opposite to the previous case, as shown in Fig.11(b).The choice of unit cells should not change the electrostatic potential in the crystal.Indeed, the difference between the two choices is compensated by the potential φ sur arisingfrom the surface layer. If the surface is terminated by the (AlO ) − layer, it remains as anextra negatively charged layer when we take [(LaO) + (AlO ) − ] as the unit cell, as shownin Fig.11(b), and the total electrostatic potential remains the same as in the [(AlO ) − (LaO) + ] unit cell case.Therefore, it is impossible to fix the direction of the dipoles without knowing thesurface termination. In other words, it is the surface that determines the stability ofthe system. So the problem can be simplified by considering the surface locally, and notby counting the number of dipoles in the material. In order to avoid the dipoles arising For simplicity, we started our discussion from a point-charge model of ions, and neglect the freecarrier distribution. More generally, the charge distribution can be described using contributions fromion cores and free carrier Wannier functions, and the ambiguity of the choice of unit cells appears in thisextended case as well [47]. , chargedistribution ρ , and associated potential φ . The filled allows indicate the orientation ofthe dipoles with size ∆ in the unit cells. (a) Taking [(AlO ) − (LaO) + ] as a unit cell.(b) Taking [(LaO) + (AlO ) − ] as a unit cell, and the total potential φ tot is the sum ofthe potential φ dip arising from the stacking of the dipoles and the potential φ sur from thesurface charge. (c) Taking a dipole-free unit cell [ (LaO) - (AlO ) - (LaO)].from the stacking of charged layers, the simplest approach is to take dipole-free unit cells.This can be achieved in any crystal [11], and in the LaAlO case, this is done by taking[ (LaO) - (AlO ) - (LaO)] (or [ (AlO ) - (LaO) - (AlO )]) as a unit cell, as shown inFig. 11(c). This is analogous to the unit cell in a type 2 model in Tasker’s classification.From group theory, it is known that spontaneous polarization can be observed only ina direction where the crystal does not have mirror symmetry. Since cubic perovskitesdo have mirror symmetry in the [001] direction (we neglect surface or interface inducedlattice polarization for now), it is useful to take dipole-free unit cells to reflect the lackof polarization in the bulk.When we take this unit cell, the stability of a polar surface or interface can be discussedby only considering the charge neutrality of each unit cell, since now there is no netdipole created by the stacking of charged layers. For example, the instability of the(AlO ) terminated surface of LaAlO is naturally derived because the top-most unit cellis [(AlO ) − (LaO) + ] − . , which violates charge neutrality, as shown in Fig. 11(c). Thusthe (001) surface of LaAlO cannot keep the bulk termination, either by an AlO or LaOlayer, and must reconstruct to compensate this charge [48, 49]. Once charge neutralityof all the unit cells is achieved, the system is free to undergo electron / hole modulationor interdiffusion / displacement of the atoms, which creates only finite dipoles and does20igure 12: Schematic illustrations of the charge structure across the two types of bulk-frozen LaAlO /SrTiO interfaces, assuming the ionic charges based on the valence statesin bulk LaAlO and SrTiO . Taking dipole-free unit cells, the interface unit cell cannotkeep charge neutrality, and the simplest neutral interface stoichiometry is written in theright. (a) LaO/TiO terminated LaAlO /SrTiO interface and (b) AlO /SrO terminatedLaAlO /SrTiO interface.not violate neutrality . These perturbations are important because they determine thereal charge structure, for example via lattice distortion close to the surface of LaAlO [48, 49].Following these arguments, the polar nature of given bulk frozen surfaces and inter-faces is clearly defined, considering local charge neutrality using dipole-free unit cells: a bulk frozen surface or interface is polar if the unit cell at the surface or interface cannotkeep charge neutrality when we take dipole-free unit cells in the bulk. /SrTiO in the local charge neutrality picture Taking dipole-free unit cells of perovskites AB O in the [001] direction, namely [ ( A O)- ( B O ) - ( A O)], the interface unit cell can be treated as a δ -dopant at the interface.As shown in Fig. 12, the interface unit cell does not keep the stoichiometry of eitherof the bulk materials, not even a simple mixture of them. This issue is actually oneof the central opportunities of the interface science of heterostructures. Namely, theLaO/TiO terminated LaAlO /SrTiO interface has La . Sr . TiO as the interface unitcell. Considering the formal electronic charges of La , Sr , and O − , the Ti ion in thisunit cell should take Ti . as a formal valence, which is the same reconstructed state as The effect of diffusion is further discussed in Section 6.1 as an example of sources of such finitedipoles. Another source of interface dipoles, the quadrupolar discontinuity , is discussed in Section 6.3. , LaAlO and La . Sr . TiO , as-suming that SrTiO and LaAlO are intrinsic semiconductors. (b) Schematic energyband diagram of the LaAlO /SrTiO interface with the δ -dopant La . Sr . TiO at theinterface.that for the previous discussion by the polar discontinuity picture.Similarly, the AlO /SrO terminated interface has La . Sr . AlO as the interface unitcell, and due to the fixed ionic charges of the elements, such a unit cell is not charge neutraland thus unstable: (La . Sr . Al O − ) . − . Instead, allowing a change of the oxygennumber in the interface unit cell, La . Sr . AlO . is charge neutral , and the decreaseof the oxygen content at the interface to stabilize a p-type LaAlO /SrTiO interfaceis naturally derived. Note in this case, these oxygen vacancies are introduced to keepcharge neutrality, and thus do not provide any free electrons. Thus, the electrons/oxygenvacancies to reconcile the polar instability of the bulk frozen LaAlO /SrTiO interfacesare supplied by the interface unit cell itself.In semiconductors, δ -doping is usually achieved in a symmetric geometry [50], that isthe dopant layer is sandwiched between the same host material. In the LaAlO /SrTiO case, by contrast, the two sandwiching materials have different band structure, withSrTiO having the narrower bandgap. Therefore, broadening of the charge distribu-tion from the δ -dopant occurs only in the SrTiO side [46]. Figure 13 shows how theLaAlO /SrTiO interface can be depicted in a semiconductor energy band diagram. This composition is not stable in bulk perovskite form, but can be considered as a mixture of thebulk compounds, LaAlO , Sr Al O , and SrAl O , stabilized epitaxially at the interface. δ -doped semi-conductor. The undepleted structure contains free carriers as well as ionized impurities.The depleted structure contains the same amount of donor and acceptor impurities (Goss-mann and Schubert [50]). When two polar discontinuities are brought in proximity to one another, coupling of thecharges can occur to minimize the total energy of the system. This is just like the couplingof dopant layers in δ -doped semiconductor heterostructures, which can be understood interms of depleted and undepleted structures. Figure 14(a) shows the band diagram ofan undepleted semiconductor with one layer of δ -dopant. Since the dopant is positivelyionized, an equal number of free electrons are left, and they screen out the potentialcreated by the δ -dopant. As a consequence, the structure is neutral, and has zero electricfield sufficiently far away from the δ -dopant layer.On the other hand, Fig. 14(b) shows the band diagram of a depleted δ -doped structure,where the number of donors is equal to that of acceptors, and they are sufficiently closein space. As a result, all the free carriers recombine and the structure is depleted. Thecritical parameters to treat electronic coupling of two δ -doping layers are the distancebetween them and the dielectric constant of the medium. When the distance betweenthem is smaller than the length scale of band bending of the undepleted structure, theycouple and the system goes to a depleted state. Note in this case, charge neutrality in theneighborhood of one dopant layer is not necessarily maintained. Superlattice calculationsusing density functional theory show that the above threshold is observed by changingthe thickness of each layer in LaAlO /SrTiO superlattices, which can be captured interms of a simple capacitor model [40, 41].In δ -doped semiconductor heterostructures, only considering modulation of free carri-23rs is sufficient to describe the coupling of the δ -dopant layers. However, when the systemallows possibilities of excess charges other than the free carriers, which come from outsideof the constructed crystal – e.g., anion or cation vacancies or foreign atoms absorbed tothe surface, the electrostatic potential can be minimized via them. For example, theinstability of the polar AlO -terminated LaAlO surface can be solved by introducingpositively charged surface oxygen vacancies [49]. The LaAlO thickness dependence ofLaO-TiO terminated LaAlO /SrTiO can be explained by a simple assumption, wherewe only consider coupling of the free electrons provided by the LaAlO /SrTiO interfaceand the surface oxygen vacancies to keep the local charge neutrality of the polar LaAlO surface. Figure 15 shows a schematic structure of the LaAlO /SrTiO heterojunction,where two polar discontinuities exist at the LaAlO /SrTiO interface and the LaAlO surface. When the LaAlO film is sufficiently thick [Fig. 15(a)] these polar discontinuitiesare decoupled and charge neutrality is preserved locally by introducing oxygen vacanciesat the surface and taking the Ti valence of 3 .
5+ at the interface unit cell. Fractionallyfilled Ti valence provides itinerant electrons, and therefore the system is metallic in thethick limit.If the two polar discontinuities are brought closer [Fig. 15(b)], the external chargesand the free carriers (surface oxygen vacancies and extra electrons in Ti valence) canrecombine in an environment with oxygen gas, and the system is depleted and the con-ductivity disappears . Here, the word “deplete” is used to mean “reduce the amountof charge other than charge bound to the crystal”, and the full depletion of the LaAlO surface means extinguishing the positively charged oxygen deficiency at the surface —i.e. the surface turns to an unreconstructed bulk frozen state. Manipulating this tran-sition appears to roughly capture the essence of writing nanoscale features [44], whichcorresponds to the “writing” of surface charge [52]. This is similar to the recombination of free electrons and holes, and the actual recombination canbe written by following Kr¨oger-Vink notation [51] as: O + 2 e ′ + V O ·· → ρ ) of the dipole-free unit cells of atom-ically abrupt LaAlO /SrTiO interfaces in (a) the thick LaAlO limit with the polardiscontinuities locally neutralized, and (b) the thin LaAlO limit with depleted polardiscontinuities. Even in systems containing only one polar discontinuity, coupling between the polardiscontinuity and the other layers can occur in analogy to modulation doping by a δ -doping layer [53], as shown in Fig. 16. At the semiconductor interface, lineup of theconduction and valence bands should be maintained, as well as a fixed chemical potential,which causes the modulation of carriers resulting in band bending.Assume an interface between two intrinsic semiconductors A (narrow bandgap) and B(wide bandgap). If the δ -dopant in B is sufficiently far from A, the flat band condition atthe interface is maintained [Fig. 16(a)]. When A is brought in proximity to the δ -dopantin B, in order to keep the conduction band lineup, the conduction band minimum of Alies at lower energy than the chemical potential in B [Fig. 16(b)]. Since the free carriersaround the δ -dopant in B have higher energy than the conduction band minimum of A,they transfer to A and band bending occurs in A [Fig. 16(c)]. As a result, the δ -dopantin B is depleted, and A is doped close to the interface.An experimental example of such charge modulation was observed from a type 3 polarsurface of LaAlO (001), which acts as a δ -dopant, to a narrow layer of the Mott insulator25igure 16: Schematic band diagram of the interface between two intrinsic semiconductorsA and B, with a layer of δ -dopant in B. (a) A is sufficiently far from the δ -dopant in B.(b) Model with no charge modulation though the distance between A and the δ -dopantis close, resulting in the mismatch of the chemical potential. (c) Charges transfer fromB to A, so as to match the chemical potential.26igure 17: (a) Schematic band diagram and crystal structure of a LaVO quantum wellembedded close to an AlO -terminated LaAlO (001) surface. (b) Illustrations showinghow reconstruction charge at the LaAlO surface is transferred to the buried LaVO quantum well. In order to solve the instability caused by the polar nature of AlO -terminated surface, positive charge is required. When the LaAlO cap is sufficientlythick (left), the LaAlO surface and the LaVO well layer are separated, and the positivecompensating charge remains at the surface. For a thinner spacing (right), the LaVO well layer accommodates this positive charge, which is energetically more favored [45, 54].LaVO with smaller bandgap, as shown in Fig. 17 [45, 54]. This system is noteworthy inthe discussion of polar discontinuity effects, in that the observed hole-doping can neitherarise from oxygen vacancies nor interdiffusion. local charge neutrality picture As discussed, the local charge neutrality picture gives a clear and self-consistent explana-tion for the various phenomena at surfaces and interfaces. The stability of given surfacesand interfaces can be simply judged by looking at the local composition. When the in-terface composition differs from that of the bulk or a simple superposition of them, adifferent electronic and/or atomic state can be expected. Enforcing local charge neu-trality at the interface unit cell naturally explains the source of the charges which canchange the stoichiometry or the electron number of the constituent atoms from those ofthe bulk. For example, in a LaO/TiO terminated LaAlO /SrTiO bulk frozen interface,the interface unit cell is La . Sr . TiO and the Ti is 3 .
5+ to achieve charge neutrality.Therefore, the difference between the Ti in the bulk SrTiO and the interface Ti . comes from the interface unit cell itself, and not from anywhere else.Another advantage of taking the local charge neutrality picture becomes clear when27e discuss the coupling of polar discontinuities. The stability of the system can bediscussed through the distance between the polar discontinuities and the screening lengthof the host material. In other words, the stability is determined by the balance of theactivation energy of the dopant and the electrostatic energy allowing polarization of themedia between the dopant layers. Therefore, the total polarizability of the media canbe considered in the calculation, and is connected to the bulk permittivity in the thicklimit. Note when the media is thin, the local effective permittivity is non-trivial since thelocal atomic displacements can be different from that in the bulk, and the local dielectricapproximation breaks down [55]. Thus far we have discussed boundary conditions based on two different choices for theunit cell. This was implicitly taking a microscopic viewpoint, since the electric field E and the total charge density ρ tot were connected by Gauss’ law: ε ∇ · E = ρ tot , where ε is the vacuum permittitivity. Since ρ tot is used (hence the atomic-scale stepped orsawtooth potentials), different choices for the unit cell were irrelevant so long as globalcharge neutrality was considered. Here we confirm the equivalence of the two picturesfrom the macroscopic electrostatic viewpoint. In media, treating ρ tot is often quite complicated, which can be simplified by using theelectric displacement D : D = ε E + P , (3)where P is the polarization. Then Gauss’ law is given by ∇ · D = ρ free , (4)28here ρ free is the free charge – the part of the macroscopic charge density due to excesscharge not intrinsic to the medium, which are designated as bound charge ρ bound . Thesedefinitions do not depend on whether the charges are localized or itinerant, and are justintroduced for practical convenience to treat the displacement of the bound charges as thedielectric response of the media to the electric field. Since the total charge is conserved( ρ tot = ρ free + ρ bound ), by taking the divergence of Eq. (3), ρ bound = −∇ · P . (5)In infinite crystals, only the divergence of E , P , and D has physical meaning, and thepolarization arising from the density of unit cell dipole moments P dipole can be neglectedsince it merely adds a constant value to P . When the crystal is finite, P dipole drops to zero at the surface. Thus the choice of unit cellis important, since it changes the nature of the discontinuity at the surface. Accordingto Eq. (5) the magnitude of the polarization discontinuity at the surface is determinedby ρ bound there. Since ρ tot at the surface does not depend on the choice of unit cell,uncertainty in the dipole moment of the unit cell is compensated by whether the chargesat the surface are defined to be free or bound. — the surface charge is bound when itbelongs to a bulk unit cell, and free when not, and this definition does not depend onthe origin of the charges [12].For simplicity, let us adopt the bulk dielectric constant ε to connect E and the inducedpolarization P ind by the field, namely P ind = ( ε − ε ) E , (6)hence P = P ind + P dipole . (7)29igure 18: Schematic illustration of macroscopic D , P dipole , P , E , ρ bound , ρ free , and ρ tot close to the (001) surface of LaAlO . (a) Taking [(AlO ) − (LaO) + ] as a unit cell and (b)taking [ (LaO) - (AlO ) - (LaO)] as a unit cell.Substituting them in Eq. (3) we obtain D = ε E + P dipole . (8)Now, let us calculate the value of E , P dipole , D , and the charge density, considering theunreconstructed AlO -terminated (001) surface of LaAlO as an example, for the twodifferent unit cell assignments previously discussed. ) − (LaO) + ] as a unit cell – the polar discontinuity picture The z -component of the vectors D , E , and P are denoted as D , E , and P . As shownin Fig. 18(a), all the ionic charges are included in the unit cells, and thus bound. Tofix the constant in E , the vacuum can be taken as the reference for E = 0. Due to30he absence of free charges, D = 0 from Eq. (4). In this case, the [(AlO ) − (LaO) + ]unit cell has a dipole moment of q a q is the elementary charge and a is thepseudocubic lattice constant of LaAlO . Considering the unit cell volume a , P dipole isgiven by P dipole = q a θ ( z ), where θ ( z ) is the step function. Equation (8) immediatelyprovides E as a function of the position, namely E = − q εa θ ( z ). The total polarizationis given by P = (cid:18) ( ε − ε ) · − q εa + q a (cid:19) θ ( z ) = ε ε q a θ ( z ), following equations (6) and(7), and from Eq. (5), ρ bound = − dd z P = − ε ε q a δ ( z ), where δ ( z ) is the Dirac δ function.This indicates that the system has bound charges of − ε ε q a at the surface. (LaO) - (AlO ) - (LaO)] as a unit cell – the local chargeneutrality picture When taking dipole-free unit cells [Fig. 18(b)], the topmost unit cell is [(AlO ) - (LaO)],which is not a bulk unit cell. Therefore, the half negative charge which belongs to thisunit cell is free, and ρ free = − q a δ ( z ). By integrating Eq. (4) and using the boundarycondition that D = 0 in the vacuum, we obtain D = − q a θ ( z ). Since P dipole = 0, E isobtained as E = D − P dipole ε = − q εa θ ( z ). According to Eq. (3), the total polarizationis P = D − ε E = ε − ε ε q a θ ( z ), and ρ bound is obtained by Eq. (5), namely ρ bound = ε − ε ε q a δ ( z ). This bound charge density appeared as a response to the electric field E arising from ρ free . Thus we confirm that ρ total is independent of the choice of unit cell. In real systems, interdiffusion of atoms across the interface is inevitable, and we shouldnote its effect on the electrostatic stability of the system. Interdiffusion is a processwhere atoms are locally exchanged, and does not change the charge neutrality conditionsaround the interface, except for the finite dipoles induced by the modulation of charges.Since the instability of polar interfaces can be derived from the lack of charge neutrality31igure 19: Schematic charge structure ρ and electrostatic potential φ of (a) an abruptinterface and (b) an interface with interdiffusion. Five dipole-free unit cells are takenas an example of a cluster covering the interface region. The dashed line in (b) is theduplication of φ in (a). (c) Difference ρ (b) − ρ (a) of the charge distributions in (a) and(b), and created potential shift φ dip by the dipole indicated by the shaded arrow.around them, simple interdiffusion in a finite region cannot compensate it. That is, localstoichiometric interdiffusion can neither create, nor remove, a potential divergence. Here,this point is emphasized by considering a simple model.Figure. 19(a) shows the charge structure ρ and the calculated electrostatic potential φ of an abrupt polar interface. By taking dipole-free unit cells, the interface unit cell hasa half positive charge per 2D unit cell, and the system does not keep charge neutralityas it is, and is unstable. Figure. 19(b) shows an example of interdiffusion, where half ofthe negatively charged layer and half of the charge neutral layer close to the interface areswapped, compared to the model in Fig. 19(a). In this case, the extra positive chargeappears at a different position, but the amount of the charge is conserved, and a similarinstability in φ arises as in the abrupt case.In order to show that interdiffusion does not change the stability or instability ofa polar interface, it is useful to consider charge neutrality in a cluster consisting of asufficiently large number of dipole-free unit cells covering the interface region. By takingdipole-free unit cells, the electrostatic structure in the bulk can be neglected to determinethe stability of the system. As an example, let us take a cluster shown in Figs. 19(a) and(b). Since the modulation of the charge to achieve the interdiffused model occurs insidethe cluster, the total amount of the charge in the cluster is conserved, and thus the same32nstability appears in both cases.For comparison of these models, let us focus on the difference of the charge distribu-tions ρ (b) − ρ (a) in the two models, as shown in Fig. 19(c), where a dipole with a finitesize appears. Since φ is calculated by spatially integrating the charge distribution twice,the difference of φ in Figs. 19(a) and (b) should be equal to the potential shift created bythe dipole in Fig. 19(c). Indeed, φ in Fig. 19(b) has a shift from that in Fig. 19(a), whichis the same amount as the dipole shift in Fig. 19(c). It should be highlighted that thisdipole shift can indeed be an interface-specific additional driving force for interdiffusion,and has been suggested to fundamentally limit the abruptness of some interfaces [24].The energy associated with a band offset, for example, can be reduced by forming thisdipole.Finally, we should note the difference between simple interdiffusion and change ofthe interface composition. In the interdiffusion process, only exchanging atoms in thefinite interface region is allowed, and therefore the total number of atoms of each elementin the region is conserved. On the other hand, interface composition can be changed,for example by inserting other materials, segregation of atoms, or creating vacancies.For example, the reconstruction model in Fig. 6(b) to compensate the instability of apolar Ge/GaAs interface cannot be achieved by a simple roughening at the interface: thenumbers of Ge, Ga, and As atoms are different compared to those in the abrupt model[Fig. 6(a)].In summary, the effect of the interdiffusion appears as an extra interface dipole mo-ment at the interface from the viewpoint of electrostatics. However, it does not changethe total number of charges at the vicinity of the interface, and thus cannot screen thecharge imbalance at polar interfaces. In order to compensate the instability of a polarinterface, therefore, introduction of a compositional change or other compensating chargeis required. 33 .2 Role of correlation effects So far we have tried to simply describe the perovskite polar discontinuity using semi-conductor language by taking dipole-free unit cells and treating the interface unit cellas a δ -dopant. It should be mentioned that in the presence of strong electron-electroncorrelations, commonly found in transition metal oxides, it is formally impossible to drawsemiconductor energy band diagrams based on the single-particle picture of independentelectrons [57, 58]. Also, in order to draw band diagrams, we should know which part ofthe charges are to be assigned as free carriers, which is non-trivial in correlated systems,such as Mott insulators. For example, a perovskite with 1 electron per unit cell can giverise to an effective carrier density ranging from ∼ × cm − to zero, depnding on thecorrelation strength.However, the local charge neutrality picture can still provide important information onthe interface charge structure of transition metal oxides, since correlations cannot changethe amount of total charge. While the distribution of this charge may be significantlymodified by correlation features in the electronic compressibility [59], the charge can stillbe determined in a cluster consisting of a sufficiently large number of dipole-free unitcells. This is because the material outside of the charge modulation region consists ofdipole-free unit cells of bulk, which are charge neutral and thus stable. Thus far, we considered the stability of polar discontinuities, and showed that the amountof charge needed at the interface can be determined by taking dipole-free unit cells andconsidering local charge neutrality in each unit cell. Of course, local charge modulationis allowed once neutrality is obtained, since it does not violate global charge neutrality.This dipole energy arising from the modulation of charges determines the real chargedistribution in the system.Here, we note that there is an intrinsic dipole shift at the interface arising from the Owing to the relatively small (still non-negligible) correlation effects in SrTiO with the Ti 3 d con-figuration and weak 2 p -3 d hybridization in the coherent state of doped SrTiO [56], the schematic banddiagram shown in Fig. 13 is still reasonably valid, approximating SrTiO to be a band semiconductor. ρ , and electrostatic potential φ of aLaAlO /SrTiO interface with a Ti . O interface layer, showing a finite shift ∆. Taking(b) [( LaO) +1 / -AlO − -( LaO) +1 / ] and (c) [( AlO ) − / -LaO +1 -( AlO ) − / ] as a unitcell. The blue and pink shaded areas in (b) and (c) show the unit cells with oppositesigns of quadrupole moment, respectively, and the green shaded area in (b) shows theunit cell with a finite dipole moment. (d) An example of dipole- and quadrupole-freeunit cells.charge stacking sequence, which should be distinguished from this charge modulation.For example, consider a LaAlO /SrTiO (001) interface with Ti . at the interface tosolve the instability of the polar interface, as shown in Fig. 20(a). Although there is nopotential divergence, a finite shift ∆ remains between the two materials when consideringthe averaged electrostatic potential on both sides.The origin of this potential shift can be understood based on the discontinuity of thequadrupole moment of the unit cells. In one dimension, the quadrupole moment density Q is defined as dd x Q = P , where P is the dipole moment density. It has the same form asthat of Gauss’ law connecting the dipole moment and the charge — i.e., the source of thequadrupole moment is the dipole moment. Therefore, following the same argument asin Section 4.1, a finite interface dipole moment appears at a quadrupolar discontinuity,when the quadrupole moment is different on the two sides.Moreover, this quadrupole moment is proportional to the potential shift at that pointin the absence of free charge, since D = 0 and thus ε E = − P . Therefore if the unitcell has a finite quadrupole moment, that means the averaged potential in the unit cellis shifted by the corresponding value. Note the quadrupole moment does not induce apotential shift outside of the unit cell, while the dipole moment does change.For example, [( LaO) +1 / -AlO − -( LaO) +1 / ], a dipole-free unit cell of LaAlO (001)35Fig. 20(b)], has a finite quadrupole moment, while the charge neutral stacking of SrTiO creates no quadrupole moment in the unit cells . This difference of quadrupole momentbetween the two sides adds a finite potential shift to the averaged potential in the unit cell.Figures 20(b) and (c) show two ways of taking dipole-free unit cells in cubic perovskites,where the sign of the quadrupole moment is opposite. The choice of unit cells cannotchange the electrostatic potential, and indeed this difference of the quadrupole moment ofthe unit cells is compensated by the dipole moment of the interface unit cell in Fig. 20(b).The ambiguity of the quadrupole moment in the unit cells is reminiscent of the un-certainty of the dipole moment, as discussed in Section 4.1, and we can take the sameprescription as we did previously — taking quadrupole-free unit cells. It is more compli-cated since these quadrupole-free unit cells should not have a dipole moment at the sametime, in order to avoid the instability of the polar discontinuity, and Fig. 20(d) showsone way to take such unit cells. Note it is always possible to take unit cells which do nothave either dipole or quadrupole moments in one direction in any bulk crystal . Oncewe take this dipole- and quadrupole-free unit cell, the size of the interface dipoles can bedetermined locally by considering the dipole moment in the interface unit cell(s), sincethere is no shift in the averaged potential in the bulk unit cells due to the absence ofquadrupole moment. As pointed out by Tasker, the order of the ionic charge stacking plays an important role forthe stability of the surfaces and interfaces, and sometimes they show a diverging potentialwhen following bulk compositions and electronic states. In semiconductors where thenumber of ionic charges are fixed, they are usually stabilized by atomic reconstructions, asshown by Harrison et al. . However, in transition metal oxides, electronic reconstructionsprovide another possibility to reconcile the instability. Exploiting this degree of freedom Absence of covalency is assumed. This can be easily proved from the periodicity of the charge structure and the macroscopic chargeneutrality of the lattice by using the intermediate-value theorem.
Acknowledgments:
We thank our many colleagues and collaborators in this field whohave helped develop these topics, and T. Yajima for pointing out the quadrupole-dipoleconnection. We acknowledge support from the Japan Science and Technology Agency,the Japan Society for the Promotion of Science, and the Department of Energy, Office ofBasic Energy Sciences, under contract DE-AC02-76SF00515 (H.Y.H.).37 eferences [1] Esaki, L. and Tsu, R. (1970). Superlattice and negative differential conductivity insemiconductors.
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