Band offsets in heterojunctions between cubic perovskite oxides

0.26 MB·

by  Alexander I. Lebedev

Band offsets in heterojunctions between cubic perovskite oxides
∗
Alexander I. Lebedev
Physics Department, Moscow State University,
Leninskie gory, 119991 Moscow, Russia
(Dated: January 7, 2014)
The band offsets for nine heterojunctions between titanates, zirconates, and niobates with the
cubic perovskite structure were calculated from first principles. The effect of strain in contacting
oxides on their energy structure, many-body corrections to the position of the band edges (calcu-
latedintheGW approximation),andthesplittingoftheconductionbandresultingfromspin-orbit
4
interaction were consistently taken into account. It was shown that the neglect of the many-body
1
effectscanleadtoerrorsindeterminationofthebandoffsetsupto0.36eV.Thefailureofthetran-
0
sitivityrule,whichisoftenusedtodeterminethebandoffsetsinheterojunctions,wasdemonstrated
2
and its cause was explained.

n
a PACSnumbers: 68.65.Cd,73.40.-c,77.84.-s,79.60.Jv
J

6
I. INTRODUCTION is one of the applications of the ferroelectric oxides. It

] requires to solve the problem of non-destructive read-
i
c Almost all electronic and optoelectronic devices out of information and to increase the packing density
s of the memory cells. For non-destructive optical read-
- contain metal–semiconductor, metal–dielectric,
l semiconductor–dielectric, or semiconductor– outmethods, the memorycellsizeislimited bythe wave
r
t semiconductor interfaces. As the energy of an electron length. For titanates with the perovskite structure, in
m which the typical band gap is ∼3 eV, the minimum cell
abruptly changes at the interface, the characteristics
. size is ∼4000 ˚A. In multiferroics, in which the informa-
t of devices that include such interfaces directly depend
a tion is stored electrically and read out magnetically, the
on the height of emerging energy barriers. Originally,
m memory cell size can be reduced to the size typical for
the concept of a heterojunction was associated with
- a contact between two semiconductors, but currently modernharddisks,∼500˚A(inhomogeneousmultiferroic
d
thin film, the physicalsize of the memory cells is limited
n its use is significantly expanded to include dielectrics.
by a rather large width of magnetic domain wall).
o For example, in solving the actual problem of replacing
c the SiO gate dielectric in silicon field-effect transistors The methods based on the electrical read-out of the
2
[ with a material with higher dielectric constant, the ferroelectric polarization are the most promising. Since

1 calculation of the tunneling current through the gate the physical size of the ferroelectric memory cells is lim-
v dielectric requires an accurate knowledge of the energy itedbythe thicknessofthe ferroelectricdomainwalland
7 diagram of the formed heterojunction. the minimum film thickness for which the ferroelectric-
5 ity still exists (both sizes are a few unit cells14–17), the
In the last decade, experimental studies have discov-
1 packing density of the memory cells in these devices is
ered a number of new physical phenomena occurring
1
maximized. These methods can be based, for example,
at the interface of two oxide dielectrics. These include
.
1 the appearance of a quasi-two-dimensional electron gas onthenonlinearcurrent-voltagecharacteristicswhichare
0 (2DEG) at the interface of two dielectrics1 and its su- reversible upon switching the polarization in the metal–
4 perconductivity;2 the magnetism at the interface of two ferroelectric–metal structures as demonstrated on thin
:1 non-magnetic oxides.3 The ability to control the con- films of PZT18 and BiFeO3.19,20 A similar behavior can
v be observed on the structures that use the tunneling
ductivity of the 2DEG with an electric field (an analog
Xi of the field effect)4,5 and so to control the temperature through a thin layer of ferroelectric.21–23
of the superconducting phase transition6 were demon- The most important physical parameters that char-
r
a strated. The strongest conductivity changes were ob- acterize an interface between two semiconductors or di-
served when one of the oxides was ferroelectric.7 When electricsarethebandoffsetsontheenergydiagramofthe
the components of heterostructures were magnetic and heterojunction. The valence band offset ∆Ev (the con-
ferroelectric oxides, the ability to control the magnetic duction band offset ∆Ec) is defined as the difference be-
propertiesofthestructurewithaswitchableferroelectric tweenthe energiesof the tops of the valence bands (bot-
polarizationwasshown.8–10 Thus,theseheterostructures toms of the conductionbands) in twocontacting materi-
acquired the properties of a multiferroic. The above- als. These band offsets determine a number of physical
mentioned and other interesting phenomena observed in properties of heterojunctions, in particular,their electri-
oxide heterostructuresopen new opportunities for devel- cal and optical properties.
oping of new multifunctional electronic devices and sug- For oxides with the perovskite structure, the re-
gesttheemergenceofanewdirectioninmicroelectronics, liable experimental data on the band offsets (ob-
the oxide electronics.11–13 tained by the photoelectron spectroscopy) exist
The development of the ferroelectric memory devices for heterojunctions SrTiO /Si,24–26 BaTiO /Si,26
3 3
2
SrTiO /GaAs,27,28 BaTiO /Ge,29 SrTiO /InN,30,31 with two rectangular windows whose lengths are deter-
3 3 3
BaTiO /InN,31 SrTiO /ZnO,31 BaTiO /ZnO,31,32 mined by the periods of components is calculated. The
3 3 3
SrTiO /SrO,33 and BaTiO /BaO.33 For heterojunc- resulting profiles of the macroscopic average of the elec-
3 3
tions between two perovskite dielectrics, the data are trostatic potential V¯(r) have flat (bulk-like) regions far
limited by the PbTiO /SrTiO ,34 SrTiO /SrZrO ,35 enough from the interface. The ∆V value is defined as
3 3 3 3
SrTiO /LaAlO ,36,37 and SrTiO /BiFeO 38 systems. In the difference energy between these plateau values. It
3 3 3 3
addition, there are data on the Schottky barrier heights shouldbenotedthatneitherthe E andE quantities,
v1 v2
for perovskite–metal structures, in which the metals nor∆V doesnothavephysicalmeaningthemselves,only
are Pt, Au, Ag, or the conductive oxides SrRuO and the sum, Eq. (1), is meaningful.
3
(La,Sr)CoO . Theconductionbandoffsetiscalculatedfromthe∆E
3 v
In this work, the band offsets for nine heterojunctions value and the difference of the band gaps in two materi-
between titanates, zirconates, and niobates with the cu- als:
bic perovskite structure are calculated from first prin-
∆E =(E −E )+∆V =(E −E )+∆E .
ciples using the density functional theory and GW ap- c c2 c1 g2 g1 v
proximation. The obtained results are compared with Roughly,thebandgapE =E −E canbeestimated
available experimental data. g c v
in the LDA approximation for the exchange-correlation
energy. However, because of the well-known band-gap
problem characteristic of this one-electron approxima-
II. METHODOLOGY
tion,moreaccuratecalculationsshouldtakeintoaccount
thecorrectionstothebandenergiesresultingfrommany-
The band offsets cannot be determined from a direct body effects. These corrections (the values ∆EQP and
comparisonoftheenergiesofthevalenceandconduction ∆EQP) are usually calculated in the quasiparticcle GW
bands obtained from the first-principles band-structure v
approximation. It is usually believed that many-body
calculations performed separately for two constituent
corrections adjust the position of the conduction band
bulk materials. This is because in the first-principles
and in this way solve the band-gap problem, however,
calculations, there is no intrinsic energy scale: the en-
the energy levels in the valence band are also corrected.
ergies correspondingto the valence band edge E and to
v In the case of well-studied materials, the ∆E value is
c
the conduction band edge E are usually measured from
c often calculatedfrom the experimentalband gaps. How-
an average of the electrostatic potential, which is an ill-
ever, if the band offset ∆E was obtained theoretically,
v
defined quantity for infinite systems. Consequently, in theproblemassociatedwiththeuncertaintyinthe∆EQP
addition to the band-structure calculations for two ma- values remains. It is often assumed that the ∆EQP vval-
terials, the lineup of the average of the electrostatic po- v
uesintwomaterialsareclose,sothattheircontributions
tential∆V betweenthemshouldalsobecalculated. The
to the band offsets cancel each other. In this work, we
latter value is determined by the dipole moment emerg-
show that this assumption in general is not true.
ingatthe heterojunctionasaresultofthe redistribution
It should be also noted that, since the position of the
of the electron density on the hybridized orbitals in the
energylevelsinacrystaldepends ontheinteratomicdis-
constituent materials. It takes into account all the fea-
tances, the calculation of E , E , E , and E should
v1 c1 v2 c2
tures typical of the interface, such as the change in the
be carried out under the same strain of materials, which
chemical composition, the structure distortions, etc.
appears in a heterojunction. Besides the strain-induced
Thus, the valence band offset can be represented as a
lifting of the degeneracy of the band edges, the band
sum of two terms:39
gap itself can vary. Moreover, the calculations should
∆E =(E −E )+∆V. (1) take into account the possible lifting of the band de-
v v2 v1
generacyresulting from spin-orbitinteraction. Although
The first term in this equation is the difference of the the structure of dielectrics—the lattice parameters and
energies corresponding to the tops of the valence bands equilibrium atomic positions—are weakly dependent on
in two bulk materials. It can be obtained from standard the spin-orbit interaction (and therefore it is usually ne-
band-structure calculations. glected in the calculations), in the band structure cal-
ThesecondterminEq.(1)isthelineupoftheaverage culations, the spin-orbit interaction strongly affects the
ofthe electrostaticpotentialthroughthe heterojunction. energypositionofthe bands,anditcannotbeneglected.
Tocalculate∆V,oneusuallystartsfromthetotalpoten- In dielectrics, the taking into account of the spin-orbit
tial (the potential of the ions plus the microscopic elec- interaction can be done a posteriori, after completion of
trostatic Hartree potential for electrons) obtained from the main first-principles calculations.
the self-consistent electron density calculation in a su-
perlattice constructed of the contacting materials. After
that, the macroscopic averaging technique40 is applied, III. THE CALCULATION TECHNIQUE
in which the electrostatic potential is first averagedover
planes parallel to the interface and then the convolution The objects of the present calculations were hetero-
of the obtained one-dimensional quasi-periodic function junctions formed between titanates and zirconates of
3
calcium, strontium, barium, and lead as well as the
KNbO /NaNbO heterojunction. They were modeled
3 3
using superlattices grownin the [001] direction and con-
sistedoftwomaterialswithequalthicknessoflayers,each
offourunitcells. The in-planelatticeparameterwasob-
tained from the condition of zero stress at the interface
(i.e., it was close to the lattice parameter of the solid
solution with the component ratio of 1:1); the period of
the superlattice and atomic displacements normalto the
interface were fully relaxed.
The equilibrium lattice parameters and atomic po-
sitions were calculated from first principles within the
density functional theory (DFT) using the ABINIT soft-
ware. The exchange-correlation interaction was de-
scribed in the localdensity approximation(LDA). Pseu-
dopotentials of the atoms were taken from Refs. 41 and
42. The maximum energy of plane waves was 30 Ha
(816eV).FortheintegrationovertheBrillouinzone,the
8×8×2Monkhorst-Packmeshwasused. Allcalculations
wereperformedforheterojunctionsbetweencubicPm3m FIG. 1. The steps of the band offsets calculation for the
phases; the possible polar and antiferrodistortive (struc- PbTiO3/BaTiO3 heterojunction. First, the changes in the
tural) distortions of the materials were neglected (they band structure of cubic phases induced by the strain in the
will be considered in a separate paper). The ∆V value heterojunctionaretakenintoaccount,thenthecorrectionsre-
was determined using the macroscopic averaging tech- sulting from the many-body effects are applied (GWA), and
nique.40 To determine the values of E , E , E , and finally, the splitting of the band edges caused by the spin-
v1 c1 v2
orbit interaction (SO) is taken into account. The bottom
E in constituent materials, similar calculations were
c2
chartshowsthechangeoftheaverageoftheelectrostaticpo-
performed for isolated crystals with the in-plane lattice
tentialintheheterojunction,whichisareferenceenergylevel
parameterequaltothatofthesuperlattice;inthenormal
for all bands.
direction the crystals were stress-free.
The calculations of the quasiparticle band gap and
the many-body corrections to the position of the band
axial strain of these compounds during the formation of
edges were carriedout in the so-calledone-shot GW ap-
the heterojunction (their in-plane lattice parameters be-
proximation.43TheKohn-Shamwavefunctionsandener-
come equal), reduces the symmetry of their unit cells
giescalculatedwithintheDFT-LDAapproachwereused
to P4/mmm. As a result, their band gaps are changed
as a zeroth-order approximation. The dielectric matrix
and the degeneracy of the energy levels at some points
ǫGG′(q,ω) was calculated for a 6×6×6 mesh of wave of the Brillouin zone are lifted. For example, the three-
vectors q from the matrix of irreducible polarizability
fold degeneracy of the conduction band at the Γ point
P0 (q,ω) calculated for 2200–2800 vectors G(G′) in
GG′ and that of the valence band at the R point are lifted
reciprocal space, 20–22 filled and 278–280 empty bands.
(Fig.2). Thesebandextremalocationsaretypicalforall
Dynamic screening was described in the Godby–Needs
compounds studied in this work except for PbTiO and
3
plasmon-pole model. The wave functions with the ener-
PbZrO . IncubicPbTiO ,thetopofthevalencebandis
3 3
gies up to 24 Ha were taken into account in the calcula-
attheX point,andthetetragonaldistortionremovesthe
tions. The energy corrections to the LDA solution were
valley degeneracy (depending on the sign of the strain,
calculatedfromthediagonalmatrixelementsof[Σ−E ]
xc the valence band edge is located at either X or Z point
operator, where Σ = GW is the self energy operator,
of the Brillouin zone of the tetragonal lattice). In cubic
E istheoperatoroftheexchange-correlationenergy,G
xc PbZrO ,theonlycompoundinwhichbothbandextrema
is the Green’s function, and W = ǫ−1v is the screened 3
are located at the X point, the strain also removes the
Coulombinteraction. IncalculatingΣ,thewavefunction
valley degeneracy,but both extrema remain at the same
with the energies up to 24 Ha were taken into account.
point of the Brillouin zone (X or Z). The energy di-
agrams of strained crystals are shown in Fig. 1 next to
thediagramsforcubicphases. Wenotethatnotonlythe
IV. RESULTS bandgap,butalsotheenergiesofthebandedges(E and
c
E ) measured from the averaged electrostatic potential
v
Fig. 1 shows the steps of the energy diagram calcu- are changed upon strain. The values of these energies in
lation for a typical heterojunction. The extreme left strained crystals are given in Tables I and II.
and right parts in the figure correspond to the individ- The calculation of the many-body corrections to the
ual compounds with the cubic Pm3m structure, whose positions of the valence band edge ∆EQP and of the
v
latticeparametercorrespondstozeroexternalstress. Bi- conduction band edge ∆EQP in the GW approximation
c
4
TABLEI.Parameters determiningthevalencebandoffset ∆Ev ontheenergy diagram ofheterojunctions (all theenergies are
in eV).
Heterojunction Ev2 ∆EvQ2P Ev1 ∆EvQ1P ∆V ∆Ev
SrTiO3/PbTiO3 13.629 −0.239 15.464 −0.315 +2.143 +0.384
BaTiO3/BaZrO3 13.422 −0.512 13.766 −0.226 +0.066 −0.564
PbTiO3/PbZrO3 12.390 −0.321 13.123 −0.239 +0.495 −0.320
PbTiO3/BaTiO3 14.291 −0.226 13.453 −0.239 −1.276 −0.425
SrTiO3/BaTiO3 14.366 −0.226 15.333 −0.315 +0.864 −0.014
SrTiO3/SrZrO3 14.391 −0.582 14.912 −0.315 +0.395 −0.393
PbZrO3/BaZrO3 13.158 −0.512 11.888 −0.321 −1.209 −0.130
SrTiO3/CaTiO3 15.631 −0.333 15.664 −0.315 +0.131 +0.080
KNbO3/NaNbO3 13.617 −0.314 14.494 −0.245 +0.944 −0.002
TABLEII.Parameters determiningtheconduction bandoffset ∆Ec ontheenergydiagram ofheterojunctions andthetypeof
theheterojunction (all theenergies are in eV).
Heterojunction Ec2 ∆EcQ2P ∆EcS2O Ec1 ∆EcQ1P ∆EcS1O ∆Ec Type
SrTiO3/PbTiO3 14.899 +1.326 −0.010 17.034 +1.431 −0.007 −0.100 I
BaTiO3/BaZrO3 16.363 +1.199 −0.026 15.143 +1.341 −0.008 +1.126 I
PbTiO3/PbZrO3 14.513 +0.266 0 14.269 +1.326 −0.010 −0.311 II
PbTiO3/BaTiO3 15.820 +1.341 −0.008 14.704 +1.326 −0.010 −0.143 II
SrTiO3/BaTiO3 15.885 +1.341 −0.008 16.879 +1.431 −0.007 −0.221 II
SrTiO3/SrZrO3 17.469 +1.283 −0.023 16.347 +1.431 −0.007 +1.353 I
PbZrO3/BaZrO3 16.116 +1.199 −0.026 14.069 +0.266 0 +1.745 I
SrTiO3/CaTiO3 17.207 +1.486 −0.007 17.237 +1.431 −0.007 +0.156 II
KNbO3/NaNbO3 15.005 +1.008 −0.038 15.823 +0.976 −0.037 +0.157 I
showsthatthemany-bodyeffectsshiftthepositionofthe oxides with the cubic perovskite structure the scatter of
conduction band up by ∼1.3 eV in all compounds stud- the ∆EQP values can be as large as 0.36 eV. This value
v
ied in this work except for PbZrO in which the shift is isa measureofthe possibleerrorindeterminationofthe
3
only 0.266 eV (Tables I and II). The valence band edge band offsets in the calculations that neglect the many-
is shifted down by 0.22–0.58 eV on taking into account body effects.45
the many-body effects. Although the absolute values of Since our crystals contain atoms with a large atomic
thesecorrectionsslowlyconvergewithincreasingnumber number,theerrorsindeterminationofthebandedgepo-
of empty bands in the GW calculations (see, for exam- sitionsresultingfromtheneglectofthespin-orbitinterac-
ple, Ref. 44), the relative shift of the difference between tioncanbequitelarge. Inthiswork,thespin-orbitsplit-
the corrections in different compounds is small. There- ting ∆ of the valence and conduction band edges was
SO
fore, if one uses the same total number of bands (300 in calculated using the full-relativistic pseudopotentials.46
our case) in the calculations of these corrections, the er- Thetestsperformedonanumberofsemiconductors(Ge,
ror in determination of the relative position of the band GaAs,CdTe),forwhichthespin-orbitsplittingoftheva-
edges in two materials is not large, ∼0.01 eV according lence bandis accuratelymeasured,showedthat the ∆
SO
to our estimates. In our calculations we have also as- values calculated in this way agree with experiment to
sumed that the many-body corrections slightly depend within ∼5%.
on the strain, and so the values calculated for the cu- Thecalculationsshowedthatthespin-orbitinteraction
bic crystals can be used. The tests have shown that the results in a splitting of the band edges at some points of
strain-induced changes of ∆EQP and ∆EQP may reach theBrillouinzone. Firstofall,itreferstotheconduction
v c
0.01–0.02eV, which gives an estimate of possible errors. band edge at the Γ point. It is interesting that despite
The energy diagrams of constituent materials after ac- the presence of heavy atoms such as Ba and Pb in our
counting for many-body effects are also shown in Fig. 1. crystals, the spin-orbit splitting is not very large. This
The calculations of many-body corrections show that is because the conduction band states at the Γ point in
the assumptionusedbymanyauthorsaboutanapproxi- these crystals are formed primarily from the d-states of
mate equality of these corrections in two contacting ma- the B atom (Ti, Zr, Nb). The values ∆ of the spin-
SO
terials, in general, is not true. It is seen that in related orbit splitting of the conduction band at the Γ point for
5
FIG. 3. Determination of the ∆V form the profile of the
average electrostatic potential V¯(x) for the SrTiO3/BaTiO3
superlattice (solid line). The dotted line shows the approxi-
mating function.
mixingofdifferentstatesandcanbeseeninFig.2. These
effects,however,donotexceed10meVandarelessthan
other systematic errors in our calculations.
In calculating the ∆V value, the profile of the elec-
trostatic potential V¯(x) obtained with the macroscopic
averagingtechniquewasapproximatedbyastepfunction
with a width of transition regions of one lattice parame-
ter(Fig.3). Thetestsshowedthatwhenthethicknessof
the individual layers in the BaTiO /SrTiO superlattice
3 3
waschangedfromthreetofiveunitcells,thevariationin
FIG.2. Effectofbiaxialstrainonthesplittingoftheconduc- the∆V valuecomputedbytheabovealgorithmwasonly
tionband(a)andofthevalenceband(b)inBaTiO3 without ∼4 meV, which gives anestimate of the errorin the ∆V
the spin-orbit interaction (lines) and with it (dots). The lat- calculation. As shown in Ref. 48, the many-body effects
tice parameter of stress-free (cubic) crystal is 3.962 ˚A.
do not influence much on the ∆V value.
The results of the band offsets calculations for nine
heterojunctionsaregiveninTablesI andII.The signsof
all materials studied except for PbZrO are given in Ta- the band offsets are defined as the energy change when
3
ble III. In PbZrO , the conduction band minimum is lo- movingfromthecompoundindicatedthefirstinthehet-
3
cated at the X point, is non-degenerate, and does not erojunction pair to the compound indicated the second.
exhibit the spin-orbit splitting. The valence band edge The energy diagrams of heterojunctions are classified as
(at R and X points) in all cubic crystals studied in this typeIheterojunctions,forwhich∆Ec and∆Ev haveop-
work does not exhibit the spin-orbit splitting too. posite signs, and type II heterojunctions, for which the
signs of ∆E and ∆E are the same. The types of the
When the spin-orbit interaction is turned on, the cen- c v
heterojunctions are given in Table II and their energy
ter ofgravityofthe splitenergylevelscoincideswiththe
diagrams are shown in Fig. 4.
position of the energy level without spin-orbit interac-
tion.47 Inallstudiedcrystals,thespin-orbitsplit-offcon-
duction band at the Γ point is always shifted to higher
energies, and so the absolute minimum of the conduc- V. DISCUSSION
tionbandatthe Γpointis shifteddownby∆ /3. This
SO
value is given in Table II and determines an additional Unfortunately, the experimental data on the band off-
shiftoftheconductionband. Thefinalenergydiagramof sets in heterojunctions between oxides with the per-
theheterojunctionobtainedaftertakingintoaccountthe ovskite structure are very limited. In Ref. 35, the band
spin-orbit interaction is shown by two internal diagrams offsets for the SrTiO /SrZrO heterojunction were ob-
3 3
in Fig. 1. We note that in these calculations, we have tained using the photoelectron spectroscopy. Accord-
neglected the weaker effects associated with the change ing to these measurements, the heterojunction is type I,
ofthebandsplitting; theyresultfromthestrain-induced and the band offsets are ∆E = −0.5 ± 0.15 eV and
v
6
TABLE III.The spin-orbit splitting ∆SO of theconduction band at theΓ point in cubic perovskites (in meV).
CaTiO3 SrTiO3 BaTiO3 PbTiO3 SrZrO3 BaZrO3 NaNbO3 KNbO3
20.7 22.0 25.3 28.5 70.2 77.5 113.7 111.0
FIG. 4. (Color online) Energy diagrams for heterojunctions between cubicperovskites studied in thiswork.
∆E = +1.9 ± 0.15 eV (in SrTiO the valence band ously affect the energy diagram of heterojunctions. This
c 3
lies higher than in SrZrO ). Our data agree well with is because the question about the character of these dis-
3
these experimental data: according to our calculations, tortions is not so simple. It is known that the charac-
the heterojunction is also type I, and the band off- ter of distortions in the perovskites may change signifi-
sets are −0.393 and +1.353 eV, respectively, for cubic cantlyunderthebiaxialstrainand,moreover,thedistor-
components.49 We note that the band offsets predicted tions in two materials usually are highly interconnected.
in this work are in much closer agreement with experi- These effects are well known for ferroelectric superlat-
mentcomparedtotheresultsofRef.50(∆E =+0.5eV, tices.42,52–54 In heterojunctions between polar materi-
v
∆E =+2.5 eV). als, the need to meet the electrical boundary conditions
c
As the experimental SrTiO /SrZrO structure35 was (equal electric displacement fields normal to the inter-
3 3
grown on SrTiO substrate, the calculated band offsets, face) causes the polarization in both constituent mate-
3
which depend on the in-plane lattice parameter, may be rials to differ from their equilibrium values. Since the
slightlydifferent(thisdependenceiswellknownforsemi- accompanying atomic displacements affect the band gap
conductor heterojunctions39,47,51). To estimate these andthe positions ofthe bandedges,inpolarheterojunc-
changes, the calculations for the SrTiO /SrZrO hetero- tions the bandoffsets canbe verydifferent fromthose in
3 3
junctionwererepeatedforthein-planelatticeparameter nonpolar structures.55 In addition, the cases are known
equal to that of SrTiO . They gave ∆E = −0.240 eV when the periodic domain structure can occur in a fer-
3 v
and ∆E = +1.230 eV, which somewhat worsened the roelectric near the interface.56 Predicting of the energy
c
agreement with experiment. diagram for such a system is particularly problematic.
It should be stressed that the results obtained in this The experimental data for the SrTiO /PbTiO het-
3 3
work are for heterojunctions formed between cubic com- erojunction34 are noticeably different from the results of
pounds. We deliberately did not take into account pos- our calculations. According to the photoelectron spec-
sible distortions of the perovskite structure, which obvi- troscopy data, this heterojunction is type II and the
7
different from that of the positive (ionic) charge.
In conclusion, we discuss the applicability of the
transitivity rule which is often used to calculate the
band offsets in heterojunctions by comparing the band
offsets in a pair of heterojunctions formed between
the components of the heterojunction under discussion
and the third common component (see, for exam-
ple, Refs. 34 and 58). From the heterojunctions we
have studied, three closed chains (contours) can be
formed in which the initial and the terminal compo-
nents are the same: SrTiO /PbTiO /BaTiO /SrTiO ,
3 3 3 3
FIG. 5. Changes of the valence band
BaTiO /PbTiO /PbZrO /BaZrO /BaTiO , and
3 3 3 3 3
position when traversing the contour
SrTiO /PbTiO /PbZrO /BaZrO /BaTiO /SrTiO .
3 3 3 3 3 3
SrTiO3/PbTiO3/PbZrO3/BaZrO3/BaTiO3/SrTiO3. The positions of the valence bands calculated from the
obtained ∆E values for one of these chains is shown
v
band offsets are ∆E = +1.1 ± 0.1 eV and ∆E = in Fig. 5. When traversing the contour, we never get
v c zero (see the figure), the deviation is from −0.027 to
+1.3±0.1 eV (in PbTiO the valence band lies higher
3 +0.539 eV. This means that the transitivity rule is
thaninSrTiO ). Accordingtoourcalculations,theband
3 not applicable. The reason for this behavior is that in
offsets are +0.384 and −0.100 eV, respectively, and the
heterojunctions, the band offsets actually depend on
heterojunction is type I. We see that in experiment and
the in-plane lattice parameter.33,39,47,51 If the lattice
calculations,thesignsof∆E arethesame,buttheirval-
v parameter in all the members of the chain would be
ues differ considerably. The fact that the crystal struc-
the same, then traversing the contour would result in
ture of PbTiO at 300 K is tetragonal cannot explain
3 zero E shift.59 However, because all heterojunctions in
sucha largediscrepancy. A possible explanationis given v
the above chains have different lattice parameters, the
below.
result is nonzero. Thus, in general, the transitivity rule
If the interface is not perfect (for example, in the case
is untenable, and the corresponding error can exceed
when the structural relaxation of strained materials oc-
0.5 eV.
curs, as one can see in Refs. 35 and 38), the dangling
bonds are created near the interface and the surface
states appear in the electronic structure. These states
VI. CONCLUSION
are electrically active and can significantly disturb the
∆V value, and so to change ∆E and ∆E . In addition,
c v
in the relaxation region where the lattice parameter de- Inthiswork,the bandoffsetsforeightheterojunctions
pendsonthecoordinate,anadditionaldriftoftheE and between titanates and zirconates of calcium, strontium,
v
Ec values (the band bending) occurs. The distortion of barium, and lead as well as for the KNbO3/NaNbO3
the energydiagramssimilarto thatarisingfromthe sur- heterojunction with the cubic perovskite structure were
face states can occur in heterojunctions between highly calculated from first principles. The effect of strain in
defective materials. Like the levels of the surface states, contacting oxides on their energy structure, many-body
defects in constituent materials can exchange electrons corrections to the position of the band edges (calculated
witheachother,whichwoulddistortthe energydiagram in the GW approximation),and the splitting of the con-
of a heterojunction. The size of the regionin which such duction band resulting from spin-orbit interaction were
anexchangeoccursisaboutthescreeninglengthinama- consistently taken into account. It was shown that the
terialandcanbequitesmall. Forexample,inadielectric neglectofthe many-bodyeffectscanleadtoerrorsinde-
with a defect concentration of 1018 cm−3, the length of terminationofthebandoffsetsupto0.36eV.Thefailure
theimpurityscreeningcanbeassmallas43˚A.57Itispos- of the transitivity rule, which is often used to determine
sible that the strong discrepancy between the calculated the band offsets in heterojunctions, was demonstrated
and experimental band offsets in the SrTiO /PbTiO and its cause was explained.
3 3
heterojunction results from the presence of defects in
materials: the band offsets observed in this heterojunc-
tioncorrespondtothecasewherethedefectlevelsintwo ACKNOWLEDGMENTS
contacting materialsare close in energy.34 Evenstronger
distortions of the band diagram can be characteristic of This work was partially supported by the Russian
heterostructures between perovskites with so-called va- Foundation for Basic Research grant No. 13-02-00724.
lencediscontinuity,suchasSrTiO /LaAlO (Ref.36and The calculations were performed on the laboratorycom-
3 3
37)andBiFeO /SrTiO (Ref. 38), inwhichthe distribu- puter cluster and the “Chebyshev” supercomputer at
3 3
tion of the quasi-two-dimensionalelectron gas density is Moscow State University.
8
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