\documentstyle[floats,aps]{revtex}

\begin{document}
\draft
\preprint{August 4, 1997}

\twocolumn[\hsize\textwidth\columnwidth\hsize\csname
@twocolumnfalse\endcsname

\title{First-principles theory of structural phase transitions
in cubic perovskites}
\author{David Vanderbilt}
\address{Department of Physics and Astronomy, Rutgers University,
Piscataway, New Jersey 08855-0849}

\date{August 4, 1997}
\maketitle

\begin{abstract}
%
A review is given of recent developments in which first-principles
electronic-structure calculations have been used to determine, with
chemical specificity, the structural properties of cubic perovskite
materials.  Direct application of these methods can be used to
determine the ground-state (zero-temperature) distorted structure,
the lattice vibrational properties in harmonic and anharmonic
order, and dielectric properties such as Born effective charges
and piezoelectric constants.  By the indirect construction of model
effective Hamiltonians and the fitting of these to the
first-principles results, one can obtain (in most cases) the
correct sequence of ferroelectric and/or antiferrodistortive
structural phase transitions, with predicted transition
temperatures typically within 30\% of the experimental ones.
Prospects for the application of these approaches to solid-solution
ferroelectrics, to the study of finite-temperature piezoelectric
properties, and to ferroelectrics in confined geometries, are
briefly discussed.
%
\end{abstract}

\vskip1pc]

%--------------------------------------------------
\section{Introduction}
%--------------------------------------------------

It is clearly of great importance to develop a fundamental
theoretical understanding of the intrinsic physical properties
of the ferroelectric perovskite materials.  The experimental and
theoretical study of these materials, and of their structural
phase transitions, has a long history; much of this is reviewed
in several standard reference works \cite{mitsui,lg,smolenski}.
However, virtually all of the early theoretical work was of an
empirical character.  Typically, a Landau model, spin model, or
empirical lattice-dynamical model is fitted to reproduce certain
experimental features, and then used to predict new features or
systematize the experimental results.  While this has been an
extremely valuable approach, the development of computational
methods in the electronic-structure community has made possible
the application of a new class of first-principles approaches.
These methods
are based upon a full solution for the quantum-mechanical ground
state of the electron system within the local-density approximation
(LDA) to Kohn-Sham density-functional theory (DFT) \cite{ks},
and in principle they take as their only inputs the atomic numbers
of the atomic constituents.  Such methods hold the promise for
providing chemically-specific information and understanding about
the structural and electronic properties of the various perovskite
materials.  In fact, the work of recent years has confirmed that
this is the case, showing that many details of the lattice dynamics
and dielectric properties, the sequence of structural phase
transitions, and even the transition temperatures can be predicted
using first-principles calculations directly, or using models
based on such calculations.

The purpose of the present manuscript is to review the development
and application of first-principles calculations for perovskite
ferroelectrics and related materials.  The discussion will focus
first on properties that can be computed directly from the
first-principles calculations, and then on indirect calculations
carried out using effective Hamiltonians that have been fitted
to a first-principles database.  The main emphasis is on the
intrinsic properties of pure bulk materials, but in the last
section some discussion will be given regarding the prospects for
application of the methods to some ``real-world'' complexities:
solid solutions, ferroelectric domain boundaries, surfaces,
and piezoelectric response.


%--------------------------------------------------
\section{Direct first-principles calculations}
%--------------------------------------------------

The simplest direct applications of the first-principles approach
involve computing the total energies, and ideally also the forces
on each atom, as a function of the atomic coordinates as the system
undergoes distortions from some reference cubic perovskite
structure.  Such distortions include displacements of one of the
ionic sublattices, linear combinations of these (``frozen
phonons''), or variations of lattice constant and other lattice
strains.  By mapping out the energy landscape using such an
approach, it is possible to check for instabilities and identify
the distorted ground-state structure (although strictly only at
$T=0$ and for a classical treatment of the ionic coordinates).

LDA-DFT calculations of this type were first carried out for FE
perovskites using the linear augmented plane-wave (LAPW) method
\cite{coh-krak,coh-nature,singh-boyer,coh-krak-2,resta-knb,%
      poster-early,singh-pbzr,singh-kta,singh-wda,cohen-li}
and shortly afterwards by ultrasoft pseudopotential (USPP)
\cite{ks-fer,ks-pr,gar-pr},
linear muffin-tin orbital (LMTO)
\cite{post-knb,post-ph1,post-ph2,post-phr},
and norm-conserving pseudopotential (NCPP) methods.
The LAPW and LMTO methods use radial basis functions inside
atom-centered spheres and other basis functions outside, while the
pseudopotential methods use a plane-wave basis throughout; the
former methods are (at least in principle) all-electron methods,
while the latter approach facilitates the efficient and accurate
calculation of forces.  Generally the experience has been that the
LAPW and USPP calculations are of high quality and are in very good
agreement with each other where they have been carefully compared,
and both are probably making errors that are small
compared to the intrinsic errors of LDA-DFT.

These calculations quickly proved their applicability to this class
of compounds by confirming the existence of FE instabilities for
such prototype materials as BaTiO$_3$
\cite{coh-krak,coh-nature,coh-krak-2,ks-fer,ks-pr},
KNbO$_3$ \cite{singh-boyer,resta-knb,poster-early,singh-wda,ks-pr,%
post-knb,post-ph1,post-ph2,post-phr},
and PbTiO$_3$ \cite{coh-nature,coh-krak-2,ks-pr,gar-pr},
and successfully predicted the correct symmetry of the
ground-state structure in each case.  Recent work has extended
these studies to a wider variety of materials, including
KTaO$_3$ \cite{singh-kta,singh-wda,post-knb,post-ph1},
PbZrO$_3$ \cite{singh-wda,ks-pr,rabe-pbzr},
LiNbO$_3$ \cite{cohen-li}, LiTaO$_3$ \cite{cohen-li},
and others \cite{ks-pr}.

Generally, the agreement between the first-principles theory and
experiment has been good, but two limitations should be noted.
First, DFT being a theory of the electronic ground state only,
one finds that band gaps of virtually all
insulators are seriously underestimated, by up to $\sim$50\%.
(This is an intrinsic limitation even in exact DFT theory, and is
not arising from the LDA.)  Thus, band gaps and optical
properties cannot be taken directly from the DFT-LDA theory.
Second, it is well known that, as a result of the LDA, the theory
tends to underestimate the equilibrium lattice constants
by roughly 1\% (see, e.g., Ref.\ \onlinecite{ks-pr}).  As a result,
quantities that depend sensitively on lattice constant, such as the
ground-state distortion amplitude or the spontaneous polarization,
can be seriously in error if one works at the theoretical equilibrium
lattice constant.  Generally, the experience of the above works is
that the results are more accurate if one works instead at the
experimental lattice constant.  This will be discussed further at
the end of the next section.

The long-range Coulomb interaction obviously plays a crucial role
in ferroelectric materials.  It is therefore critical to be able
to calculate the spontaneous (i.e., zero-field) electric polarization
$P$ as a function of structural degrees of freedom.  Recently it has
been understood how to calculate $P$ directly as a Berry phase of
the Bloch states \cite{ksv,resta-rmp}, and when this calculation
is carried out for the experimentally determined ground-state
structure, one obtains satisfactory agreement with the experimental
spontaneous polarization \cite{z-lo}.  As for the dependence of
$P$ upon structure, the first-order variation of $P$ with atomic
displacements is given by the dynamical effective charge
$Z^*$, which can be computed either directly using linear-response
methods \cite{baroni}, or numerically by finite differences.
The latter method was applied to KNbO$_3$ by the authors of
Ref.\ \cite{resta-knb}, with the remarkable result that the
$Z^*=9.13$ for Nb is significantly larger than its nominal ionic
value $Z=5$ (with a corresponding anomalous $Z^*$ for oxygen motion
along the Nb--O bond).  This pattern of anomalous $Z^*$ values was
found to be a universal feature of virtually all of the
ferroelectric perovskites \cite{z-lo}, and its origin was traced to
the borderline ionic-covalent character and specifically to the
hybridization between O $2p$ and B-atom $3d$ or $4d$ orbitals in
the ABO$_3$ material \cite{resta-knb,poster-early,z-lo}.
Subsequent work has more carefully characterized these anomalous
$Z^*$ values, especially their dependence upon structural
parameters and their band-by-band decomposition
\cite{gonze-z-ba,ghosez-wo3,resta-bigz,krak-zdep,yu-zstar}.
Such investigations have also provided insight into the origin of
the FE instability and its relation to the long-range Coulomb
interactions \cite{ghosez-decomp} and to
many-body electron correlation effects
\cite{resta-corr}.

Having characterized the strength of the long-range Coulomb
interactions through the $Z^*$'s, one can move on to a study of
the lattice dynamics.  Actually, the transverse optic (TO) modes
at $\Gamma$ can be computed without $Z^*$; some early
frozen-phonon studies
\cite{coh-krak,singh-boyer,coh-krak-2,singh-kta,singh-wda,ks-pr,post-ph1,%
post-ph2,post-phr}
were able to identify the eigenvectors and frequencies of the TO modes,
and especially to provide detailed knowledge of the FE soft mode.
With the additional knowledge of the $Z^*$ values, it is then fairly
straightforward to obtain also the corresponding longitudinal
optic (LO) modes at $\Gamma$ \cite{z-lo}.

While zone-boundary phonons and phonons at other rational wavevectors
are accessible in principle by applying the frozen-phonon approach
to supercells, linear-response methods \cite{baroni} are the
methods of choice for computing phonon properties on a mesh of
wavevectors.  Developed in the NCPP context \cite{baroni},
this approach was then extended to LAPW methods by
Krakauer and coworkers \cite{krak-lr}.  In most cases, the analysis
of the phonon spectrum in perovskites has been carried out in
the reference ideal cubic structure, so that the FE or other
instabilities show up as imaginary phonon frequencies.  For
example, applying such an approach to KNbO$_3$ \cite{yu-zstar,krak-knbo3}
one finds that the portion of the Brillouin zone at which imaginary
frequencies (potential FE instabilities) occur comprises a set
of three ``sheets'' surrounding the $x-y$, $x-z$, and $y-z$ planes,
consistent with a picture in which the FE dipoles associated with each
unit cell are strongly correlated in longitudinal chains along the
Cartesian axes.  Similar results have been reported for the similar
material BaTiO$_3$ \cite{ghosez-ba}.
On the other hand, for PbZrO$_3$ \cite{wagh-thesis}
and SrTiO$_3$ \cite{lasota-sr}, the same approach demonstrates that
both FE and antiferrodistortive instabilities are simultaneously
present in the cubic reference structure, and nicely characterizes
the portions of the Brillouin zone where each kind of instability
is strongest.

Of course it is also of interest to
first identify the $T=0$ ground state structure by a careful
total-energy minimization, and then apply the linear-response
calculation to study the phonons in the ground state.
A good example of this kind is a recent study of the rhombohedral
ground-state structure of KNbO$_3$ \cite{krak-zdep}.

%--------------------------------------------------
\section{Indirect modeling via effective Hamiltonians}
%--------------------------------------------------

The methods discussed so far are essentially restricted to study
the zero-temperature properties of the materials.  Clearly it is
of the first importance to see whether one can understand such
features as the phase transition sequence and transition temperatures
on a material-specific basis.  A very successful approach to this
problem has been the effective Hamiltonian approach
\cite{rabe-heff,rabe-lattwann,z-ba}.  Here, one defines a reduced
number of degrees of freedom per unit cell (typically, a FE mode
vector and a displacement vector in each unit cell), and constructs
a model Hamiltonian, written as a function of these reduced degrees
of freedom, that reproduces the spectrum of low-energy excitations
(FE soft modes and strains) for the given material as obtained
from the ab-initio DFT-LDA calculations.  One then arrives at a
model, typically containing 10-20 parameters, that can be subjected
to Monte Carlo (MC)
\cite{z-ba,z-sr,vz-long,rabe-pbzr,rabe-mc}
or molecular dynamics (MD) \cite{krak-md}
simulations in order
to determine the finite-temperature properties of the material.

This approach has been applied with considerable success to the
FE materials BaTiO$_3$ \cite{z-ba}, PbTiO$_3$ \cite{rabe-mc},
and KNbO$_3$ \cite{krak-md}, with the theory correctly reproducing the
experimental phase transition sequence in each case, as summarized
in Table I.  It can be seen that the theoretical values tend to
be too low, typically by 20\% or so, but in the worst case by
almost 50\%.  Given that the calculations are based almost entirely
on first principles, taking no input from experiment (except for the
experimental lattice constant), this is overall a satisfying level
of agreement.  Moreover, the calculations give other useful information
about the nature (soft-mode vs.\ order-disorder) of the transitions
and their latent heats, and the correlations, fluctuations, and dynamics
\cite{krak-md} in the various phases.  They can also be helpful in
identifying the role of strain coupling in the FE transitions
\cite{rabe-strain}.

For more complex materials such as SrTiO$_3$
\cite{z-sr,vz-long}, PbZrO$_3$ \cite{rabe-pbzr}, CaTiO$_3$ \cite{vz-long},
and NaNbO$_3$ \cite{vz-long}, one has antiferrodistortive instabilities
competing with the FE ones.  In SrTiO$_3$ one has reasonable agreement
(130K theory vs.\ 105K experiment) for the cubic-to-tetragonal
antiferrodistortive transition, but the simulation also shows FE phases
below 70K \cite{z-sr,vz-long}; however, when quantum fluctuations are
taken into account, the former transition changes to 110K, and the
FE phases disappear \cite{z-qfluct}, in excellent agreement with
experiment.  In the more complex materials PbZrO$_3$, CaTiO$_3$,
and NaNbO$_3$, the behavior is more complicated, and the
predicted sequence of phase transitions does not always exactly follow
the experimental one \cite{rabe-pbzr,vz-long}.  Further work is
needed to understand where the limitations of the theory lie, e.g.,
whether it is the LDA approximation to DFT itself that is
inadequate, or whether the formulation of the effective Hamiltonian
was oversimplified.

\begin{table}
\caption{Comparison of theoretical and experimental ferroelectric
transition temperatures (Kelvin). Symbols C, T, O and R refer to cubic,
tetragonal, orthorhombic, and rhombohedral phases, respectively.}
\begin{tabular}{llcc}
Material  & Transition & $T_{\rm c}$ (theo.) & $T_{\rm c}$ (expt.)  \\
\tableline
BaTiO$_3$ (Ref.\ \onlinecite{z-ba})
          & C--T & 290 & 403 \\
          & T--O & 230 & 278 \\
          & O--R & 197 & 183 \\
% \tableline
KNbO$_3$ (Ref.\ \onlinecite{krak-md})
          & C--T & 370 & 710 \\
          & T--O & 260 & 488 \\
          & O--R & 210 & 210 \\
% \tableline
PbTiO$_3$ (Ref.\ \onlinecite{rabe-mc})
          & C--T & 660 & 763 \\
\end{tabular}
\label{tab:bulk_par}
\end{table}

As mentioned earlier, the theories tend to underestimate the lattice
constants with respect to experiment by approximately 1\%.  In the
above works, this was corrected by working at a negative fictitious
pressure \cite{z-ba}, by artificially dropping the term linear in
strain in the effective Hamiltonian \cite{rabe-mc}, or by a
fortuitous choice of exchange-correlation potential
\cite{krak-md}.  These effects are not small; in BaTiO$_3$, e.g.,
the C--T transition temperature falls to $\sim$160K if the negative
fictitious pressure is removed \cite{z-ba}.  The necessity of fixing
the lattice constant with reference to experiment
clearly constitutes an unsatisfying retreat from the
goal of a completely first-principles approach.  In a very
encouraging development, Singh has recently
shown that use of the the weighted-density approximation
(WDA) in place of the LDA appears to give good systematic improvement
at the expense of only modest computational cost
\cite{singh-wda}.

%--------------------------------------------------
\section{Prospects}
%--------------------------------------------------

In the coming years, the first-principles theories will increasingly
need to address some of the complexities associated with applications
of FE materials.  For example, the behavior of the material
under conditions of applied electric fields and strains is often of
primary importance.  Thus, the study of the piezoelectric properties
of these materials is now beginning, with a recent calculation on
PbTiO$_3$ illustrating one way to proceed \cite{saghi-piezo}.  Many of
the materials of interest for applications are solid solutions,
e.g., PZT (PbZr$_{1-x}$Ti$_x$O$_3$), BST (Ba$_{1-x}$Sr$_x$TiO$_3$),
and PMN (PbMn$_{1/3}$Nb$_{2/3}$O$_3$). Studies of the cation ordering
and its consequences in such compounds
are just beginning \cite{saghi-ord,rappe-pzt},
and eventually the program of construction of an effective
Hamiltonian and its use in MC or MD simulations will have to
be carried through for such materials.  Finally, the
real materials are frequently ceramics or films, and a critical
role is often played by boundaries and defects of various kinds,
including grain boundaries, surfaces, interfaces, contacts,
and FE domain walls.  The 180$^\circ$ domain
wall in BaTiO$_3$ has recently been studied using the
effective-Hamiltonian approach \cite{pad-dw}, and some first
studies of BaTiO$_3$ free surfaces have also appeared
\cite{coh-surf,pad-surf}.

Clearly there is a great deal that remains to be done.  In the
real world, the material properties are frequently dominated by
effects that have not yet even been mentioned above.  For example,
in order to understand hysteresis and fatigue effects, much more
knowledge is needed regarding the role of grain boundaries,
dislocations, impurities, oxygen vacancies, stoichiometric
variations, electronic carriers, pinning of domain walls, etc.
While these are certainly hard problems, it can be expected that
clever and judicious applications of first-principles methods
will contribute to solving many of these problems over the coming
years.

%--------------------------------------------------
\begin{references}
%--------------------------------------------------

% Standard books
% --------------

\bibitem{mitsui}
T. Mitsui, I. Tatsuzaki and E. Nakamura,
{\sl An Introduction to the Physics of Ferroelectrics,}
Gordon and Breach, New York, 1976.

\bibitem{lg}
M.E. Lines and A.M. Glass,
{\sl Principles and Applications of Ferroelectrics and Related Materials,}
Clarendon Press, Oxford, 1977.

\bibitem{smolenski}
G.A. Smolenski, editor,
{\sl Ferroelectrics and Related Materials,}
Gordon and Breach, New York, 1984.

% LDA-DFT
% -------

\bibitem{ks}
W. Kohn and L.J. Sham,
% ``Self-consistent equations including exchange and correlation effects,''
Phys. Rev. {\bf 140}, A1133 (1965).

% ----------------------------------------------
% Total-energy, force, and electronic structure of perov
% ferroelectrics, including frozen phonons
% ----------------------------------------------

% LAPW
% ----

\bibitem{coh-krak} 
R.E. Cohen and H. Krakauer,
% ``Lattice dynamics and origin of ferroelectricity in BaTiO3: Linearized
% augmented plane wave total energy calculations,''
Phys. Rev. B {\bf 42}, 6416 (1990).

\bibitem{coh-nature}
R.E. Cohen,
% ``Origin of ferroelectricity in oxide ferroelectrics and the
% difference in ferroelectric behavior of BaTiO3 and PbTiO3,''
Nature {\bf 358}, 136 (1992).

\bibitem{singh-boyer}
D.J. Singh and L.L. Boyer,
% ``First principles analysis of vibrational modes in KNbO$_3$,''
Ferroelectrics {\bf 136}, 95 (1992).

\bibitem{coh-krak-2}
R.E. Cohen and H. Krakauer,
% ``Electronic structure studies of the differences in ferroelectric
% behavior of BaTiO3 and PbTiO3,''
Ferroelectrics {\bf 136}, 65 (1992).

\bibitem{resta-knb}
R. Resta, M. Posternak, and A. Baldereschi,
% ``Towards a quantum theory of polarization in ferroelectrics:
% The case of KNbO$_3$,''
Phys. Rev. Lett. {\bf 70}, 1010 (1993).

\bibitem{poster-early}
M. Posternak, R. Resta, and A. Baldereschi,
% ``Role of covalent bonding in the polarization of perovskite oxides:
% The case of KNbO$_3$,''
Phys. Rev. B {\bf 50}, 8911 (1994).

\bibitem{singh-pbzr}
D.J. Singh,
% ``Structure and energetics of PbZrO3,''
Phys. Rev. B {\bf 52}, 12559 (1995).

\bibitem{singh-kta}
D.J. Singh,
% ``Stability and phonons in KTaO$_3$,''
Phys. Rev. B {\bf 53}, 176 (1996).

\bibitem{singh-wda}
D.J. Singh,
% ``Density functional studies of PbZrO3, KTaO3, and KNbO3,''
Ferroelectrics {\bf 194}, 299 (1997).

\bibitem{cohen-li}
I. Inbar and R.E. Cohen,
% ``Origin of ferroelectricity in LiNbO3 and LiTaO3,''
Ferroelectrics {\bf 194}, 83 (1997).

% USPP
% ----

\bibitem{ks-fer}
R.D. King-Smith and D. Vanderbilt,
% ``A first-principles pseudopotential investigation of ferroelectricity
% in barium titanate,''
Ferroelectrics {\bf 136}, 85 (1992).

\bibitem{ks-pr}
R.D. King-Smith and D. Vanderbilt,
% ``First-principles investigation of ferroelectricity
% in perovskite compounds,''
Phys. Rev. B {\bf 49}, 5828 (1994).

\bibitem{gar-pr}
A. Garc\'{\i}a and D. Vanderbilt,
% ``First-principles study of stability and vibrational
% properties of tetragonal PbTiO$_3$,''
Phys. Rev. B {\bf 54}, 3817 (1996).

% LMTO
% ----

\bibitem{post-knb}
A.V. Postnikov, T. Neumann, G. Borstel, and M. Methfessel,
% ``Ferroelectric structure of KNbO$_3$ and KTaO$_3$
% from first-principles,''
Phys. Rev. B {\bf 48}, 5910 (1993).

\bibitem{post-ph1}
A.V. Postnikov, T. Neumann, and G. Borstel,
% ``Phonon properties of KNbO$_3$ and KTaO$_3$ from first-principles
% calculations,''
Phys. Rev. B {\bf 50}, 758 (1994).

\bibitem{post-ph2}
A.V. Postnikov and G.Borstel,
% ``$\Gamma$ phonons and microscopic structure of orthorombic KNbO$_3$
% from first-principles calculations,''
Phys. Rev. B  {\bf 50}, 16403 (1994).

\bibitem{post-phr}
A.V.Postnikov, G.Borstel,
% ``Refined geometry and frozen phonons in KNbO$_3$,''
Ferroelectrics {\bf 194}, 69 (1997).

\bibitem{rabe-pbzr}
U.V. Waghmare and K.M. Rabe,
% ``Lattice Instabilities, anharmonicity and phase transitions
% in PbZrO3 from first principles,''
Ferroelectrics {\bf 194}, 135 (1997).

% ----------------------------------------------
% polarization
% ----------------------------------------------

\bibitem{ksv}
R.D. King-Smith and D. Vanderbilt,
% ``Theory of polarization of crystalline solids,''
Phys. Rev. B {\bf 47}, 1651 (1993);
%
% \bibitem{vks}
D. Vanderbilt and R.D. King-Smith,
% % ``Electric Polarization as a Bulk Quantity and its Relation
% to Surface Charge,''
Phys. Rev. B {\bf 48}, 4442 (1993).

\bibitem{resta-rmp}
R. Resta, Rev. Mod. Phys. {\bf 66}, 899 (1994).

\bibitem{z-lo}
W. Zhong, R.D. King-Smith, and D. Vanderbilt,
% ``Giant LO--TO splittings in perovskite ferroelectrics,''
Phys. Rev. Letters {\bf 72}, 3618 (1994).

\bibitem{baroni}
S. Baroni, P. Giannozzi, and A. Testa,
% ``Green's-function approach to linear response in solids,''
Phys. Rev. Lett. {\bf 58}, 1861 (1987);
P. Giannozzi, S. de Gironcoli, P. Pavone, and S. Baroni,
% ``Ab-initio calculation of phonon dispersion in semiconductors,''
Phys. Rev. B {\bf 43}, 7231 (1991).

% ----------------------------------------------
% Focus on Z*
% ----------------------------------------------

\bibitem{gonze-z-ba}
Ph. Ghosez, X. Gonze and J.-P. Michenaud,
% ``First principle calculations of dielectric and effective charge tensors in
% barium titanate,''
Ferroelectrics {\bf 153}, 91 (1994);
%
% \bibitem{gonze-z-ba2}
Ph. Ghosez, X. Gonze, Ph. Lambin and J.-P. Michenaud,
% ``Born effective charges of barium titanate : band by band decomposition and
% sensitivity to structural features,''
Phys. Rev. B {\bf 51}, 6765 (1995).

\bibitem{ghosez-wo3}
F. Detraux, Ph. Ghosez and X. Gonze,
% ``Giant born effective charges in WO3,''
Phys. Rev. B {\bf 56}, 983 (1997).

\bibitem{resta-bigz}
R. Resta, M. Posternak, and A. Baldereschi,
% ``Quantum mechanism of polarization in perovskites,''
Ferroelectrics {\bf 164}, 153 (1995).

\bibitem{krak-zdep}
C.-Z. Wang, R. Yu and H. Krakauer,
% ``Polarization dependence of Born effective charge and
% dielectric constant in KNbO3,''
Phys. Rev. B {\bf 54}, 11161 (1996).

\bibitem{yu-zstar}
C.-Z. Wang, R. Yu, and H. Krakauer,
% ``Born effective charges, dielectric constants, and lattice
% dynamics of KNbO$_3$,''
Ferroelectrics {\bf 194}, 97 (1997).

% ----------------------------------------------
% Origins of FE: SR-LR decomp
% ----------------------------------------------

\bibitem{ghosez-decomp}
%
% \bibitem{ghosez-decomp-1}
Ph. Ghosez, X. Gonze and J.-P. Michenaud,
% ``Coulomb interaction and ferroelectric instability of BaTiO3,''
Europhys. Letters {\bf 33}, 713 (1996);
%
% \bibitem{ghosez-decomp-2}
% Ph. Ghosez, X. Gonze and J.-P. Michenaud,
% ``Role of the Coulomb interaction on the ferroelectric instability of Barium
% Titanate,''
Ferroelectrics {\bf 186}, 73 (1996).

% ----------------------------------------------
% Origins of FE: Correlation effects
% ----------------------------------------------

\bibitem{resta-corr}
R. Resta and S. Sorella,
% ``Many-body effects on polarization and dynamical charges in a partly
% covalent polar insulator,''
Phys. Rev. Lett. {\bf 74}, 4738 (1995);
%
% \bibitem{resta-corr-fer}
R. Resta,
% ``Role of covalence and of correlation in the dielectric polarization
% of oxides,''
Ferroelectrics {\bf 194}, 1 (1997).

% ----------------------------------------------
% Phonons from linear response
% ----------------------------------------------

\bibitem{krak-lr}
R. Yu and H. Krakauer,
% ``Linear response calculations of lattice dynamics within the
% linearized augmented plane wave method,''
Phys. Rev. B {\bf 49}, 4467 (1994);
%
% \bibitem{krak-lapwlr}
H. Krakauer, R. Yu, and C.-Z. Wang,
% ``$Ab$ $initio$ LAPW Linear Response Method for Ferroelectrics,''
Ferroelectrics {\bf 151}, 39 (1994).

% KNBO3

\bibitem{krak-knbo3}
%
% \bibitem{krak-chains}
R. Yu and H. Krakauer,
% ``First-principles determination of chain-structure instability in
% KNbO3,''
Phys. Rev. Lett. {\bf 74}, 4067 (1995);
%
% \bibitem{yu-knbo3}
R. Yu, C.-Z. Wang, and H. Krakauer,
% ``Lattice dynamics of ferroelectrics using the LAPW linear response method:
% Application to KNbO3,''
Ferroelectrics {\bf 164}, 161 (1995);
%
% \bibitem{krak-knbo3}
H. Krakauer, R. Yu, and C.-Z. Wang,
% ``Wavevector dependent instabilities in KNbO3,''
J. Phys. Chem. Solids {\bf 57}, 1409 (1996).

% BATIO3

\bibitem{ghosez-ba}
%
% \bibitem{ghosez-ba-instabil}
Ph. Ghosez, X. Gonze and J.-P. Michenaud,
% ``Lattice dynamics and ferroelectric instability of Barium Titanate,''
Ferroelectrics {\bf 194}, 39 (1997);
%
% \bibitem{ghosez-ba-fer}
% Ph. Ghosez, X. Gonze and J.-P. Michenaud,
% ``Ab initio phonon dispersion curves and interatomic force constants of
% barium titanate,''
% Ferroelectrics, in press.
ibid., in press.

% PBZRO3

\bibitem{wagh-thesis}
U.V. Waghmare,
{\sl Ab initio statistical mechanics of structural phase transitions,}
PhD Thesis, Department of Applied Physics, Yale University, 1996.

% PBTIO3

\bibitem{lasota-sr}
C. LaSota, C.-Z. Wang, R. Yu and H. Krakauer,
% ``Ab initio linear response study of SrTiO$_3$,''
Ferroelectrics {\bf 194}, 109 (1997).

% ----------------------------------------------
% MC or MD based on $H_{\rm eff}$
% ----------------------------------------------

\bibitem{rabe-heff}
K.M. Rabe and U.V. Waghmare,
% ``First-principles model Hamiltonians for ferroelectric
% phase transitions,''
Ferroelectrics {\bf 136}, 147 (1992).

\bibitem{rabe-lattwann}
K.M. Rabe and U.V. Waghmare,
% ``Localized basis for effective
% lattice Hamiltonians: Lattice Wannier functions,''
Phys. Rev. B {\bf 52}, 13236 (1995).

\bibitem{z-ba}
W. Zhong, D. Vanderbilt, and K.M. Rabe,
% ``Phase transitions in BaTiO$_3$ from first principles,''
Phys. Rev. Lett. {\bf 73}, 1861 (1994);
%
% \bibitem{z-prb}
% W. Zhong, D. Vanderbilt, and K.M. Rabe,
% ``First-principles theory of ferroelectric phase transitions for perovskite
% compounds: The case of BaTiO$_3$,''
Phys. Rev. B {\bf 52}, 6301 (1995).

\bibitem{z-sr}
W. Zhong and D. Vanderbilt,
% ``Competing structural instabilities in cubic perovskites,''
Phys. Rev. Lett. {\bf 74}, 2587 (1995).

\bibitem{vz-long}
D. Vanderbilt and W. Zhong,
% ``First-principles theory of structural phase transitions for
% perovskites: Competing instabilities,''
Ferroelectrics, in press.

\bibitem{rabe-mc}
%
% \bibitem{rabe-mc-proc}
K.M. Rabe and U.V. Waghmare,
% ``Ferroelectric phase transitions from first principles,''
J. Phys. Chem. Sol. {\bf 57}, 1397 (1996);
% 
% \bibitem{rabe-mc}
U.V. Waghmare and K.M. Rabe,
% ``Ab initio statistical mechanics of the
% ferroelectric phase transition in PbTiO3,''
Phys. Rev. B {\bf 55}, 6161 (1997).

\bibitem{krak-md}
H. Krakauer, R. Yu, C.-Z. Wang and C. LaSota,
% ``Precursor structures in ferroelectrics from first principles calculations,''
Ferroelectrics, in press.

% ROLE OF STRAIN
% --------------

\bibitem{rabe-strain}
K.M. Rabe and U.V. Waghmare,
% ``Strain coupling in the PbTiO3 ferroelectric transition,''
Phil. Trans. Roy. Soc. Lond. {\bf A354}, 2897 (1996).

% Q FLUCT
% -------

\bibitem{z-qfluct}
W. Zhong and D. Vanderbilt,
% ``Effect of quantum fluctuations on structural phase transitions in
% SrTiO$_3$ and BaTiO$_3$,''
Phys. Rev. B {\bf 53}, 5047 (1996).

% ==============================================
% New Directions
% ==============================================

% ----------------------------------------------
% Strain coupling and piezoelectricity
% ----------------------------------------------

\bibitem{saghi-piezo}
G. Saghi-Szabo, R.E. Cohen, and H. Krakauer,
% ``First principles study of piezoelectricity in tetragonal PbTiO3,''
Ferroelectrics, in press.

% ----------------------------------------------
% Ordering in solid solutions
% ----------------------------------------------

\bibitem{saghi-ord}
G. Saghi-Szabo and R.E. Cohen,
% ``Long range order effects in PbZr$_{1/2}$Ti$_{1/2}$O$_3$,''
Ferroelectrics {\bf 194}, 287 (1997).

\bibitem{rappe-pzt}
N.J. Ramer, E.J. Mele, and A.M. Rappe,
% ``Theoretical examination of stress fields in Pb(Zr$_{0.5}$Ti$_{0.5}$)O$_3$,
Ferroelectrics, in press.

% ----------------------------------------------
% Domain walls, dislocations, surfaces, etc
% ----------------------------------------------

\bibitem{pad-dw}
J. Padilla, W. Zhong, and D. Vanderbilt,
% ``First-principles investigation of 180$^\circ$ domain walls in BaTiO$_3$,''
Phys. Rev. B {\bf 53}, R5969 (1996).

\bibitem{coh-surf}
%
% \bibitem{coh-surf-1}
R.E. Cohen,
% ``Periodic slab LAPW computations for ferroelectric BaTiO3,''
J. Phys. Chem. Solids {\bf 57}, 1393 (1996);
%
% \bibitem{coh-surf-2}
% R.E. Cohen,
% ``Surface effects in ferroelectrics: Periodic slab computations for
% BaTiO3,''
Ferroelectrics {\bf 194}, 323 (1997).

\bibitem{pad-surf}
J. Padilla and D. Vanderbilt,
% ``Ab-initio study of BaTiO$_3$ surfaces,''
Phys. Rev. B {\bf 56}, 1625 (1997).

\end{references}

\end{document}
