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Phonon-Electric Field Coupling: Born Effective Charges

In polar crystals, long range macroscopic electric fields arise that are associated with long wave longitudinal optical phonons. These electric fields are a result of the long range character of the Coulomb interaction [73], and are responsible for the well-known phenomenon of LO-TO splitting, that is, the shift in frequency between longitudinal optical and transverse optical phonons at the Brillouin zone centre.

The origin of this splitting can be understood most readily if one considers an argument due to Yu and Cardona [95]. Consider a long wave TO phonon propagating along the [111] direction in a zinc-blende polar crystal. The positive and negative ions lie on separate planes perpendicular to the direction of propagation. Under excitation of a TO mode, these planes slide past each other, which is analogous to the oppositely charged plates of a capacitor moving past each other whilst their separation is fixed. Now consider the situation when the ions are excited by a LO mode. In this case, the planes move apart, as can be seen in figure 3.1. In the capacitor, this increase in separation of the charged plates will be accompanied by an extra force due to the electric field between the plates. Similarly, in the crystal, the LO mode is accompanied by an extra restoring force due to the Coulomb interaction. This additional force then leads to the frequency change.

Figure 3.1: Illustration of the displacements of atoms within an ionic crystal during a long-wave longitudinal optical phonon. F is the restoring force due to the displacements of the charges shown. It is this that leads to the LO-TO splitting. Note the similarity to the situation in a parallel-plate capacitor.
\includegraphics[scale=0.8,angle=0]{born_charge.eps}

This coupling between optical phonons and electric fields is quantified by the Born effective charge, which is defined through


\begin{displaymath}
Z^{\star}_{\kappa,\beta\alpha} = \Omega_{0}\frac{\partial\ma...
...c,\beta}}{\partial\tau_{\kappa\alpha}(\mathbf{q}=\mathbf{0})},
\end{displaymath} (3.95)

i.e. the Born effective charge is the coefficient of proportionality between a change in macroscopic polarisation in direction $\beta$ caused by an atomic displacement in direction $\alpha $ under conditions of zero external field.

If one considers the electric enthalpy, defined through


\begin{displaymath}
\tilde{E}[u_{n\mathbf{k}};\mathbf{\varepsilon}] = E_{KS} - \...
...a_{0}\sum_{\alpha}\mathcal{P}_{mac,\alpha}\varepsilon_{\alpha}
\end{displaymath} (3.96)

it can be readily seen that


\begin{displaymath}
\mathcal{P}_{mac,\alpha} = -\frac{1}{\Omega_{0}}\frac{\partial\tilde{E}}{\partial\varepsilon_{\alpha}}
\end{displaymath} (3.97)

allowing the effective charge to be written as


\begin{displaymath}
Z^{\star}_{\kappa,\alpha\beta} = -\frac{\partial^{2}\tilde{E...
...partial\varepsilon_{\beta}}\bigg\vert _{\tau_{\kappa\alpha}=0}
\end{displaymath} (3.98)

and thus an equivalent definition is as the coefficient of proportionality relating a change in the force in direction $\alpha $ due to an homogeneous electric field applied in direction $\beta$ with nuclei clamped in place.

The effective charge may be decomposed into ionic and electronic contributions:


\begin{displaymath}
Z^{\star}_{\kappa,\alpha\beta} = Z_{\kappa}\delta_{\alpha\beta} + Z^{el}_{\kappa,\alpha\beta}
\end{displaymath} (3.99)

where the first term is simply the ionic charge. In the following, discussion will centre around how to calculate the electronic contribution to the effective charge. The three definitions of the effective charge are formally equivalent, but lead to different approaches to the calculation of the effective charge, which will now be considered.



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Next: Berry Phase Approach Up: Density Functional Perturbation Theory Previous: The Dielectric Permittivity   Contents
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