量子输运笔记
\[ i\hbar\frac{\partial\Psi(\bm{r},t)}{\partial t} = H\Psi(\bm{r},t);\quad\quad H = -\frac{\hbar^2}{2m}\nabla^2 + U(\bm{r},t)
\]
\[U(\bm{r},t) = U(\bm{r})
\]
\[\Psi(\bm{r},t) = e^{-iEt/\hbar}\psi(\bm{r})\\
E\psi(\bm{r}) = H\psi(\bm{r}) = \left[-\frac{\hbar^2}{2m}\nabla^2 + U(\bm{r})\right]\psi(\bm{r})
\]
\[U(\bm{r}) = -\frac{Zq^2}{4\pi\varepsilon_0r}
\]
\[E_n = -\frac{Z^2}{n^2}E_0
\]
\[ H\psi(x) = \left[-\frac{\hbar^2}{2m}\frac{d^2}{d x^2} + U(x)\right]\psi(x)
\]
\[\frac{d^2\psi}{dx^2}\bigg|_i = \frac{\psi_{i-1}-2\psi_{i} + \psi_{i+1}}{a^2}
\]
\[t_0= \frac{\hbar^2}{2ma^2}
\]
\[[-t_0\psi_{i-1} + (U_i+2t_0)\psi_i - t_0\psi_{i+1}] = E\psi_i\\
(i=1,2,3,...,N)
\]
\[\begin{pmatrix}
2t_0+U_1 & -t_0 & & & & \\
-t_0 & 2t_0+U_2 & -t_0 & & & \\
& -t_0 & 2t_0+U_3 & -t_0 & & \\
& & \ddots & \ddots & \ddots & \\
& & & -t_0 & 2t_0+U_{N-1} &-t_0\\
& & & & -t_0 & 2t_0+U_N
\end{pmatrix}
\begin{pmatrix}
\psi_1\\
\psi_2\\
\psi_3\\
\vdots \\
\psi_{N-1}\\
\psi_{N}
\end{pmatrix}
= E\begin{pmatrix}
\psi_1\\
\psi_2\\
\psi_3\\
\vdots \\
\psi_{N-1}\\
\psi_{N}
\end{pmatrix}
\]
\[H=\begin{pmatrix}
2t_0 & -t_0 & & & & -t_0\\
-t_0 & 2t_0 & -t_0 & & & \\
& -t_0 & 2t_0 & -t_0 & & \\
& & \ddots & \ddots & \ddots & \\
& & & -t_0 & 2t_0 &-t_0\\
-t_0 & & & & -t_0 & 2t_0
\end{pmatrix}_{N\times N}
\]
\[E_{\alpha} = 2t_0 - t_0\cos\frac{\alpha\pi}{N+1}, (\alpha= 1,2,...,N)
\]
\[\psi=\frac{2}{\sqrt{N+1}}\begin{pmatrix}
\sin\frac{\pi}{N+1} & \sin\frac{2\pi}{N+1} &\sin\frac{3\pi}{N+1}&\cdots&\cdots& \sin\frac{N\pi}{N+1} \\
\sin\frac{2\pi}{N+1}& \sin\frac{4\pi}{N+1}&\sin\frac{6\pi}{N+1}&\cdots&\cdots& \sin\frac{2N\pi}{N+1} \\
\sin\frac{3\pi}{N+1}& \sin\frac{6\pi}{N+1}&\sin\frac{9\pi}{N+1}&\cdots&\cdots& \sin\frac{3N\pi}{N+1}\\
\vdots&\vdots& \vdots & \ddots & \ddots &\vdots\\
\vdots&\vdots&\vdots & \ddots & \ddots &\vdots\\
\sin\frac{N\pi}{N+1}&\sin\frac{2N\pi}{N+1}&\sin\frac{3N\pi}{N+1}&\cdots&\cdots& \sin\frac{N^2\pi}{N+1}
\end{pmatrix}_{N\times N}
\]
\[-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{d x^2}=E\psi(x)
\]
\[\psi_{\alpha}(x) \sim \sin(k_{\alpha}x)\\
k_{\alpha} = \frac{{\alpha}\pi}{L},\quad (\alpha= 1,2,...,N)
\]
\[E_{\alpha} = \frac{\hbar^2k_{\alpha}^2}{2m} = t_0(k_{\alpha}a)^2
\]
\[\psi_{\alpha}(x_i)=\frac{2}{\sqrt{N+1}}\sin\frac{\alpha\pi ia}{(N+1)a}= \frac{2}{\sqrt{N+1}}\sin(k_{\alpha}x_i)\\ (\alpha, i = 1,2,...,n)
\]
\[E_{\alpha} = 2t_0 - t_0\cos\frac{\alpha\pi a}{(N+1)a} = 2t_0 - t_0\cos(k_{\alpha}a)\\
k_{\alpha} = \frac{{\alpha}\pi}{L},\quad (\alpha= 1,2,...,N)
\]
\[k_{\alpha}a = \frac{\alpha \pi a}{L} \ll 1\\
\cos(k_{\alpha}a) \approx 1-\frac{(k_{\alpha}a)^2}{2}
\]
\[1e^{ik_1x}\\
re^{ik_1x}\\
te^{ik_2x}
\]
\[[-t_0\psi_{0} + (U_1+2t_0)\psi_1 - t_0\psi_{2}] = E\psi_1\\
[-t_0\psi_{N-1} + (U_N+2t_0)\psi_{N} - t_0\psi_{N+1}] = E\psi_{N}
\]
\[x = 0,\quad [-t_0\psi_{-1} + (U_0+2t_0)\psi_0 - t_0\psi_{1}] = E\psi_0\\
[-t_0\psi_{N} + (U_{N+1}+2t_0)\psi_{N+1} - t_0\psi_{N+2}] = E\psi_{N+1}
\]
\[\psi(x) = e^{ik_1x} + re^{-ik_1x}, \quad x\leq 0\\
\psi(x=0) = \psi_0 = 1 + r\\
\psi(x=-a) = \psi_{-1} = e^{-ik_1a} + re^{ik_1a}
\]
\[\psi_{-1} = e^{-ik_1a} + (\psi_0 -1)e^{ik_1a} = \psi_0e^{ik_1a} - (e^{ik_1a}-e^{-ik_1a})
\]
\[\begin{align}
x=0,\quad &[t_0 (e^{ik_1a}-e^{-ik_1a}) + (U_0+2t_0 -t_0e^{ik_1a})\psi_0 - t_0\psi_{1}] = E\psi_0\\
x=1,\quad & [-t_0\psi_{0} + (U_1+2t_0)\psi_1 - t_0\psi_{2}] = E\psi_1\\
x = i,\quad & [-t_0\psi_{i-1} + (U_i+2t_0)\psi_i - t_0\psi_{i+1}] = E\psi_i\\
\vdots&\\
x= N,\quad &[-t_0\psi_{N-1} + (U_N+2t_0)\psi_{N} - t_0\psi_{N+1}] = E\psi_{N}\\
x = N+1,\quad &[t_0 (e^{ik_2a}-e^{-ik_2a}) -t_0\psi_{N} + (U_{N+1}+2t_0 - t_0e^{ik_2a})\psi_{N+1}] = E\psi_{N+1}
\end{align}
\]
\[[EI-H-\Sigma_1 - \Sigma_2]\psi = s = i\gamma
\]
\[H =
\begin{pmatrix}
2t_0+U_0 & -t_0 & & & & \\
-t_0 & 2t_0+U_1 & -t_0 & & & \\
& -t_0 & 2t_0+U_2 & -t_0 & & \\
& & \ddots & \ddots & \ddots & \\
& & & -t_0 & 2t_0+U_{N} &-t_0\\
& & & & -t_0 & 2t_0+U_{N+1}
\end{pmatrix}_{(N+2) \times(N+2)}
\]
\[\Sigma_1 =
\begin{pmatrix}
-t_0e^{ik_1a} & & & & & \\
& & & & & \\
& & & & & \\
& & & & & \\
& & & & & \\
& & & & &
\end{pmatrix}
\]
\[\Sigma_2 =
\begin{pmatrix}
& & & & & \\
& & & & & \\
& & & & & \\
& & & & & \\
& & & & & \\
& & & & &-t_0e^{ik_2a}
\end{pmatrix}
\]
\[\gamma =
\begin{pmatrix}
-[\Sigma_1(1,1)-\Sigma_1^{*}(1,1)]\\
0\\
0\\
\vdots\\
\vdots\\
-[\Sigma_2(N,N)-\Sigma_2^{*}(N,N)]
\end{pmatrix}
\]
\[\gamma =
\begin{pmatrix}
t_0(e^{ik_1a}-e^{-ik_1a})\\
0\\
0\\
\vdots\\
\vdots\\
t_0(e^{ik_2a}-e^{-ik_2a})
\end{pmatrix}
\]
\[\psi =
\begin{pmatrix}
\psi_0\\
\psi_1\\
\psi_2\\
\\
\psi_N\\
\psi_{N+1}
\end{pmatrix}
\]
\[\psi = G^Rs\\
G^R = [EI-H-\Sigma_1-\Sigma_2]^{-1}
\]
\[\bm{G^n = G^R\Sigma^{in}G^A}\\
\bm{\rho = \int\frac{G^n(E)}{2\pi}dE}
\]
\[H,\quad U, \quad \Sigma_1,\quad \Sigma_2,\quad \mu_1,\quad \mu_2\\
U \quad\rightarrow\quad \rho\\
\nabla\cdot(\varepsilon_r\nabla U) = -\frac{\rho}{\varepsilon_0}\\
\rho \quad\rightarrow\quad U
\]
\[\bar{I}_1 = \frac{q}{h}\mathbf{Trace}(\Sigma_1^{in}A-\Gamma_1 G^n)\\
\bar{I}_2 = \frac{q}{h}\mathbf{Trace}(\Sigma_2^{in}A-\Gamma_2 G^n)\\
\Sigma_1^{in} = \Gamma_1 f_1\\
\Sigma_2^{in} = \Gamma_2 f_2 \\
\bm{G^n = G^R\Sigma^{in}G^A = G^R(\Gamma_1 f_1 + \Gamma_2 f_2)G^A}\\
\bm{A = i(G^R-G^A) = G^R (\Gamma_1 + \Gamma_2) G^A}\\
\bm{\Gamma = i(\Sigma-\Sigma^{\dagger})}\\
\Gamma_1 = i(\Sigma_1 -\Sigma_1^{*})\\
\Gamma_2 = i(\Sigma_2 -\Sigma_2^{*})\\
I = \int \hat{I}(E) dE = \frac{q}{h}\int T(E) [f_1- f_2] dE = \frac{q}{h}\int \mathbf{Trace}(\Gamma_1 G^R\Gamma_2 G^A) [f_1- f_2] dE\\
\bm{I = \frac{q}{h}\int \mathbf{Trace}(\Gamma_1 G^R\Gamma_2 G^A) [f_1- f_2] dE}\\
\bm{G^R = [EI- H- U-\Sigma_1 - \Sigma_2]^{-1}}\\
\]
