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ISSN 1063 7834, Physics of the Solid State, 2014, Vol. 56, No. 5, pp. 1039­1047. © Pleiades Publishing, Ltd., 2014. Original Russian Text © A.I. Lebedev, 2014, published in Fizika Tverdogo Tela, 2014, Vol. 56, No. 5, pp. 1000­1008.

SURFACE PHYSICS AND THIN FILMS

Band Offsets in Heterojunctions Formed by Oxides with Cubic Perovskite Structure
A. I. Lebedev
Moscow State University, Moscow, 119991 Russia e mail: swan@scon155.phys.msu.ru
Received November 11, 2013

Abstract--A number of recent discoveries on heterostructures formed by oxides suggest the emergence of a new direction in microelectronics, the oxide electronics. In the present work, band offsets in nine heterojunc tions formed by titanates, zirconates, and niobates with the cubic perovskite structure are calculated from first principles. The effect of strain in contacting oxides on their energy structure; the GW corrections to the band edge positions resulting from many body effects; and the conduction band edge splitting resulting from spin­ orbit coupling are consistently taken into account. It is shown that the neglect of the many body effects can cause errors in the determination of the band offsets, reaching 0.36 eV. The fundamental inapplicability of the transitivity rule often used to determine the band offsets in heterojunctions by comparing the band offsets in a pair of heterojunctions formed by the components of the heterojunction under study with a third common component is demonstrated. The cause of the inapplicability is explained. DOI: 10.1134/S106378341405014X

1. INTRODUCTION Almost all electronic and optoelectronic devices contain metal­semiconductor, metal­dielectric, semiconductor­ferroelectric, or semiconductor­ semiconductor interfaces. Since the electron energy changes abruptly at the interface, the characteristics of devices containing such interfaces directly depend on the heights of emerging energy barriers. Although the concept of a heterojunction was first introduced for the contact of two semiconductors, its use is currently significantly extended and includes dielectrics. For example, in solving the important problem of the sub stitution of the SiO2 gate dielectric in silicon field effect transistors with a material with a higher dielec tric constant, the calculation of the tunneling current through the gate dielectric requires an accurate knowl edge of the energy band diagram of the formed hetero junction. In the last decade, experimental studies have dis covered a number of new physical phenomena occur ring at the interface of two oxide dielectrics: the for mation of a quasi two dimensional electron gas at the heterointerface [1]; the appearance of magnetism at the interface of two non magnetic oxides [2]; the superconductivity of the quasi two dimensional elec tron gas [3], and the possibility of controlling the superconducting transition temperature by an electric field [4]. In experiments [5, 6], the possibility of con trolling the conductivity in a quasi two dimensional layer by an electric field was demonstrated (an ana logue of the field effect). The strongest effect was observed if a ferroelectric was used as one of oxides [7].

When the heterostructure components were magnetic and ferroelectric oxides, it was possible to control the magnetic properties of a magnet and the magnetore sistive effect observed in it by a switchable ferroelectric polarization [8­10]. Thus, these heterostructures acquired the properties of multiferroics. The above mentioned and other new phenomena discovered in oxide heterostructures form a basis for developing new multifunctional electronic devices and suggest the emergence of a new direction in microelectronics, the oxide electronics [11­13]. One of the applications of ferroelectric oxides is the ferroelectric memory. The development of such devices requires solving the problems of nondestruc tive information read out and the increase in the packing density of memory cells. When using nonde structive optical read out methods, the cell sizes are limited by the used wavelength. The typical band gap of titanates with the perovskite structure is ~3 eV; therefore, the minimum cell size is ~0.4 m. When using multiferroics, in which the information is stored electrically and read out magnetically, the cell sizes can be decreased to sizes typical of modern hard disks, i.e., ~500 å (when using homogeneous multiferroic thin films, the physical size of the memory cells is lim ited by a rather large thickness of the magnetic domain wall). The methods based on electrical read out of the ferroelectric polarization seem the most promising. For example, the nonlinear current­voltage charac teristics reversible upon switching the polarization direction, which were recently observed in metal­fer

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roelectric­metal structures formed on single crystals and thin films of BiFeO3 [14, 15], can be used for non destructive information read out from the memory cells. Structures that use the tunneling through an ultrathin ferroelectric layer can find similar applica tion [16­18]. Since the physical limit of the memory cell size in the ferroelectric memory with electrical read out is the ferroelectric domain wall thickness and the minimum film thickness at which the ferroelec tricity still exists (both sizes are a few unit cells [19­ 22]), the packing density in such memory devices will be maximum. The most important physical parameters charac terizing an interface between two semiconductors or dielectrics are the band offsets in the energy band dia gram of a heterojunction. The valence band offset Ev (the conduction band offset Ec) is defined as the dif ference between the positions of the tops of the valence bands (bottoms of the conduction bands) in two con tacting materials. These band offsets control many physical properties of heterojunctions, in particular, their electrical and optical properties. For oxides with the perovskite structure there are experimental data on the band offsets for perovskite/Si [23, 24], SrTiO3/SrO and BaTiO3/BaO heterojunc tions [25], but the data for heterojunctions formed by two perovskite dielectrics are very limited [26­29]. In addition, there are data on the Schottky barrier heights for perovskite­metal structures in which Pt, Au, Ag, and conductive SrRuO3 and (La, Sr)CoO3 oxides are used as a metal. In this work, the band offsets in heterojunctions formed by titanates, zirconates, and niobates with the cubic perovskite structure are calculated from first principles using the density functional theory and GW approximation. The obtained results are compared with available experimental data. 2. CALCULATION METHODOLOGY The band offsets cannot be determined by simple comparison of corresponding band edge energies obtained in independent first principles calculations of the band structure of two bulk materials. This is due to the absence of intrinsic energy scale in such calcu lations: the energies corresponding to the valence band edge Ev and the conduction band edge Ec are usually measured from an average of the electrostatic potential which is a poorly defined quantity in infinite systems. Therefore, in addition to calculations of the band structure of two contacting materials, the change in the average of the electrostatic potential V at the interface of two materials should also be calculated. This quantity is determined by the dipole moment emerging at the heterointerface due to the electron density redistribution at hybridized orbitals in con tacting materials, and accounts for all features inher

ent to the interface, such as variations of chemical composition, structure distortions, and so on. Thus, the band offset in the valence band can be written as a sum of two terms [30] Ev = ( E
v2

­ Ev 1 ) + V .

(1 )

The first term in this formula is the difference of ener gies corresponding to the tops of the valence bands, which are determined from standard band structure calculations of bulk materials. The second term is the change of the average of the electrostatic potential through the heterojunction. To calculate V, one usually starts from the total potential (the potential of ions plus the microscopic electrostatic Hartree potential for electrons) obtained from the self consistent electron density calculation in the superlattice constructed of the contacting materi als. Then the macroscopic averaging technique [31] is used, in which the electrostatic potential is first aver aged over planes parallel to the interface, and then the obtained quasi periodic one dimensional function is convoluted with two rectangular filters whose lengths are determined by the periods of components. The obtained profiles of the average electrostatic potential V ( r ) contain flat (bulk like) regions sufficiently far from the heterojunction. The quantity V is defined as the energy difference between these plateaus. It should be noted that neither the quantities Ev1 and Ev2 them selves nor V have physical meaning; only their sum (1) is physically meaningful. The offset in the conduction band is calculated from Ev and the difference of band gaps in two mate rials, Ec = ( Ec 2 ­ Ec 1 ) + V = ( Eg 2 ­ Eg 1 ) + Ev . The band gap Eg = Ec ­ Ev can be roughly esti mated in the LDA approximation for the exchange correlation energy. However, because of the well known band gap problem characteristic of this one electron approach, more accurate calculations require to take into account the corrections to the band edges positions resulting from many body effects. These QP QP corrections to the self energy ( E c and E v ) are usually calculated within the quasiparticle GW approximation. It is usually believed that many body corrections adjust the conduction band position, thus solving the band gap problem; however, the energy lev els in the valence band become also subjected to the correction. In the case of well studied materials, Ec is often calculated using experimental band gaps. However, if the band offset Ev was obtained theoretically, the problem associated with the uncertainty in E
QP QP v

remains. It is often assumed that the E v values in two materials are close, so that their contributions
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cancel each other. In this work, it will be shown that in the general case this assumption is not true. It should also be kept in mind that Ev1, Ec1, Ev2, and Ec2 should be calculated at the same materials strains as in the heterojunction itself, since the energy levels in a crystal depend on interatomic distances. In this case, in addition to possible splitting of degenerate levels corresponding to the band edges, the band gap can also vary. Moreover, the calculations should take into account the possible lifting of the band edge degener acy resulting from spin­orbit coupling. Although such physical properties of dielectrics as lattice parameters and equilibrium positions of atoms depend weakly on spin­orbit coupling (that is why spin­orbit coupling is usually neglected in calculating these values), spin­ orbit coupling significantly affects the energy position of band edges and the band gap, and it cannot be neglected in the band structure calculations. The spin­orbit coupling effects in dielectrics can be con sidered a posteriori, i.e., after completing the main first principles calculations. 3. CALCULATION TECHNIQUE The objects of the present calculations were hetero junctions formed by titanates and zirconates of cal cium, strontium, barium, and lead, and the KNbO3/NaNbO3 heterojunction. The heterojunc tions were modeled using superlattices grown in the [001] direction and constructed of two materials with equal thickness of layers, each of four perovskite unit cells. The in plain lattice parameter was determined from the condition of zero strain in the layers plane (i.e., it was close to the lattice parameter of the solid solution with the component ratio of 1:1). The atomic displacements normal to the interface were com pletely relaxed and the superlattice period was deter mined in the zero strain condition in this direction. The equilibrium lattice parameters and atomic coordinates in the superlattices were calculated from first principles within the density functional theory using the ABINIT software. The exchange­correla tion interaction was described in the local density approximation (LDA). Atomic pseudopotentials were taken from [32, 33]. The maximum plane wave energy was 30 Ha (816 eV). Integration over the Brillouin zone was performed using the 8 â 8 â 2 Monkhorst­ Pack mesh. All calculations were performed for het erojunctions formed by cubic Pm3m phases; the effect of the possible polar and structural distortions of materials on the band offsets will be considered else where. The value of V was determined using the mac roscopic averaging technique [31]. The values of Ev1, Ec1, Ev2, and Ec2 in contacting materials were obtained from similar calculations for isolated crystals with the in plane lattice parameter equal to the lattice parame ter of the superlattice under study; in the third dimen sion, crystals were considered to be stress free.
PHYSICS OF THE SOLID STATE Vol. 56 No. 5

Cubic Strained GWA SO E Ec Ev
c

SO GWA Strained Cubic

Ev

PbTiO3 V dV

BaTiO3

Fig. 1. Steps of the band offsets calculation in heterojunc tions. First, changes in the band structure of cubic phases caused by strain in the heterojunction are taken into account; then, corrections for many body effects (GWA) and finally the band splitting caused by spin­orbit (SO) coupling are considered. The lower diagram shows the change through the heterojunction of the average of the electrostatic potential from which all energy levels are measured.

The quasiparticle band gap and the many body corrections to the band edge positions were calculated in the one shot GW approximation [34].1 The Kohn­ Sham wave functions and energies calculated using the density functional theory in the LDA approximation were used as the zeroth order approximation. The dielectric matrix GG'(q, ) was calculated for the 6 â 6 â 6 mesh of wave vectors q from the irreducible 0 polarizability matrix P GG' ( q, ) calculated for 2200­ 2800 vectors G(G') in reciprocal space, 20­22 filled and 278­280 empty bands. Dynamic screening was described in the Godby­Needs plasmon pole model. Wave functions with energies up to 24 Ha were taken into account in the calculations. The energy correc tions to the LDA solution were calculated from the diagonal matrix elements of the [ ­ Exc] operator, where = GW is the self energy operator (the mass operator), Exc is the exchange­correlation energy operator, G is the Green's function, and W = ­1v is the screened Coulomb interaction. In the calculation of , the wave functions with energies up to 24 Ha were taken into account. 4. RESULTS Figure 1 shows the steps of the heterojunction energy band diagram calculation. The leftmost and rightmost diagrams in the figure relate to individual
1

The results of these calculations will be described elsewhere in more detail.

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0

-40 -80

at the Z point of the Brillouin zone of the tetragonal lattice). In cubic PbZrO3, the only compound in which both band extrema are at the X point, the strain also removes the valley degeneracy, but both extrema remain at the same point of the Brillouin zone (X or Z). The energy diagrams of strained crystals are shown in Fig. 1 near the diagrams of initial cubic phases. We note that not only the band gap, but also the energy positions of the Ec and Ev band edges measured from the average of the electrostatic potential are changed upon strain. These energies in strained crystals are given in Tables 1 and 2. The calculation of the corrections to the positions QP of the valence band edge E v and the conduction
4.05

3.85

3.90

E, meV 80 40 0 -40 -80 3.85 3.90

3.95 a, å

4.00

(b)

3.95 a, å

4.00

4.05

Fig. 2. Effect of biaxial strain of cubic BaTiO3 on the (a) conduction and (b) valence band edges splitting without (curves) and with (dots) taking into account spin­orbit coupling. The calculated lattice parameter of the stress free crystal is 3.962 å.

compounds with the cubic Pm3m structure whose lat tice parameters correspond to zero external stress. The biaxial strain of these materials during formation of a heterojunction, when their in plane lattice parameters become equal, reduces the symmetry of their unit cells to P4/mmm. As a result, the band gaps of the materials are changed and the degeneracy at some points of the Brillouin zone is lifted. For example, the threefold degeneracy of the conduction band at the point and that of the valence band at the R point are lifted (Fig. 2). These positions of extrema at the and R points are characteristic of all compounds considered in this work, except for PbTiO3 and PbZrO3. In cubic PbTiO3, the valence band extremum is at the X point, and the tetragonal distortion removes the valley degeneracy (depending on the strain sign, the valence band edge extremum is located either at the X point or

band edge E c within the GW approximation shows that the many body effects shift the conduction band edge upward in energy by ~1.3 eV in all compounds considered in this work, except for PbZrO3 in which the shift is only 0.266 eV (Tables 1 and 2). The valence band corrected for many body effects shifts downward by 0.22­0.58 eV. Although the absolute values of the shifts under consideration slowly converge with increasing number of empty bands taken into account in the GW calculations (see, e.g., [35]), the relative drift of the difference between these shifts in different compounds is small. Therefore, if the same total num ber of bands (300 in our calculations) is used in the calculations of the corrections, the error in the deter mination of the relative position of band edges in two materials will be small, ~0.01 eV according to our esti mates. Moreover, in our calculations it was assumed that the many body corrections depend weakly on strain induced structure distortions, and the values calculated for cubic crystals were used. The tests have shown that additional strain induced changes of QP QP E v and E c can reach 0.01­0.02 eV, which pro vides some insight into possible errors. The energy dia grams of contacting materials after taking into account many body effects are also shown in Fig. 1. The calculations of many body corrections show that the assumption used by many authors about an approximate equality of these corrections in two con tacting materials is not valid in the general case. It is QP seen that the spread in the E v values reaches 0.36 eV in related oxides with the cubic perovskite structure. This value is a measure of the possible error in the determination of the band offsets in calculations that neglect the many body effects.2 Since our crystals contain atoms with high enough nucleus charge, the errors in the determination of the band edges positions resulting from the neglect of
2

QP

More detailed studies show that in a wide class of oxides, fluo rides, and nitrides the E v variation range reaches 3 eV. These results and their explanation will be published elsewhere. Vol. 56 No. 5 2014
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Table 1. Parameters determining the valence band offset Ev in the energy band diagram of studied heterojunctions (all energies are in eV) Heterojunction SrTiO3/PbTiO3 BaTiO3/BaZrO3 PbTiO3/PbZrO3 PbTiO3/BaTiO3 SrTiO3/BaTiO3 SrTiO3/SrZrO3 PbZrO3/BaZrO3 SrTiO3/CaTiO3 KNbO3/NaNbO E
v2

E

QP v2

E

v1

E

QP v1

V +2.143 +0.066 +0.495 ­1.276 +0.864 +0.395 ­1.209 +0.131 +0.944

E

v

3

13.629 13.422 12.390 14.291 14.366 14.391 13.158 15.631 13.617

­0.239 ­0.512 ­0.321 ­0.226 ­0.226 ­0.582 ­0.512 ­0.333 ­0.314

15.464 13.766 13.123 13.453 15.333 14.912 11.888 15.664 14.494

­0.315 ­0.226 ­0.239 ­0.239 ­0.315 ­0.315 ­0.321 ­0.315 ­0.245

+0.384 ­0.564 ­0.320 ­0.425 ­0.014 ­0.393 ­0.130 +0.080 ­0.002

Table 2. Parameters determining the conduction band offset Ec in the energy band diagram of heterojunctions and their types (all energies are in eV) Heterojunction SrTiO3/PbTiO3 BaTiO3/BaZrO3 PbTiO3/PbZrO3 PbTiO3/BaTiO3 SrTiO3/BaTiO3 SrTiO3/SrZrO3 PbZrO3/BaZrO3 SrTiO3/CaTiO3 KNbO3/NaNbO E
c2

E

QP c2

E

SO c2

E

c1

E

QP c1

E

SO c1

E

c

Type I I II II II I I II I

3

14.899 16.363 14.513 15.820 15.885 17.469 16.116 17.207 15.005

+1.326 +1.199 +0.266 +1.341 +1.341 +1.283 +1.199 +1.486 +1.008

­0.010 ­0.026 0 ­0.008 ­0.008 ­0.023 ­0.026 ­0.007 ­0.038

17.034 15.143 14.269 14.704 16.879 16.347 14.069 17.237 15.823

+1.431 +1.341 +1.326 +1.326 +1.431 +1.431 +0.266 +1.431 +0.976

­0.007 ­0.008 ­0.010 ­0.010 ­0.007 ­0.007 0 ­0.007 ­0.037

­0.100 +1.126 ­0.311 ­0.143 ­0.221 +1.353 +1.745 +0.156 +0.157

Table 3. Spin­orbit splittings of states at the point of the conduction band in cubic perovskites (in meV) CaTiO3 20.7 SrTiO3 22.0 BaTiO3 25.3 PbTiO3 28.5 SrZrO3 70.2 BaZrO3 77.5 NaNbO3 113.7 KNbO3 111.0

spin­orbit coupling can be rather large. In this work, the spin­orbit splitting SO of the valence and conduc tion band edges was calculated using the fully relativ istic pseudopotentials [36]. The tests performed for a number of semiconductors (Ge, GaAs, CdTe), for which the spin­orbit splitting of the valence band is accurately determined experimentally, showed that the results of these calculations agree with experiment with an accuracy of ~5%. Calculations show that the spin­orbit coupling results in the band edge splitting at certain points of the Brillouin zone. First of all, this is true for the con duction band edge at the point. It is interesting that, despite the presence of such heavy atoms as Ba and Pb in our crystals, the spin­orbit splitting is not so large. This is because the conduction band states at the
PHYSICS OF THE SOLID STATE Vol. 56 No. 5

point in the perovskite structure are mostly formed of d states of the B atom (Ti, Zr, Nb). The spin­orbit splittings SO of the conduction band edge at the point for all studied materials except for PbZrO3 are given in Table 3. In PbZrO3, the conduction band minimum is at the X point, it is non degenerate, and is not subjected to spin­orbit splitting. The valence band edge (at R and X points) in all cubic crystals studied in this work is not split if the spin­orbit coupling is taken into account. Since the centroid of the energy levels spin­orbit coupling coincides with the level calculated without spin­orbit coupling [37] spin­orbit split off conduction band at the always shifted to higher energies in all studied the conduction band minimum at the point split by position and the point is crystals, appears

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5. DISCUSSION Unfortunately, the experimental data on the band offsets in heterojunctions between oxides with the per ovskite structure is very limited. In [27], the band off sets in the SrTiO3/SrZrO3 heterojunction were studied by photoelectron spectroscopy. According to the mea surements, this heterojunction is type I, and the band offsets are Ev = ­0.5 ± 0.15 eV and Ec = +1.9 ± 0.15 eV (the top of the valence band in SrTiO3 is higher than that in SrZrO3). The data of the present calcula tions are in good agreement with these experimental data: according to our data, the heterojunction is also type I; the band offsets are ­0.393 and +1.353 eV, respectively. In our opinion, the cause of a large dis crepancy between the experimental and calculated values of Ec is the fact that SrZrO3 at 300 K has a dis torted (orthorhombic) structure in which the band gap is larger than in the cubic phase. One more cause of disagreement can be the fact that the structures [27] were grown on SrTiO3 substrates; hence, the calcu lated band offsets, which depend on the in plane lat tice parameter, can be slightly different (this depen dence is well known for semiconductor heterojunc tions [30, 37, 39]). To test the possible changes, the calculations were repeated for the SrTiO3/SrZrO3 het erojunction with the in plane lattice parameter equal to the lattice parameter of SrTiO3; these calculations yielded Ev = ­0.240 eV and Ec = +1.230 eV, which slightly worsened the agreement with experiment. The experimental data for the SrTiO3/PbTiO3 het erojunction [26] differ appreciably from the results of our calculations. According to the photoelectron spectroscopy data, it is a type II heterojunction, and the band offsets are Ev = +1.1 ± 0.1 eV and Ec = +1.3 ± 0.1 eV (the top of the valence band in PbTiO3 is higher than in SrTiO3). According to our calculations, the band offsets are +0.384 and ­0.100 eV, respec tively, and the heterojunction is type I. Thus, the signs of Ev in the calculations and experiment are identi cal, but the values themselves differ appreciably. The fact that at 300 K the crystal structure of PbTiO3 is tet ragonal rather than cubic cannot explain such a large discrepancy. Another possible explanation will be dis cussed below. It should be kept in mind that the present results refer to heterojunctions formed by cubic crystals. We deliberately neglected possible distortions of the per ovskite structure, which can obviously affect the het erojunction energy band diagram. The point is that the question about the character of these distortions is not so simple as it can seem first. It is known that the char acter of distortions in these compounds can vary strongly under the biaxial strain, and distortions in two materials are usually tightly coupled with each other. These effects are well known in ferroelectric superlat tices [33, 40­42]. In the case of heterojunctions that include polar materials, the necessity of satisfying the
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30

Fig. 3. Determination of V from the profile of the average of the electrostatic potential V ( x ) for the SrTiO3/BaTiO3 superlattice (solid curve). The dashed curve is the approx imating function.

shifted downward by SO/3. This value determines an additional shift of the conduction band edge and is given in Table 2. The final energy band diagram of the heterojunction obtained after taking into account the spin­orbit coupling is shown by two internal diagrams in Fig. 1. We note that in these calculations we neglected the weaker effects associated with changes in the band splitting resulting from the strain induced mixing of the spin­orbit split states, which can be seen in Fig. 2. These effects do not exceed 10 meV and are smaller than other systematic errors in our calcula tions. In calculating V, the averaged electrostatic poten tial profile V ( r ) obtained using the macroscopic aver aging technique was approximated by a step function with transition regions of one lattice parameter (Fig. 3). The tests showed that when the individual layer thickness in the BaTiO3/SrTiO3 superlattice was changed from three to five unit cells, the variation of V calculated using the described algorithm was only ~4 meV; this makes an estimate of the error in the V determination. As shown in [38], the many body effects have a weak influence on V. The results of the band offsets calculation for nine heterojunctions are given in Tables 1 and 2. The signs of the band offsets are defined as the energy change in going from the compound indicated the first in the heterojunction pair to the compound indicated the second. Depending on the energy band diagram, the heterojunctions are classified as type I, for which the signs of Ec and Ev are opposite, and type II, for which the signs of Ec and Ev are identical. The types of the heterojunctions are given in Table 2 and their energy band diagrams are shown in Fig. 4.

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v

v

v

v

v

v

v

v

v

Fig. 4. Energy band diagrams of all heterojunctions studied in this work.

electrical boundary conditions (the equality of the electric displacement field components normal to the interface) results in that the polarization in each of contacting materials differs from the equilibrium polarization. Since the atomic displacements affect the band gap and the band edge positions, the band offsets in polar heterojunctions can be very different from those in nonpolar structures.3 Moreover, cases are known where even a periodic domain structure can appear in a ferroelectric near the interface [43]. It is especially difficult to predict the energy band diagram for such a system. If the interface is not perfect (e.g., in the case of structural relaxation of strained materials as occurred in [27, 29]), the dangling bonds are formed at the interface, and the surface states appear in the elec tronic structure. These states are electrically active and can alter substantially V, and so can affect Ec and Ev . In addition, an extra drift of Ev and Ec (the band bending) can occur in the relaxation region in which the lattice parameter depends on the coordinate. The
3

energy band diagram distortions similar to those caused by surface states can also appear in heterojunc tions between strongly defective materials. Like the surface states, defects in contacting materials can exchange electrons with each other; this will distort the heterojunction energy band diagram. In this case, the sizes of the regions in which this exchange occurs can be rather small. For example, the impurity screen ing radius can be as small as 43 å at the defect density of 1018 cm­3 [44]. It is possible that the above dis cussed large discrepancy between calculations and experiment for the SrTiO3/PbTiO3 heterojunction is due to defects in materials: the band offsets observed in this heterojunction correspond exactly to the case when the levels of defects in two materials are close in energy. In the case of heterojunctions formed by a pair of perovskites with the "valence discontinuity" such as SrTiO3/LaAlO3 [28] or BiFeO3/SrTiO3 [29], the energy band diagram can be additionally altered by the appearance of quasi two dimensional electron gas at the interface. Finally, we discuss the applicability of the transitiv ity rule which is often used to calculate the band offsets in heterojunctions by comparing the band offsets for a pair of heterojunctions formed by the components of

The same reasoning is also applicable to semiconductor hetero junctions such as GaN/AlN with the wurtzite structure, in which contacting materials have nonzero spontaneous polariza tion. PHYSICS OF THE SOLID STATE Vol. 56 No. 5

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the heterojunction under study with a third common component (see, e.g., [26, 45]). The application of the calculated band offsets Ev to closed chains SrTiO3/PbTiO3/BaTiO3/SrTiO3, BaTiO3/PbTiO3/PbZrO3/BaZrO3/BaTiO3, and SrTiO3/PbTiO3/PbZrO3/BaZrO3/BaTiO3/SrTiO3 to test the transitivity rule shows that we never obtain zero by contour traversing; the deviation is from 0.027 to +0.539 eV. Such a behavior is caused by the dependence of the band offsets on the in plane lattice parameter in a heterojunction [25, 30, 37, 39]. If the lattice parameter would be identical for all heterojunc tions entering the chain, the contour traversing would yield zero.4 However, since the lattice parameter is dif ferent for all heterojunctions entering the chains, the result is nonzero. Thus, the transitivity rule appears inapplicable in the general case, and the error can exceed 0.5 eV. 6. CONCLUSIONS The band offsets for nine heterojunctions formed by titanates, zirconates, and niobates with the cubic perovskite structure were calculated from first princi ples. The effect of strain in contacting oxides on their energy structure; the GW corrections to the band edge positions resulting from the many body effects; and the conduction band edge splitting resulting from spin­orbit coupling were consistently taken into account. It was shown that the neglect of the many body effects can cause errors in the determination of the band offsets, reaching 0.36 eV. The fundamental inapplicability of the transitivity rule which is often used to determine the band offsets in heterojunctions was demonstrated. The cause of this inapplicatibility is the dependence of the band offsets on the in plane lat tice parameter in a heterojunction. The calculations presented in this work were per formed on the laboratory computer cluster and the SKIF MGU "Chebyshev" supercomputer. ACKNOWLEDGMENTS This work was supported by the Russian Founda tion for Basic Research (project no. 13 02 00724). REFERENCES
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4

4.

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

This is indeed observed for a set of three almost isoperiodic het erojunction pairs Ge/GaAs, Ge/ZnSe, and GaAs/ZnSe [45].

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Translated by A. Kazantsev

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2014