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次へ: use transfer matrix 上へ: surface Green function, orthogonal 戻る: surface Green function, orthogonal

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Green function

\begin{displaymath}
(\omega -H) G = I
\end{displaymath} (7)

consider periodic layers with an edge. There exist transfer integrals only between neighboring layers. The layer 0 has the edge.


\begin{displaymath}
H = \left[ \begin{array}{ccccc}
H_{00} & H_{01} & 0 & 0 & 0...
...& \ddots \\
0 & 0 & 0 & \ddots & \ddots
\end{array} \right]
\end{displaymath} (8)

By definition, \( H_{01}=H_{i,i+1}=H_{i+1,i}^\dag , H_{00}=H_{ii}=H_{i+1,i+1} \).

Green function

\begin{displaymath}
G = \left[ \begin{array}{ccccc}
G_{00} & G_{01} & G_{02} & ...
... & & & G_{33} \\
\vdots & & & & \ddots
\end{array} \right]
\end{displaymath} (9)

Use $G_{i0}$ column,

$\displaystyle (\omega -H_{00} )G_{00} - H_{01} G_{10}$ $\textstyle =$ $\displaystyle I$ (10)
$\displaystyle -H_{10} G_{00} + (\omega -H_{11}) G_{10} -H_{12} G_{20}$ $\textstyle =$ $\displaystyle 0$ (11)
$\displaystyle -H_{21} G_{10} + (\omega -H_{22}) G_{20} -H_{23} G_{30}$ $\textstyle =$ $\displaystyle 0$ (12)
$\displaystyle \vdots$     (13)

eq.(10) to eq.(12) is generalized as

\begin{displaymath}
(\omega -H_{nn}) G_{n0} = H_{n,n-1} G_{n-1,0} + H_{n,n+1} G_{n+1,0}
\end{displaymath} (14)

eq.(10) and eq.(14) describes relation between the neighboring layer. rewrite eq.(10) and eq.(14) as

$\displaystyle (\omega -\epsilon _0^s) G_{00}$ $\textstyle =$ $\displaystyle I + \alpha _0 G_{10}$ (15)
$\displaystyle (\omega -\epsilon _0 ) G_{n0}$ $\textstyle =$ $\displaystyle \beta _0 G_{n-1,0} + \alpha _0 G_{n+1,0}$ (16)

The relationship between the second neighboring layer corresponding to eq.(15) is

$\displaystyle (\omega -\epsilon _0^s) G_{00}$ $\textstyle =$ $\displaystyle I + \alpha _0(\omega -\epsilon _0)^{-1} ( \beta _0 G_{0,0} + \alpha _0 G_{2,0} )$  
$\displaystyle \{ \omega - (\epsilon _0^s + \alpha _0(\omega -\epsilon _0)^{-1} \beta _0) \} G_{00}$ $\textstyle =$ $\displaystyle I +
\alpha _0(\omega -\epsilon _0)^{-1} \alpha _0 G_{2,0}$  
$\displaystyle (\omega -\epsilon _1^s)G_{00}$ $\textstyle =$ $\displaystyle I + \alpha _1 G_{20}$ (17)

The relationship between the second neighboring layer corresponding to eq.(16) is

$\displaystyle (\omega -\epsilon _0 ) G_{n0}$ $\textstyle =$ $\displaystyle \beta _0 (\omega -\epsilon _0)^{-1} (
\beta _0 G_{n-2,0} + \alpha _0 G_{n,0} )$  
    $\displaystyle + \alpha _0 (\omega -\epsilon _0)^{-1} (
\beta _0 G_{n,0} + \alpha _0 G_{n+2,0} )$  
$\displaystyle \{ \omega - ( \epsilon _0 + \beta _0 (\omega -\epsilon _0)^{-1} \alpha _0 + \alpha _0 (\omega -\epsilon _0)^{-1} \beta _0 \} G_{n0}$ $\textstyle =$ $\displaystyle \beta _0 (\omega -\epsilon _0)^{-1} \beta _0 G_{n-2,0} + \alpha _0 (\omega -\epsilon _0)^{-1} \alpha _0 G_{n+2,0}$ (18)
$\displaystyle (\omega - \epsilon _1 ) G_{n0}$ $\textstyle =$ $\displaystyle \beta _1 G_{n-2,0} + \alpha _1 G_{n+2,0}$ (19)

From eq.(15), eq.(16), eq.(17), eq.(19), the relation between $2^i$th neighboring layer is

$\displaystyle (\omega -\epsilon _i^s)G_{00}$ $\textstyle =$ $\displaystyle I + \alpha _i G_{2^i,0}$ (20)
$\displaystyle (\omega - \epsilon _i ) G_{n0}$ $\textstyle =$ $\displaystyle \beta _i G_{n-2^i,0} + \alpha _i G_{n+2^i,0}$ (21)

$\epsilon _i^s,\epsilon _i , \alpha _i, \beta _i$ can be evaluated iteratively

$\displaystyle \epsilon _{i+1}^s$ $\textstyle =$ $\displaystyle \epsilon _i^s + \alpha _i ( \omega -\epsilon _i )^{-1} \beta _i$ (22)
$\displaystyle \epsilon _{i+1}$ $\textstyle =$ $\displaystyle e_{i} + \beta _i ( \omega -\epsilon _i )^{-1} \alpha _i + \alpha _i ( \omega -\epsilon _i )^{-1} \beta _i$ (23)
$\displaystyle \alpha _{i+1}$ $\textstyle =$ $\displaystyle \alpha _i ( \omega -\epsilon _i )^{-1} \alpha _i$ (24)
$\displaystyle \beta _{i+1}$ $\textstyle =$ $\displaystyle \beta _i ( \omega -\epsilon _i )^{-1} \beta _i$ (25)

The initial values are

$\displaystyle \epsilon _{0}^s$ $\textstyle =$ $\displaystyle H_{00}$ (26)
$\displaystyle \epsilon _{0}$ $\textstyle =$ $\displaystyle H_{00}$ (27)
$\displaystyle \alpha _{0}$ $\textstyle =$ $\displaystyle H_{01}$ (28)
$\displaystyle \beta _{0}$ $\textstyle =$ $\displaystyle H_{10}$ (29)

if $\vert\alpha_i\vert\rightarrow 0$ for $i\rightarrow \infty$, 1 and also use eq.(22),

\begin{displaymath}
G_{00} = (\omega - \epsilon _i^s)^{-1} \vert_{i\rightarrow \infty}
\end{displaymath} (30)


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次へ: use transfer matrix 上へ: surface Green function, orthogonal 戻る: surface Green function, orthogonal
kino 平成18年4月17日