スピントロニクス理論の基礎/8-11 のバックアップソース(No.3)

更新 
[[[前の章へ]>スピントロニクス理論の基礎/8-10]] <<<<　
[[スピントロニクス理論の基礎]]（目次）　>>>>
[[[次の章へ]>スピントロニクス理論の基礎/9-1A]]

* 8-11 不純物散乱のもとでの lesser Green 関数 [#f9ddef59]

(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 &math(g_{\bm k,\bm k',\omega}\propto \delta_{\bm k,\bm k'}); 
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})
);

&math(\delta_\Sigma(\ )); はフェルミレベルのぼけによりなまったδ関数である。

* 質問・コメント [#q187aaf8]

#article_kcaptcha
Counter: 4311 (from 2010/06/03), today: 1, yesterday: 0