スピントロニクス理論の基礎/8-11 の履歴(No.3)
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8-11 不純物散乱のもとでの lesser Green 関数†
(8.111) を波数表示に直すと、
(8.145), (8.114) より
&math( &g_{\bm k,\bm k',\omega}^<=\delta_{\bm k,\bm k'}g_{0\bm k,\bm k',\omega}^<
- \sum_{\bm q}\big[
g_{0\bm k,\bm k',\omega}^rv_i(\bm q)g_{\bm k+\bm q,\bm k',\omega}^< +g_{0\bm k,\bm k',\omega}^<v_i(\bm q)g_{\bm k+\bm q,\bm k',\omega}^a
\big] \\&= \delta_{\bm k,\bm k'}g_{0\bm k,\bm k',\omega}^<+\\ &\sum_{\bm q}\big[g_{0\bm k,\bm k',\omega}^r v_i(\bm q) \big( \delta_{\bm k+\bm q,\bm k'}g_{0\bm k+\bm q,\bm k',\omega}^< +\sum_{\bm q'}\big[ g_{0\bm k+\bm q,\bm k',\omega}^rv_i(\bm q')g_{\bm k+\bm q+\bm q',\bm k',\omega}^< +g_{0\bm k+\bm q,\bm k',\omega}^<v_i(\bm q')g_{\bm k+\bm q+\bm q',\bm k',\omega}^a \big]\big)\\
&\hspace{4mm}+g_{0\bm k,\bm k',\omega}^< v_i(\bm q) \big( \delta_{\bm k+\bm q,\bm k'}g_{0\bm k+\bm q,\bm k',\omega}^a +\sum_{\bm q'} g_{0\bm k+\bm q,\bm k',\omega}^av_i(\bm q')g_{\bm k+\bm q+\bm q',\bm k',\omega}^a \big)
\big] \\&= \delta_{\bm k,\bm k'}g_{0\bm k,\bm k',\omega}^<+ \sum_{\bm q}\Big[g_{0\bm k,\bm k',\omega}^r v_i(\bm q) \delta_{\bm k+\bm q,\bm k'}g_{0\bm k+\bm q,\bm k',\omega}^< +g_{0\bm k,\bm k',\omega}^< v_i(\bm q) \delta_{\bm k+\bm q,\bm k'}g_{0\bm k+\bm q,\bm k',\omega}^a\Big]+\\
&\sum_{\bm q,\bm q'}\Big[g_{0\bm k,\bm k',\omega}^r v_i(\bm q) g_{0\bm k+\bm q,\bm k',\omega}^rv_i(\bm q')g_{\bm k+\bm q+\bm q',\bm k',\omega}^< + g_{0\bm k,\bm k',\omega}^r v_i(\bm q) g_{0\bm k+\bm q,\bm k',\omega}^<v_i(\bm q')g_{\bm k+\bm q+\bm q',\bm k',\omega}^a\\
&\hspace{4mm}+g_{0\bm k,\bm k',\omega}^< v_i(\bm q) g_{0\bm k+\bm q,\bm k',\omega}^av_i(\bm q')g_{\bm k+\bm q+\bm q',\bm k',\omega}^a
\Big] \\&= \delta_{\bm k,\bm k'}g_{0\bm k,\bm k',\omega}^<+ \sum_{\bm q}\langle v_i(\bm q) \rangle_i\delta_{\bm k+\bm q,\bm k'}\Big[g_{0\bm k,\bm k',\omega}^r g_{0\bm k+\bm q,\bm k',\omega}^< +g_{0\bm k,\bm k',\omega}^< g_{0\bm k+\bm q,\bm k',\omega}^a\Big]+\\
&\sum_{\bm q,\bm q'}\langle v_i(\bm q)v_i(\bm q') \rangle_i\Big[g_{0\bm k,\bm k',\omega}^r g_{0\bm k+\bm q,\bm k',\omega}^rg_{\bm k+\bm q+\bm q',\bm k',\omega}^< + g_{0\bm k,\bm k',\omega}^r g_{0\bm k+\bm q,\bm k',\omega}^<g_{\bm k+\bm q+\bm q',\bm k',\omega}^a +g_{0\bm k,\bm k',\omega}^< g_{0\bm k+\bm q,\bm k',\omega}^ag_{\bm k+\bm q+\bm q',\bm k',\omega}^a
\Big] \\&= \delta_{\bm k,\bm k'}g_{0\bm k,\bm k',\omega}^<+ \sum_{\bm q}0\cdot\delta_{\bm k+\bm q,\bm k'}\Big[g_{0\bm k,\bm k',\omega}^r g_{0\bm k+\bm q,\bm k',\omega}^< +g_{0\bm k,\bm k',\omega}^< g_{0\bm k+\bm q,\bm k',\omega}^a\Big]+\\
&\frac{n_iv_i^2}{N}\sum_{\bm q,\bm q'}\delta_{\bm q+\bm q',\bm 0}\Big[g_{0\bm k,\bm k',\omega}^r g_{0\bm k+\bm q,\bm k',\omega}^rg_{\bm k+\bm q+\bm q',\bm k',\omega}^< + g_{0\bm k,\bm k',\omega}^r g_{0\bm k+\bm q,\bm k',\omega}^<g_{\bm k+\bm q+\bm q',\bm k',\omega}^a +g_{0\bm k,\bm k',\omega}^< g_{0\bm k+\bm q,\bm k',\omega}^ag_{\bm k+\bm q+\bm q',\bm k',\omega}^a
\Big] \\&= \delta_{\bm k,\bm k'}g_{0\bm k,\bm k',\omega}^<+ \frac{n_iv_i^2}{N}\sum_{\bm q}\Big[g_{0\bm k,\bm k',\omega}^r g_{0\bm k+\bm q,\bm k',\omega}^rg_{\bm k,\bm k',\omega}^< + g_{0\bm k,\bm k',\omega}^r g_{0\bm k+\bm q,\bm k',\omega}^<g_{\bm k,\bm k',\omega}^a +g_{0\bm k,\bm k',\omega}^< g_{0\bm k+\bm q,\bm k',\omega}^ag_{\bm k,\bm k',\omega}^a
\Big] \\&= \delta_{\bm k,\bm k'}g_{0\bm k,\bm k',\omega}^<+ \Big[g_{0\bm k,\bm k',\omega}^r \Sigma^r g_{\bm k,\bm k',\omega}^< +g_{0\bm k,\bm k',\omega}^r \Sigma^< g_{\bm k,\bm k',\omega}^a +g_{0\bm k,\bm k',\omega}^< \Sigma^a g_{\bm k,\bm k',\omega}^a
\Big] );
繰り返し代入すると が得られることから、
(8.124)
&math( &g_{\bm k,\omega}^< = g_{0\bm k,\bm k',\omega}^<+ \Big[
g_{0\bm k,\omega}^r \Sigma^r g_{\bm k,\omega}^< +g_{0\bm k,\omega}^r \Sigma^< g_{\bm k,\omega}^a +g_{0\bm k,\omega}^< \Sigma^a g_{\bm k,\omega}^a
\Big] );
を得る。
(8.148)
&math( \Sigma^\alpha(\hbar\omega)\equiv n_iv_i^2\frac{1}{N}\sum_{\bm k}g_{0\bm k,\omega}^\alpha );
(8-10.9) より
(8.149)
&math( \frac{g_{\bm k,\omega}^r-g_{0\bm k,\omega}^r}{g_{\bm k,\omega}^r} = \Sigma^r g_{0\bm k,\omega}^r );
&math( \frac{g_{\bm k,\omega}^a-g_{0\bm k,\omega}^a}{g_{0\bm k,\omega}^a} = \Sigma^a g_{\bm k,\omega}^a );
(8.150)
式を整理すると、
&math( &g_{\bm k,\omega}^< = g_{0\bm k,\omega}^<+ \Big[
g_{0\bm k,\omega}^r \Sigma^r g_{\bm k,\omega}^< +g_{0\bm k,\omega}^r \Sigma^< g_{\bm k,\omega}^a +g_{0\bm k,\omega}^< \Sigma^a g_{\bm k,\omega}^a
\Big] \\&= g_{0\bm k,\omega}^<+ \Big[
\frac{g_{\bm k,\omega}^r-g_{0\bm k,\omega}^r}{g_{\bm k,\omega}^r} g_{\bm k,\omega}^< +g_{0\bm k,\omega}^r \Sigma^< g_{\bm k,\omega}^a +\frac{g_{\bm k,\omega}^a-g_{0\bm k,\omega}^a}{g_{0\bm k,\omega}^a} g_{0\bm k,\omega}^<
\Big] );
&math( \frac{g_{0\bm k,\omega}^r}{g_{\bm k,\omega}^r}g_{\bm k,\omega}^<=
g_{0\bm k,\omega}^r \Sigma^< g_{\bm k,\omega}^a +\frac{g_{\bm k,\omega}^a}{g_{0\bm k,\omega}^a} g_{0\bm k,\omega}^<
);
&math( &g_{\bm k,\omega}^<=
g_{\bm k,\omega}^r \Sigma^< g_{\bm k,\omega}^a +\frac{g_{\bm k,\omega}^r}{g_{0\bm k,\omega}^r}\frac{g_{\bm k,\omega}^a}{g_{0\bm k,\omega}^a} g_{0\bm k,\omega}^<
\\&=
g_{\bm k,\omega}^r \Sigma^< g_{\bm k,\omega}^a +\frac{\hbar\omega-\varepsilon_{\bm k}+i0}{\hbar\omega-\varepsilon_{\bm k}-\Sigma^r} \frac{\hbar\omega-\varepsilon_{\bm k}-i0}{\hbar\omega-\varepsilon_{\bm k}-\Sigma^a} 2\pi i f(\hbar\omega)\delta(\hbar\omega-\varepsilon_{\bm k})
\\&=
g_{\bm k,\omega}^r \Sigma^< g_{\bm k,\omega}^a +2\pi i f(\hbar\omega) \frac{(\hbar\omega-\varepsilon_{\bm k})^2\delta(\hbar\omega-\varepsilon_{\bm k})} {(\hbar\omega-\varepsilon_{\bm k}-\Sigma^r)(\hbar\omega-\varepsilon_{\bm k}-\Sigma^a)}
\\&=
g_{\bm k,\omega}^r \Sigma^< g_{\bm k,\omega}^a
=
\frac{\Sigma^<} {(\hbar\omega-\varepsilon_{\bm k}-\Sigma^r)(\hbar\omega-\varepsilon_{\bm k}-\Sigma^a)}
);
ここで、(8.148), (8.91) より
&math( &\Sigma^< = \frac{n_iv_i^2}{N}\sum_{\bm k} f_{\bm k}(\hbar\omega)(g_{0\bm k,\omega}^a-g_{0\bm k,\omega}^r) \\&= f_{\bm k}(\hbar\omega) (\Sigma^a-\Sigma^r) \\&= f_{\bm k}(\hbar\omega) \left( \frac{i\hbar}{\tau} \right) );
したがって、
&math( &g_{\bm k,\omega}^<=
\frac{f_{\bm k}(\hbar\omega)(\Sigma^a-\Sigma^r)} {(\hbar\omega-\varepsilon_{\bm k}-\Sigma^r)(\hbar\omega-\varepsilon_{\bm k}-\Sigma^a)}
\\&=f_{\bm k}(\hbar\omega)\left[
\frac{1}{\hbar\omega-\varepsilon_{\bm k}-\Sigma^a} -\frac{1}{\hbar\omega-\varepsilon_{\bm k}-\Sigma^r}
\right] \\&=f_{\bm k}(\hbar\omega)\left[g_{\bm k,\omega}^a-g_{\bm k,\omega}^r\right] \\&=2\pi i f_{\bm k}(\hbar\omega)\delta_\Sigma(\hbar\omega-\varepsilon_{\bm k}) );
はフェルミレベルのぼけによりなまったδ関数である。