Non-analytical term of force constant

LO and LA modes in semiconductor or insulator may have long range interaction when $ \mathbf{k}\to \mathbf{0}$. On contrary to these modes, TO and TA modes may have short range interaction. Phonon calculation technique which fropho employs can treat TO, LA and TA mode, but can not treat LO mode since the long range the dipole-dipole interaction of LO mode breaks the periodic boundary condition. To correct this error, non-analytical term is added to dynamical matrix [17,18,19] as follows,

$\displaystyle D_{\alpha\beta}(jj',\mathbf{k}\to \mathbf{0}) = D_{\alpha\beta}^{...
...beta}]} {\sum_{\alpha\beta}k_{\alpha}\epsilon_{\alpha\beta}^{\infty} k_{\beta}}$ (3.17)

where this is given in atomic unit. $ D^{\mathrm{N}}(\mathbf{k})$ is the normal dynamical matrix, $ Z_{j}^{*}$ is the Born effective charge tensor of the $ j$-th atom in the unit cell, $ \Omega_0$ is the volume of the unit cell, and $ \epsilon_{\alpha\beta}^{\infty}$ is the high frequency dielectric contant tensor.

In fropho this is implemented as follows. The non-analytical term is multiplied with a function that is rapidly decreasing with getting away from $ \mathbf{k} = \mathbf{0}$. Then the dynamical matrix of the general $ \mathbf{k}$ is written in,

$\displaystyle D_{\alpha\beta}(jj',\mathbf{k}) = D_{\alpha\beta}^{\mathrm{N}}(jj...
...ta}^{\infty} k_{\beta}} \times \exp(-a\frac{\vert\mathbf{k}\vert^2}{\sigma^2}),$ (3.18)

where $ a$ and $ \sigma$ are the adjustable damping factors. They should be chosen as the damping function becomes close to zero at Brillouin zone surface.

togo 2009-02-12
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