Berry phases and finite electric fields: Difference between revisions

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Central to the modern theory of polarization is the proposition of Resta<ref name="Resta92"/> to write the electronic contribution to the change in polarization due to a finite adiabatic change in the Kohn-Sham Hamiltonian of the crystalline solid, as
Central to the modern theory of polarization is the proposition of Resta<ref name="Resta92"/> to write the electronic contribution to the change in polarization due to a finite adiabatic change in the Kohn-Sham Hamiltonian of the crystalline solid, as


<span id="\label{PolarizationChange1">
<span id="PolarizationChange1">
:<math>
:<math>
\Delta \mathbf{P}_{e}= \int^{\lambda_{2}}_{\lambda_{1}}{\partial \mathbf{P}_{e}
\Delta \mathbf{P}_{e}= \int^{\lambda_{2}}_{\lambda_{1}}{\partial \mathbf{P}_{e}
Line 32: Line 32:
King-Smith and Vanderbilt<ref name="Vanderbilt93I"/> have cast this expression in a form in which the conduction band states &psi;<sup>(&lambda;)</sup><sub>m'''k'''</sub> no longer explicitly appear, and they show that [[#PolarizationChange1|the change in polarization]] along an arbitrary path, can be found from only a knowledge of the system at the end points
King-Smith and Vanderbilt<ref name="Vanderbilt93I"/> have cast this expression in a form in which the conduction band states &psi;<sup>(&lambda;)</sup><sub>m'''k'''</sub> no longer explicitly appear, and they show that [[#PolarizationChange1|the change in polarization]] along an arbitrary path, can be found from only a knowledge of the system at the end points


<math>\label{PolarizationChange2}
<span id="PolarizationChange2">
\Delta \mathbf{P}_{e}= \mathbf{P}^{\left(\lambda_{2} \right)}_{e} - {\bf
:<math>
P}^{\left(\lambda_{1}\right)}_{e}
\Delta \mathbf{P}_{e}= \mathbf{P}^{\left(\lambda_{2} \right)}_{e} - \mathbf{P}^{\left(\lambda_{1}\right)}_{e}
</math>
</math>
</span>


\noindent with
with


<math>\label{Polarization}
<span id="Polarization">
\mathbf{P}^{\left(\lambda\right)}_{e}=-{if|e|\over 8\pi^{3}} \sum^{M}_{n=1}
:<math>
\int_{BZ} d^{3}k \langle u^{\left(\lambda\right)}_{n\mathbf{k}} |\>
\mathbf{P}^{\left(\lambda\right)}_{e}=-{if|e|\over 8\pi^{3}} \sum^{M}_{n=1}\int_{BZ} d^{3}k \langle u^{\left(\lambda\right)}_{n\mathbf{k}} | \nabla_{\mathbf{k}} | u^{\left(\lambda\right)}_{n\mathbf{k}} \rangle
\nabla_{\mathbf{k}} \>| u^{\left(\lambda\right)}_{n\mathbf{k}} \rangle
</math>
</math>
</span>


\noindent where $f$ is the occupation number of the states in the valence
where ''f'' is the occupation number of the states in the valence bands, u<sup>(&lambda;)</sup><sub>n'''k'''</sub> is the cell-periodic part of the Bloch function &psi;<sup>(&lambda;)</sup><sub>n'''k'''</sub>, and the sum ''n'' runs over all ''M'' occupied bands.
bands, $u^{\left(\lambda\right)}_{n\mathbf{k}}$ is the cell-periodic part of the
Bloch function $\psi^{\left(\lambda\right)}_{n\mathbf{k}}$, and the sum $n$ runs
over all $M$ occupied bands.


The physics behind Eq.\ (\ref{Polarization}) becomes more transparent when this
The physics behind [[#Polarization|the equation above]] becomes more transparent when this expression is written in terms of the Wannier functions of the occupied bands,
expression is written in terms of the Wannier functions of the occupied bands,


<math>\label{Wannier}
<span id="Wannier">
:<math>
\mathbf{P}^{\left(\lambda\right)}_{e}=-{f |e| \over \Omega_{0}} \sum^{M}_{n=1}
\mathbf{P}^{\left(\lambda\right)}_{e}=-{f |e| \over \Omega_{0}} \sum^{M}_{n=1}
\langle W^{\left(\lambda\right)}_{n} | \mathbf{r} |W^{\left(\lambda\right)}_{n}
\langle W^{\left(\lambda\right)}_{n} | \mathbf{r} |W^{\left(\lambda\right)}_{n}
\rangle
\rangle
</math>
</math>
</span>
where W<sub>n</sub> is the Wannier function corresponding to valence band ''n''.


\noindent where $W_{n}$ is the Wannier function corresponding to valence band
This shows the change in polarization of a solid, induced by an adiabatic change in the Hamiltonian, to be proportional to the displacement of the charge centers '''r'''<sub>n</sub>=&lang; W<sup>(&lambda;)</sup><sub>n</sub>|'''r'''| W<sup>(&lambda;)</sup><sub>n</sub>&rang;, of the Wannier functions corresponding to the valence bands.
$n$.


Eq.\ (\ref{Wannier}) shows the change in polarization of a solid, induced by an
It is important to realize that the [[#Polarization|polarization in terms of Bloch]] or [[#Wannier|Wannier]] functions, and consequently the [[#PolarizationChange2|change in polarization]], is only well-defined modulo ''fe'''''R'''/&Omega;<sub>0</sub>, where '''R''' is a lattice vector. This indeterminacy stems from the fact that the charge center of a Wannier function is only invariant modulo '''R''', with respect to the choice of phase of the Bloch functions.
adiabatic change in the Hamiltonian, to be proportional to the displacement of
the charge centers $\mathbf{r}_{n} = \langle W^{\left(\lambda\right)}_{n} |
\mathbf{r} | W^{\left(\lambda\right)}_{n} \rangle$, of the Wannier functions
corresponding to the valence bands.


It is important to realize that the polarization as given by Eq.\
In practice one is usually interested in polarization changes |&Delta;'''P'''<sub>e</sub>| << |''fe'''''R'''<sub>1</sub>/&Omega;<sub>0</sub>|, where '''R'''<sub>1</sub> is the shortest nonzero lattice vector. An arbitrary term ''fe'''''R'''/&Omega;<sub>0</sub> can therefore often be removed by simple inspection of the results. In cases where |&Delta;'''P'''<sub>e</sub>| is of the same order of magnitude as ''fe'''''R'''<sub>1</sub>/&Omega;<sub>0</sub> any uncertainty can always be removed by dividing the total change in the Hamiltonian &lambda;<sub>1</sub>&rarr;&lambda;<sub>2</sub> into a number of intervals.
(\ref{Polarization}) or (\ref{Wannier}), and consequently the change in
polarization as given by Eq.\ (\ref{PolarizationChange2}) is only well-defined
modulo $fe\mathbf{R}/\Omega_{0}$, where $\mathbf{R}$ is a lattice vector. This
indeterminacy stems from the fact that the charge center of a Wannier function
is only invariant modulo $\mathbf{R}$, with respect to the choice of phase of
the
Bloch functions.


In practice one is usually interested in polarization changes $|\Delta{\bf
P}_{e}| \ll |fe\mathbf{R}_{1}/\Omega_{0}|$, where $\mathbf{R}_{1}$ is the
shortest
nonzero lattice vector.  An arbitrary term $fe\mathbf{R}/\Omega_{0}$ can
therefore often be removed by simple inspection of the results. In cases where
$|\Delta\mathbf{P}_{e}|$ is of the same order of magnitude as $|fe{\bf
R}_{1}/\Omega_{0}|$ any uncertainty can always be removed by dividing the total
change in the Hamiltonian $\lambda_{1} \rightarrow \lambda_{2}$ into a number
of intervals.
== Self-consistent response to finite electric fields ==
== Self-consistent response to finite electric fields ==



Revision as of 14:58, 6 March 2011

Berry phase expression for the macroscopic polarization

Calculating the change in dipole moment per unit cell under PBC's, is a nontrivial task. In general one cannot define it as the first moment of the induced change in charge density δ(r), through

without introducing a dependency on the shape of Ω0, the chosen unit cell.[1]

Recently King-Smith and Vanderbilt[2], building on the work of Resta[3], showed that the electronic contribution to the difference in polarization ΔPe, due to a finite adiabatic change in the Hamiltonian of a system, can be identified as a geometric quantum phase or Berry phase of the valence wave functions. We will briefly summarize the essential results (for a review of geometric quantum phases in polarization theory see the papers of Resta[4][5]).

Central to the modern theory of polarization is the proposition of Resta[3] to write the electronic contribution to the change in polarization due to a finite adiabatic change in the Kohn-Sham Hamiltonian of the crystalline solid, as

with

where me and e are the electronic mass and charge, N is the number of unit cells in the crystal, Ω0 is the unit cell volume, M is the number of occupied bands, p is the momentum operator, and the functions ψ(λ)nk are the usual Bloch solutions to the crystalline Hamiltonian. Within Kohn-Sham density-functional theory, the potential V(λ) is to be interpreted as the Kohn-Sham potential V(λ)KS, where λ parameterizes some change in this potential, for instance due to the displacement of an atom in the unit cell.

King-Smith and Vanderbilt[2] have cast this expression in a form in which the conduction band states ψ(λ)mk no longer explicitly appear, and they show that the change in polarization along an arbitrary path, can be found from only a knowledge of the system at the end points

with

where f is the occupation number of the states in the valence bands, u(λ)nk is the cell-periodic part of the Bloch function ψ(λ)nk, and the sum n runs over all M occupied bands.

The physics behind the equation above becomes more transparent when this expression is written in terms of the Wannier functions of the occupied bands,

where Wn is the Wannier function corresponding to valence band n.

This shows the change in polarization of a solid, induced by an adiabatic change in the Hamiltonian, to be proportional to the displacement of the charge centers rn=⟨ W(λ)n|r| W(λ)n⟩, of the Wannier functions corresponding to the valence bands.

It is important to realize that the polarization in terms of Bloch or Wannier functions, and consequently the change in polarization, is only well-defined modulo feR0, where R is a lattice vector. This indeterminacy stems from the fact that the charge center of a Wannier function is only invariant modulo R, with respect to the choice of phase of the Bloch functions.

In practice one is usually interested in polarization changes |ΔPe| << |feR10|, where R1 is the shortest nonzero lattice vector. An arbitrary term feR0 can therefore often be removed by simple inspection of the results. In cases where |ΔPe| is of the same order of magnitude as feR10 any uncertainty can always be removed by dividing the total change in the Hamiltonian λ1→λ2 into a number of intervals.

Self-consistent response to finite electric fields

Related Tags and Sections

LBERRY, IGPAR, NPPSTR, LPEAD, IPEAD, LCALCPOL, LCALCEPS, EFIELD_PEAD, DIPOL

References

  1. P. Vogl, J. Phys. C: Solid State Phys. 11, 251 (1978).
  2. a b R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993).
  3. a b R. Resta, Ferroelectrtics 136, 51 (1992).
  4. R. Resta, Rev. Mod. Phys. 66, 899 (1994).
  5. R. Resta, in Berry Phase in Electronic Wavefunctions, Troisième Cycle de la Physique en Suisse Romande, Ann%eacute;e Academique 1995-96, (1996).

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