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\begin{document}
\title{First-principles calculations of atomic and
electronic structure of SrTiO$_3$
(001) and (011) surfaces}
\author{R. I. Eglitis, and
David Vanderbilt}
\affiliation{Department of Physics and Astronomy,
Rutgers University, 136 Frelinghuysen Road, Piscataway,
New Jersey 08854-8019, USA}

\date{February 28, 2008}

\begin{abstract}
We present and discuss the results of calculations of surface
relaxation and rumpling on  SrTiO$_3$ (001) and (011) surfaces.
We consider both SrO and TiO$_2$ terminations of the (001) surface,
and three terminations (Sr, TiO and O) of the polar (011) surface.
The calculations are based on hybrid
Hartree-Fock and density-functional theory exchange functionals, using
Becke's three-parameter method, combined with the nonlocal correlation
functionals of Perdew and Wang.  We find that
all top-layer atoms for TiO$_2$ and SrO-terminated SrTiO$_3$ (001) surfaces
relax inwards, with the exception of SrO-terminated surface O atoms, whereas
all second-layer atoms relax outwards.
The surface rumpling for the  TiO-terminated SrTiO$_3$ (011) surface,
11.28\% of the bulk lattice constant, is considerably larger than the
relevant surface rumplings for SrO and TiO$_2$-terminated
(001) surfaces. The surface rumplings for the SrO and TiO$_2$-terminated
(001) surfaces are in an excellent agreement with relevant
LEED and RHEED experimental data,
and the surface relaxation energies on both surfaces are similar.
In contrast, the different terminations of the
(011) surface lead to large differences in relaxation energies.
The O-terminated (011) surface has the lowest surface relaxation
energy ($-$1.32\,eV). The TiO-terminated  (011) surface has much higher
surface relaxation energy of $-$1.55\,eV, while the Sr-terminated  (011)
surface has the highest surface relaxation energy ($-$1.95\,eV).
Our calculations indicate a considerable increase of the Ti-O bond covalency
(0.130$e$) near the TiO-terminated (011) surface relative to the bulk
(0.088$e$), much larger than that for the (001) surface (0.118$e$). The
Ti-O bond populations are considerably larger in the direction
perpendicular to the TiO-terminated (011) surface (0.188$e$) than
in the plane (0.130$e$).
\end{abstract}

\pacs{68.35.Ct, 68.35.Md, 68.47.Gh}

\maketitle

\section{INTRODUCTION}

Oxide perovskites are promising for many device applications because of
their diverse physical properties \cite{scott}, both in bulk
and thin-film form \cite{scott,ring}. In  particular, titanate
perovskites are of great interest for their ferroelectric and
piezoelectric properties, their electrochemical behavior, and their
use in electrodes and sensors.  SrTiO$_3$ is among the best studied
and most important of the perovskite titanates, as it is widely
used as a dielectric and as a substrate for growth of thin films
or superlattices of other functional perovskite or related
(e.g., high-$T_c$) materials.  For these reasons, a detailed understanding
of the surface structure and electronic properties are of primary
importance.

It is not surprising that this high technological importance has motivated
several {\it ab initio}
\cite{kim,zhi,her,nat,kub,heif1,heif2,kar,eg,pi,her1,lan,lii,chen,pad,mpv}
and classical shell-model
\cite{heif,rav} studies of the (001) surface of SrTiO$_3$.
The (001) surface relaxation and rumpling have also been
studied experimentally by means of low energy electron
diffraction (LEED), reflection high-energy electron diffraction
(RHEED), medium energy ion scattering (MEIS), and surface x-ray
diffraction (SXRD) measurements \cite{bic,hik,kudo,kido,ike,char}.
The most recent experimental studies on the SrTiO$_3$ surfaces
include a combination of XPS, LEED, and time-of-flight scattering
and recoil spectrometry (TOF-SARS) \cite{heid} as well as
metastable impact electron spectroscopy \cite{maus}. In these
recent studies, well-resolved 1$\times$1 LEED patterns were
obtained for the TiO$_2$-terminated SrTiO$_3$ (001) surface.
Simulations of the TOF-SARS azimuthal scans indicate that the
O atoms are situated 0.1\,\AA\  above the Ti layer (surface
plane) in the case of the TiO$_2$-terminated SrTiO$_3$
(001) surface.
There is general agreement between theory and
LEED and RHEED experiments on the larger
rumpling for the SrO termination. However, there is a disagreement
about the direction of surface O atom displacements on the
TiO$_2$-terminated SrTiO$_3$ (001) surface, probably due to
neglect in the theory of the anharmonic vibrations of the surface
atoms, especially Ti. On the other hand, several diffraction
experiments clearly contradict each other, most likely because of
differences in sample preparation or
different interpretations of indirect experimental data on the
atomic surface relaxations \cite{pad}.
Resonance photoemission \cite{cou} from Ti\,3$d$ states above the Ti\,3$p$
absorption threshold has been used to extract regions of Ti\,3$d$-state
hybridization in the O\,2$p$ valence band of bulk SrTiO$_3$ and of
the surface of SrTiO$_3$ (001). An enhanced covalent mixing in
the surface is found \cite{cou}.

ABO$_3$ perovskite (011) surfaces in general, and SrTiO$_3$ (011)
surfaces in particular, are considerably less well studied than the
corresponding (001) surfaces. The first {\it ab initio} calculations
of the SrTiO$_3$ (011) surface was performed by Bottin {\it et al.}
\cite{bot}, who carried out a systematic first-principles study
of the electronic and atomic structure of several (1$\times$1)
terminations of the (011) surface.  The
electronic structures of the stoichiometric SrTiO and O$_2$
terminations were characterized by marked differences with
respect to the bulk, as a consequence of the polarity compensation.
One year later, Heifets {\it et al.} \cite{heif3}
performed {\it ab initio} Hartree-Fock calculations for four
possible terminations (TiO, Sr, and two kinds of O terminations) of
the SrTiO$_3$ (011) surface.
Heifets {\it et al.} \cite{heif4} also investigated the atomic
structure and charge redistribution for different terminations of
the BaZrO$_3$ (011) surfaces using density-functional theory, finding that
while the O-terminated (011) surface had the
smallest cleavage energy among (011) surfaces, this value was
still twice as large as for the formation of a pair of
complimentary (001) surfaces.
Finally, Eglitis and Vanderbilt
\cite{egvan} performed {\it ab initio} calculations for the
technologically important BaTiO$_3$ and PbTiO$_3$ (011) surfaces.

In this paper, we analyze in detail the structural and electronic
properties of the SrTiO$_3$ (001) and (011) surfaces.
For the better-studied (001) surfaces, we address in particular
the contradictory experimental results for this surface,
and we independently check the reports of enhanced Ti-O covalent
bonding near the surface.\cite{cou}
Then, since only two {\it ab initio} studies have been reported up
to now dealing with the  SrTiO$_3$ (011) surfaces
\cite{bot,heif3}, we perform
detailed predictive calculations for several terminations of this surface,
with an emphasis on the effect of the surface relaxation and
rumpling, surface energies, and the charge redistributions and
changes in bond strength that occur at the surface.
We chose the hybrid B3PW functional for our current study
because it yields excellent results for the SrTiO$_3$, BaTiO$_3$, and
PbTiO$_3$ bulk lattice constant and bulk modulus \cite{heif2,pis}.
For example, for bulk SrTiO$_3$ our calculated lattice
constant of 3.904\,\AA\ is in excellent agreement with the
experimental value of 3.89\,\AA, whereas LDA is usually\cite{explan-bot}
about 1\% too small \cite{ksv,zhong,lasota,sai}
and Hartree-Fock (3.93\,\AA) is about 1\% too large.\cite{heif3}


\section{COMPUTATIONAL METHOD}


To perform the first-principles DFT-B3PW calculations, we used the
CRYSTAL-2003 computer code \cite{crys}. This code employs
Gaussian-type functions (GTFs) localized on atoms as the basis for
an expansion of the crystalline orbitals. The features of the CRYSTAL-2003
code that are most important for this study are its ability
to calculate the electronic structure of materials
within both Hartree-Fock (HF)
and Kohn-Sham Hamiltonians, and its implementation of an isolated 2D
slab model without artificial repetition along the $z$-axis.
However, in order to employ the LCAO-GTF method, it is desirable to
have optimized basis sets. Such an optimized basis set for SrTiO$_3$
was developed and discussed in Ref.~[\onlinecite{pis}]. In the present work
we have adopted this new basis set, which differs from that used in previous
calculations \cite{heif1,heif2} by inclusion of polarizable
$d$-orbitals on the O ions. It was shown \cite{pis} that this leads
to better agreement of the calculated lattice constant and bulk modulus
with experimental data.

Our calculations were performed using the hybrid exchange-correlation
B3PW functional involving a mixture of non-local Fock exact exchange,
LDA exchange, and Becke's gradient corrected exchange
\cite{bec}, combined with the nonlocal gradient-corrected correlation
potential by Perdew and Wang \cite{per123}. The Hay-Wadt
small-core effective core pseudopotentials (ECP) were adopted for
Ti and Sr atoms \cite{hay123}. The small-core ECPs replace
only the inner core orbitals, while orbitals for sub-valence electrons as
well as for valence electrons are calculated self-consistently.
Oxygen atoms were treated with an all-electron basis set.

The reciprocal space integration was performed by  sampling the
Brillouin zone with an 8$\times$8
Pack-Monkhorst mesh \cite{monk}.
To achieve high
accuracy, large enough tolerances of 7, 8, 7, 7, and 14 were
chosen for the Coulomb overlap, Coulomb penetration, exchange
overlap, first exchange pseudo-overlap, and
second exchange pseudo-overlap, respectively \cite{crys}.

\begin{figure}
\includegraphics[width=2.0in]{Fig1.eps}
\caption{(Color online.)
Side view of SrTiO$_3$ (001) surfaces.
(a) SrO-terminated surface.
(b) TiO$_2$-terminated surface, with definitions of
surface rumpling $s$ and the near-surface interplanar
separations $d_{12}$ and $d_{23}$.}
\end{figure}

The  SrTiO$_3$ (001) surfaces were modeled with
two-dimensional (2D) slabs consisting of several planes
perpendicular to the [001] crystal direction, as illustrated
in Fig.~1. The CRYSTAL-2003
code allowed us to avoid artificial periodicity along the $z$
direction and to perform simulations for stand-alone 2D slabs.
To simulate SrTiO$_3$ (001) surfaces, we used slabs consisting
of seven alternating TiO$_2$ and SrO layers, with a mirror symmetry
through the middle of the slab.  One of these slabs was terminated
by SrO planes and contained 17 atoms in the supercell, while the
second was terminated by TiO$_2$ planes and contained 18 atoms.
These slabs are non-stoichiometric, with unit cell formulae
Sr$_4$Ti$_3$O$_{10}$ and Sr$_3$Ti$_4$O$_{11}$ respectively.
The sequences of layers for the two SrTiO$_3$ (001) surfaces are
shown in Fig.~1.

Turning next to the polar SrTiO$_3$ (011) surface, we note that
the crystal is composed of charged O-O or SrTiO planes, as shown
in Fig.~2.  If one assumes nominal ionic charges of Sr$^{2+}$,
Ti$^{4+}$, and O$^{2-}$, the O-O and SrTiO layers have charges
of $\mp4$ respectively.  Thus, a simple cleavage terminating on
an O-O layer would leave a net negative surface charge, while a
termination on a SrTiO layer would leave a net positive charge.
This would lead to a large dipole for an asymmetric slab like
that of Fig.~3(a), or a net charge in the supercell in the
case of Figs.~3(b-c).
The surfaces might become metallic in order to avoid the
infinite electrostatic energy arising from such surface charges,
but in any case the surface energy would be expected to be quite
high.\cite{nog,tas,poj,bot}  It is much more likely that the
surface would reconstruct in order to restored the neutrality of
the surface layers.

\begin{figure}
\includegraphics[width=2.4in]{Fig2.eps}
\caption{(Color online.)
Sketch of the cubic SrTiO$_3$ perovskite structure showing two
(011) cleavage planes that give rise to charged SrTiO
and O$_2$ (011) surfaces.}
\end{figure}

We thus construct surface slab models as follows.
Starting from the symmetric SrTiO-terminated slab of Fig.~3(c),
we can remove the Sr atom from each surface to obtain the
7-layer (16-atom) TiO-terminated slab shown in Fig.~3(d).
We can alternatively remove TiO units from each surface and obtain the
7-layer (14-atom) Sr-terminated slab shown in Fig.~3(e).
Finally, we can also start from the symmetric O$_2$-terminated
slab of Fig.~3(b) and remove one of each two surface O atoms
to obtain the 7-layer (15-atom) O-terminated slab model of
Fig.~3(f).  We use the slab models of Figs.~3(d-f) for
our subsequent calculations.  Note that the O-terminated slab
in Fig.~3(f) is the only one of the three that is stoichiometric,
but all have symmetry through the middle of the slab and have
non-polar surface terminations. For a more in-depth discussion
of stoichiometric and non-stoichiometric (011) surface terminations
of this crystal, see Ref.~[\onlinecite{bot}].

\begin{figure}
\includegraphics[width=3.2in]{Fig3.eps}
\caption{(Color online.)
Possible (011) surface slab models considered in the text.
(a-c) Slabs obtained by simple cleavage, yielding
mixed, O-terminated, and SrTiO-terminated polar surfaces, respectively.
(d-f) Slabs with non-polar TiO-terminated, Sr-terminated, and O-terminated
surfaces, respectively.}
\end{figure}



\section{RESULTS OF CALCULATIONS}

\subsection{SrTiO$_3$ bulk atomic and electronic structure}

As a starting point for our calculations, we calculated the
SrTiO$_3$  bulk lattice constant to be 3.904\,\AA,
in almost perfect agreement with the experimental result extrapolated
to 0\,K (3.89\,\AA).\cite{fer} We used the theoretical
SrTiO$_3$ bulk lattice constant in the following surface structure calculations.
To characterize the chemical bonding and covalency effects, we used a
standard Mulliken population analysis for the effective atomic charges $Q$
and other local electronic-structure properties (bond orders, atomic
covalencies and full valencies) as described, for example, in
Refs.~[\onlinecite{cat,boc}].
Our calculated static effective charges for bulk SrTiO$_3$
are 1.871$e$ for the Sr atom, 2.351$e$ for the Ti atom,
and $-$1.407$e$ for the O atom, while the population of the chemical
bond between Ti and O atoms is 0.088$e$ (see Table I).

% \begin{table}[b]
% \caption{Computed effective charges $Q$ and bond populations $P$ of
% atoms in bulk SrTiO$_3$.}
% \begin{ruledtabular}
% \begin{tabular}{ccccc}
% ~&Ion & Property & B3PW&~\\
% \hline
% ~&Sr  & $Q$ & +1.871 &~  \\
% ~&~   & $P$ & -0.010 &~  \\
% ~&O   & $Q$ & -1.407 &~  \\
% ~&~   & $P$ & +0.088 &~  \\
% ~&Ti  & $Q$ & +2.351 &~  \\
% \end{tabular}
% \end{ruledtabular}
% \end{table}

\begin{table}[b]
\caption{Computed effective charges $Q$ and bond populations $P$ of
atoms in bulk SrTiO$_3$.}
\begin{ruledtabular}
\begin{tabular}{ccr}
Ion & Property & B3PW\\
\hline
Sr  & $Q$ &    1.871 \\
    & $P$ & $-$0.010 \\
O   & $Q$ & $-$1.407 \\
    & $P$ &   0.088 \\
Ti  & $Q$ &   2.351 \\
\end{tabular}
\end{ruledtabular}
\end{table}



\def\pp{$\phantom{-}$}
\def\mm{$-$}
%
\begin{table*}
\caption{Calculated atomic relaxation (in percent of bulk lattice
constant) for SrO and TiO$_2$ terminated  SrTiO$_3$ (001) surfaces.
Positive (negative) values refer to displacements outwards from
(inwards to) the surface.}
\begin{ruledtabular}
\begin{tabular}{ccccccccccc}
\multicolumn{6}{c}{SrO terminated} &
\multicolumn{5}{c}{TiO$_2$ terminated} \\
\hline
N&Ion&This work&SM\cite{heif}&LDA\cite{pad}&LDA\cite{chen}&Ion&This work&SM\cite{heif}&LDA\cite{pad}&LDA\cite{chen}\\
\hline
1&Sr&\mm 4.84&\mm 7.10&\mm 5.7&\mm 6.66&Ti&\mm 2.25&\mm 2.96&\mm 3.4&\mm 1.79\\
~&O &\pp 0.84&\pp 1.15&\pp 0.1&\pp 1.02&O &\mm 0.13&\mm 1.73&\mm 1.6&\mm 0.26\\
2&Ti&\pp 1.75&\pp 1.57&\pp 1.2&\pp 1.79&Sr&\pp 3.55&\pp 3.46&\pp 2.5&\pp 4.61\\
~&O &\pp 0.77&\pp 0.87&0.0 &\pp 0.26&O &\pp 0.57&\mm 0.21&\mm 0.5&\pp 0.77\\
3&Sr&     &\mm 1.42&\mm 1.2&\mm 1.54&Ti&     &\mm 0.60&\mm 0.7&\mm 0.26\\
~&O &     &\pp 0.70&\mm 0.1&\pp 0.26&O &     &\mm 0.29&\mm 0.5&\pp 0.26\\
\end{tabular}
\end{ruledtabular}
\end{table*}

\begin{table*}
\caption{Surface rumpling $s$ and relative displacements
$\Delta$$d_{ij}$ (in percent of bulk lattice constant) for
the three near-surface planes of SrO and TiO$_2$ terminated
SrTiO$_3$ (001) surfaces.
}
\begin{ruledtabular}
\begin{tabular}{ccccccc}
&
\multicolumn{3}{c}{SrO terminated} &
\multicolumn{3}{c}{TiO$_2$ terminated}  \\
\hline
~                 & $s$  & $\Delta$$d_{12}$ & $\Delta$$d_{23}$ & $s$ & $\Delta$$d_{12}$ & $\Delta$$d_{23}$ \\
\hline
This study        & 5.66 & $-$6.58            &  1.75            & 2.12& $-$5.79            & 3.55             \\
{\it Ab initio} \cite{pad} & 5.8  & $-$6.9  & 2.4 & 1.8 & $-$5.9 & 3.2         \\
{\it Ab initio} \cite{chen} & 7.7  & $-$8.6  & 3.3 & 1.5 & $-$6.4 & 4.9 \\
Shell model \cite{heif}    & 8.2  & $-$8.6  & 3.0 & 1.2 & $-$6.4 & 4.0 \\
LEED experiment \cite{bic} & 4.1$\pm$2&$-$5$\pm$1&2$\pm$1&2.1$\pm$2&1$\pm$1&$-$1$\pm$1 \\
RHEED experiment \cite{hik} & 4.1  &  2.6  & 1.3 & 2.6 &  1.8 & 1.3 \\
MEIS experiment \cite{ike} &  ~   &   ~   &  ~  & 1.5$\pm$0.2 & 0.5$\pm$0.2& ~  \\
SXRD experiment \cite{char} &1.3$\pm$12.1  &$-$0.3$\pm$3.6&$-$6.7$\pm$2.8& 12.8$\pm$8.5 & 0.3$\pm1$ & ~ \\
\end{tabular}
\end{ruledtabular}
\end{table*}

\subsection{SrTiO$_3$ (001) surface atomic and electronic structure}

In the present SrTiO$_3$ (001) calculations using the
hybrid B3PW method, we allowed atoms
in the two outermost surface layers to relax along the $z$ axis
(by symmetry the atoms have no forces along the $x$ or $y$ axis).
The resulting atomic displacements for the TiO$_2$- and
SrO-terminated (001) surfaces are shown in Table II.
For the  TiO$_2$-terminated surface we find that both the Ti and O
atoms in the first surface layer relax inwards (i.e., towards
the bulk), whereas in the case of the SrO-terminated
surface the upper-layer Sr atoms relax inwards while the upper-layer
O atoms relax outwards. For both terminations, outward relaxation of
all atoms in the second layer is found. 
A comparison with the surface atomic displacements obtained by other
theoretical methods is also given in Table II.  This shows that
the direction of the atomic displacements calculated by
quite different {\it ab-initio} and classical shell-model methods
is always the same for both first-layer atoms,
as well as for the second-layer Ti atoms, but the
displacement magnitudes are quite different.


\def\pp{$\phantom{-}$}
\def\mm{$-$}
%
\begin{table*}
\caption{Calculated absolute magnitudes of atomic displacements $D$
(in \AA), the effective atomic charges $Q$ (in $e$) and the bond
populations $P$ between nearest Me-O atoms (in $e$) for the TiO$_2$
and SrO terminated SrTiO$_3$
(001) surfaces.
}
\begin{ruledtabular}
\begin{tabular}{cccccc}
Layer & Property& Ion&TiO$_2$-terminated&Ion&SrO-terminated \\
\hline
1& $D$ &Ti &\mm 0.088& Sr & \mm 0.189 \\
~& $Q$ & ~ &\pp 2.291& ~  & \pp 1.846 \\       
~& $P$ & ~ &\pp 0.118& ~  & \mm 0.006 \\
~& $D$ & O &\mm 0.005& O  & \pp 0.033 \\
~& $Q$ & ~ &\mm 1.296& ~  & \mm 1.522 \\
~& $P$ & ~ &\mm 0.014& ~  & \pp 0.074 \\
2& $D$ & Sr&\pp 0.139& Ti & \pp 0.068 \\
~& $Q$ & ~ &\pp 1.850& ~  & \pp 2.363 \\
~& $P$ &~  &\mm 0.008& ~  & \pp 0.078 \\
~& $D$ &O  &\pp 0.022&~O  & \pp 0.030 \\
~& $Q$ &~  &\mm 1.365&~   & \mm 1.450 \\
~& $P$ &~  &\pp 0.080&~   & \mm 0.010 \\
3& $Q$ &Ti &\pp 2.348&Sr  &\pp 1.875  \\
~& $P$ &~  &\pp 0.096&~   &\mm 0.012  \\
~& $Q$ &O  &\mm 1.384&O~  & \mm 1.429 \\
~& $P$ &~  &\mm 0.010& ~  & \pp 0.084 \\
\end{tabular}
\end{ruledtabular}
\end{table*}

In order to compare the calculated SrTiO$_3$ (001) surface
structures with available experimental results, the amplitudes of
the surface rumpling $s$ (the relative displacement of oxygen with
respect to the metal atom in the surface layer) and the changes
in the interlayer distances $\Delta$$d_{ij}$ ($i$ and $j$ are the
layer numbers)  are presented in Table III. Our calculations of the
interlayer distances are based on the positions of relaxed metal
ions, which are known to be much stronger electron scatterers
than oxygen ions \cite{bic}. The agreement is quite good for
all theoretical methods, which give the same sign for both the
surface rumpling and changes of the interlayer distances. The
amplitude of the surface rumpling for the SrO-terminated SrTiO$_3$
(001) surface is predicted to be much larger than that for
TiO$_2$-terminated surface.  As one can see from Table III,
both SrO and TiO$_2$ terminated SrTiO$_3$ (001) surfaces display
a reduction of the interlayer distance $\Delta$$d_{12}$ and
an expansion of $\Delta$$d_{23}$.

The calculated surface rumpling amplitudes $s$ for both (001)
surface terminations are in qualitative agreement with
available LEED, RHEED, MEIS, and SXRD experimental results
\cite{bic,hik,ike,char}.  Unfortunately, the calculated changes
in interlayer distances $\Delta d_{ij}$ are in disagreement with
LEED experimental data \cite{bic} for the TiO$_2$ terminated (001)
surface, which show an expansion of the interlayer distance
$\Delta d_{12}$ and a reduction of the interlayer distance
$\Delta d_{23}$, while on the contrary all {\it ab initio} and
classical shell-model calculations predict a reduction of the
interlayer distance $\Delta d_{12}$ and an expansion of
$\Delta d_{23}$.  Moreover, as can be seen from the table,
the experiments contradict each other regarding
the sign of $\Delta d_{12}$ and $\Delta d_{23}$ for the SrO-terminated
surface, as well as for $\Delta d_{23}$ of the TiO$_2$-terminated
surface.

Another discrepancy between theory and experiment is that the LEED, RHEED and
MEIS experiments \cite{bic,hik,ike} demonstrate that the topmost layer oxygen
atoms  always move outwards from the surface, whereas all {\it ab initio}
and classical shell-model calculations for the TiO$_2$-terminated SrTiO$_3$
(001) surface predict that the oxygen atoms relax towards the bulk.
It is important to note also the contradiction between the LEED, RHEED and
MEIS experiments \cite{bic,hik,ike} and a recent SXRD \cite{char}
experiment, where oxygen atoms are predicted to move inwards for both
surface terminations, reaching a
very large rumpling amplitude up to 12.8\,\% of the bulk lattice constant for
the TiO$_2$-terminated surface. The reasons for such discrepancies between
the different experimental data are not clear, but the matter is
comprehensively discussed in Refs.~[\onlinecite{pad}] and
[\onlinecite{char}].  In any case,
we conclude that the disagreement in some cases between theoretical
{\it ab initio} and shell model results on one side, and experimental
results on the other, should not be taken too seriously until the
internal inconsistencies in the experimental results are resolved.

The atomic displacements $D$, effective static atomic charges $Q$,
and bond populations $P$ between nearest metal and oxygen atoms for the
SrTiO$_3$ (001) surfaces are given in Table IV. The major
effect observed here is a strengthening of the Ti-O chemical bond
near the TiO$_2$-terminated (001) surface.  Note that the Ti and
O effective charges in bulk SrTiO$_3$ of 2.351$e$ and $-$1.407$e$
respectively (see Table I)
are much smaller than those expected in an ionic model.
The Ti-O bond population for the TiO$_2$-terminated (001) surface is
0.118$e$ (see Table IV),
which is considerably larger than the
value of 0.088$e$ in the bulk. In
contrast, the Sr-O bond populations are very small. The lack
of covalency in the Sr-O bond
is also seen in the Sr effective charges of 1.871$e$ in the bulk
and 1.846$e$ on the SrO-terminated (001) surface, which are
close to the formal ionic charge of 2$e$.

\subsection{SrTiO$_3$ (011) surface atomic structure}

Our calculated atomic relaxations for the SrTiO$_3$ (011) surfaces
are shown in Table V.  An idea of the nature of the relaxed (011) surfaces can
be obtained from Figs.~3(d)-(f).
The first-layer metal atoms for the TiO- and Sr-terminated
(011) surfaces relax strongly inwards, by 0.0769$a$ for
Ti and, even more strongly, by 0.1281$a$ for Sr, whereas
the O atoms on the TiO-terminated (011) surface relax
outwards by 0.0102$a$. (Here $a$ is the bulk lattice constant.)
The O atoms in the top layer of
the O-terminated (011) surface also move inwards
by 0.0661$a$. Results calculated using the classical shell model \cite{heif}
for the TiO, O, and Sr-terminated (011) surface
upper layers display the same atomic displacement directions
as our calculations, but in most cases
the atomic displacement magnitudes are considerably larger.
Also for the second- and third-layer atoms on the
TiO and Sr-terminated (011) surface, the directions of the
atomic displacements calculated using the hybrid B3PW method
coincide in all cases with those calculated using the classical
shell model \cite{heif}, but the displacements calculated from
the latter method are almost always larger.  Only the third-layer
O atom displacement for the Sr-terminated (011) surface
(0.0108$a$) calculated using
the hybrid B3PW method is larger than the
displacement (0.0025$a$) obtained by the means
of the classical shell model \cite{heif}.

\begin{table*}
\caption{Atomic relaxation of the SrTiO$_3$  (011) surface (in
per cent of the bulk lattice constant) for the three terminations calculated
by means of the {\it ab initio} B3PW method. A positive sign corresponds to
outward atomic displacements.}
\begin{ruledtabular}
\begin{tabular}{cccccc}
 Layer & Ion & $\Delta$z & $\Delta$y & $\Delta$z (SM) & $\Delta$y (SM) \\
\hline
\multicolumn{6}{c}{TiO terminated SrTiO$_3$ (011) surface} \\
 1 & Ti &\mm 7.69&  &\mm 5.99 &  \\
 1 & O  &\pp 3.59&  &\pp 8.48 &  \\
 2 & O  &\mm 0.51&  &\mm 1.72 &  \\
 3 & Sr &\mm 2.10&  &\mm 6.96 &  \\
 3 & O  &\mm 2.56&  &\mm 4.10 &  \\
 3 & Ti &\pp 0.16&  &\pp 2.14 &  \\
\multicolumn{6}{c}{Sr terminated SrTiO3 (011) surface} \\
1 & Sr &\mm 12.81&  &\mm 19.07&  \\
2 & O  &\pp 1.02 &  & \pp 3.18&  \\
3 & Ti &\mm 0.04 &  & \mm 0.89&  \\
3 & O  &\mm 1.08 &  & \mm 0.25&  \\
3 & Sr &\pp 0.26 &  & \pp 4.67&  \\
\multicolumn{6}{c}{O terminated SrTiO$_3$ (011) surface}  \\
1 & O &\mm 6.61&\mm 0.14&\mm 14.20&\mm 8.54 \\
2 & Ti&\mm 1.02&\mm 4.35&\mm 2.37 &\mm 8.27 \\
2 & Sr&\mm 1.18&\pp 0.85&\pp 4.10 &\mm 10.79\\
2 & O &\pp 1.79&\pp 6.40&\pp 5.71 &\pp 8.20 \\
3 & O &\mm 0.79&\pp 2.10&\mm 11.06&\mm 11.01\\
\end{tabular}
\end{ruledtabular}
\end{table*}

For the O-terminated (011) surface, the atomic
displacement directions calculated using the hybrid
B3PW method are mostly the same as those
calculated  by the
shell-model method \cite{heif}, but in same cases
there are also qualitative differences.
For example,
according to our B3PW results,
the Sr atoms in the second layer of this surface move
along the surface by 0.0085$a$, and also slightly
inwards by 0.0118$a$. In contrast, the same atom, according
to the shell-model calculation \cite{heif}, moves along
the surface in the opposite direction by 0.1079$a$, and also
outwards by 0.0410$a$. The atomic  displacements in the third
plane from the surface for all three terminations of
the (011) surface are still large. This is in sharp
contrast with our results for the neutral (001)
surfaces in Table II, where the atomic displacements
converged very quickly and were already negligible  in the
third layer.

Our calculated surface rumpling $s$ for the TiO-terminated (011)
surface, and the relative displacements of the three top layers
$\Delta d_{12}$ and $\Delta d_{23}$ for TiO and O-terminated
(011) surfaces, are listed in Table VI.
For the TiO-terminated surface, our computed B3PW surface rumplings
(11.28\%) and those computed from the shell model
\cite{heif} (14.47\%) are comparable and very large.
This arises, according to the results of our calculations, from
a combination of a strong  O atom outward displacement by 3.59\% and
an even stronger Ti atom inward displacement by 7.69\%.
This (011) surface rumpling
is much larger than that found for the (001) surfaces.
Our B3PW-calculated reduction of relative distances $\Delta d_{12}$
between the first and second layer for the TiO and O-terminated
(011) surfaces ($-$7.18\% and $-$5.59\%, respectively)
are more that ten times larger than the reduction of relative
distances $\Delta d_{23}$ between the second and third
layer ($-$0.67\% and $-$0.23\%).
The corresponding interlayer distance reductions computed from the
classical shell model \cite{heif} are also large  and
comparable with our {\it ab initio} results.
There is also one quantitative difference between our
{\it ab initio} B3PW calculations and the
shell-model results, namely, that the latter
\cite{heif} predicts an expansion of the interlayer
distance $\Delta d_{23}$ for the O-terminated (011) surface,
while our calculations predict a reduction of the same 
interlayer distance.

\begin{table*}
\caption{Surface rumpling $s$ and relative displacements
$\Delta d_{ij}$ (in percent of the bulk lattice constant)
for three near-surface planes of the TiO and O-terminated
SrTiO$_3$ (011) surfaces.
}
\begin{ruledtabular}
\begin{tabular}{cccccc}
\multicolumn{1}{c}{~} &
\multicolumn{3}{c}{TiO terminated} &
\multicolumn{2}{c}{O terminated} \\
\hline
~& $s$ & $\Delta$$d_{12}$ & $\Delta$$d_{23}$ & $\Delta$$d_{12}$ & $\Delta$$d_{23}$ \\
\hline
This study & \pp 11.28 & \mm 7.18 & \mm 0.67 & \mm 5.59 & \mm 0.23 \\
Shell model \cite{heif} & \pp 14.47 & \mm 4.27 & \mm 3.86 & \mm 11.83 & \pp 8.69 \\
\end{tabular}
\end{ruledtabular}
\end{table*}

\subsection{SrTiO$_3$ surface energies}

In order to calculate the SrTiO$_3$ (001) surface energy, we started
with the cleavage energy for unrelaxed SrO and TiO$_2$-terminated
(001) surfaces.  Surfaces  with both terminations arise simultaneously under
(001) cleavage of the crystal, and we adopt the convention that the
cleavage energy is distributed equally between the created surfaces.
In our calculations, the 7-layer SrO-terminated (001) slab with
17 atoms and the TiO$_2$-terminated one with 18 atoms represent,
together, seven bulk unit cells (35 atoms), so that
%
\begin{eqnarray}
E_{\rm surf}^{(\rm unr)}(\Lambda)={\frac{1}{4}}[E_{\rm slab}^{(\rm unr)}({\rm SrO})
+E_{\rm slab}^{\rm (unr)}({\rm TiO_2}) - 7E_{\rm bulk}],
\end{eqnarray}
%
where $\Lambda$ denotes SrO or TiO$_2$,
$E_{\rm slab}^{\rm (unr)}(\Lambda)$ are the unrelaxed energies of the
SrO- or  TiO$_2$-terminated (001) slabs, $E_{\rm bulk}$ is the
energy per bulk unit cell, and the factor of 4 comes from the fact that
we create four surfaces upon the cleavage procedure. According to the results
of our hybrid B3PW calculations, the cleavage results in a surface
energy of 1.39\,eV.
Next, we can calculate the relaxation energies for each of the SrO and
TiO$_2$ terminations, when both sides of the slabs relax, according to
%
\begin{eqnarray}
E_{\rm rel}(\Lambda)={\frac{1}{2}}
       [E_{\rm slab}^{(rel)}(\Lambda)-E_{\rm slab}^{\rm (unr)}(\Lambda)],
\end{eqnarray}
%
where $E_{\rm slab}^{\rm (rel)}(\Lambda)$ is the slab energy after
relaxation (and again $\Lambda$ = SrO
or TiO$_2$). According to the results of our calculations, the SrO-
and TiO$_2$-terminated surfaces relax by 0.24\,eV and 0.16\,eV
respectively.  The surface energy is then defined as the sum
of the cleavage and relaxation energies:
%
\begin{eqnarray}
E_{\rm surf}(\Lambda) = E_{\rm surf}^{\rm (unr)}(\Lambda) + E_{\rm rel}(\Lambda).
\end{eqnarray}
%
Our calculated surface energy for the SrO termination then comes to
1.15\,eV, slightly smaller than the computed surface energy of 1.23\,eV
for the TiO$_2$ termination.  The results are summarized in
Table VII, where it can also be seen that we obtain slightly smaller
surface energies than those obtained from the classical shell model
\cite{heif} (1.32\,eV and 1.36\,eV for the SrO and TiO$_2$
terminations, respectively).

In order to calculate the SrTiO$_3$ (011) surface energies for
the TiO and Sr-terminated surfaces, we consider the cleavage
of six bulk unit cells (30 atoms) to result in the TiO- and
Sr-terminated slabs, containing 16 and 14 atoms respectively,
shown in Figs.~3(d-e).  We again divide the cleavage energy
equally between these two surfaces and obtain
%
\begin{eqnarray}
E_{\rm surf}^{\rm (unr)}(\Lambda)={\frac{1}{4}}
       [E_{\rm slab}^{\rm (unr)}({\rm Sr})
      + E_{\rm slab}^{\rm (unr)}({\rm TiO}) - 6E_{\rm bulk}],
\end{eqnarray}
%
where $\Lambda$ denotes Sr or TiO, $E_{\rm slab}^{\rm (unr)}(\Lambda)$
is the energy of the unrelaxed Sr or TiO terminated (011) slab,
and $E_{\rm bulk}$ is the SrTiO$_3$ energy per bulk unit cell.
Our calculated cleavage energy for the Sr- or TiO-terminated (011)
surface is 4.61\,eV.
Next, we calculated the relaxation energies
$E_{\rm rel}(\Lambda)$ using Eq.~(2) for each of the Sr- and
TiO-terminated surfaces, when both sides of slabs are allowed to relax.
According to the results of our calculations, the relaxation
energy of the Sr-terminated (011) surface is 1.95\,eV, while that of the
TiO-terminated surface is 1.55\,eV.  Thus, the surface relaxation energies
are roughly ten times larger for the (011) surfaces than for the (001)
surfaces.  The surface energies are then obtained from
Eq.~(3), and the results are again summarized in Table VII.

\begin{table}
\caption{Calculated cleavage, relaxation, and surface energies
for  SrTiO$_3$ (001) and (011) surfaces (in eV
per surface cell). SM indicated comparative results from the
shell-model calculation of Ref.~[\onlinecite{heif}]. In both cases, three
near-surface planes were relaxed.
}
\begin{ruledtabular}
\begin{tabular}{cccccc}
Surface        &  Termination & E$_{\rm cleav}$& E$_{\rm rel}$ 
               & E$_{\rm surf}$ & E$_{\rm surf}$(SM)  \\
\hline
SrTiO$_3$ (001)& TiO$_2$      & 1.39       & $-$0.16     & 1.23       & 1.36            \\
               & SrO          & 1.39       & $-$0.24     & 1.15       & 1.32            \\
SrTiO$_3$ (011)& TiO          & 4.61       & $-$1.55     & 3.06       & 2.21            \\
               & Sr           & 4.61       & $-$1.95     & 2.66       & 3.04           \\
               & O            & 3.36       & $-$1.32     & 2.04       & 1.54           \\
\end{tabular}
\end{ruledtabular}
\end{table}

Finally, when we cleave the SrTiO$_3$ crystal in another way,
we obtain identical O-terminated (011) surface slabs containing
15 atoms each.  This allows for us to simplify the
calculations, since the unit cell of the 7-plane O-terminated (011) slab
contains three bulk unit cells. Therefore, the relevant surface
energy is
%
\begin{eqnarray}
E_{\rm surf}({\rm O}) = {\frac{1}{2}}[E_{\rm slab}^{\rm (rel)}({\rm O})
   - 3E_{\rm bulk}],
\end{eqnarray}
%
where $E_{\rm surf}({\rm O})$ and $E_{\rm slab}^{\rm (rel)}({\rm O})$
are the surface energy and the
relaxed  slab total energy for the O-terminated (011) surface.
Table VII gives our calculated surface energies.

Unlike for the (001) surface,  we can see
that different terminations of the (011) surface lead to large
differences in the surface energies. Here, the lowest
energy, according to our hybrid B3PW calculations,
is 2.04\,eV for the O-terminated surface.
Our calculated surface energy of 3.06\,eV for the TiO-terminated
(011) surface is larger than that of the
Sr-terminated (011) surface (2.66\,eV).
Note that, according to the classical shell-model calculations,\cite{heif}
the O-terminated surface has the lowest energy (1.54\,eV) of all three
SrTiO$_3$ (011) surfaces. As we can see from Table VII, the O-terminated
(011) surface energy is comparable with the (001) surface energies,
both from our hybrid B3PW and from classical shell model \cite{heif}
calculations.


\subsection{SrTiO$_3$ (011) surface electronic structure}

\begin{table}
\caption{The A-B bond populations $P$ (in $e$) and the
relevant interatomic distances $R$ (in \AA) for three different
SrTiO$_3$ (011) terminations.  Symbols I-IV denote the number of
each plane enumerated from the surface.  The nearest-neighbor Ti-O
distance in unrelaxed bulk SrTiO$_3$ is 1.952\,\AA.
}
\begin{ruledtabular}
\begin{tabular}{lccc}
Atom A&Atom B&$P$&$R$\\
\hline
\multicolumn{4}{l}{TiO-terminated (011) surface} \\
~~~Ti(I)   & O(I)  &\pp 0.130& 2.001 \\
     &   O(II)  &\pp 0.188& 1.765 \\
~~~O(II)  &Ti(III)&\pp 0.110& 1.933 \\
     &   Sr(III)&\pp 0.000& 2.792 \\
     &   O(III) &\mm 0.024& 2.801 \\
~~~Ti(III)&Sr(III)&\pp 0.000& 3.382 \\
     &    O(III)&\pp 0.106& 1.955 \\
     &    O(IV) &\pp 0.080& 1.956 \\
~~~Sr(III)& O(III)&\mm 0.010& 2.760 \\
     &    O(IV) &\mm 0.014& 2.720 \\
~~~O(III) & O(IV) &\mm 0.028& 2.712 \\
\multicolumn{4}{l}{Sr-terminated (011) surface} \\
~~~Sr(I)  & O(II) &\mm 0.044& 2.534 \\
~~~O(II)  &Sr(III)&\mm 0.012& 2.775 \\
    &   Ti(III)&\pp 0.064& 1.981 \\
    &   O(III) &\mm 0.042& 2.802 \\
~~~Sr(III)&O(III) &\mm 0.012& 2.761 \\
    &   O(IV)  &\mm 0.010& 2.765 \\
~~~Ti(III)&O(III) &\pp 0.064& 1.952 \\
    &   Sr(III)&\pp 0.000& 3.381 \\
    &   O(IV)  &\pp 0.092& 1.951 \\
~~~O(III) &O(IV)  &\mm 0.048& 2.739 \\
\multicolumn{4}{l}{O-terminated (011) surface} \\
~~~O(I)   &Sr(II) &\mm 0.012& 2.641\\
    &   Ti(II) &\pp 0.146& 1.682\\
     &   O(II)  &\mm 0.026&2.755\\
~~~Sr(II) &O(II)  &\mm 0.042&2.546 \\
    &   Ti(II) &\pp 0.000&3.217 \\
~~~Ti(II)  &O(II)  &\pp 0.080&2.000 \\
     &   O(III) &\pp 0.100&1.776 \\
~~~O(II)  &O(III) &\pp 0.002&2.898 \\
~~~Sr(II)  &O(III) &\mm 0.008&2.778\\
~~~O(III)  &O(IV)  &\mm 0.036&2.787 \\
     &Ti(IV) &\pp 0.060&1.989 \\
     &Sr(IV) &\mm 0.016&2.705 \\
\end{tabular}
\end{ruledtabular}
\end{table}

The interatomic bond populations for the three possible SrTiO$_3$ (011) surface
terminations are given in Table VIII. The most important effect observed
here is a strong increase of the Ti-O chemical bonding near the surface
as compared to already large Ti-O bonding in the SrTiO$_3$ bulk.
The most significant  increase of the Ti-O chemical bonding occurs near the
TiO-terminated (011) surface (0.130$e$), which is much stronger,
than the relevant Ti-O chemical bonding value near the  TiO$_2$-terminated
(001) surface (0.118$e$), and in the bulk (0.088$e$).
For the O-terminated (011) surface, the O(I)-Ti(II) bond population
is even larger (0.146$e$).  The largest chemical
bond population we found is between the Ti(I) and O(II) atoms
in the surface-normal direction on the TiO-terminated (011) surface;
it is roughly 50\% larger than the
Ti-O bond population near the TiO$_2$-terminated
(001) surface, and slightly more than twice as large as the
Ti-O bond
population in the bulk.
From Table VIII we can see that for the TiO-terminated (011)
surface, the Ti(I)-O(II) bond populations in the direction
perpendicular to the surface (0.188$e$) are larger than those
of the T(I)-O(I) chemical bond populations (0.130$e$) in the
in-plane direction.

Table IX shows our calculated Mulliken effective charges $Q$, and
their changes $\Delta Q$ with respect to bulk SrTiO$_3$, for the
three (011) terminations.  The charge of the surface Ti atoms in the
TiO-terminated (011) surface is reduced by 0.14$e$. Metal atoms
in the third layer lose much less charge, with Sr and Ti atoms losing
0.028$e$ and 0.018$e$ respectively.  The
O ions in the first, second and third layer, except
the central one, also have charges that are reduced by 0.102$e$,
0.247$e$, and 0.074$e$, respectively (i.e., they become
less negative).  In contrast, the central-layer O ions slightly
increase their charges by 0.022$e$.
The largest change is observed for subsurface O atoms (0.247$e$),
which add up to contribute a large positive change of
0.494$e$ in the subsurface layer.

\begin{table}
\caption{Calculated Mulliken atomic charges $Q$ (in $e$) and
changes in atomic charges $\Delta $Q with respect to the bulk
charges (in $e$) for the
three  SrTiO$_3$ (011)
surface terminations. The  Mulliken atomic
charges in the SrTiO$_3$ bulk are
2.351$e$ for Ti,
$-$1.407$e$ for O, and 1.871$e$ for Sr.
}
\begin{ruledtabular}
\begin{tabular}{lcc}
Atom(layer)&$Q$&$\Delta Q$\\
\hline
\multicolumn{3}{l}{TiO-terminated (011) surface} \\
~~~Ti(I)&\pp 2.211& \mm 0.140 \\
~~~O(I)&\mm 1.305&\pp 0.102\\
~~~O(II)&\mm 1.160&\pp 0.247\\
~~~Sr(III)&\pp 1.843&\mm 0.028\\
~~~Ti(III)&\pp 2.333&\mm 0.018\\
~~~O(III)&\mm 1.333&\pp 0.074\\
~~~O(IV)&\mm 1.429&\mm 0.022\\
\multicolumn{3}{l}{Sr-terminated (011) surface} \\
~~~Sr(I)&\pp 1.766&\mm 0.105\\
~~~O(II)&\mm 1.560&\mm 0.153\\
~~~Sr(III)&\pp 1.874&\pp 0.003\\
~~~Ti(III)&\pp 2.362&\pp 0.011\\
~~~O(III)&\mm 1.486&\mm 0.079\\
~~~O(IV)&\mm 1.396&\pp 0.011\\
\multicolumn{3}{l}{O-terminated (011) surface} \\
~~~O(I)&\mm 1.172&\pp 0.235\\
~~~Sr(II)&\pp 1.851&\mm 0.020\\
~~~Ti(II)&\pp 2.240&\mm 0.111\\
~~~O(II)&\mm 1.461&\mm 0.054\\
~~~O(III)&\mm 1.394&\pp 0.013\\
~~~Sr(IV)&\pp 1.867&\mm 0.004\\
~~~Ti(IV)&\pp 2.332&\mm 0.019\\
~~~O(IV)&\mm 1.433&\mm 0.026\\
\end{tabular}
\end{ruledtabular}
\end{table}

In the case of the Sr-terminated (011) surface, negative changes
in the charge are observed for all atoms except for oxygen in the
central layer and metal atoms in the third layer. The largest
changes occur for the subsurface O ion ($-$0.153$e$) and for the
surface Sr ion ($-$0.105$e$).  For the O-terminated (011) surface,
the negative charge on the surface oxygen is decreased ($\Delta Q$
= 0.235$e$). The net charge change in the second layer is negative
($-$0.185$e$) and comes mostly from the Ti ion ($-$0.111$e$).  The O
ion charge of the third layer is almost unchanged ($\Delta Q$
= 0.013$e$).  The net charge change of the central-layer atoms
is again negative, but about four times smaller ($-$0.049$e$) than
for the second layer, and now comes mostly from the charge change
on the O ion ($-$0.026$e$).


\section{Conclusions}

According to the results of our {\it ab initio} hybrid B3PW calculations,
all upper layer atoms for the TiO$_2$- and SrO-terminated
SrTiO$_3$ (001) surface, with the exception of the O atoms on the
SrO-terminated surface, relax inwards, whereas all second-layer atoms
for both terminations
relax outwards. The inward displacement of the Sr on
the SrO-terminated surface is about twice as large as that of the
Ti atom on the TiO$_2$-terminated surface.  Our computed
surface rumpling for the SrO-terminated (001) surface
is much larger than of the TiO$_2$-terminated (001) surface, and is in
excellent agreement with
LEED \cite{bic} and RHEED \cite{hik} experiment results.
Our calculations predict a compression of the distance between the
first and second planes and an expansion for the second and third planes,
in agreement with the results of previous {\it ab initio} and shell
model calculations.
Our calculations, as well as all previous {\it ab initio} and
shell-model calculations, agree
with the LEED experiments regarding the compression of the distance
between the first and second planes for the SrO-terminated
surface, but disagree with RHEED experiments. For the TiO$_2$-terminated
surface, just the opposite is the case: our calculations and all
previous {\it ab initio} and shell-model calculations
agree with the RHEED\cite{hik} experiments regarding the sign
of the interlayer relaxation between the second and third
planes, but disagree with LEED results.\cite{bic} The reason
for this discrepancy is not clear, but it is discussed in
Refs.~[\onlinecite{pad}] and [\onlinecite{char}].
Thus, we conclude that in some cases,
the disagreement between theoretical and experimental results
should not be taken too seriously until the conflict between
different experimental results is resolved.

For the SrTiO$_3$ (011) surface, we found that the relaxation magnitudes
for the upper-layer metal atoms are considerably larger on the Sr and
TiO-terminated surfaces than they are on the (001) surface
upper-layer atoms.  Whereas the metal atoms on the Sr- and TiO-terminated
(011) surface relax strongly inwards, the upper-layer oxygen atoms
on the TiO-terminated
(011) surface relax outwards by 3.59$\%$ of the lattice
constant $a$. The atomic displacements in the third plane from the surface
for the three (011) terminations are still large.
This is in sharp contrast contrast with our results for the (001)
surfaces, where the atomic displacements converge very quickly and are already
small in the third layer. Our calculated surface rumpling for the
TiO-terminated (011) surface is considerably larger than that of the
SrO- or TiO$_2$ terminated (001) surfaces.

The SrO- and TiO$_2$-terminated SrTiO$_3$ (001) surfaces have
comparable but small relaxation energies, so that the surface
energies assigned to them are similar (1.15\,eV and 1.23\,eV,
respectively).  On the other hand, the
different terminations of the (011) surface have large cleavage
and relaxation energies, and large differences in the
surface energies. The O-terminated surface has the lowest surface
energy of the three (011) surfaces, although at 2.04\,eV it is
still significantly higher that that of the (001) terminations.
The calculated surface energies of Sr-terminated (2.66\,eV) and
TiO-terminated (3.06\,eV) (011) surfaces are both more than twice
as large as those of the (001) surfaces.

Our {\it ab initio} calculations indicate a considerable increase
of the Ti-O bond covalency near the (011)
surface relative to the bulk, much larger
than that of the (001) surface. The Ti-O bond
populations are larger in the direction perpendicular to the
TiO-terminated (011) surface than in plane.

% This is a crude but successway way to force some white space aove
% the "Acknowledgments" header:
%
$\phantom{a}$

\acknowledgments

The present work was supported by
Deutsche Forschungsgemeinschaft (DFG)
and by ONR Grant No. N00014-05-1-0054.

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\end{thebibliography}

\end{document}
