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The Berry Phase Technique.

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

$\displaystyle \Delta {\bf P}_{e}= \int^{\lambda_{2}}_{\lambda_{1}}{\partial {\bf P}_{e} \over\partial \lambda} d\lambda$ (A.2)

with

$\displaystyle {\partial {\bf P}_{e} \over \partial \lambda}= {i \vert e\vert \h...
...bda\right)}_{n{\bf k}}-\epsilon^{\left(\lambda\right)}_{m{\bf k}} \right)^{2}}+$   c.c. (A.3)

where $ m_{e}$ and $ e$ are the electronic mass and charge, $ N$ is the number of unit cells in the crystal, $ \Omega_{0}$ is the unit cell volume, $ M$ is the number of occupied bands, $ {\bf p}$ is the momentum operator, and the functions $ \psi^{\left(\lambda\right)}_{n{\bf k}}$ are the usual Bloch solutions to the crystalline Hamiltonian. Within Kohn-Sham density-functional theory, the potential $ V^{\left(\lambda\right)}$ is to be interpreted as the Kohn-Sham potential $ V^{\left(\lambda\right)}_{KS}$, where $ \lambda$ 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 $ \psi^{\left(\lambda\right)}_{m{\bf k}}$ no longer explicitly appear, and they show that the change in polarization given by Eq. (2) along an arbitrary path, can be found from only a knowledge of the system at the end points

$\displaystyle \Delta {\bf P}_{e}= {\bf P}^{\left(\lambda_{2} \right)}_{e} - {\bf P}^{\left(\lambda_{1}\right)}_{e}$ (A.4)

with

$\displaystyle {\bf P}^{\left(\lambda\right)}_{e}=-{if\vert e\vert\over 8\pi^{3}...
...}} \vert\> \nabla_{{\bf k}} \>\vert u^{\left(\lambda\right)}_{n{\bf k}} \rangle$ (A.5)

where $ f$ is the occupation number of the states in the valence bands, $ u^{\left(\lambda\right)}_{n{\bf k}}$ is the cell-periodic part of the Bloch function $ \psi^{\left(\lambda\right)}_{n{\bf k}}$, and the sum $ n$ runs over all $ M$ occupied bands.

The physics behind Eq. (5) becomes more transparent when this expression is written in terms of the Wannier functions of the occupied bands,

$\displaystyle {\bf P}^{\left(\lambda\right)}_{e}=-{f \vert e\vert \over \Omega_...
...eft(\lambda\right)}_{n} \vert {\bf r} \vert W^{\left(\lambda\right)}_{n}\rangle$ (A.6)

where $ W_{n}$ is the Wannier function corresponding to valence band $ n$.

Eq. (6) 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 $ {\bf r}_{n} = \langle W^{\left(\lambda\right)}_{n} \vert {\bf r} \vert 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. (5) or (6), and consequently the change in polarization as given by Eq. (4) is only well-defined modulo $ fe{\bf R}/\Omega_{0}$, where $ {\bf R}$ is a lattice vector. This indeterminacy stems from the fact that the charge center of a Wannier function is only invariant modulo $ {\bf R}$, with respect to the choice of phase of the Bloch functions.

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


next up previous
Next: Computational Aspects. Up: Modern Polarization Theory Previous: Modern Polarization Theory
Georg Kresse
2001-03-23