mirror of https://github.com/abinit/abinit.git
1605 lines
55 KiB
TeX
1605 lines
55 KiB
TeX
\documentclass[11pt,fleqn]{article}
|
|
|
|
\usepackage{graphicx}
|
|
|
|
\hoffset=-0.06\textwidth
|
|
\textwidth=1.12\textwidth
|
|
\voffset=-0.06\textheight
|
|
\textheight=1.12\textheight
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\def\n{\noindent}
|
|
\def\dis{\displaystyle}
|
|
|
|
\def\bea{\begin{eqnarray}}
|
|
\def\nn{\nonumber\\}
|
|
\def\eea{\end{eqnarray}}
|
|
\def\beq{\begin{equation}}
|
|
\def\eeq{\end{equation}}
|
|
|
|
|
|
\def\wt#1{\widetilde{#1}}
|
|
%\def\wtt#1{\widetilde{\widetilde{#1}}}
|
|
|
|
\def\P{{\bf P}}
|
|
\def\a{{\bf a}}
|
|
\def\E{{\cal E}}
|
|
\def\D{{\bf D}}
|
|
\def\G{{\bf G}}
|
|
\def\u{{\bf u}}
|
|
\def\r{{\bf r}}
|
|
\def\k{{\bf k}}
|
|
|
|
\def\eps{\epsilon}
|
|
\def\epso{\eps_0}
|
|
|
|
\def\O{\Omega}
|
|
\def\Oo{\Omega_0}
|
|
|
|
\def\bc{_{\rm c}}
|
|
\def\veps{\varepsilon}
|
|
|
|
\def\half{{\textstyle{1\over2}}}
|
|
|
|
\def\ABINIT{{{\tt ABINIT}}}
|
|
\def\ANADDB{{\tt ANADDB}}
|
|
\def\DDB{{\tt DDB}}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\begin{document}
|
|
|
|
\begin{center}
|
|
{\Large\bf
|
|
Systematic second-order perturbation theory\\
|
|
\baselineskip 20pt
|
|
for displacements, strains, and electric fields
|
|
}
|
|
\large
|
|
\par\bigskip
|
|
Notes relevant to the {\tt ANADDB} module of {\tt ABINIT}
|
|
\par\medskip
|
|
D. Vanderbilt\\
|
|
\today
|
|
\end{center}
|
|
|
|
|
|
\bigskip
|
|
These notes were originally written to explain the background,
|
|
establish the notation, and provide an explanation of certain
|
|
features added to the \ANADDB\ \cite{abi} package in an upgrade implemented by
|
|
Xifan Wu in January 2004, providing for improved flexibility in the
|
|
computation of dielectric, elastic, piezoelectric, and related
|
|
tensors. However, I believe they may prove helpful more generally,
|
|
e.g., as a starting point for the definition and computation of
|
|
higher-order derivatives, or of properties computed under conditions
|
|
of finite electric fields.
|
|
|
|
These notes will normally reside in the file {\sl `vanderbilt-anaddb-notes.pdf'}
|
|
in the {\sl `/Infos'} subdirectory of the \ABINIT\ distribution \cite{abi}.
|
|
Regarding the definitions of elastic tensors, especially under
|
|
conditions of finite pressure or stress, please also consult
|
|
the notes {\sl `elasticity-oganov.pdf'} \cite{oganov} written by A.~Oganov
|
|
and located in the same subdirectory.
|
|
|
|
I would like to thank X.~Wu, D.R.~Hamann, and K.M.~Rabe for input
|
|
and helpful comments.
|
|
|
|
%==============================================================
|
|
\section{Introduction}
|
|
%==============================================================
|
|
|
|
The purpose of these notes is to provide a systematic framework for
|
|
the discussion of response tensors that can be defined in terms of
|
|
three kinds of homogeneous perturbations in crystalline insulators:
|
|
\begin{itemize}
|
|
\item Zone-center phonons
|
|
\item Homogeneous electric fields
|
|
\item Homogeneous strains
|
|
\end{itemize}
|
|
In particular, the notes provide a set of definitions of many of
|
|
the elastic, dielectric, piezoelectric, and other tensors that
|
|
can be defined in terms of the system response to these
|
|
perturbations, and clarifies the connections between them.
|
|
In addition, the notes make a direct connection with the quantities
|
|
(i.e., the matrices of second derivatives) that are calculated and
|
|
stored in the ``derivative database'' (\DDB) of the
|
|
\ABINIT\ computer package, and with the analysis
|
|
of these carried out by the \ANADDB\ package of \ABINIT\
|
|
\cite{abi}.
|
|
|
|
%==============================================================
|
|
\section{Units}
|
|
%==============================================================
|
|
|
|
SI units are used throughout. This is the convention of most
|
|
textbooks and review articles such as Lines and Glass \cite{lines},
|
|
Nye \cite{nye}, Ballato \cite{ballato}, etc. However, Waghmare's
|
|
article \cite{waghmare}, and much of the usual electronic
|
|
structure literature, uses Gaussian units. We recall here that in
|
|
SI units, potential = $(1/4\pi\epso)\,q_1q_2/r$, energy density =
|
|
$(\epso/2)\E^2$, and $D=\epso\,\E+P$.
|
|
|
|
%==============================================================
|
|
\section{Notation}
|
|
%==============================================================
|
|
\label{sec:notation}
|
|
|
|
Consider an insulating crystal with $N$ atoms per primitive cell.
|
|
We choose a reference state in which the lattice vectors are
|
|
${\bf a}_1$, ${\bf a}_2$, and ${\bf a}_3$ and the atomic coordinates
|
|
are $R_m^{(0)}$. Here $m$ is a composite label (atom and displacement
|
|
direction) running over $1,...,3N$.
|
|
%
|
|
\bea
|
|
u_m &=& \hbox{displacements of atoms from positions $R_m^{(0)}$
|
|
($m=1,...,3N$)}
|
|
\nonumber\\
|
|
\eta_j &=& \hbox{strain in Voigt notation ($j=1,...,6$)}
|
|
\nonumber\\
|
|
\E_\alpha &=& \hbox{electric field ($\alpha=x,y,z$)}
|
|
\eea
|
|
%
|
|
Cartesian coordinates are used throughout, with the exception of
|
|
some \ABINIT\ internal representations discussed in
|
|
Sec.~\ref{sec:abinit}. I will try to
|
|
consistently use $m$, $n$, ... for the $3N$ atomic displacement labels
|
|
(e.g., forces will be $F_m$); $j$, $k$, ... for Voigt labels (e.g.,
|
|
stresses will be $\sigma_j$); and $\alpha$, $\beta$, ... for
|
|
Cartesian labels (e.g., polarizations will be $P_\alpha$). Also
|
|
note that the Voigt notation is a bit tricky for shear strains and
|
|
stresses, e.g., $\sigma_6=\sigma_{xy}$ but $\eta_6=2\eta_{xy}$,
|
|
etc. This is explained more carefully in Sec.~\ref{sec:voigt}
|
|
(see also Nye, Ref.~\cite{nye}).
|
|
|
|
Throughout these notes, we are only going to be interested in
|
|
atomic displacements that preserve the primitive unit cell. In the
|
|
phonon language, this means we are considering only phonon modes
|
|
with $q$-vectors at the Brillouin zone (BZ) center.
|
|
|
|
Also, we will restrict ourselves entirely to zero temperature.
|
|
Thus, entropy will never enter, and the distinction between
|
|
thermodynamic functions written in terms of temperature vs.\ in
|
|
terms of entropy will never arise.
|
|
|
|
The cell volume in the reference configuration (i.e., at zero strain)
|
|
is by definition $\Oo={\bf a_1}\cdot{\bf a_2}\times{\bf a_3}$, so that
|
|
the cell volume at strain $\eta_j$ is $\O=\Oo\,\det[\eta]$
|
|
(interpreted as the determinant of the Cartesian
|
|
$\eta_{\alpha\beta}$ corresponding to Voigt $\eta_j$).
|
|
|
|
Energies $E$ (and other related thermodynamic energy functions)
|
|
will be understood to be defined as the {\it energy per unit
|
|
undeformed volume}. That is, the energy $E$ is really an energy
|
|
density, with units of J/m$^3$, defined as the energy per primitive
|
|
cell of the {\it strained} crystal divided by the volume of the
|
|
{\it unstrained} crystal. This is a tricky point that is often swept
|
|
under the rug in many textbooks, leaving confusion. You can find
|
|
an explicit discussion in Landau and Lifshitz, Theory of Elasticity
|
|
(3rd Edition), p. 8: ``The following remark...'' (Many texts
|
|
will say that $E$ is the energy per unit volume, but then this
|
|
turns out to be inconsistent with what they do later. Think
|
|
about a crystal at its equilibrium volume, where the stress should
|
|
be zero. The meaning of equilibrium is really that the energy
|
|
{\it per unit cell} (let's call it $\bar{E}_{\rm c}$) is stationary.
|
|
But since the volume is {\it not} stationary with respect to
|
|
strains, defining $\sigma$ as the derivative of the quantity
|
|
energy per unit volume is incorrect.) In summary, $H$, $H_0$,
|
|
etc.\ will have units of J/m$^3$, as
|
|
though it were an energy per unit volume, but it is really an
|
|
energy per {\it undeformed} unit volume.
|
|
|
|
We may sometimes use a collective notation
|
|
%
|
|
\[ x_a= ( \; u_m \;,\; \eta_j \;,\; \E\alpha \; ) \]
|
|
%
|
|
where $a$ runs over the $(3N+6+3)$ coordinates describing the complete
|
|
set of displacements, the strain, and the electric field. Then, for
|
|
example, we may write an expansion
|
|
%
|
|
\beq
|
|
H=H_0 + A_a\,x_a + \half B_{ab}\,x_a\,x_b
|
|
\label{eq:Hexpan}
|
|
\eeq
|
|
%
|
|
(implied sum notation) which is short-hand for
|
|
%
|
|
\bea
|
|
H = H_0 &+& A_m\,u_m+A_j\,\eta_j+A_\alpha\,\E_\alpha
|
|
\nonumber\\
|
|
&+& \half B_{mn}\,u_m\,u_n + \half B_{jk}\,\eta_j\,\eta_k
|
|
+ \half B_{\alpha\beta}\,\E_\alpha\,\E_\beta
|
|
\nonumber\\
|
|
&+& B_{mj}\,u_m\,\eta_j + B_{m\alpha}\,u_m\,\E_\alpha
|
|
+ B_{j\alpha}\,\eta_j\,\E_\alpha
|
|
\;\;.
|
|
\label{eq:Hdetail}
|
|
\eea
|
|
%
|
|
In this expansion, the first-order coefficients have the meaning
|
|
%
|
|
\bea
|
|
A_m&=&-F_m/\Oo, \quad F_m = \hbox{force (N)}
|
|
\nonumber\\
|
|
A_j&=&+\sigma_j, \quad \sigma_j = \hbox{stress (J/m$^3$)}
|
|
\nonumber\\
|
|
A_\alpha&=&-P_\alpha, \quad P_\alpha = \hbox{polarization (C/m$^2$)}
|
|
\;\;.
|
|
\eea
|
|
%
|
|
The second-order coefficients are the three diagonal-block tensors
|
|
%
|
|
\bea
|
|
\label{eq:dblock}
|
|
B_{mn}&=&K_{mn}/\Oo, \quad K_{mn} = \hbox{force-constant matrix}
|
|
\nn
|
|
B_{jk}&=&C_{jk}, \quad C_{jk} = \hbox{elastic-constant matrix}
|
|
\nn
|
|
B_{\alpha\beta}&=&-\chi_{\alpha\beta},
|
|
\quad \chi_{\alpha\beta} = \hbox{dielectric susceptibility matrix }
|
|
\eea
|
|
%
|
|
and the three off-diagonal-block tensors
|
|
%
|
|
\bea
|
|
\label{eq:odblock}
|
|
B_{mj}&=&-\Lambda_{mj}/\Oo, \quad \Lambda_{mj} =
|
|
\hbox{``internal-strain tensor''}
|
|
\nn
|
|
B_{m\alpha}&=&-Z_{m\alpha}/\Oo, \quad Z_{m\alpha} =
|
|
\hbox{Born dynamical charge tensor}
|
|
\nn
|
|
B_{j\alpha}&=&-e_{j\alpha}, \quad e_{j\alpha} =
|
|
\hbox{piezoelectric tensor}
|
|
\;\;.
|
|
\eea
|
|
%
|
|
In other words, the ``big gradient vector'' and ``big Hessian matrix''
|
|
are
|
|
%
|
|
\[
|
|
A= \pmatrix { -F/\Oo \cr \sigma \cr -P \cr }
|
|
\qquad\qquad
|
|
B = \pmatrix { K/\Oo & -\Lambda/\Oo & -Z/\Oo \cr
|
|
-\Lambda^T/\Oo & C & -e \cr
|
|
-Z^T/\Oo & -e^T & -\chi \cr } \;\;. \]
|
|
|
|
The quantity $F$ above is the force computed at $u_m=\eta_j=\E_\alpha=0$,
|
|
but the variation of the force with these variables is given by
|
|
%
|
|
\beq
|
|
F_m(u_m,\eta_j,\E\alpha) = -\Oo\,
|
|
\left( A_m + B_{mn} u_n + B_{mj}\eta_j + B_{m\alpha}\E_\alpha \right)
|
|
\;\;.
|
|
\eeq
|
|
%
|
|
Defining $\Delta F_m$ to be the change in the force from the reference
|
|
crystal, and doing a similar analysis for $\Delta\sigma_j$ and
|
|
$\Delta P_\alpha$, we find
|
|
%
|
|
\bea
|
|
\Delta F_m &=&
|
|
- K_{mn}\,u_n + \Lambda_{mj}\,\eta_j + Z_{m\alpha}\,\E_\alpha
|
|
\nn
|
|
\Delta\sigma_j &=&
|
|
- \Oo^{-1}\Lambda_{mj}\,u_m + C_{jk}\,\eta_k - e_{j\alpha}\,\E_\alpha
|
|
\nn
|
|
\Delta P_\alpha &=&
|
|
- \Oo^{-1}\,Z_{m\alpha}\,u_m + e_{j\alpha}\,\eta_j +
|
|
\chi_{\alpha\beta}\,\E_\beta
|
|
\;\;.
|
|
\label{eq:resp}
|
|
\eea
|
|
|
|
The following points should be kept in mind regarding the definitions
|
|
of the various quantities above:
|
|
|
|
\begin{itemize}
|
|
|
|
\item
|
|
All the second-derivative tensors $K$, $C$, $\chi$,
|
|
$\Lambda$, $Z$ and $e$ are ``bare'' quantities calculated
|
|
{\it at fixed} $u$, $\eta$, and $\E$. The formulation of the
|
|
``relaxed-ion'' or ``dressed'' quantities will be discussed
|
|
later.
|
|
|
|
\item
|
|
The terminology regarding the ``internal-strain tensor''
|
|
does not seem to be well established in the literature.
|
|
Our $\Lambda_{mj}$ is the tensor that expresses the first-order change in
|
|
the force on an atom resulting from a first-order strain; for this
|
|
reason, we will sometimes refer to it as the ``force-response
|
|
internal-strain tensor'' when confusion might otherwise arise.
|
|
Similarly, there is a ``displacement-response internal-strain
|
|
tensor'' $\Gamma_{mj}$ expressing the first-order change of the
|
|
relaxed atomic displacements resulting from a first-order strain;
|
|
this will be discussed in Sec.~\ref{sec:internal}.
|
|
|
|
\item
|
|
Regarding the piezoelectric coefficient, our definition is really the
|
|
transpose of the one most commonly found in the literature; that is,
|
|
what we call `$e_{51}$' is more commonly referred to as `$e_{15}$'
|
|
in the literature.
|
|
|
|
\item
|
|
All quantities defined
|
|
here and calculated by \ABINIT\ use the Voigt notation for stress-
|
|
or strain-linked indices. However, all such quantities appearing
|
|
above -- that is, $C$, $\Lambda$, $e$, and $\sigma$ -- have the
|
|
property that no factors of 2 are needed to relate them to true
|
|
tensor quantities. That is, $\sigma_4=\sigma_{yz}$,
|
|
$C_{14}=C_{xx,yz}$, $C_{44}=C_{yz,yz}$, etc. This is explained
|
|
further in Sec.~\ref{sec:voigt}, where it becomes evident that
|
|
any object that can be defined as a (first, second, or higher)
|
|
derivative of $H$ with respect to strain is free of such
|
|
conversion factors.
|
|
|
|
\end{itemize}
|
|
|
|
% %--------------------------------------------------------------
|
|
% \subsection{Things to check or clarify}
|
|
% %--------------------------------------------------------------
|
|
%
|
|
% This subsection will disappear after we check these things:
|
|
%
|
|
% \begin{itemize}
|
|
% \item Check that my notes are {\it internally consistent} as
|
|
% regards signs, volume factors, transpose vs. non-transpose, etc.
|
|
% \item Check that the definitions regarding volume factors and
|
|
% signs are the conventional ones used most frequently in the
|
|
% literature. I tried to follow Waghmare regarding these
|
|
% conventions because I think he used conventional definitions.
|
|
% \item Items that might need clarification:
|
|
% Proper piezo; Voigt; etc.
|
|
% \end{itemize}
|
|
|
|
%==============================================================
|
|
\section{Connection to \ABINIT}
|
|
%==============================================================
|
|
\label{sec:abinit}
|
|
|
|
%--------------------------------------------------------------
|
|
\subsection{Introduction}
|
|
%--------------------------------------------------------------
|
|
\label{sec:ab-intro}
|
|
|
|
\ABINIT\ stores the following information in the
|
|
\DDB\ database:
|
|
%
|
|
\begin{itemize}
|
|
\item Geometry of reference structure: $\{R_m^{(0)}\}$,
|
|
${\bf a}_1$, ${\bf a}_2$, ${\bf a}_3$.
|
|
\item Masses $M_m$ and bare ionic charges $Z_m^{\rm ion}$.
|
|
\item Hessian tensors entering $B$: $K$, $C$, $\chi$, $\Lambda$, $Z$, and $e$.
|
|
\item Gradients $F$ and $\sigma$ are {\it not} included. See
|
|
discussion below.
|
|
\item The polarization $\P$ is currently not included, but
|
|
we may wish to think about revising \ABINIT\ so that it is
|
|
included in the future.
|
|
\end{itemize}
|
|
|
|
Regarding $F$ and $\sigma$,
|
|
the philosophy is that the structural degrees of freedom,
|
|
including strain, should have been relaxed already in the main
|
|
run. Thus, any remaining forces and strains are assumed to be
|
|
``computational noise'' associated with numerical roundoff and
|
|
non-zero tolerances. So, the job of \ANADDB\ should be to compute
|
|
the properties {\it as though} all the forces and stresses are
|
|
zero.
|
|
|
|
Other information is stored in the \DDB\ as well. The header part
|
|
of the file includes various information to help identify the run
|
|
that produced it, and some other information, such as force-constant
|
|
matrix elements at non-zero $q$ and certain third derivatives, is
|
|
also included in the main part. For the present purposes,
|
|
we will ignore this other information.
|
|
|
|
% We probably ought to keep in mind that the \DDB\ information could
|
|
% come from a source other than the \ABINIT\ first-principles
|
|
% pseudopotential calculations. For example, somebody might
|
|
% someday write a package based on an atomistic interatomic potential,
|
|
% or a bond-charge model, or something, that will store the same
|
|
% raw second-derivative information. However, let's not worry
|
|
% about this for now; an interface to such a code could presumably
|
|
% be built at a later time if needed.
|
|
|
|
%--------------------------------------------------------------
|
|
\subsection{Detailed description of \ABINIT\ inputs}
|
|
%--------------------------------------------------------------
|
|
\label{sec:ab-detail}
|
|
|
|
In this section, we attempt to specify precisely how the
|
|
second-derivative information is stored in the \DDB\ file.
|
|
This information corresponds to partial second derivatives of the
|
|
energy with respect to displacements, electric fields, and strains,
|
|
and thus there is a direct correspondence with the information
|
|
in the tensors $K$, $C$, $\chi$, $\Lambda$, $Z$ and $e$.
|
|
However, the conventions for these quantities in the \DDB\ file
|
|
differ from those in the present notes by factors associated
|
|
with units conversions, reduced vs.~Cartesian coordinates,
|
|
cell volume factors, etc.
|
|
|
|
%--------------------------------------------------------------
|
|
\subsubsection{Lattice vectors and cell volume}
|
|
%--------------------------------------------------------------
|
|
|
|
The real space lattice vectors are $\a_\mu$ and henceforth
|
|
$\mu$, $\nu$, ... will be used to label the three lattice
|
|
directions. The reciprocal-space lattice vectors are defined via
|
|
\linebreak
|
|
$\G_\mu\cdot\a_\nu=2\pi\,\delta_{\mu\nu}$, but we introduce instead
|
|
the ``reduced'' reciprocal-space lattice vectors
|
|
$\G'_\mu=\G_\mu/2\pi$, so that
|
|
%
|
|
\beq
|
|
\G'_\mu\cdot\a_\nu=\delta_{\mu\nu}
|
|
\eeq
|
|
%
|
|
Recall that $\Oo={\bf a_1}\cdot{\bf a_2}\times{\bf a_3}$.
|
|
We will use the subscript ``c'' to denote an energy per unit cell.
|
|
Thus,
|
|
%
|
|
\beq
|
|
H\bc=\Oo\,H
|
|
\eeq
|
|
|
|
%--------------------------------------------------------------
|
|
\subsubsection{Reduced displacements and forces}
|
|
%--------------------------------------------------------------
|
|
|
|
In Sec.~\ref{sec:notation}, the displacements were defined in
|
|
Cartesian coordinates as $u_m$, $m=1,3N$. Recall that $m$ is
|
|
a composite index, $m\rightarrow\tau\alpha$, where $\tau=1,N$
|
|
is a label for the atom in the unit cell and $\alpha$ is a
|
|
Cartesian axis label.
|
|
|
|
The \ABINIT\ code internally uses ``reduced'' coordinates (also
|
|
sometimes called ``lattice-vector'' or ``canonical'' or ``internal''
|
|
coordinates) for displacements, defined via
|
|
$s_{\tau\mu}=\G'_\mu\cdot\u_\tau$ and
|
|
$\u_\tau=s_{\tau\mu}\a_\mu$ or, using index notation,
|
|
%
|
|
\beq
|
|
s_{\tau\mu}=G'_{\mu\alpha}\,u_{\tau\alpha}
|
|
\eeq
|
|
%
|
|
and
|
|
%
|
|
\beq
|
|
u_{\tau\alpha}=s_{\tau\mu}\,a_{\mu\alpha}
|
|
\eeq
|
|
%
|
|
Application of the chain rule shows that derivatives behave like
|
|
%
|
|
\beq
|
|
{\partial\over\partial s_{\tau\mu}}=a_{\mu\alpha}\,
|
|
{\partial\over\partial u_{\tau\alpha}}
|
|
\;\;.
|
|
\eeq
|
|
%
|
|
Thus, if $F_{\tau\alpha}=-\partial H\bc/\partial u_{\tau\alpha}$
|
|
is the force vector in Cartesian coordinates, then the ``force
|
|
vector in reduced coordinates'' is
|
|
%
|
|
\beq
|
|
f_{\tau\mu}=-\,{\partial H\bc\over \partial s_{\tau\mu}}
|
|
= a_{\mu\alpha}\,F_{\tau\alpha}
|
|
\eeq
|
|
%
|
|
where $f$ has dimensions of energy.
|
|
|
|
%--------------------------------------------------------------
|
|
\subsubsection{Reduced electric fields and polarizations}
|
|
%--------------------------------------------------------------
|
|
|
|
Let
|
|
%
|
|
\beq
|
|
\veps_\mu=|q_e|\;\E\cdot \a_\mu \;\;,
|
|
\eeq
|
|
%
|
|
be the electric field in reduced coordinates, where $|q_e|$ is the
|
|
proton charge. Thus, $\veps_\mu$, having dimensions of energy, is
|
|
the work done on a charge quantum $|q_e|$ to translate it by a lattice
|
|
vector $\a_\mu$. Similarly, we can introduced the ``reduced
|
|
polarization''
|
|
%
|
|
\beq
|
|
p_\mu=|q_e|^{-1}\,\Oo\;\G'_\mu\cdot\P
|
|
\;\;.
|
|
\eeq
|
|
%
|
|
Note that $p_\mu$ (dimensionless) is defined modulo 1
|
|
(that is, the ``quantum of
|
|
polarization'' is just unity for each component $p_\mu$), and
|
|
that the electronic contribution to it is just
|
|
%
|
|
\beq
|
|
p_\mu^{\rm elec}=-(2\pi)^{-1}\sum_n\phi_{n,\mu}
|
|
\label{eq:pelec}
|
|
\eeq
|
|
%
|
|
where $\phi_{n,\mu}$ is the Berry phase of band $n$ (defined modulo
|
|
$2\pi$) as in Ref.~\cite{dv-piezo}. The energy per primitive cell
|
|
arising from the $\E$-field is converted from Cartesian to internal
|
|
coordinates as
|
|
%
|
|
\beq
|
|
\Oo\,\E\cdot\P=\veps_\mu\,p_\mu
|
|
\label{eq:epterm}
|
|
\eeq
|
|
%
|
|
where both sides have dimensions of energy.
|
|
|
|
%--------------------------------------------------------------
|
|
\subsubsection{Definitions of perturbations in \ABINIT}
|
|
%--------------------------------------------------------------
|
|
|
|
\ABINIT\ computes and stores the
|
|
derivatives of the wavefunctions with respect to {\it reduced}
|
|
displacements, {\it reduced} electric fields, and {\it Cartesian}
|
|
strains, and its energy units are Hartree (atomic units).
|
|
The wavefunctions are expanded as $\psi_{n\k}(\r)=\sum_l c_{n\k,l}\,
|
|
\exp[i(\G_l+\k)\cdot\r]$ (where $l$ represents a triplet of integers)
|
|
and the $c_{n\k,l}$ are dimensionless, so what is stored is
|
|
%
|
|
\begin{itemize}
|
|
\item For displacements: $d c_{n\k}/d s_\mu$ (dimensionless)
|
|
\item For $\E$-fields: $d c_{n\k,l}/d \veps_\mu$ (inverse energy)
|
|
\item For strains: $d c_{n\k,l}/d \eta_j$ (Cartesian Voigt;
|
|
dimensionless)
|
|
\end{itemize}
|
|
%
|
|
Similarly, forces are initially computed in reduced form
|
|
as $f_{\tau\mu}=-\,dH\bc/ds_{\tau\mu}$, while stresses are
|
|
computed as $\sigma_j=\Oo^{-1}\,dH\bc/d\eta_j$.
|
|
|
|
%--------------------------------------------------------------
|
|
\subsubsection{Second derivative information in the \DDB}
|
|
%--------------------------------------------------------------
|
|
\label{sec:ddbinfo}
|
|
|
|
Similarly, the quantities that are stored in the second-derivative
|
|
\DDB\ are as follows.
|
|
%
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{itemize}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\item Displacement--and--Displacement:
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%
|
|
The quantity that is computed and stored in the \DDB\ is
|
|
%
|
|
\beq
|
|
{d^2H\bc\over ds_{\tau\mu}\,ds_{\tau'\nu}}
|
|
\eeq
|
|
%
|
|
in terms of which the force-constant matrix is
|
|
%
|
|
\beq
|
|
K_{\tau\alpha,\tau'\beta}=\,G'_{\mu\alpha}\,G'_{\nu\beta}\,
|
|
{d^2H\bc \over ds_{\tau\mu}\,ds_{\tau'\nu}}
|
|
\;\;.
|
|
\eeq
|
|
%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\item Strain--and--Strain:
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%
|
|
The quantity that is computed and stored in the \DDB\ is
|
|
%
|
|
\beq
|
|
{d^2 H\bc \over d\eta_j\,d\eta_k}
|
|
\eeq
|
|
%
|
|
in terms of which the elastic-constant tensor is
|
|
%
|
|
\beq
|
|
C_{jk}= \Oo^{-1}\,\,{d^2 H\bc \over d\eta_j\,d\eta_k}
|
|
\eeq
|
|
%
|
|
(recall that $H\bc=\Oo\,H$).
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\item Field--and--Field:
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%
|
|
The quantity that is computed and stored in the \DDB\ is
|
|
%
|
|
\beq
|
|
{d^2H\bc \over d\veps_\mu\,d\veps_\nu}
|
|
\eeq
|
|
%
|
|
having dimensions of inverse Hartree,
|
|
in terms of which the susceptibility is
|
|
%
|
|
\beq
|
|
\chi_{\alpha\beta}=-\,q_e^2\, \Oo^{-1}\, a_{\mu\alpha}\,a_{\nu\beta}\,
|
|
{d^2H\bc \over d\veps_\mu\,d\veps_\nu}
|
|
\eeq
|
|
%
|
|
and $\eps_{\alpha\beta}=\delta_{\alpha\beta}+4\pi\,\chi_{\alpha\beta}$.
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\item Displacement--and--Strain:
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%
|
|
The quantity that is computed and stored in the \DDB\ is
|
|
%
|
|
\beq
|
|
{d^2H\bc \over ds_{\tau\mu}\,d\eta_j}
|
|
\eeq
|
|
%
|
|
in terms of which the force-response internal-strain tensor is
|
|
%
|
|
\beq
|
|
\Lambda_{\tau\alpha,j}= -\,G'_{\mu\alpha}\,
|
|
{d^2H\bc \over ds_{\tau\mu}\,d\eta_j}
|
|
\;\;.
|
|
\eeq
|
|
%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\item Displacement--and--Field:
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%
|
|
The quantity that is computed and stored in the \DDB\ is
|
|
the purely electronic contribution
|
|
%
|
|
\beq
|
|
{d^2H\bc \over ds_{\tau\mu}\,d\veps_\nu}
|
|
\eeq
|
|
%
|
|
to the dynamical charge tensor. The ionic contribution must be
|
|
added to it. Thus, the full Born dynamical charge tensor is
|
|
%
|
|
\beq
|
|
Z_{\tau\alpha,\beta}=-\,|q_e|\,G'_{\mu\alpha}\,a_{\nu\beta}\,
|
|
{d^2H\bc \over ds_{\tau\mu}\,d\veps_\nu}
|
|
+ Z^{\rm ion}_\tau\,\delta_{\alpha\beta}
|
|
\eeq
|
|
%
|
|
which has dimensions of charge.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\item Strain--and--Field:
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%
|
|
The quantity that is computed and stored in the \DDB\ is
|
|
%
|
|
\beq
|
|
{d^2H\bc \over d\eta_j\,d\veps_\nu}
|
|
\label{eq:piezo-store}
|
|
\eeq
|
|
%
|
|
in terms of which the piezoelectric tensor is
|
|
%
|
|
\beq
|
|
e_{j\alpha}=-\,|q_e|\,\Oo^{-1}\,a_{\nu\alpha}\,
|
|
{d^2H\bc \over d\eta_j\,d\veps_\nu}
|
|
\label{eq:piezo-convert}
|
|
\eeq
|
|
%
|
|
|
|
Note that $dH\bc/d\veps_\nu$ has the interpretation
|
|
of being a reduced polarization $p_\nu^{\rm elec}$ in the sense
|
|
of Eqs.~(\ref{eq:pelec}-\ref{eq:epterm}). Thus, the quantity
|
|
in Eq.~(\ref{eq:piezo-store}) has the interpretation of
|
|
$dp_\nu^{\rm elec}/d\eta_j$, i.e., a derivative of the {\it reduced
|
|
polarization} with respect to strain.
|
|
This corresponds \cite{dv-piezo} to the ``proper'' and not
|
|
the ``improper'' piezoelectric tensor. Thus, it is (a reduced
|
|
version of) the ``proper'' piezoelectric tensor that is stored
|
|
in the \DDB.
|
|
|
|
Note that a finite-difference calculation
|
|
of derivatives of unreduced polarization $\bf P$ with respect
|
|
to strain would yield the ``improper'' piezoelectric tensor.
|
|
On the other hand, the quantity in Eq.~(\ref{eq:piezo-store})
|
|
corresponds (up to factors of $-2\pi$, see Eq.~(\ref{eq:pelec}))
|
|
to numerical derivatives of the {\it Berry phases} with
|
|
respect to strain components. As explained in Ref.~\cite{dv-piezo},
|
|
when converted via an equation like Eq.~(\ref{eq:piezo-convert}),
|
|
these yield the ``proper'' piezoelectric tensor.
|
|
|
|
Thankfully, the ionic contribution to the ``proper'' piezoelectric
|
|
tensor is identically zero \cite{dv-piezo}, so we don't have to
|
|
include it in the \DDB.
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\end{itemize}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%--------------------------------------------------------------
|
|
\subsection{Electric field}
|
|
%--------------------------------------------------------------
|
|
|
|
Extensions to {\tt ABINIT} to allow non-zero electric fields are in
|
|
progress, based on the approach of Souza, \'I\~niguez and
|
|
Vanderbilt \cite{souza-ef}. For the time being, however, we will
|
|
assume that the reference calculation is always at $\E=0$, i.e., in
|
|
vanishing electric field.
|
|
|
|
% At the moment,
|
|
% \ABINIT\ is only capable of handling the zero-field case, but this
|
|
% might change with a possible future implementation of the
|
|
% Souza-Iniguez-Vanderbilt scheme \cite{souza-ef}. Also, non-zero $\E$
|
|
% might be allowed if some package other than \ABINIT\ is used to
|
|
% compute the \DDB\ (see previous subsection). But let's agree to
|
|
% worry about this later.
|
|
|
|
%--------------------------------------------------------------
|
|
\subsection{Symmetry reduction}
|
|
%--------------------------------------------------------------
|
|
|
|
Often it may be possible to block the vectors $A_a$ and the
|
|
tensors $B_{ab}$ into irreducible representations (``irreps'') according
|
|
to the crystal symmetry. In this case, it may be possible to
|
|
focus on just one irrep and calculate all the derived quantities just
|
|
for that irrep. However, at least for now, the philosophy of the
|
|
ANADDB implementation is to compute the full tensors, and to handle
|
|
any symmetry reduction by hand at the stage of analyzing the results.
|
|
|
|
% Ideally, the package should eventually be able to handle
|
|
% this case somehow, but let's leave that for the future for now.
|
|
|
|
%==============================================================
|
|
\section{Computation of derived quantities}
|
|
%==============================================================
|
|
\label{sec:deriv}
|
|
|
|
We are in the process of implementing, in the {\tt ANADDB} module
|
|
of {\tt ABINIT}, a package that will be capable of computing and
|
|
reporting a large number of derived quantities based on the input
|
|
of information from the \DDB. In this section, we list some of
|
|
these quantities and give the formulas necessary for computing them.
|
|
|
|
% In some cases we will provide the formal background for the manipulation,
|
|
% if it can be done briefly. More subtle or complicated derivations
|
|
% will be deferred to later sections, where we will also introduce
|
|
% the different thermodynamic energies that can be used to define and
|
|
% relate various quantities.
|
|
|
|
The starting point will be the bare quantities
|
|
$K$, $C$, $\chi$, $\Lambda$, $Z$ and $e$. It will be assumed that
|
|
the information needed to construct these tensors has been read from
|
|
the \DDB\ file, and any needed conversions (units, reduced vs.~
|
|
Cartesian, cell volume factors, etc.) have already been applied
|
|
as described in Sec.~\ref{sec:ab-detail}.
|
|
|
|
%--------------------------------------------------------------
|
|
\subsection{Elastic compliance tensor}
|
|
%--------------------------------------------------------------
|
|
\label{sec:compliance}
|
|
|
|
The clamped-ion elastic compliance tensor $S$ is simply obtained
|
|
by inverting the elastic-tensor matrix $C$:
|
|
%
|
|
\beq
|
|
S_{jk}=(C^{-1})_{jk} .
|
|
\eeq
|
|
%
|
|
Warning messages
|
|
are issued if the matrix $C_{jk}$ is found to be nearly singular.
|
|
|
|
%--------------------------------------------------------------
|
|
\subsection{Internal-strain tensors}
|
|
%--------------------------------------------------------------
|
|
\label{sec:internal}
|
|
|
|
As noted at the end of Sec.~\ref{sec:notation}, $\Lambda_{mj}$
|
|
is the ``force-response internal-strain tensor'' giving the
|
|
force on atom $m$ due to strain $j$. If instead one wants the
|
|
``displacement-response internal-strain tensor,'' which gives
|
|
displacement of atom $m$ due to strain $j$, it is given by
|
|
the product
|
|
%
|
|
\beq
|
|
\Gamma_{mj}=(K^{-1})_{mn}\,\Lambda_{nj} .
|
|
\eeq
|
|
%
|
|
|
|
%--------------------------------------------------------------
|
|
\subsection{Relaxed-ion quantities}
|
|
%--------------------------------------------------------------
|
|
|
|
%--------------------------------------------------------------
|
|
\subsubsection{Formulation}
|
|
%--------------------------------------------------------------
|
|
\label{sec:formulation}
|
|
|
|
The tensors $C$, $\chi$, and $e$ are ``clamped-ion''
|
|
quantities defined at fixed $u$. We define the corresponding
|
|
``relaxed-ion'' or ``dressed'' response tensors $\wt{C}$,
|
|
$\wt{\chi}$, and $\wt{e}$ as follows. Let
|
|
%
|
|
\beq
|
|
\wt{H}(\eta_j,\E_\alpha)=\min_{\{u_m\}} H(u_m,\eta_j,\E_\alpha)
|
|
\;\;.
|
|
\eeq
|
|
%
|
|
Referring back to Eq.~(\ref{eq:Hdetail}) and setting $\partial
|
|
H/\partial u_n=0$, and assuming that the reference configuration is
|
|
one in which the forces vanish, we find
|
|
%
|
|
\[ 0=B_{nm}\,u_m+B_{nj}\,\eta_j+B_{n\alpha}\,\E_\alpha \]
|
|
%
|
|
from which it follows that
|
|
%
|
|
\[ u_m=-(B^{-1})_{mn} \,[\,B_{nj}\,\eta_j+B_{n\alpha}\,\E_\alpha] \]
|
|
%
|
|
or
|
|
%
|
|
\beq
|
|
\wt{H}(\eta_j,\E_\alpha)=\half\wt{B}_{jk}\,\eta_j\,\eta_k +
|
|
\half\wt{B}_{\alpha\beta}\,\E_\alpha\,\E_\beta +
|
|
\wt{B}_{j\alpha}\,\eta_j\,\E_\alpha
|
|
\eeq
|
|
%
|
|
where
|
|
%
|
|
\bea
|
|
\wt{B}_{jk} &=& B_{jk}-B_{mj}\,(B^{-1})_{mn}\,B_{nk}
|
|
\nn
|
|
\wt{B}_{\alpha\beta} &=& B_{\alpha\beta}-B_{m\alpha}\,(B^{-1})_{mn}\,B_{n\beta}
|
|
\nn
|
|
\wt{B}_{j\alpha} &=& B_{j\alpha}-B_{mj}\,(B^{-1})_{mn}\,B_{n\alpha}
|
|
\;\;.
|
|
\eea
|
|
%
|
|
Using the definitions given in Eqs.~(\ref{eq:dblock}) and (\ref{eq:odblock}),
|
|
and making use of the fact that $B$, and thus $\wt{B}$, are real
|
|
symmetric matrices, these become
|
|
%
|
|
\bea
|
|
\wt{C}_{jk} &=& C_{jk}-\Oo^{-1}\,\Lambda_{mj}\,(K^{-1})_{mn}\,\Lambda_{nk}
|
|
\label{eq:ctilde}
|
|
\\
|
|
\wt{\chi}_{\alpha\beta} &=& \chi_{\alpha\beta}
|
|
+ \Oo^{-1}\,Z_{m\alpha}\,(K^{-1})_{mn}\,Z_{n\beta}
|
|
\label{eq:chitilde}
|
|
\\
|
|
\wt{e}_{j\alpha} &=& e_{j\alpha}+\Oo^{-1}\,\Lambda_{mj}\,(K^{-1})_{mn}\,Z_{n\alpha}
|
|
\label{eq:etilde}
|
|
\eea
|
|
%
|
|
|
|
Note that $\wt{C}$ and $\wt{e}$ are the ``physical'' elastic constant
|
|
and piezoelectric tensors, respectively, since it is virtually
|
|
impossible to measure these quantities without allowing the atomic
|
|
coordinates to respond.
|
|
The first and second terms on the right-hand sides of the above
|
|
equations are usually referred to as the ``clamped-ion'' (or
|
|
``purely electronic'') and ``relaxed-ion'' (or ``lattice'')
|
|
contributions, respectively.
|
|
|
|
%--------------------------------------------------------------
|
|
\subsubsection{Pseudo-inverse of the force-constant matrix}
|
|
%--------------------------------------------------------------
|
|
\label{sec:pseudoinverse}
|
|
|
|
The force-constant matrix $K$ should have three vanishing
|
|
eigenvalues as a result of translational symmetry. Moreover,
|
|
in soft-mode systems, other eigenvalues may be close to zero.
|
|
Thus, for the many operations that use $K^{-1}$, there is a
|
|
danger of numerical instabilities arising from attempting to
|
|
invert a singular, or nearly singular, matrix. For this reason,
|
|
we have implemented a careful procedure for obtaining the
|
|
``pseudo-inverse'' of $K$; throughout these notes, whenever
|
|
we refer to $K^{-1}$, we really mean the pseudo-inverse.
|
|
|
|
Briefly, we begin by identifying the three-dimensional space of
|
|
acoustic modes (i.e., uniform translations), and project the
|
|
matrix $K$ onto the complementary subspace of dimension $3N-3$.
|
|
We then invert in this subspace, reporting a warning message if
|
|
any eigenvalues are very small, and then transform back to the
|
|
full space. Thus, by construction, the resulting pseudo-inverse
|
|
is zero in the subspace of translational modes, and is the inverse
|
|
of the original matrix in the complementary subspace.
|
|
|
|
As a result, any time $K^{-1}$ is multiplied by another tensor,
|
|
a pre-projection onto the complementary subspace of dimension $3N-3$
|
|
is effectively carried out. In other words, ``the acoustic
|
|
sum rule is effectively enforced'' in any operation involving
|
|
$K^{-1}$.
|
|
|
|
% %--------------------------------------------------------------
|
|
% \subsubsection{Elastic tensor and stability analysis}
|
|
% %--------------------------------------------------------------
|
|
%
|
|
% As noted just above, $K$ should be checked for stability.
|
|
% In addition, the bare elastic constant tensor $C$
|
|
% should also be diagonalized and checked for stability
|
|
% (near-zero or negative eigenvalues).
|
|
%
|
|
% However, note that even if both $K$ and $C$ each indicate
|
|
% stability separately, it is still possible for the crystal
|
|
% to be unstable overall (at fixed $\E$) if the internal strain
|
|
% tensor elements are large enough. This can be checked in
|
|
% two ways:
|
|
%
|
|
% \begin{itemize}
|
|
%
|
|
% \item
|
|
% The elastic tensor $\wt{C}_{jk}$ can be diagonalized
|
|
% and checked for stability (near-zero or negative eigenvalues).
|
|
% In the case that it is not stable, the program should refuse
|
|
% to compute subsequent properties that depend on the compliance
|
|
% tensor $\wt{C}^{-1}$.
|
|
%
|
|
% \item
|
|
% An alternative approach to the stability analysis would be to
|
|
% diagonalize the enlarged matrix corresponding to both strain
|
|
% and displacement degrees of freedom (that is, regarding the
|
|
% $B$ matrix as 3-blocks-by-three-blocks, diagonalize a
|
|
% 2-block-by-two-block sub-matrix involving $K$, $C$, $\Lambda$,
|
|
% and $\Lambda^\dagger$). Again, a near-zero or
|
|
% negative eigenvalue would indicate marginal stability or instability,
|
|
% respectively. The advantage
|
|
% to this approach is that it would not get confused in the case
|
|
% of eigenvalues of $K$ being very close to zero.
|
|
%
|
|
% \end{itemize}
|
|
|
|
%--------------------------------------------------------------
|
|
\subsubsection{Dielectric tensor}
|
|
%--------------------------------------------------------------
|
|
\label{sec:eps-free-stress}
|
|
|
|
The dielectric response can be measured in three frequency
|
|
regimes:
|
|
%
|
|
\begin{itemize}
|
|
\item At frequencies much greater than any phonon frequency (but less
|
|
than electronic frequencies). In this case only the electrons respond
|
|
and
|
|
%
|
|
\bea
|
|
\eps^{(\infty)}_{\alpha\beta}=\delta_{\alpha\beta}+4\pi\chi_{\alpha\beta}
|
|
\;\;.
|
|
\eea
|
|
%
|
|
\item At frequencies low compared to any phonon frequency, but above
|
|
mechanical resonance frequencies,
|
|
%
|
|
\bea
|
|
\eps^{(0)}_{\alpha\beta}=\delta_{\alpha\beta}+4\pi\wt{\chi}_{\alpha\beta}
|
|
\;\;.
|
|
\eea
|
|
%
|
|
\item At frequencies low compared to mechanical resonance frequencies of
|
|
the macroscopic sample, the strain can also relax,
|
|
%
|
|
\bea
|
|
\eps^{(\sigma)}_{\alpha\beta}=\delta_{\alpha\beta}+
|
|
4\pi\wt{\chi}^{(\sigma)}_{\alpha\beta}
|
|
\label{eq:eps-sig}
|
|
\eea
|
|
%
|
|
as discussed below and in Sec.~\ref{sec:piezo-tens}.
|
|
\end{itemize}
|
|
|
|
Note that the \ANADDB\ module of \ABINIT\ has been capable of computing
|
|
$\eps^{(0)}$ for some time. Actually, it is capable of computing
|
|
$\eps_{\alpha\beta}(\omega)$ in the regime of lattice frequencies, and
|
|
in particular, if this is evaluated for $\omega=0$, $\eps^{(0)}$
|
|
results. However, the revised version of the \ANADDB\ is being
|
|
written to compute $\eps^{(0)}$ and $\eps^{(\sigma)}_{\alpha\beta}$
|
|
independently, using the pseudo-inverse of Sec.~\ref{sec:pseudoinverse}.
|
|
|
|
The derivation for the
|
|
free-stress susceptibility $\eps^{(\sigma)}$ starts from the
|
|
free-stress energy functional
|
|
%
|
|
\beq
|
|
\wt{H}^{(\sigma)}(\E_\alpha)=\min_{\{\eta_j\}} \wt{H}(\eta_j,\E_\alpha)
|
|
\eeq
|
|
%
|
|
Following a line of reasoning similar to what was done for the
|
|
elimination of displacement degrees of freedom in
|
|
Sec.~\ref{sec:formulation}, we obtain
|
|
%
|
|
\beq
|
|
\wt{\chi}^{(\sigma)}_{\alpha\beta} = \wt{\chi}_{\alpha\beta}
|
|
+ \wt{e}_{j\alpha}\,(\wt{C}^{-1})_{jk}\,\wt{e}_{k\beta}
|
|
\label{eq:chifs}
|
|
\eeq
|
|
%
|
|
in terms of which the free-stress dielectric
|
|
tensor $\eps^{(\sigma)}_{\alpha\beta}$ is given as in
|
|
Eq.~(\ref{eq:eps-sig}).
|
|
Note that Eq.~(\ref{eq:chifs}) is equivalent to Eq.~(\ref{eq:epsfs})
|
|
appearing later in Sec.~\ref{sec:piezo-tens}.
|
|
|
|
%--------------------------------------------------------------
|
|
\subsection{Change of notation}
|
|
%--------------------------------------------------------------
|
|
|
|
For the remainder of Sec.~\ref{sec:deriv}, all quantities will
|
|
be assumed to be relaxed-ion quantities. Up to now such quantities
|
|
have been denoted by a tilde (e.g., $\wt{\chi}$), but henceforth
|
|
the tildes will be dropped.
|
|
|
|
%--------------------------------------------------------------
|
|
\subsection{Elastic tensors}
|
|
%--------------------------------------------------------------
|
|
\label{sec:elas}
|
|
|
|
From Eq.~(\ref{eq:resp}), the elastic tensor $C$ is defined
|
|
as the partial derivative of $\sigma_j$ with respect to $\eta_k$
|
|
at fixed macroscopic electric field $\E$; to emphasize this,
|
|
we will optionally write a superscript $(\E)$:
|
|
%
|
|
\beq
|
|
C_{jk}^{(\E)}={\partial\sigma_j\over\partial\eta_k}\,\Big\vert_\E
|
|
\;\;.
|
|
\eeq
|
|
%
|
|
For some purposes, we may be interested instead in the elastic
|
|
constant tensor at {\it fixed displacement field} $D$:
|
|
%
|
|
\beq
|
|
C_{jk}^{(D)}={\partial\sigma_j\over\partial\eta_k}\,\Big\vert_D
|
|
\;\;.
|
|
\eeq
|
|
%
|
|
The inverse of the elastic constant tensor is the compliance
|
|
tensor. It can be defined either at fixed $\E$ or $D$:
|
|
%
|
|
\beq
|
|
S_{jk}^{(\E)}={\partial\eta_j\over\partial\sigma_k}\,\Big\vert_\E
|
|
\;\;,
|
|
\eeq
|
|
%
|
|
\beq
|
|
S_{jk}^{(D)}={\partial\eta_j\over\partial\sigma_k}\,\Big\vert_D
|
|
\;\;.
|
|
\eeq
|
|
|
|
The definitions of $C$ and $S$ are shown schematically on the left-hand
|
|
side of Fig.~1. The meaning of the arrow pointing from $\eta$ to
|
|
$\sigma$ and labeled by $C$ is that
|
|
%
|
|
\[ \delta\sigma_j=C_{jk}\,\delta\eta_k \]
|
|
%
|
|
(either at fixed $\E$ or fixed $D$), and similarly the downgoing
|
|
arrow labeled $S$ indicates that
|
|
%
|
|
\[ \delta\eta_j=S_{jk}\,\delta\sigma_k \;\;. \]
|
|
|
|
\begin{figure}
|
|
\begin{center}
|
|
\includegraphics[width=7.6cm,angle=0]{response}
|
|
\end{center}
|
|
\caption{Definitions of response functions.}
|
|
\label{fig:response}
|
|
\end{figure}
|
|
|
|
The elastic tensors and compliances are related as inverses,
|
|
%
|
|
\beq
|
|
S^{(\E)}=(C^{(\E)})^{-1}
|
|
\eeq
|
|
%
|
|
%
|
|
\beq
|
|
S^{(D)}=(C^{(D)})^{-1} \;.
|
|
\eeq
|
|
%
|
|
The relation between $C^{(D)}$ and $C^{(\E)}$,
|
|
and between $S^{(D)}$ and $S^{(\E)}$, will be given
|
|
at the end of Sec.~\ref{sec:piezo-tens}.
|
|
|
|
Note that the compliance matrices are defined above in Voigt
|
|
notation, and in this case {\it there are factors of 2 and 4}
|
|
needed to make connection with true tensor quantities:
|
|
$S_{14}=2S_{xx,yz}$, $S_{44}=4S_{yz,yz}$, etc. This is explained
|
|
more fully in Sec.~\ref{sec:voigt}.
|
|
|
|
Finally, note that various different definitions can be given of
|
|
the elastic constants under conditions of nonzero hydrostatic
|
|
pressure or, more generally, under nonzero stress. In this case,
|
|
the experimentally relevant tensors (e.g., for seismic waves in
|
|
the interior of the earth) {\it do not} necessarily correspond to the ones
|
|
computed directly by \ABINIT. For a discussion of these issues,
|
|
please see \cite{oganov}.
|
|
|
|
% %--------------------------------------------------------------
|
|
% \subsection{Dielectric tensors}
|
|
% %--------------------------------------------------------------
|
|
% \label{sec:diel}
|
|
%
|
|
% The definitions of the dielectric tensor $\eps$ and the
|
|
% inverse dielectric tensor $\beta$ are shown schematically on
|
|
% the right-hand side of Fig.~1. These quantities can either
|
|
% be defined at fixed strain (explicitly with superscript $(\eta)$
|
|
% or implicitly if no superscript appears), or at fixed stress
|
|
% (if superscript $(\sigma)$ appears):
|
|
%
|
|
% \bigskip
|
|
%
|
|
% {\sl [TO BE WRITTEN. WRITE IN A WAY THAT IS ROUGHLY PARALLEL TO THE
|
|
% PREVIOUS SECTION.]}
|
|
%
|
|
% \bigskip
|
|
%
|
|
% {\sl [Actually, this section is really redundant with
|
|
% Sec.~\ref{sec:eps-free-stress},
|
|
% where we already talked about $\eps^{(\sigma)}$ in a different notation.
|
|
% But maybe it doesn't matter? Or remove earlier section?]}
|
|
|
|
%--------------------------------------------------------------
|
|
\subsection{Piezoelectric tensors}
|
|
%--------------------------------------------------------------
|
|
\label{sec:piezo-tens}
|
|
|
|
The piezoelectric tensors are defined schematically by the lines
|
|
crossing horizontally and diagonally in Fig.~(1). (All piezoelectric
|
|
tensors are the ``proper'' ones -- see the last part of
|
|
Sec.~\ref{sec:ddbinfo} and Ref.~\cite{dv-piezo} for a discussion.)
|
|
The interpretation of the arrows is
|
|
%
|
|
\bea
|
|
\delta\eta_j &=& d_{j\alpha}\,\delta\E_\alpha
|
|
\nn
|
|
\delta D_\alpha &=& d_{j\alpha}\,\delta\sigma_j
|
|
\nn
|
|
\delta\sigma_j &=& -\,e_{j\alpha}\,\delta\E_\alpha
|
|
\nn
|
|
\delta D_\alpha &=& e_{j\alpha}\,\delta\eta_j
|
|
\;\;.
|
|
\eea
|
|
%
|
|
The first of the four equations above is sometimes said to describe
|
|
the ``converse'' piezoelectric effect, while the second one
|
|
describes the ``direct'' piezoelectric effect, and the third and/or
|
|
the fourth describe the ``inverse'' piezoelectric effect. Thus,
|
|
one sometimes refers to $d$ as the coefficient of the ``direct
|
|
piezoelectric effect'' while $e$ (denoted as $c$ almost as often as
|
|
$e$ -- these notations are equivalent) is the coefficient of the
|
|
``inverse piezoelectric effect.'' Restating, the piezoelectric
|
|
coefficients may be defined via
|
|
%
|
|
\beq
|
|
e_{j\alpha}={\partial D_\alpha\over\partial\eta_j} \Big\vert_\E
|
|
={\partial P_\alpha\over\partial\eta_j} \Big\vert_\E
|
|
\qquad {\rm or} \qquad
|
|
e_{j\alpha}=-\,{\partial \sigma_j\over\partial\E_\alpha} \Big\vert_\eta
|
|
\label{eq:e-def}
|
|
\eeq
|
|
%
|
|
and
|
|
%
|
|
\beq
|
|
d_{j\alpha}= {\partial\eta_j\over\partial\E_\alpha} \Big\vert_\sigma
|
|
\qquad {\rm or} \qquad
|
|
d_{j\alpha}= {\partial D_\alpha\over\partial\sigma_j} \Big\vert_\E
|
|
= {\partial P_\alpha\over\partial\sigma_j} \Big\vert_\E
|
|
\;\;.
|
|
\label{eq:d-def}
|
|
\eeq
|
|
%
|
|
The equivalence between the two expressions for $e$, and similarly between
|
|
the two expressions for $d$, comes from thermodynamic relations
|
|
as discussed in the Appendix of Ballato
|
|
\cite{ballato} and in Nye \cite{nye}.
|
|
|
|
The $e$ and $d$ tensors are related by
|
|
%
|
|
\beq
|
|
e_{j\alpha}=C^{(\E)}_{jk}\,d_{k\alpha}
|
|
\qquad \hbox{or, equivalently,} \qquad
|
|
d_{j\alpha}=S^{(\E)}_{jk}\,e_{k\alpha}
|
|
\;\;.
|
|
\eeq
|
|
%
|
|
|
|
One also sometimes defines tensors $g$ and $h$ via
|
|
%
|
|
\label{eq:gh-def}
|
|
\bea
|
|
\delta\eta_j &=& g_{j\alpha}\,\delta D_\alpha
|
|
\nn
|
|
\delta\E_\alpha &=& -\,g_{j\alpha}\,\delta\sigma_j
|
|
\nn
|
|
\delta\sigma_j &=& -\,h_{j\alpha}\,\delta D_\alpha
|
|
\nn
|
|
\delta\E_\alpha &=& -\,h_{j\alpha}\,\delta\eta_j
|
|
\;\;.
|
|
\eea
|
|
%
|
|
(I am not aware of any standard names for these piezoelectric
|
|
tensors.) They are related to $e$ and $d$ via
|
|
%
|
|
\bea
|
|
g_{j\alpha}&=&\beta^{(\sigma)}_{\alpha\beta}\,d_{j\beta}
|
|
\\
|
|
\label{eq:h-relat}
|
|
h_{j\alpha}&=&\beta^{(\eta)}_{\alpha\beta}\,e_{j\beta}
|
|
\;\;.
|
|
\eea
|
|
%
|
|
|
|
The Voigt notation introduces no factors of 2 for shear components
|
|
of $e$ or $h$, but there {\it are} such factors for $d$ and $g$:
|
|
$d_{51}=2d_{xz,x}$, etc.~(see Sec.~\ref{sec:voigt}). Also, note that
|
|
it is more common to find the indices reversed in the literature;
|
|
e.g., this piezoelectric component is more usually referred to as
|
|
`$d_{15}$'.
|
|
|
|
The relations between the elastic tensors defined at fixed $\E$
|
|
and fixed $\D$ in Sec.~\ref{sec:elas} are
|
|
%
|
|
\bea
|
|
C^{(D)}_{jk}&=&C^{(\E)}_{jk}+h_{j\alpha}\,e_{k\alpha}
|
|
\\
|
|
S^{(D)}_{jk}&=&S^{(\E)}_{jk}-g_{j\alpha}\,d_{k\alpha}
|
|
\eea
|
|
%
|
|
and between the dielectric tensors defined at fixed $\eta$
|
|
and fixed $\sigma$ in Sec.~\ref{sec:eps-free-stress} are
|
|
%
|
|
\bea
|
|
\eps^{(\sigma)}_{\alpha\beta}&=&
|
|
\eps^{(\eta)}_{\alpha\beta}+e_{j\alpha}\,d_{j\beta}
|
|
\label{eq:epsfs}
|
|
\\
|
|
\beta^{(\sigma)}_{\alpha\beta}&=&
|
|
\beta^{(\eta)}_{\alpha\beta}-g_{j\alpha}\,h_{j\beta}
|
|
\;\;.
|
|
\eea
|
|
%
|
|
Note that Eq.~(\ref{eq:epsfs}) is equivalent to Eq.~(\ref{eq:chifs}).
|
|
|
|
% %--------------------------------------------------------------
|
|
% \subsection{Electromechanical coupling constants}
|
|
% %--------------------------------------------------------------
|
|
% \label{sec:coup-factor}
|
|
%
|
|
% In the case in which there is a single kind of strain that
|
|
% couples only to a single Cartesian direction of the electric
|
|
% field, so that $C$, $\eps$, and $e$ are simply constants,
|
|
% it is conventional to define
|
|
% %
|
|
% \beq
|
|
% k^2={e^2\over C\eps}
|
|
% \eeq
|
|
% %
|
|
% which is variously known as an {\it electromechanical coupling constant}
|
|
% or a {\it piezoelectric coupling factor}. (A case in point would be
|
|
% a tetragonal ferroelectric such as PbTiO$_3$, in which $C=C_{11}$,
|
|
% $\eps=\eps_{xx}$, and $e=e_{1x}$ for tetragonal axis along $x$.)
|
|
%
|
|
% Note that $k$ is a dimensionless measure of the strength of the
|
|
% piezoelectric coupling. The meaning of $k$ can also be
|
|
% seen by ... instability ... efficiency ... {\sl [ FINISH ]}
|
|
%
|
|
% In the general case, it is possible to compute the electromechanical
|
|
% coupling tensor as
|
|
% %
|
|
% \beq
|
|
% k_{j\alpha}=(C^{-1/2})_{jk}\;(\eps^{-1/2})_{\alpha\beta}\;e_{k\beta}
|
|
% \;\;.
|
|
% \eeq
|
|
% %
|
|
% In a simple case, like that of tetragonal PbTiO$_3$, this tensor may
|
|
% have just a single non-zero element, which will be the $k$ discussed
|
|
% above. However, in general, one can decompose it into ``active
|
|
% channels'' via the singular value decomposition
|
|
% %
|
|
% \beq
|
|
% k=U\,\kappa\,V^\dagger
|
|
% \eeq
|
|
% %
|
|
% where $U$ is a 6$\times$6 unitary matrix, $\kappa$ is a positive
|
|
% real diagonal 3$\times$6 matrix (that is, the only non-zero
|
|
% elements are $\kappa_{11}$, $\kappa_{22}$, $\kappa_{33}$, normally
|
|
% abbreviated to just $\kappa_1$ etc., and these are all real and
|
|
% positive), and $V$ is a 3$\times$3 unitary matrix. There are
|
|
% standard mathematical library subroutines for doing this
|
|
% decomposition. Then the diagonal elements of $\kappa$, which are
|
|
% known as the ``singular values,'' are the relevant
|
|
% electromechanical coupling constants. It may also be of interest
|
|
% to report, for each $\kappa_i$, the corresponding mechanical strain
|
|
% channel (i.e., print out the $i$'th column of of $U$) and electric
|
|
% field channel (i.e., print out the $i$'th column of of $V$) that
|
|
% are electromechanically coupled by $\kappa_i$.
|
|
%
|
|
% %==============================================================
|
|
% \section{More on phonon modes and lattice dielectric response}
|
|
% %==============================================================
|
|
% \label{sec:ir-phonons}
|
|
%
|
|
% %==============================================================
|
|
% \section{Thermodynamic energy functions}
|
|
% %==============================================================
|
|
% \label{sec:thermo}
|
|
%
|
|
% {\sl To be written.}
|
|
|
|
%==============================================================
|
|
\section{Voigt notation}
|
|
%==============================================================
|
|
\label{sec:voigt}
|
|
|
|
%--------------------------------------------------------------
|
|
\subsection{Basic formulation}
|
|
%--------------------------------------------------------------
|
|
|
|
There are two systems that are commonly in use to index quantities
|
|
that depend on strains or stresses: the ``true tensor notation'' in which
|
|
all Cartesian indices are written explicitly, and the ``Voigt notation''
|
|
in which a reduced index is used:
|
|
$xx\rightarrow 1$, $yy\rightarrow 2$, $zz\rightarrow 3$,
|
|
$yz\rightarrow 4$, $xz\rightarrow 5$, $xy\rightarrow 6$.
|
|
The stress elements are defined simply by making this index
|
|
replacement:
|
|
%
|
|
\bea
|
|
\label{eq:voigt-sig}
|
|
&& \sigma_1=\sigma_{xx} \qquad
|
|
\sigma_4=\sigma_{yz} \nonumber \\
|
|
&& \sigma_2=\sigma_{yy} \qquad
|
|
\sigma_5=\sigma_{xz} \\
|
|
&& \sigma_3=\sigma_{zz} \qquad
|
|
\sigma_6=\sigma_{xy} \nonumber
|
|
\eea
|
|
%
|
|
However, the {\it strain elements have factors of two inserted in
|
|
the definition of the shear elements}:
|
|
%
|
|
\bea
|
|
\label{eq:voigt-eta}
|
|
&& \eta_1=\eta_{xx} \qquad
|
|
\eta_4=2\eta_{yz} \nonumber \\
|
|
&& \eta_2=\eta_{yy} \qquad
|
|
\eta_5=2\eta_{xz} \\
|
|
&& \eta_3=\eta_{zz} \qquad
|
|
\eta_6=2\eta_{xy} \nonumber
|
|
\eea
|
|
|
|
The context here is that $\eta$ is a symmetric
|
|
tensor, i.e., $\eta_{xy}=\eta_{yx}$, defined via
|
|
$\eta_{\alpha\beta}=\half(u_{\alpha,\beta}+u_{\beta,\alpha})$.
|
|
Here $u_{\alpha,\beta}=\partial u_\alpha/\partial x_\beta$ is
|
|
the unsymmetrized tensor defined in term of spatial derivatives
|
|
$\partial/\partial x_\beta$ of the medium displacement $u_\alpha$.
|
|
Thus, we could
|
|
alternatively have written $\eta_6=\eta_{xy}+\eta_{yx}$, etc.
|
|
|
|
The reason for the introduction of these factors of 2 is explained
|
|
very nicely in the book by Nye, Ref.~\cite{nye}
|
|
(see, e.g., Secs.~VII.2-3 and VIII.2 therein). I give a brief
|
|
discussion based on the thermodynamic energy function
|
|
$H$ of Sec.~\ref{sec:notation}. The change $dH$ of the energy
|
|
should be the same in both frameworks. In the true tensor
|
|
framework, $H$ is a a function of 9 variables, and we can write
|
|
%
|
|
\beq
|
|
dH= \sigma_{xx} \, d\eta_{xx} + ...
|
|
+ \sigma_{yz} \, d\eta_{yz} + ...
|
|
+ \sigma_{zy} \, d\eta_{zy} + ...
|
|
\eeq
|
|
%
|
|
where each `$...$' indicates two more terms obtained by cyclic
|
|
permutation of $(xyz)$.
|
|
On the other hand, in the Voigt framework, the same $dH$ can
|
|
be written as a sum of only six terms as
|
|
%
|
|
\beq
|
|
dH= \sigma_1 \, d\eta_1 + ...
|
|
+ \sigma_4 \, d\eta_4 + ...
|
|
\eeq
|
|
%
|
|
where each `$...$' indicates two more terms obtained by cyclic
|
|
permutation $(123)$ or $(456)$.
|
|
Comparing these two equations, which must be equal, and using that
|
|
$\eta_{yz}=\eta_{zy}$, it is clear that a factor of two must be inserted
|
|
into the defining connection either between $\sigma_{yz}$ and $\sigma_4$,
|
|
or else between $\eta_{yz}$ and $\eta_4$. The Voigt notation arises
|
|
from making the second choice, i.e., Eqs.~(\ref{eq:voigt-sig}) and
|
|
(\ref{eq:voigt-eta}).
|
|
|
|
In practice, this means that if you want to introduce a Voigt strain
|
|
$\eta_6=t$ into an \ABINIT\ calculation (say in order to check, by
|
|
finite differences, the computation of the stress or elastic tensor),
|
|
then one should set $\eta_{xy}=\eta_{yx}=t/2$ when constructing the
|
|
unit cell. For example, if the original cell vectors describe
|
|
a simple cubic lattice, then the new lattice vectors could be
|
|
${\bf a}_1'=a(1,t/2,0)$,
|
|
${\bf a}_2'=a(t/2,1,0)$,
|
|
and ${\bf a}_3'=a(0,0,1)$;
|
|
or they could be
|
|
${\bf a}_1'=a(1,t,0)$,
|
|
${\bf a}_2'=a(0,1,0)$,
|
|
and ${\bf a}_3'=a(0,0,1)$; etc.
|
|
(In the first case, $u_{x,y}=u_{y,x}=t/2$;
|
|
in the second case, $u_{x,y}=0$ and $u_{y,x}=t$; in both cases,
|
|
$\eta_{xy}=t/2$ and $\eta_6=t$.)
|
|
Then, if $H$ is expanded in powers of $t$, the coefficient of the
|
|
linear term is just $\sigma_6$, the coefficient of the
|
|
quadratic term is just $C_{66}$, etc.
|
|
|
|
%--------------------------------------------------------------
|
|
\subsection{Systematics}
|
|
%--------------------------------------------------------------
|
|
|
|
We can understand when to insert, or not to insert, factors of 2
|
|
in the relations between Voigt and Cartesian notations by following
|
|
the following rules:
|
|
|
|
\begin{itemize}
|
|
|
|
\item
|
|
Any inserted factor of two makes the Voigt object larger: $Q_{...4...}=
|
|
2Q_{...yz...}$
|
|
|
|
\item
|
|
If there is a factor of $\eta$ in the numerator of a derivative,
|
|
{\it do insert} a factor of 2 for each shear component.
|
|
|
|
\item
|
|
If there is a factor of $\sigma$ in the numerator of a derivative,
|
|
no factors are needed.
|
|
|
|
\item
|
|
If there is a derivative with respect to $\eta$, no factors are needed.
|
|
|
|
\item
|
|
If there is a derivative with respect to $\sigma$, {\it do insert}
|
|
a factor of 2 for each shear component.
|
|
|
|
\end{itemize}
|
|
|
|
\noindent
|
|
Thus, there is no need to insert factors of two when interpreting
|
|
most of the objects we have introduced, including $\sigma$, $C$, $e$,
|
|
$\Lambda$, and $\Gamma$. However, we {\it do} need factors for $S$;
|
|
since $S_{ij}=d\eta_i/d\sigma_j$, we need $S_{14}=2S_{xx,yz}$,
|
|
$S_{44}=4S_{yz,yz}$, etc.
|
|
As for the piezoelectric
|
|
tensors, it is clear from Eqs.~(\ref{eq:e-def}-\ref{eq:h-relat})
|
|
and the rules given above that factors of 2
|
|
{\it are} needed for $d$ and $g$, but not for $e$ and $h$.
|
|
|
|
Finally, in the linear-response calculation, one needs to calculate
|
|
and store objects such the first-order derivatives of Bloch
|
|
wavefunctions with respect to strains, i.e., $d\vert u_{n{\bf k}}\rangle
|
|
/d\eta_j$. This is a derivative with respect to strain, so by
|
|
the above rules, there is no need to worry about a factor of two
|
|
in the definition of this object.
|
|
|
|
%--------------------------------------------------------------
|
|
\subsection{A word about terminology}
|
|
%--------------------------------------------------------------
|
|
|
|
Nye (Ref.~\cite{nye}) refers to the arrays of Voigt elements as
|
|
``matrices'' while the arrays of Cartesian-labeled elements are
|
|
called ``tensors.'' In his terminology, the term ``tensor'' is
|
|
reserved for objects that transform under rotations by the
|
|
application of the associated 3$\times$3 rotation matrix to each
|
|
Cartesian index. Thus, for example, the rotated piezoelectric
|
|
tensor is given by
|
|
%
|
|
\beq
|
|
e_{\beta\gamma,\alpha}'= R_{\beta\mu} \, R_{\gamma\nu}\,
|
|
R_{\alpha\tau}\,e_{\mu\nu,\tau}
|
|
\eeq
|
|
%
|
|
where $R$ is the rotation matrix and there are implicit sums over
|
|
repeated indices. Since the the $6\times3$ matrix
|
|
of $e_{j\alpha}$ elements in the Voigt notation does {\it not}
|
|
transform in a similar way, Nye takes pains to call the
|
|
Voigt $e_{j\alpha}$ a ``matrix'' and not a ``tensor.''
|
|
|
|
While Nye's point is very well taken, the habit of
|
|
referring to the Voigt elements $e_{j\alpha}$ as elements of the
|
|
``piezoelectric tensor'' is by now rather widely ingrained, and in my
|
|
opinion it is rather too pedantic to insist on the narrow definition.
|
|
In these notes, therefore, I generally use the term ``tensor'' to
|
|
refer indiscriminately to either the Voigt or the fully Cartesian
|
|
notations. However, in deference to Nye, I do sometimes use the
|
|
extended phase ``true tensor notation'' to refer to the fully
|
|
Cartesian notation.
|
|
|
|
|
|
%==============================================================
|
|
\section{Summary}
|
|
%==============================================================
|
|
|
|
As of January 2004, the calculation of the following quantities
|
|
has been implemented for the proposed update of \ANADDB\ to version 4.3:
|
|
|
|
\begin{itemize}
|
|
%
|
|
\item All six ``bare'' tensors $K$, $C$, $\chi$,
|
|
$\Lambda$, $Z$ and $e$, Sec.~\ref{sec:notation}.
|
|
%
|
|
\item The clamped-ion compliance tensor $S$ of Sec.~\ref{sec:compliance}.
|
|
%
|
|
\item The displacement-response internal-strain tensor
|
|
$\Gamma$ of Sec.~\ref{sec:internal}.
|
|
%
|
|
\item The relaxed-ion elastic, dielectric, and piezoelectric
|
|
tensors ($\wt{C}$, $\wt{\chi}$, and $\wt{e}$) of Sec.~\ref{sec:formulation},
|
|
and the relaxed-ion compliance tensor $S^{(\E)}$ of Sec.~\ref{sec:elas}.
|
|
%
|
|
\end{itemize}
|
|
|
|
\noindent
|
|
It is our intention to include the ability to calculate additional
|
|
tensors, including many of the ones defined in Secs.~\ref{sec:elas}
|
|
and \ref{sec:piezo-tens}, in a future release of the \ANADDB\ module
|
|
of \ABINIT.
|
|
|
|
|
|
%==============================================================
|
|
\begin{thebibliography}{99}
|
|
%==============================================================
|
|
|
|
\bibitem{abi} {\sf ABINIT} is a common project of the
|
|
Universit\'{e} Catholique de Louvain,
|
|
Corning Incorporated, and other contributors (https://www.abinit.org).
|
|
X. Gonze, J.-M. Beuken, R. Caracas, F. Detraux, M. Fuchs, G.-M. Rignanese,
|
|
L. Sindic, M. Verstraete, G. Zerah, F. Jollet, M. Torrent, A. Roy,
|
|
M. Mikami, Ph. Ghosez, J.-Y. Raty, D.C. Allan,
|
|
Comput. Mater. Sci. {\bf 25}, 478-492 (2002).
|
|
|
|
\bibitem{oganov}
|
|
A set of notes on elastic constants can be found in the file
|
|
{\sl `elasticity-oganov.pdf'} written by A.~Oganov and located in
|
|
the {\sl /Infos} subdirectory of the {\sf ABINIT} distribution.
|
|
|
|
\bibitem{lines} M. E. Lines and A. M. Glass, {\it Principles and
|
|
Applications of Ferroelectrics and Related Materials,} (Clarendon
|
|
Press, Oxford, 1977).
|
|
|
|
\bibitem{nye} J. F. Nye, {\it Physical properties of crystals}
|
|
(Oxford U.P., Oxford 1985).
|
|
|
|
\bibitem{ballato} A.~Ballato, IEEE Transac. Ultrason. Ferro. and
|
|
Freq. Control {\bf 42}, 916 (1995).
|
|
|
|
\bibitem{waghmare} U.~Waghmare, unpublished.
|
|
|
|
\bibitem{baroni87} S.~Baroni, P. ~Giannozzi and A.~Testa,
|
|
Phys. Rev. Lett. {\bf 78}, 1861 (1987).
|
|
|
|
\bibitem{degir91} P. Giannozzi, S. de Gironcoli, P. Pavone, and S. Baroni,
|
|
Phys. Rev. B {\bf 43}, 7231 (1991).
|
|
|
|
\bibitem{gonze95a} X.~Gonze, Phys. Rev. A {\bf 52}, 1096 (1995).
|
|
|
|
\bibitem{gonze97a} X.~Gonze,
|
|
Phys. Rev. B {\bf 55}, 10337 (1997).
|
|
|
|
\bibitem{gonze97b} X.~Gonze and C. ~Lee, Phys. Rev. B {\bf 55}, 10355 (1997).
|
|
|
|
\bibitem{rmp-baroni} S. Baroni, S. de Gironcoli, A. Dal Corso,
|
|
and P. Giannozzi, Rev. Mod. Phys. 73, 515 (2001).
|
|
|
|
\bibitem{ksv} R.D.~King-Smith and D.~Vanderbilt,
|
|
Phys. Rev. B {\bf 47}, 1651 (1993).
|
|
|
|
\bibitem{rmp-resta} R. Resta, Rev. Mod. Phys. 66, 899 (1994).
|
|
|
|
\bibitem{nunes01} R. W. Nunes and X. Gonze, Phys. Rev. B {\bf 63}, 155107
|
|
(2001).
|
|
|
|
\bibitem{souza-ef} I.~Souza, J.~\'I\~niguez, and D.~Vanderbilt,
|
|
Phys. Rev. Lett. {\bf 89}, 117602 (2002).
|
|
|
|
\bibitem{dv-piezo} D. Vanderbilt, J. Phys. Chem. Solids {\bf 61}, 147 (2000).
|
|
|
|
\end{thebibliography}
|
|
|
|
|
|
|
|
\end {document}
|