normal

Green function

$$(\omega -H) G = I$$ (7)

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

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

By definition, .

Green function

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

Use $G_{i0}$ column,

$$(\omega -H_{00} )G_{00} - H_{01} G_{10} = I$$ (10)

$$-H_{10} G_{00} + (\omega -H_{11}) G_{10} -H_{12} G_{20} = 0$$ (11)

$$-H_{21} G_{10} + (\omega -H_{22}) G_{20} -H_{23} G_{30} = 0$$ (12)

$$\vdots$$ (13)

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

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

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

$$(\omega -\epsilon _0^s) G_{00} = I + \alpha _0 G_{10}$$ (15)

$$(\omega -\epsilon _0 ) G_{n0} = \beta _0 G_{n-1,0} + \alpha _0 G_{n+1,0}$$ (16)

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

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

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

$$(\omega -\epsilon _1^s)G_{00} = I + \alpha _1 G_{20}$$ (17)

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

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

$$+ \alpha _0 (\omega -\epsilon _0)^{-1} ( \beta _0 G_{n,0} + \alpha _0 G_{n+2,0} )$$

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

$$(\omega - \epsilon _1 ) G_{n0} = \beta _1 G_{n-2,0} + \alpha _1 G_{n+2,0}$$ (19)

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

$$(\omega -\epsilon _i^s)G_{00} = I + \alpha _i G_{2^i,0}$$ (20)

$$(\omega - \epsilon _i ) G_{n0} = \beta _i G_{n-2^i,0} + \alpha _i G_{n+2^i,0}$$ (21)

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

$$\epsilon _{i+1}^s = \epsilon _i^s + \alpha _i ( \omega -\epsilon _i )^{-1} \beta _i$$ (22)

$$\epsilon _{i+1} = e_{i} + \beta _i ( \omega -\epsilon _i )^{-1} \alpha _i + \alpha _i ( \omega -\epsilon _i )^{-1} \beta _i$$ (23)

$$\alpha _{i+1} = \alpha _i ( \omega -\epsilon _i )^{-1} \alpha _i$$ (24)

$$\beta _{i+1} = \beta _i ( \omega -\epsilon _i )^{-1} \beta _i$$ (25)

The initial values are

$$\epsilon _{0}^s = H_{00}$$ (26)

$$\epsilon _{0} = H_{00}$$ (27)

$$\alpha _{0} = H_{01}$$ (28)

$$\beta _{0} = H_{10}$$ (29)

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

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