スピントロニクス理論の基礎/8-11 のバックアップ(No.1)

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目次はこちら >> スピントロニクス理論の基礎

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}^<

  1. \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 () g_{\bm k,\bm k',\omega}\propto \delta_{\bm k,\bm k'} が得られ、

(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 );

である。

(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)

&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)}

);


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