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A. Appendix: Electron-phonon coefficients

The electron-phonon coefficients g are defined as

g$\scriptstyle \bf q$$\scriptstyle \nu$($\displaystyle \bf k$, i, j) = $\displaystyle \left(\vphantom{{\hbar\over 2M\omega_{{\bf q}\nu}}}\right.$$\displaystyle {\hbar\over 2M\omega_{{\bf q}\nu}}$$\displaystyle \left.\vphantom{{\hbar\over 2M\omega_{{\bf q}\nu}}}\right)^{{1/2}}_{}$$\displaystyle \langle$$\displaystyle \psi_{{i,{\bf k}}}^{}$|$\displaystyle {dV_{SCF}\over d {\hat u}_{{\bf q}\nu}}$ . $\displaystyle \hat{\epsilon}_{{{\bf q}\nu}}^{}$|$\displaystyle \psi_{{j,{\bf k}+{\bf q}}}^{}$$\displaystyle \rangle$. (1)
The phonon linewidth $ \gamma_{{{\bf q}\nu}}^{}$ is defined by

$\displaystyle \gamma_{{{\bf q}\nu}}^{}$ = 2$\displaystyle \pi$$\displaystyle \omega_{{{\bf q}\nu}}^{}$$\displaystyle \sum_{{ij}}^{}$$\displaystyle \int$$\displaystyle {d^3k\over \Omega_{BZ}}$| g$\scriptstyle \bf q$$\scriptstyle \nu$($\displaystyle \bf k$, i, j)|2$\displaystyle \delta$(e$\scriptstyle \bf q$, i - eF)$\displaystyle \delta$(e$\scriptstyle \bf k$+$\scriptstyle \bf q$, j - eF), (2)
while the electron-phonon coupling constant $ \lambda_{{{\bf q}\nu}}^{}$ for mode $ \nu$ at wavevector $ \bf q$ is defined as

$\displaystyle \lambda_{{{\bf q}\nu}}^{}$ = $\displaystyle {\gamma_{{\bf q}\nu} \over \pi\hbar N(e_F)\omega^2_{{\bf q}\nu}}$ (3)
where N(eF) is the DOS at the Fermi level. The spectral function is defined as

$\displaystyle \alpha^{2}_{}$F($\displaystyle \omega$) = $\displaystyle {1\over 2\pi N(e_F)}$$\displaystyle \sum_{{{\bf q}\nu}}^{}$$\displaystyle \delta$($\displaystyle \omega$ - $\displaystyle \omega_{{{\bf q}\nu}}^{}$)$\displaystyle {\gamma_{{\bf q}\nu}\over\hbar\omega_{{\bf q}\nu}}$. (4)
The electron-phonon mass enhancement parameter $ \lambda$ can also be defined as the first reciprocal momentum of the spectral function:

$\displaystyle \lambda$ = $\displaystyle \sum_{{{\bf q}\nu}}^{}$$\displaystyle \lambda_{{{\bf q}\nu}}^{}$ = 2$\displaystyle \int$$\displaystyle {\alpha^2F(\omega) \over \omega}$d$\displaystyle \omega$. (5)

Note that a factor M-1/2 is hidden in the definition of normal modes as used in the code.

McMillan:

Tc = $\displaystyle {\Theta_D \over 1.45}$exp$\displaystyle \left[\vphantom{
{-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}}\right.$$\displaystyle {-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}$$\displaystyle \left.\vphantom{
{-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}}\right]$ (6)
or (better?)

Tc = $\displaystyle {\omega_{log}\over 1.2}$exp$\displaystyle \left[\vphantom{
{-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}}\right.$$\displaystyle {-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}$$\displaystyle \left.\vphantom{
{-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}}\right]$ (7)
where

$\displaystyle \omega_{{log}}^{}$ = exp$\displaystyle \left[\vphantom{ {2\over\lambda} \int {d\omega\over\omega}
\alpha^2F(\omega) \mbox{log}\omega }\right.$$\displaystyle {2\over\lambda}$$\displaystyle \int$$\displaystyle {d\omega\over\omega}$$\displaystyle \alpha^{2}_{}$F($\displaystyle \omega$)log$\displaystyle \omega$$\displaystyle \left.\vphantom{ {2\over\lambda} \int {d\omega\over\omega}
\alpha^2F(\omega) \mbox{log}\omega }\right]$ (8)


next up previous contents
Next: About this document ... Up: User's Guide for the Previous: 6 Troubleshooting   Contents
Layla Martin-Samos Colomer 2012-05-10