Hybrid DFT calculations of the atomic and electronic structure for ABO3 perovskite (001) surfaces

Surface Science 575 (2005) 75–88

www.elsevier.com/locate/susc

Hybrid DFT calculations of the atomic and electronicstructure for ABO3 perovskite (001) surfaces

S. Piskunov a,b,*, E.A. Kotomin b,c, E. Heifets d, J. Maier c,R.I. Eglitis a, G. Borstel a

a Fachbereich Physik, Universitat Osnabruck, D-49069 Osnabruck, Germanyb Institute of Solid State Physics, University of Latvia, Kengaraga 8, LV-1063 Riga, Latvia

c Max-Planck-Institut fur Festkorperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germanyd California Institute of Technology, MS 139-74, Pasadena, CA 91125, USA

Received 15 June 2004; accepted for publication 4 November 2004

Available online 24 November 2004

Abstract

We present the results of first-principles calculations on two possible terminations of the (001) surfaces of SrTiO3,

BaTiO3, and PbTiO3 perovskite crystals. Atomic structure and the electronic configurations were calculated for differ-

ent 2D slabs, both stoichiometric and non-stoichiometric, using hybrid (B3PW) exchange-correlation technique and

re-optimized basis sets of atomic (Gaussian) orbitals. Results are compared with previous calculations and available

experimental data. The electronic density distribution near the surface and covalency effects are discussed in details

for all three perovskites. Both SrTiO3 and BaTiO3 (001) surfaces demonstrate reduction of the optical gap with respect

to the bulk, especially for the TiO2-terminated surfaces, whereas PbTiO3 surfaces show a slight increase of the gap. The

top of the PbTiO3 valence band lies at X point of the Brillouin Zone, unlike the M point for SrTiO3 and BaTiO3.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Strontium titanate; Barium titanate; Lead titanate; Low index single crystal surfaces; Surface relaxation and reconstruction;

Surface electronic phenomena; Density functional calculations

0039-6028/$ - see front matter � 2004 Elsevier B.V. All rights reserv

doi:10.1016/j.susc.2004.11.008

* Corresponding author. Address: Institute of Solid State

Physics, University of Latvia, Kengaraga 8, LV-1063 Riga,

Latvia. Tel.: +371 718 7480; fax: +371 713 2778.

E-mail address: [email protected] (S. Piskunov).

1. Introduction

The ABO3 perovskite surfaces are important for

many technological applications, e.g. SrTiO3 is

widely used as a substrate for epitaxial growth

of high-Tc superconducting thin films, BaTiO3,

PbTiO3 and their solid solutions are promising

ed.

mailto:[email protected]

76 S. Piskunov et al. / Surface Science 575 (2005) 75–88

for non-volatile memory cells, as electro-optical

materials, and for piezoelectrical devices [1–4].

Due to intensive development and progressive

miniaturization of electronic devices, the electronic

properties and atomic structure of the ABO3

perovskite thin films were extensively studied dur-

ing the last years. The SrTiO3(001) surface struc-

ture has been analyzed by means of low energy

electron diffraction (LEED) [5], reflection high

energy electron diffraction (RHEED), X-ray pho-

toelectron spectroscopy (XPS) and ultraviolet elec-

tron spectroscopy (UPS) [6], medium energy ion

scattering (MEIS) [7], and surface X-ray diffrac-tion (SXRD) [8]. The most recent experimental

studies on the SrTiO3(001) include a combination

of XPS, LEED, and time-of-flight scattering and

recoiling spectrometry (TOF-SARS) [9], and UPS

and metastable impact electron spectroscopy

(MIES) [10]. The BaTiO3 and PbTiO3 surfaces

are less studied.

The first ab initio calculations of the ABO3

perovskite surfaces has been presented by Kimura

et al. in 1995 [11]. Since then a number of calcula-

tions were performed, using different methods,

e.g. linearized augmented plane waves (LAPW)

[12,13], PW ultrasoft-pseudopotential [14–17],

and Hartree–Fock (HF) [18]. Recently, new studies

on the SrTiO3(001) surface relaxation were per-

formed using density functional theory (DFT) withplane wave (PW) basis set (DFT-PW method) [19],

and classical shell model (SM) [20,21]. It is well-

known that DFT considerably underestimate the

band gap. On the other hand, band gap obtained

through the HF calculations is usually overesti-

mated [22]. To solve this problem, the functionals

containing ‘‘hybrid’’ of the non-local HF exchange,

DFT exchange, and generalized gradient approxi-mation (GGA) correlation functionals, known as

B3LYP and B3PW could be used. These function-

als are quite popular in quantum chemistry of mol-

ecules [23,24]. Recently, periodic-structure hybrid

DFT calculations were carried out for a wide range

of crystalline materials [25], including perovskites

and their surfaces [26,27]. In all cases the hybrid

functional technique shows the best agreementwith experimental data for both bulk geometry

and optical properties of materials under investiga-

tion. In this paper, we continue our recent theoret-

ical studies of the surfaces of perovskite materials.

Our studies were carried out using both semi-

empirical SM [28] and ab initio HF and DFTmeth-

ods [27,29–31] and were dedicated mostly to the

SrTiO3(001) surfaces. We studied the effects of dif-ferent type of Hamiltonians (varied from DFT-

LDA to HF with a posteriori corrections) on

surface properties and atomic structure. This anal-

ysis allowed us to choose the B3PW functional for

calculations of bulk properties of all three perovsk-

ites under consideration [26] as well as of their sur-

face properties [27]. Therefore, B3PW is adopted in

present study to compare atomic structure andelectronic properties of the (001) surfaces of three

similar perovskites, SrTiO3, BaTiO3, and PbTiO3.

In our simulations, the (001) surfaces of perovsk-

ites are modelled using a single slab model.

The paper is organized as follows: in Section 2

we present the technical details of calculations.

In Section 3 we discuss the surface structure and

the electronic properties of the (001) surfaces.Our conclusions are summarized in Section 4.

2. Computational details

To perform the first-principles DFT-B3PW cal-

culations, we used the CRYSTAL-98 computer

code [22,32–34]. This code employs the Gauss-ian-type functions (GTF) localized at atoms as

the basis for an expansion of the crystalline orbi-

tals. Features of the CRYSTAL code, which are

most important for this study, are its ability to cal-

culate the electronic structure of materials within

both HF and Kohn–Sham (KS) Hamiltonians

and implementation of isolated 2D slab model

without its artificial repetition along the z-axis.However, in order to employ the LCAO-GTF

method, it is desirable to have optimized basis sets

(BS). Such BS�s optimization for all three perovsk-

ites was developed and discussed in Ref. [26]. In

this paper, we used this new BS which differs from

our previous calculations [27,29–31] by inclusion

of polarizable d-orbitals on O ions. We demon-

strated that this leads to better agreement of calcu-lated optical gaps with experimental data.

Our calculations were performed using the hy-

brid exchange-correlation B3PW functional involv-

S. Piskunov et al. / Surface Science 575 (2005) 75–88 77

ing a hybrid of non-local Fock exact exchange,

LDA exchange and Becke�s gradient corrected ex-

change functional [35], combined with the non-

local gradient corrected correlation potential by

Perdew and Wang [36–38]. The Hay–Wadt small-core effective core pseudopotentials (ECP) were

adopted for Ti, Sr, and Ba atoms [39–41]. The

‘‘small-core’’ ECP�s replace only inner core orbi-

tals, but orbitals for sub-valence electrons as

well as for valence electrons are calculated self-

consistently. Light oxygen atoms were treated with

the all-electron BS. The BSs were adopted in the

following forms: O—8-411(1d)G, Ti—411(311d)G,Sr and Ba—311(1d)G; see Ref. [26] for more

details.

The reciprocal space integration was performed

by the sampling the Brillouin zone of the unit cell

with the 8 · 8 · 1 Pack–Monkhorst net [42], that

provides the balanced summation in direct and re-

ciprocal spaces [43]. To achieve high accuracy,

large enough tolerances of 7, 8, 7, 7, 14, werechosen for the Coulomb overlap, Coulomb pene-

tration, exchange overlap, the first exchange

Fig. 1. On top: schematic view of the modelling for ABO3(001) surf

both sides, (c) asymmetrical termination (AO atop and TiO2 on botto

outermost surface layers relaxation: (d) SrTiO3, (e) BaTiO3, (f) PbTi

termination, right panels—TiO2. Dashed lines represent positions of

pseudo-overlap, and for the second exchange

pseudo-overlap, respectively [32].

The ABO3(001) surfaces were modelled with 2D

slabs, consisting several crystal planes perpendicu-

lar to (001) crystal direction. The slabs were ro-tated to make them perpendicular to Oz axis. The

CRYSTAL code allowed us to avoid artificial peri-

odicity along Oz direction and to perform simula-

tions for stand-alone 2D slabs. The symmetrical

(with respect to the mirror plane) 7-plane slabs

are either AO- or TiO2-terminated. These slabs

are non-stoichiometric, with unit cell formulas

A4B3O10 and A3B4O11, respectively. An alternativeasymmetrical slab is AO- and TiO2-terminated

(from each side, respectively) and stoichiometric

with unit cell formula A4B4O12, as shown schemat-

ically in Fig. 1(top). The slabs containing seven

planes (symmetrical) and eight planes (asymmetri-

cal) can be treated as thick enough since the conver-

gence of the calculated slab total energy per ABO3

unit is achieved. The total energies differ less than5 · 10�4 Hartree when 7- and 9-layered (or 8- and

10-layered for asymmetrical termination) slabs

aces: (a) AO-terminated on both sides, (b) TiO2-terminated on

m). On bottom: schematic illustration of the relaxation for two

O3. The view in the [010] direction. Left panels in d–f are AO

the same layers in unrelaxed slabs.


are considered. The symmetrical slabs have advan-

tage of no macroscopic dipole moment perpendicu-

lar to the slab. However, their non-stoichiometry

can potentially affect the electronic density distri-

butions and thus atomic displacements. On theother hand, since the effective atomic charges of

Ti and O ions differ from the ionic charges +4e,

�2e (due to the covalency contribution in Ti–O

chemical bonding), the alternating TiO2 and AO

planes are slightly charged, what produces in the

asymmetrical slab certain dipole moment perpen-

dicular to the slab. This dipole moment is cancelled

by the electronic density redistribution near the sur-face, which also can affect the optimized atomic dis-

placements. That is why a critical comparison of

these two slab models is important for making reli-

able conclusions.

In order to compare properties of three per-

ovskites under the same conditions and to reduce

Table 1

Atomic displacements with respect to atomic positions on unrelaxed

Termination N At. SrTiO3 Ba

This

study

SM

[28]

LDA

PW [15]

LDA

PW [19]

Th

stu

AO 1 A �4.84 �7.10 �5.7 �6.66 �1

O 0.84 1.15 0.1 1.02 �0

2 Ti 1.75 1.57 1.2 1.79 1

O2 0.77 0.87 0.0 0.26 1

3 A �1.42 �1.2 �1.54

O 0.7 �0.1 0.26

TiO2 1 Ti �2.25 �2.96 �3.4 �1.79 �3

O2 �0.13 �1.73 �1.6 �0.26 �0

2 A 3.55 3.46 2.5 4.61 2

O 0.57 �0.21 �0.5 0.77 0

3 Ti �0.6 �0.7 �0.26

O2 �0.29 �0.5 0.26

Asymmetrical 1 A �5.22 �2

O 0.39 �0

2 Ti 1.55 1

O2 0.61 1

– – – –

– – – –

– – – –

7 O �0.64 �0

A �3.74 �2

8 O2 0.15 0

Ti 2.27 3

Symbol A stands for Sr, Ba, or Pb atom. Positive displacements are ou

slab center.

computational efforts, we consider here only the

high-symmetry cubic (Pm3m) phases of these crys-

tals. The calculated bulk lattice constants (in A)

are: a = 3.90 for SrTiO3, a = 4.01 for BaTiO3,

and a = 3.93 for PbTiO3, which demonstrate quitegood agreement with experiment [44,45].

3. Results and discussions

3.1. Surface atomic structure

In present surface structure simulations we al-lowed atoms of the two outermost surface layers

to relax along the z-axis (surfaces of perfect cubic

crystals by symmetry have no forces along the

x- and y-axes). Displacements of the third layer

atoms were found negligibly small in our calcula-

tions and thus are not treated. The optimization

ABO3(001) surfaces (in percent of bulk lattice constant)

TiO3 PbTiO3

is

dy

SM

[28]

LDA

PW [14]

SM

[20]

LAPW

[13]

This

study

LDA

PW [16]

.99 �3.72 �2.79 �0.72 �3.82 �4.36

.63 1.00 �1.40 �1.09 �0.31 �0.46

.74 1.25 0.92 1.70 3.07 2.39

.40 0.76 0.48 2.75 2.30 1.21

�0.51 0.53 �0.69 �1.37

0.16 0.26 �0.28 �0.20

.08 �2.72 �3.89 �4.14 �2.81 �3.40

.35 �0.94 �1.63 �2.74 0.31 �0.34

.51 2.19 1.31 2.36 5.32 4.53

.38 �0.17 �0.62 �0.50 1.28 0.43

�0.33 �0.75 �0.81 �0.92

�0.01 �0.35 �0.72 �0.27

.09 �1.8 �4.28 �4.02

.69 �1.98 �3.26 �0.24

.70 3.01

.50 1.95

– – –

– – –

– – –

.37 �1.22

.54 �5.44

.34 3.52 2.68 �0.74

.03 4.72 4.79 2.84

twards (to the vacuum), negative displacements mean inward the


of atomic coordinates was done through the slab

total energy minimization using our own computer

code which implements conjugated gradients opti-

mization technique [46] with numerical computa-

tion of derivatives.Our calculated atomic displacements are pre-

sented in Table 1 and are schematically illustrated

in Fig. 1(bottom). A comparison with the surface

atomic displacements obtained by other theoretical

calculations is also done in Table 1. The relaxation

of surface metal atoms is much larger than that of

oxygen ions what leads to a considerable rumpling

of the outermost plane. Atoms of the first surfacelayer relax inwards, i.e. towards the bulk. The only

two exceptions are the top oxygen ions of SrTiO3

SrO-terminated and of PbTiO3 TiO2-terminated

surfaces. (At the latter point, our calculations dis-

agree with the DFT-PW-pseudopotential calcula-

tions [16], however the magnitudes of calculated

displacements are relative small, 0.31% and

�0.34% of lattice constant, respectively, which isclose to the limit of method accuracy.) The outward

relaxation of all atoms in the second layer is found

for all three perovskites and both terminations. The

displacements obtained for asymmetrically termi-

nated slabs are practically the same as for those

symmetrically terminated. This means that both

symmetrical and asymmetrical slabs are reliable

for the calculations of the (001) neutral surfaces.

Table 2

Calculated and experimental surface rumpling s, and relative displac

terminated surfaces Ddij (in percent of lattice constant)

AO-terminated

s Dd12 D

SrTiO3 This study 5.66 (5.61) �6.58 (�6.79) 1

LDA PW [15] 5.8 �6.9 2

LDA PW [19] 7.7 �8.6 3

SM [28] 8.2 �8.6 3

LEED expt. [5] 4.1 ± 2 �5 ± 1 2

RHEED expt. [6] 4.1 2.6 1

BaTiO3 This study 1.37 (1.40) �3.74 (�3.79) 1

LDA PW [14] 1.39 �3.71 0

SM [20] 0.37 �2.42 2

SM [28] 4.72 �4.97 1

PbTiO3 This study 3.51 (3.78) �6.89 (�7.03) 3

LDA PW [16] 3.9 �6.75 3

Results for asymmetric slabs are given in brackets.

In order to compare the calculated surface

structures with experimental results, the surface

rumpling s (the relative displacement of oxygen

with respect to the metal atom in the surface layer)

and the changes in interlayer distances Dd12 andDd23 (1, 2, and 3 are the numbers of near-surface

layers) are presented in Table 2. Our calculations

of the interlayer distances are based on the posi-

tions of relaxed metal ions (Fig. 1), which are

known to be much stronger electron scatters than

oxygen ions [5]. Qualitative agreement between all

theoretical methods is observed. The amplitude of

surface rumpling of SrO-terminated SrTiO3 is pre-dicted much larger than that for TiO2-terminated

SrTiO3 surface, whereas the rumpling of BaTiO3

TiO2-terminated surface is predicted to exceed by

a factor of two that for BaO-terminated surface.

Lastly, PbTiO3 demonstrates practically equal

rumpling for both terminations. From Table 2

one can see that all surfaces show the reduction

of interlayer distance d12 and expansion of d23.The calculated surface rumpling agrees quite well

with LEED and RHEED experiments [5,6] (which

are available so far only for SrTiO3 surfaces).

Theory agrees qualitatively also with the LEED re-

sults for the Dd12 and Dd23. However, the LEED

and RHEED experiments demonstrate that the

topmost oxygen always move outwards the sur-

faces whereas all calculations predict for the

ements of the three near-surface planes for the AO- and TiO2-

TiO2-terminated

d23 s Dd12 Dd23

.75 (1.55) 2.12 (2.43) �5.79 (�6.02) 3.55 (3.74)

.4 1.8 �5.9 3.2

.3 1.5 �6.4 4.9

.0 1.2 �6.4 4.0

± 1 2.1 ± 2 1 ± 1 �1 ± 1

.3 2.6 1.8 1.3

.74 (1.70) 2.73 (2.69) �5.59 (�5.57) 2.51 (2.54)

.39 2.26 �5.2 2.06

.39 1.4 �6.5 3.17

.76 1.78 �4.91 2.52

.07 (3.01) 3.12 (3.58) �8.13 (�8.28) 5.32 (5.44)

.76 3.06 �7.93 5.45

Table 3

Calculated surface energies (in eV per surface cell)

SrTiO3 BaTiO3 PbTiO3

SrO TiO2 Asymmetrical BaO TiO2 Asymmetrical PbO TiO2 Asymmetrical

This study 1.15 1.23 1.19 1.19 1.07 1.13 0.83 0.74 0.85

SM [28] 1.32 1.36 1.45 1.40

LDA PW [19] 1.21 1.19

SM [21] 1.17

LDA PW [16] 1.26 1.24 0.97

Results for previous ab initio calculations [16,19,21] are averaged over AO- and TiO2-terminated surfaces.


TiO2-terminated SrTiO3 surface that oxygen goes

inwards. Moreover, Table 2 shows also that LEED

and RHEED experiments contradict each other in

the sign of Dd12 for SrO-terminated surface andDd23 of TiO2-terminated surface. Up to now the

reason for such discrepancies between the different

experimental data is not clear and still discussed

(e.g. see [15]). Thus, experimental check of our pre-

dictions at the moment is prevented by a conflict

between different experimental results. New de-

tailed experimental studies are important for

resolving this contradiction.The calculated surface energies of the relaxed

surfaces, presented in Table 3, were computed

using the method described in Ref. [29]. The ener-

gies calculated for AO- and TiO2-terminated sur-

faces demonstrate only a small difference, that

means both terminations could co-exist. Neverthe-

less, the energy computed for TiO2-terminated

SrTiO3 surface is a little bit larger than that forSrO-termination, in contrast to BaTiO3 and

PbTiO3 crystals where TiO2-terminated surface is

a little bit energetically more favorable.

3.2. Electronic charge redistribution

We begin discussion of the electronic structure

of surfaces with the analysis of charge redistribu-tion in near-surface planes. The effective atomic

charges (calculated using Mulliken population

analysis) and dipole atomic moments characteriz-

ing atomic deformation along the z-axis are pre-

sented for AO-, TiO2-surfaces in Table 4. The

differences in charge densities in the (001) planes

in ABO3 bulk crystals and on the (001) surfaces

are analyzed in Table 5.

First of all, note that the effective charges of Sr

and Ba are close to the +2e formal charges,

whereas that of Pb is considerably smaller. Ti

and O charges are also much smaller than formalcharges, similarly to the bulk [26], what results

from the Ti–O covalent bonding. The AO-termi-

nated surfaces of SrTiO3 and BaTiO3 show similar

behavior. The charges of top layer cations are

smaller with respect to the relevant bulk charges,

the oxygen ions attract an additional electron

charge and become more negative. Charges of

the titanium ions in the second layer are slightly in-creased. Oxygen ions in the same layer again be-

came more negative due to additional electron

charge transfer. Changes in atomic charges in dee-

per layers become very small and practically equal

zero in the center of the slabs. Unlike SrTiO3 and

BaTiO3, charges of the surface oxygen ions on

PbO-terminated surface becomes smaller, Ti in

the second layer shows practically no changes inthe effective charges.

The charge redistribution in the TiO2-termi-

nated surfaces (Table 4) of all three perovskites

demonstrates quite similar behavior. All cations

in both topmost layers demonstrate charge reduc-

tion. For surface Ti it is a little bit larger then for

A ions in subsurface layer. Changes of charges for

ions in the asymmetrically terminated slabs arepractically the same as for the symmetrically AO-

terminated and TiO2-terminated slabs, as it should

be when slabs are thick enough and its surfaces

do not interact. However, charge redistribution

makes the ABO3(001) surfaces to be polar with

the dipole moment perpendicular to the surface

[47]. Such surfaces are potentially unstable, be-

cause they bear infinite charges or dipole moments

Table 4

The calculated Mulliken effective charges and dipole moments for the AO and TiO2 terminations

N Ion SrTiO3 BaTiO3 PbTiO3

Q, e d, ea.u. Q, e d, ea.u. Q, e d, ea.u.

AO-terminated surfaces

1 A 1.84 �0.22 1.75 �0.46 1.28 �0.48

(�0.03) (�0.04) (�0.07)

O �1.52 �0.03 �1.47 �0.05 �1.13 0.02

(�0.12) (�0.09) (+0.09)

2 Ti 2.36 0.01 2.38 0.01 2.33 �0.02

(+0.01) (+0.01) (0.00)

O2 �1.45 �0.02 �1.42 0.02 �1.26 �0.01

(�0.04) (�0.03) (�0.03)

3 A 1.87 �0.02 1.8 �0.04 1.35 �0.05

(0.00) (+0.01) (+0.01)

O �1.43 0 �1.41 �0.01 �1.26 �0.01

(�0.02) (�0.03) (�0.03)

4 Ti 2.34 0 2.39 0 2.34 0

(�0.01) (0.00) (+0.01)

O2 �1.41 0 �1.39 0 �1.23 0

(0.00) (�0.01) (�0.01)

TiO2-terminated surfaces

1 Ti 2.31 0.08 2.3 0.08 2.28 0.1

(�0.04) (�0.06) (�0.06)

O2 �1.32 0.04 �1.28 0.02 �1.18 �0.03

(+0.08) (+0.11) (+0.04)

2 A 1.85 0.04 1.76 0.09 1.27 0.1

(�0.02) (�0.03) (�0.07)

O �1.36 �0.04 �1.34 �0.03 �1.17 �0.01

(+0.05) (+0.04) (+0.06)

3 Ti 2.37 0.01 2.36 0.01 2.33 0.02

(+0.04) (0.00) (0.00)

O2 �1.39 �0.02 �1.37 �0.01 �1.2 �0.02

(+0.02) (+0.02) (+0.02)

4 A 1.87 0 1.79 0 1.34 0

(0.00) (0.00) (�0.01)

O �1.4 0 �1.38 0 �1.22 0

(+0.01) (+0.01) (+0.01)

Numbers in brackets are deviations from bulk values. Bulk charges (in e); SrTiO3: Sr = 1.87, Ti = 2.35, O = �1.41, BaTiO3: Ba = 1.79,

Ti = 2.36, O = �1.39, PbTiO3: Pb = 1.34, Ti = 2.33, O = �1.23 [26].


what cause spurious electric field. As a result of

self-consistent calculations, the additional charge

densities (Table 5) localized mainly on two upper

layers whereas the central layers practically retain

the bulk charge density. Such the electron charge

density redistribution accompanied by atomic dis-

placements on the surfaces allows to compensate

the surface dipole moment even for the asymmetri-

cally terminated slabs. Indeed, as one can see, the

sum of changes in charge densities for three top-

most layers of the asymmetrically terminated slab

(on both sides) of all three perovskites approxi-

mately equals half the charge density of the corre-

sponding bulk plane, which is necessary condition

to remove the macroscopic dipole moment [47].

Both (001) surfaces of AIIBIVO3 perovskites

Table 5

Calculated charge densities in the (001) planes in the bulk perovskites (in e, per TiO2 or AO unit), data are taken from [26]) and in four

top planes of the AO-, TiO2-terminated and asymmetrical slabs

Termination N Unit SrTiO3 BaTiO3 PbTiO3

Bulk AO 0.46 0.41 0.18

TiO2 �0.46 �0.41 �0.18

AO 1 AO 0.32 0.28 0.15

(�0.14) (�0.13) (0.03)

2 TiO2 �0.53 �0.46 �0.18

(�0.07) (�0.05) (�0.06)

3 AO 0.45 0.39 0.1

(�0.02) (�0.02) (�0.02)

4 TiO2 �0.49 �0.4 �0.12

(�0.02) (0.01) (0.00)

TiO2 1 TiO2 �0.33 �0.25 �0.08

(0.13) (0.16) (0.03)

2 AO 0.49 0.42 0.1

(0.03) (0.01) (�0.01)

3 TiO2 �0.39 �0.38 �0.08

(0.07) (0.03) (0.04)

4 AO 0.47 0.41 0.12

(0.01) (0.00) (0.00)

Asymmetrical 1 AO 0.32 0.28 0.12

(�0.14) (�0.13) (0.01)

2 TiO2 �0.54 �0.46 �0.17

(�0.07) (�0.05) (�0.05)

3 AO 0.45 0.39 0.09

(�0.01) (�0.02) (�0.02)

4 TiO2 �0.46 �0.42 �0.12

(0.00) (0.00) (0.00)

5 AO 0.47 0.41 0.11

(0.00) (0.00) (0.00)

6 TiO2 �0.42 �0.38 �0.07

(0.05) (0.03) (0.04)

7 AO 0.48 0.42 0.11

(0.02) (0.01) (�0.01)

8 TiO2 �0.3 �0.25 �0.08

(0.16) (0.16) (0.04)

Deviations of charge density with respect to the bulk are given in brackets.


correspond to stable surfaces revealing the weak-

polarity [2,48] due to partly covalent nature of

perovskite chemical bonding discussed in the

next section.The atomic dipole moments (Table 4) character-

ize atomic deformation and polarization along the

z-axis perpendicular to the surface [32]. On the

AO-terminated surfaces of all three perovskites

the cations have the negative dipole moments, di-

rected inwards, to the slab center, whereas on the

TiO2 terminated surfaces polarization of cations

has a positive sign, as well as for the oxygen ions

in SrTiO3 and BaTiO3. In contrast, oxygen ions

on PbO surface have negative dipole moment. Cat-

ions of subsurface layers for all three perovskitesreveal the positive dipole moments whereas those

of oxygen ions are negative.

The Mulliken bond populations between atoms

in surface layers (which arise due to covalency)

exhibit the largest effect for PbTiO3 crystal: the

population of the Pb–O bond (54me) on the

top layer increases a factor of three, as compared


to the bulk (16me). The partly covalent nature of

Pb–O bond in lead titanate crystal due to hybrid-

ization of Pb 6s AO state with the O 2p AO is

already pronounced in the bulk [26], but due to

bond shortening (caused by the surface relaxa-tion) its covalency is increased. This effect is also

observed on the TiO2 surface, for the bond pop-

ulation between surface O and Pb in the second

plane. Unlike PbTiO3, in SrTiO3 and BaTiO3

there is no indication on the Sr-, Ba-bonding

with O atoms. The Ti–O bonds of all three per-

ovskites on the TiO2 terminated surfaces increase

their covalency due to bond shortening (causedby surface relaxation) and breaking O surface

bonds.

We calculated also the total and difference

electron density maps (with respect to the super-

position density of spherical A2+, Ti4+ and

O2� ions) for SrTiO3, BaTiO3 and PbTiO3 sur-

faces. Total density maps might be useful for

comparison with future experiments. In Fig. 2we present results for the PbTiO3, as an example.

The density maps demonstrate considerable elec-

Ti (4)

Ti (2)

O (3)

O (1)

Pb (3)

Pb (1)

PTO PbO

0

0

0

0

Ti (3)

Ti (1)

O (4)

O (2)

Pb (4)

Pb (2)

PTO TiO2

0

0

Pb (7)

Pb (5)

Pb (3)

Pb (1)

PTO

0

0

0

0

(a)

Fig. 2. The PbTiO3 difference electron density maps in the cross sec

PbO-, TiO2- and asymmetrical terminations (a). For asymmetrical term

curves are drawn from �0.05 to +0.05ea.u.�3 with an increment of 0

tron charge density redistribution near the perov-

skite surfaces and are entirely consistent with the

above-discussed Mulliken charges and bond pop-

ulation analysis. For all three perovskites the

excess of electron density (the solid isodensitycurves) is observed for the Ti–O bonds, which

corresponds to the bond covalency. For all termi-

nations nearest to the surface Ti–O bond becomes

stronger, while the next nearest bond becomes

weaker. The A cations on the AO-terminated sur-

faces demonstrate considerable polarization, as it

was predicted above from dipole moment calcula-

tions (Table 4). Nevertheless, the electron densitymaps demonstrate no clear trace of the covalent

bonding (zero dot-dashed curves in area between

A cations and Ti–O pairs) between A cations and

oxygen, even for PbTiO3, despite the Pb–O bond

population was calculated as 54me. This means,

in reality the covalent contribution in Pb–O bond

in PbTiO3 is quite weak. Our calculated Pb–O

bond population can be interpreted only as an in-crease of the electron attraction between Pb and

O ions.

O (7)

O (5)

O (3)

Ti (8)

Ti (6)

Ti (4)

O (1)

Ti (2)

asymmetrical termination

0

O (7)

O (5)

O (3)

Ti (8)

Ti (6)

Ti (4)

O (1)

Ti (2)

Pb (7)

Pb (5)

Pb (3)

Pb (1)

(b)

tion perpendicular to the (001) surface ((110) plane) with the

ination total electron density map is also shown (b). Isodensity

.0025ea.u.�3.

Fig. 3. The calculated electronic band structure for BaTiO3

bulk and surfaces.


3.3. Density of states and band structures

The calculated band structures for SrTiO3 and

BaTiO3 bulk and surfaces (Fig. 3) are quite simi-

lar. The band structures for the bulk perovskiteswere calculated using an unit cell which is fourfold

extended along the z-axis. Such a supercell is sim-

ilar to the eight-layer ‘‘slab’’ periodically repeated

in 3D space and provides also the most natural

comparison with the surface band structures. In

the bulk band structure the bands are plotted using

the C–X–M–C directions of the typical ‘‘surface’’

Brillouin zone. The highest valence bands (VB)for the SrTiO3 and BaTiO3 bulk are quite flat, with

the top at M point and also flat between M and X

points.

The main contribution into the highest VB

comes from O 2px and 2py as it is well seen from

the calculated density of states (DOS) projected

onto the corresponding atomic orbitals (AOs).

The bottom of lowest conduction band (CB) liesat the C point with quite flat fragment between

the C and X points and consist of Ti 3d threefold

degenerated T 2g level. The optical band gaps for

surfaces and bulk of all three perovskites as calcu-

lated by means of the hybrid DFT technique are

presented in Table 6. One can see good agreement

with experiment. We should stress here remarkable

agreement of the bulk gap with the experiment forSrTiO3 (3.6eV vs 3.3eV). This is in a sharp con-

trast with the typical HF overestimate of the gap

and DFT underestimate (e.g. 1.8eV for SrTiO3

and BaTiO3 [14,15]).

The optical band gap for the SrO-terminated

surface becomes smaller with respect to the band

gap of the bulk SrTiO3. The narrow indirect gap

between the C and M points is 3.3eV, whereasthe narrowest gap in the bulk is 3.63eV, i.e. surface

gap is reduced by 0.3eV (see Table 6 for details).

Analysis of the DOS calculated for the SrO-termi-

nated surface demonstrates no contribution of the

surface O 2p states into the top of VB which

mainly consists of 2p AOs of the oxygen ions from

the central plane. The main contribution into the

CB bottom comes from the Ti 3d which are inthe second layer.

The band structure calculated for another,

TiO2-terminated surface of SrTiO3 has not so flat

Table 6

The calculated optical gap (in eV) for the bulk [26] and surface-terminated perovskites

Optical gap SrTiO3 BaTiO3 PbTiO3

Bulk SrO TiO2 Asymmetrical Bulk BaO TiO2 Asymmetrical Bulk PbO TiO2 Asymmetrical

Direct

C–C 3.96 3.72 3.95 3.03 3.55 3.49 2.96 2.73 4.32 3.58 3.18 3.08

(4.43) (4.12) (3.78) (4.43) (3.61) (3.77)

X–X 4.53 4.37 4.04 4.09 4.39 4.22 3.63 3.72 3.02 3.79 3.10 3.28

(5.08) (4.70) (4.38) (3.21) (3.82) (3.12)

M–M 5.70 5.62 5.17 4.66 5.39 5.40 4.17 4.17 5.55 5.37 5.01 4.88

(6.45) (5.94) (5.04) (5.80) (6.02) (4.89)

R–R 6.47 6.12 5.98

(7.18)

Indirect

X–C 4.39 3.55 3.92 3.41 4.20 3.49 3.41 3.18 2.87 2.96 2.98 2.78

(3.18) (3.03) (3.12)

M–C 3.71 3.30 3.17 2.31 3.60 3.32 2.33 2.10 3.66 3.55 3.19 2.96

(4.23) (3.71) (3.09) (3.85) (4.05) (2.99)

R–C 3.63 3.50 3.66

(4.16)

LDA-DFT

PW (Ref. [16])

1.85 1.86 1.13 1.79 1.80 0.84 1.54 1.53 1.61

Experiment 3.75—Direct gap 3.2 3.4

3.25—Indirect gap Ref. [49] Ref. [50]

Ref. [51]

The numbers in brackets are from Ref. [27] for SrTiO3 and Ref. [52] for PbTiO3. Both these calculations had no d-orbitals on O, and

the later one is done with B3LYP functional. The last row contains experimental data.


VB top, as that for the the SrO-termination. The

indirect optical band gap (M–C) 3.17eV becomes

by �0.46eV narrower as compared with the bulk.

For the TiO2-terminated SrTiO3 surface the main

contribution into the top of VB is made by O

2px and 2py AOs which are perpendicular to Ti–

O–Ti bridge. The main contribution to the CB bot-

tom comes from the 3d AOs of Ti in the thirdlayer, energy levels of surface Ti lie a little bit high-

er in the energy. The calculated SrTiO3 DOS are in

a good agreement with MIES and UPS spectra

recently measured for the TiO2-terminated

SrTiO3(001) surface [10]. Moreover, our calcu-

lated position of the VB top for TiO2-terminated

SrTiO3 with respect to the vacuum (5.9eV) practi-

cally coincides with the experimentally observedvalue of 5.7 ± 0.2eV (Ref. [10]).

The band structure calculated for the asymmet-

rically terminated slab demonstrates the mixture of

band structures obtained for the two symmetri-

cally terminated slabs discussed above. The SrTiO3

band gap becomes narrower (2.31eV). The VB top

consists mainly of O 2p from TiO2-terminated slab

surface, and the CB bottom from 3d AOs of Ti

from a subsurface layer (II in top of Fig. 1(c)).

The split of the upper VB (�0.8eV) is well pro-

nounced in asymmetrical SrTiO3 slab. The band

structure of the BaTiO3(001) surfaces demon-strates practically the same behavior as SrTiO3

does (Table 6). Nevertheless, the split of the VB

upper band is pronounced more for the TiO2-ter-

minated BaTiO3, as compared with the SrTiO3

surface. Due to hybridization of Pb 6s and O 2p

orbitals in PbTiO3, the calculated PbTiO3 band

structure and DOS slightly differ from those calcu-

lated for SrTiO3 and BaTiO3 (see Fig. 4, and Table6). The narrowest gap of both, bulk and surface

band structures corresponds to the transition be-

tween C and X points of the Brillouin zone. In

the bulk, the VB top is formed significantly by

Fig. 4. The calculated electronic band structure for PbTiO3

bulk and surfaces.


Pb 6s AOs (which also make the main contribution

to the VB bottom). The bulk CB bottom consist of

Ti 3d AOs, as in two other perovskites. Surpris-

ingly, the optical band gaps in PbO-terminated

surfaces are not smaller, as in BaTiO3 and SrTiO3,but even a little bit increases (up to 2.96eV with re-

spect to 2.87 in the bulk). The VB top in the PbO-

terminated surface consists of a mixture of Pb 6s

and O 2p AOs from the third layer, whereas the

CB bottom is formed by Ti 3d AOs from the sub-

surface layer.

The VB top for the TiO2-terminated

PbTiO3(001) surface at the X point consists of amixture of the O 2p and Pb 6s AOs from both sur-

face and central layers. Moreover, the main contri-

bution comes from the orbitals of atoms from the

central layer. The CB bottom for the TiO2-termi-

nated PbTiO3 consists mainly of the Ti 3d AOs

from a third layer.

The band structure calculated for the asymmet-

rical PbTiO3 slab shows a mixture of the bandstructures of the two symmetrical slabs, as it was

observed for SrTiO3 and BaTiO3.

4. Conclusions

Using hybrid DFT approach, we calculated the

surface relaxation and the electronic structure ofthe two possible terminations of the (001) surfaces

for SrTiO3, BaTiO3 and PbTiO3 perovskite crys-

tals. The data obtained for the surface structure

are in good agreement with available results of the-

oretical ab initio calculations and with experimen-

tal data. Our study shows that the surfaces are

quite stable, in agreement with ideas of a ‘‘weak

polarity’’ [2,47,48] and existing experiments.The calculated difference electron density maps

and bond populations demonstrate an increase of

the Ti–O bond covalency near the surfaces, and

additionally a weak covalency (polarization) of

the Pb–O bond on the PbO terminated surface.

The bulk band gap calculated with our new ba-

sis set is very close to the experimental data, an

agreement is much better than for typical DFT cal-culations (Table 6). The observed absence of the

surface electronic states which would split off the

upper VB top for the AO-terminated (001) sur-


faces and considerable (0.5eV) reduction of the

gap for the TiO2-terminated SrTiO3 and BaTiO3

surfaces are important factor for the future treat-

ment of the electronic structure of surface defects

on perovskite surfaces, as well as adsorption andsurface diffusion of atoms and small molecules,

which is relevant for catalysis, fuel cells, and

microelectronics.

Acknowledgments

SP was partly supported by DFG whereas EKand EH by German- Israeli Foundation (GIF

grant No. G-703.41.10). Authors are indebted to

R. Evarestov for numerous discussions.

References

[1] M.E. Lines, A.M. Glass, Principles and Applications of

Ferroelectrics and Related Materials, Clarendon Press,

Oxford, 1977.

[2] C. Noguera, Physics and Chemistry at Oxide Surfaces,

Cambridge University Press, New York, 1996.

[3] V.E. Henrick, P.A. Cox, The Surface Science of Metal

Oxides, Cambridge University Press, New York, 1994.

[4] J.F. Scott, Ferroelectric Memories, Advanced Microelec-

tronics 3, Springer, Berlin, 2000.

[5] N. Bickel, G. Schmidt, K. Heinz, K. Muller, Phys. Rev.

Lett. 62 (17) (1989) 2009.

[6] T. Hikita, T. Hanada, M. Kudo, M. Kawai, Surf. Sci. 287/

288 (1993) 377.

[7] A. Ikeda, T. Nishimura, T. Morishita, Y. Kido, Surf. Sci.

433–435 (1999) 520.

[8] G. Charlton, S. Brennan, C.A. Muryn, R. McGrath, D.

Norman, T.S. Turner, G. Thorthon, Surf. Sci. 457 (2000)

L376.

[9] P.A.W. van der Heide, Q.D. Jiang, Y.S. Kim, J.W.

Rabalais, Surf. Sci. 473 (2001) 59.

[10] W. Maus-Friedrichs, M. Frerichs, A. Gunhold, S. Kris-

chok, V. Kempter, G. Bihlmayer, Surf. Sci. 515 (2002) 499.

[11] S. Kimura, J. Yamauchi, M. Tsukada, S. Watanabe, Phys.

Rev. B 51 (16) (1995) 11049.

[12] R.E. Cohen, J. Phys. Chem. Solids 57 (10) (1996) 1393.

[13] R.E. Cohen, Ferroelectrics 194 (1997) 323.

[14] J. Padilla, D. Vanderbilt, Phys. Rev. B 56 (3) (1997) 1625.

[15] J. Padilla, D. Vanderbilt, Surf. Sci. 418 (1998) 64.

[16] B. Meyer, J. Padilla, D. Vanderbilt, Faraday Discussions

114: The Surface Science of Metal Oxides, Royal Society of

Chemistry, London, 1999, Ch. Theory of PbTiO3, BaTiO3,

and SrTiO3 surfaces, p. 395.

[17] X.Y. Xue, C.L. Wang, W.L. Zhong, Surf. Sci. 550 (2004)

73.

[18] F. Cora, C.R.A. Catlow, Faraday Discussions 114: The

Surface Science of Metal Oxides, Royal Society of Chem-

istry, London, 1999, Ch. QM investigations on perovskite-

structured transition metal oxides: bulk, surfaces and

interfaces, p. 421.

[19] C. Cheng, K. Kunc, M.H. Lee, Phys. Rev. B 62 (15) (2000)

10409.

[20] S. Tinte, M.D. Stachiotti, AIP Conf. Proc. 535 (2000)

273.

[21] S. Tinte, M.D. Stachiotti, Phys. Rev. B 64 (2001) 235403.

[22] C. Pisani (Ed.), Quantum-Mechanical Ab-initio Calcu-

lations of the Properties of Crystalline Materials, Vol.

67 of Lecture Notes in Chemistry, Springer, Berlin,

1996.

[23] L.A. Curtiss, K. Raghavachari, P.C. Redfern, J.A. Pople,

J. Chem. Phys. 106 (3) (1997) 1063.

[24] L.A. Curtiss, P.C. Redfern, K. Raghavachari, J.A. Pople,

J. Chem. Phys. 109 (1) (1998) 42.

[25] J. Muscat, A. Wander, N.M. Harrison, Chem. Phys. Lett.

342 (2001) 397.

[26] S. Piskunov, E. Heifets, R.I. Eglitis, G. Borstel, Comp.

Mat. Sci. 29 (2004) 165.

[27] E. Heifets, R.I. Eglitis, E.A. Kotomin, J. Maier, G.

Borstel, Surf. Sci. 513 (2002) 211.

[28] E. Heifets, E.A. Kotomin, J. Maier, Surf. Sci. 462 (2000)

19.

[29] E. Heifets, R.I. Eglitis, E.A. Kotomin, J. Maier, G.

Borstel, Phys. Rev. B 64 (2001) 235417.

[30] E.A. Kotomin, R.I. Eglitis, J. Maier, E. Heifets, Thin

Solid Films 400 (2001) 76.

[31] G. Borstel, R.I. Eglitis, E.A. Kotomin, E. Heifets, Phys.

State Sol. (b) 236 (2) (2003) 253.

[32] V.R. Saunders, R. Dovesi, C. Roetti, M. Causa, N.M.

Harrison, R. Orlando, C.M. Zicovich-Wilson, CRYS-

TAL�98 User�s Manual, Universita di Torino, Torino,

1998.

[33] Available from: <http://www.chimifm.unito.it/teorica/crys-

tal/crystal.html>.

[34] Available from: <http://www.cse.clrc.ac.uk/cmg/crystal>.

[35] A.D. Becke, J. Chem. Phys. 98 (7) (1993) 5648.

[36] J.P. Perdew, Y. Wang, Phys. Rev. B 33 (12) (1986) 8800.



[39] P.J. Hay, W.R. Wadt, J. Chem. Phys. 82 (1) (1984) 270.



[42] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (12) (1976)

5188.

[43] T. Bredow, R.A. Evarestov, K. Jug, Phys. Stat. Solidi (b)

222 (2000) 495.

[44] K.H. Hellwege, A.M. Hellwege (Eds.), Ferroelectrics and

Related Substances, Vol. 3 of New Series, Landolt–

Bornstein, Springer Verlag, Berlin, 1969, group III.

[45] B.G. Shirane, R. Repinsky, B.C. Frazer, Acta Cryst. 9

(1956) 131.

http://www.chimifm.unito.it/teorica/crystal/crystal.html

http://www.chimifm.unito.it/teorica/crystal/crystal.html

http://www.cse.clrc.ac.uk/cmg/crystal


[46] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flan-

nery, Numerical Recipies in Fortran77, second ed., Cam-

bridge University Press, Cambridge, MA, 1997.

[47] P.W. Tasker, J. Phys. C: Solid State Phys. 12 (1979) 4977.

[48] J. Goniakowski, C. Noguera, Surf. Sci. 365 (1996) L657.

[49] S.H. Wemple, Phys. Rev. B 2 (7) (1970) 2679.

[50] C.H. Peng, J.F. Chang, S. Desu, Mater. Res. Soc. Symp.

Proc. 243 (1992) 21.

[51] K. van Benthem, C. Elsaesser, R.H. French, J. Appl. Phys.

90 (12) (2001) 6156.

[52] S. de Lasaro, E. Longo, J.R. Sambrano, A. Beltran, Surf.

Sci. 552 (2004) 149.

Hybrid DFT calculations of the atomic and electronic structure for ABO3 perovskite (001) surfaces

Documents

Transcript of Hybrid DFT calculations of the atomic and electronic structure for ABO3 perovskite (001) surfaces