A tutorial on the NEGF method for electron transport in devices and defective materials

Abstract

A tutorial on non-equilibrium Green’s function theory and its applications on nanoscale devices is presented. A stepwise tutorial presentation, starting from the concept of Green’s function to its application on nanoscale FETs is presented in this work. The mathematical implementation of the retarded and advanced Green’s function on the device channel is shown in detail. Also, the partitioning of Green’s function into a source Green’s function, a drain Green’s function, and a device channel Green’s function are shown. The construction of the Hamiltonian matrix by applying a non-interacting tight-binding methodology is shown for device channels with rectangular cross sections and line defects. Mathematical expressions for the transmission function and the terminal currents are obtained. Finally, a brief explanation of the concept of scattering contacts and the NEGF mode space extension is presented. This work is useful for understanding the application of Green’s function on nanoscale devices.

Graphical abstract

Appendices

Appendix A: Evaluation of retarded and advanced Green’s function

The differential equation with an impulse excitation at the origin can be given as,

$$ \left( {\frac{{{\text{d}}^{2} }}{{{\text{d}}z^{2} }} + k^{2} } \right)G\left( z \right) = \delta \left( z \right). $$

(A1)

The equation above Eq. (A1) can be written in the $k$-space as

$$ \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^{\infty } {{\text{e}}^{izs} \left( { - s^{2} + k^{2} } \right)} \tilde{g}\left( s \right){\text{d}}s = \frac{1}{2\pi }\int\limits_{ - \infty }^{\infty } {{\text{e}}^{izs} } {\text{d}}s. $$

(A2)

From (A2), $G\left( z \right)$ can be extracted as

$$ G\left( z \right) = \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^{\infty } {\tilde{g}\left( s \right)} {\text{e}}^{izs} {\text{d}}s = \frac{1}{2\pi }\int\limits_{ - \infty }^{\infty } {{\text{d}}s\frac{{{\text{e}}^{izs} }}{{k^{2} - s^{2} }}} . $$

(A3)

From (A3), the expression for Green’s functions, $G^{{\text{R}}} \left( z \right)$, and $G^{{\text{A}}} \left( z \right)$ are evaluated as

$$ G^{{\text{R}}} \left( z \right) = - \frac{i}{2k}\exp \left( {ikz} \right) $$

(A4)

and

$$ G^{{\text{A}}} \left( z \right) = + \frac{i}{2k}\exp \left( { - ikz} \right). $$

(A5)

Appendix B: Evaluation of the matrix $\left[ {G_{{\text{D}}}^{{\text{R}}} } \right]$, $\left[ {g_{{\text{s}}}^{{\text{R}}} } \right]$, and $\left[ {g_{{\text{d}}}^{{\text{R}}} } \right]$

Here, the matrix for $\left[ {G^{{\text{R}}} } \right]$ is given as

$$ \left[ {G^{{\text{R}}} } \right] = \left[ {\begin{array}{*{20}c} {G_{{\text{s}}} } & {G_{{{\text{sD}}}} } & 0 \\ {G_{{{\text{Ds}}}} } & {G_{{\text{D}}} } & {G_{{{\text{Dd}}}} } \\ 0 & {G_{{{\text{dD}}}} } & {G_{{\text{d}}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\left( {E + i\eta } \right)I - H_{{\text{s}}} } & { - \tau_{{\text{s}}}^{ + } } & 0 \\ { - \tau_{{\text{s}}} } & {EI - H_{{\text{D}}} } & { - \tau_{{\text{d}}} } \\ 0 & { - \tau_{{\text{d}}}^{ + } } & {\left( {E + i\eta } \right)I - H_{{\text{d}}} } \\ \end{array} } \right]^{ - 1} . $$

(B1)

Now, the above matrix equation can be rewritten as

$$ \begin{aligned} & \left[ {\begin{array}{*{20}c} {\left( {E + i\eta } \right)I - H_{{\text{s}}} } & { - \tau_{{\text{s}}}^{ + } } & 0 \\ { - \tau_{{\text{s}}} } & {EI - H_{{\text{D}}} } & { - \tau_{{\text{d}}} } \\ 0 & { - \tau_{{\text{d}}}^{ + } } & {\left( {E + i\eta } \right)I - H_{{\text{d}}} } \\ \end{array} } \right]\\&\left[ {\begin{array}{*{20}c} {G_{{\text{s}}} } & {G_{{{\text{sD}}}} } & 0 \\ {G_{{{\text{Ds}}}} } & {G_{{\text{D}}} } & {G_{{{\text{Dd}}}} } \\ 0 & {G_{{{\text{dD}}}} } & {G_{{\text{d}}} } \\ \end{array} } \right] \\ & \quad = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]. \\ \end{aligned} $$

(B2)

Now, by only taking the source and the device channel as the system in order to understand the procedure and avoid the tedious algebra, the matrix for $\left[ {G^{{\text{R}}} } \right]$ under this becomes,

$$ \left[ {G^{{\text{R}}} } \right] = \left[ {\begin{array}{*{20}c} {G_{{\text{s}}} } & {G_{{{\text{sD}}}} } \\ {G_{{{\text{Ds}}}} } & {G_{{\text{D}}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\left( {E + i\eta } \right)I - H_{{\text{s}}} } & { - \tau_{{\text{s}}}^{ + } } \\ { - \tau_{{\text{s}}} } & {EI - H_{{\text{D}}} } \\ \end{array} } \right]^{ - 1} $$

(B3)

that can be written as,

$$ \left[ {\begin{array}{*{20}c} {\left( {E + i\eta } \right)I - H_{{\text{s}}} } & { - \tau_{{\text{s}}}^{ + } } \\ { - \tau_{{\text{s}}} } & {EI - H_{{\text{D}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {G_{{\text{s}}} } & {G_{{{\text{sD}}}} } \\ {G_{{{\text{Ds}}}} } & {G_{{\text{D}}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right]. $$

(B4)

Now, from (B4),

$$ \left[ {\left( {E + i\eta } \right)I - H_{{\text{s}}} } \right]G_{{{\text{SD}}}} - \tau_{{\text{s}}}^{ + } G_{{\text{D}}} = 0 $$

(B5)

$$ \left[ { - \tau_{{\text{s}}} } \right]G_{{{\text{SD}}}} + \left[ {EI - H_{{\text{D}}} } \right]G_{{\text{D}}} = \left[ I \right]. $$

(B6)

Now, from (B5), $G_{{{\text{SD}}}}$ becomes,

$$ \begin{aligned} G_{{{\text{SD}}}} & = \left[ {\left( {E + i\eta } \right)I - H_{{\text{s}}} } \right]^{ - 1} \tau_{{\text{s}}}^{ + } G_{{\text{D}}} \\ & = \left[ {g_{{\text{s}}}^{{\text{R}}} } \right]\tau_{{\text{s}}}^{ + } G_{{\text{D}}} . \\ \end{aligned} $$

(B7)

Now, from (B6) and (B7), the retarded Green’s function for the device channel becomes,

$$ \begin{aligned} & \left[ { - \tau_{{\text{s}}} } \right]\left[ {g_{{\text{s}}}^{{\text{R}}} } \right]\left[ {\tau_{{\text{s}}}^{ + } } \right]G_{{\text{D}}} + \left[ {EI - H_{{\text{D}}} } \right]G_{{\text{D}}} = \left[ I \right] \\ & \left[ {EI - H_{{\text{D}}} - \tau_{{\text{s}}} g_{{\text{s}}}^{{\text{R}}} \tau_{{\text{s}}}^{ + } } \right]\left[ {G_{{\text{D}}} } \right] = \left[ I \right] \\ & \left[ {G_{{\text{D}}} } \right] = \left[ {EI - H_{{\text{D}}} - \tau_{{\text{s}}} g_{{\text{s}}}^{{\text{R}}} \tau_{{\text{s}}}^{ + } } \right]^{ - 1} . \\ \end{aligned} $$

(B8)

Now, the above procedure can be generalized for obtaining the retarded Green’s function $\left[ {G_{{\text{D}}}^{{\text{R}}} } \right]$ for the device with source and drain terminals as

$$ \left[ {G_{{\text{D}}}^{{\text{R}}} } \right] = \left[ {EI - H_{{\text{D}}} - \tau_{{\text{s}}} g_{{\text{s}}}^{{\text{R}}} \tau_{{\text{s}}}^{ + } - \tau_{{\text{d}}} g_{{\text{d}}}^{{\text{R}}} \tau_{{\text{d}}}^{ + } } \right]^{ - 1} . $$

(B9)

Appendix C: Derivation of Eqs. (34) and (35) from Eqs. (36) and (37)

The Schrodinger equation for source/drain contacts when they are not coupled to the device channel can be given as

$$ \left[ {EI - H_{{{\text{s}},{\text{d}}}} } \right]\left| {\Phi_{{{\text{s}},{\text{d}}}} } \right\rangle = 0. $$

(C1)

Here, $\left[ {H_{{{\text{s}},{\text{d}}}} } \right]$ is the Hamiltonian for the source/drain contacts. Now, Eq. (C1) can be further written as,

$$ \left[ {EI - H_{{{\text{s}},{\text{d}}}} + i\eta } \right]\left| {\Phi_{{{\text{s}},{\text{d}}}} } \right\rangle = \left\{ {S_{{{\text{s}},{\text{d}}}} } \right\}. $$

(C2)

Here, the term $\left[ {i\eta } \right]\left| {\Phi_{{{\text{s}},{\text{d}}}} } \right\rangle$ represents the extraction of electrons from source/drain contacts and the term $\left\{ {S_{{{\text{s}},{\text{d}}}} } \right\}$ represents the reinjection of the electrons into the source/drain contacts. Now, from Eqs. (C1) and (C2), Eqs. (36) and (37) are obtained as

$$ \begin{aligned} & \left[ {EI - H_{{\text{s}}} + i\eta } \right]\left| {\Phi_{{\text{s}}} } \right\rangle = \left\{ {S_{{\text{s}}} } \right\} \\ & \left| {\Phi_{{\text{s}}} } \right\rangle = \left[ {EI - H_{{\text{s}}} + i\eta } \right]^{ - 1} \left\{ {S_{{\text{s}}} } \right\} \\ & \left| {\Phi_{{\text{s}}} } \right\rangle = \left[ {g_{{\text{s}}}^{{\text{R}}} } \right]\left\{ {S_{{\text{s}}} } \right\} \\ \end{aligned} $$

(C3)

and

$$ \begin{aligned} & \left[ {EI - H_{{\text{d}}} + i\eta } \right]\left| {\Phi_{{\text{d}}} } \right\rangle = \left\{ {S_{{\text{d}}} } \right\} \\ & \left| {\Phi_{{\text{d}}} } \right\rangle = \left[ {EI - H_{{\text{d}}} + i\eta } \right]^{ - 1} \left\{ {S_{{\text{d}}} } \right\} \\ & \left| {\Phi_{{\text{d}}} } \right\rangle = \left[ {g_{{\text{d}}}^{{\text{R}}} } \right]\left\{ {S_{{\text{d}}} } \right\}. \\ \end{aligned} $$

(C4)

Now, the overall two-terminal system consisting of the source, the drain, and the device channel as shown in Fig. 3 can be represented as a matrix (Eq. 33)

$$ \left[ {\begin{array}{*{20}c} {\left[ {EI - H_{{\text{s}}} + i\eta } \right]} & {\left[ { - \tau_{{\text{s}}}^{ + } } \right]} & {\left[ 0 \right]} \\ {\left[ { - \tau_{{\text{s}}} } \right]} & {\left[ {EI - H_{{\text{D}}} } \right]} & {\left[ { - \tau_{{\text{d}}} } \right]} \\ {\left[ 0 \right]} & {\left[ { - \tau_{{\text{d}}}^{ + } } \right]} & {\left[ {EI - H_{{\text{d}}} + i\eta } \right]} \\ \end{array} } \right]\left( {\begin{array}{*{20}c} {\left| {\Phi_{{\text{s}}} } \right\rangle + \left| {\chi_{{\text{s}}} } \right\rangle } \\ {\left| \psi \right\rangle } \\ {\left| {\Phi_{{\text{d}}} } \right\rangle + \left| {\chi_{{\text{d}}} } \right\rangle } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\left\{ {S_{{\text{s}}} } \right\}} \\ {\left\{ 0 \right\}} \\ {\left\{ {S_{{\text{d}}} } \right\}} \\ \end{array} } \right). $$

(C5)

Now, from Eqs. (C3) and (C5),

$$ \begin{aligned} & \left[ {EI - H_{{\text{s}}} + i\eta } \right]\left| {\Phi_{{\text{s}}} } \right\rangle \\ & \quad + \left[ {EI - H_{{\text{s}}} + i\eta } \right]\left| {\chi_{{\text{s}}} } \right\rangle + \left[ { - \tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle = \left\{ {S_{{\text{s}}} } \right\} \\ & \left[ {EI - H_{{\text{s}}} + i\eta } \right]\left| {\chi_{{\text{s}}} } \right\rangle = \left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle \\ & \left| {\chi_{{\text{s}}} } \right\rangle = \left[ {EI - H_{{\text{s}}} + i\eta } \right]^{ - 1} \left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle \\ & \left| {\chi_{{\text{s}}} } \right\rangle = \left[ {g_{{\text{s}}}^{{\text{R}}} } \right]\left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle . \\ \end{aligned} $$

(C6)

On similar lines, from Eqs. (C4) and (C5),

$$ \begin{aligned} & \left[ { - \tau_{{\text{d}}}^{ + } } \right]\left| \psi \right\rangle + \left[ {EI - H_{{\text{d}}} + i\eta } \right]\left| {\Phi_{{\text{d}}} } \right\rangle \\ & \quad + \left[ {EI - H_{{\text{d}}} + i\eta } \right]\left| {\chi_{{\text{d}}} } \right\rangle = \left\{ {S_{{\text{d}}} } \right\} \\ & \left[ {EI - H_{{\text{d}}} + i\eta } \right]\left| {\chi_{{\text{d}}} } \right\rangle = \left[ {\tau_{{\text{d}}}^{ + } } \right]\left| \psi \right\rangle \\ & \left| {\chi_{{\text{d}}} } \right\rangle = \left[ {EI - H_{{\text{d}}} + i\eta } \right]^{ - 1} \left[ {\tau_{{\text{d}}}^{ + } } \right]\left| \psi \right\rangle \\ & \left| {\chi_{{\text{d}}} } \right\rangle = \left[ {g_{{\text{d}}}^{{\text{R}}} } \right]\left[ {\tau_{{\text{d}}}^{ + } } \right]\left| \psi \right\rangle . \\ \end{aligned} $$

(C7)

Hence, Eqs. (34) and (35) are obtained from Eqs. (36) and (37).

Appendix D: Evaluation of the terminal currents (part 1)

From Eq. (44) of Sect. 5,

$$ \begin{aligned} & i\hbar \frac{{\text{d}}}{{{\text{d}}t}}\left( {\begin{array}{*{20}c} {\left| {\Phi_{{\text{s}}} } \right\rangle + \left| {\chi_{{\text{s}}} } \right\rangle } \\ {\left| \psi \right\rangle } \\ {\left| {\Phi_{{\text{d}}} } \right\rangle + \left| {\chi_{{\text{d}}} } \right\rangle } \\ \end{array} } \right) = \left[ {\begin{array}{*{20}c} {\left[ {H_{{\text{s}}} - i\eta } \right]} & {\left[ {\tau_{{\text{s}}}^{ + } } \right]} & {\left[ 0 \right]} \\ {\left[ {\tau_{{\text{s}}} } \right]} & {\left[ {H_{{\text{D}}} } \right]} & {\left[ {\tau_{{\text{d}}} } \right]} \\ {\left[ 0 \right]} & {\left[ {\tau_{{\text{d}}}^{ + } } \right]} & {\left[ {H_{{\text{d}}} - i\eta } \right]} \\ \end{array} } \right]\left( {\begin{array}{*{20}c} {\left| {\Phi_{{\text{s}}} } \right\rangle + \left| {\chi_{{\text{s}}} } \right\rangle } \\ {\left| \psi \right\rangle } \\ {\left| {\Phi_{{\text{d}}} } \right\rangle + \left| {\chi_{{\text{d}}} } \right\rangle } \\ \end{array} } \right) \\ & \quad \Rightarrow i\hbar \frac{{{\text{d}}\left( {\left| {\Phi_{{\text{s}}} } \right\rangle + \left| {\chi_{{\text{s}}} } \right\rangle } \right)}}{{{\text{d}}t}} = \left[ {H_{{\text{s}}} - i\eta } \right]\left( {\left| {\Phi_{{\text{s}}} } \right\rangle + \left| {\chi_{{\text{s}}} } \right\rangle } \right) + \left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle \\ & \quad \Rightarrow i\hbar \frac{{{\text{d}}\left( {\left| {S_{{\text{s}}} } \right\rangle } \right)}}{{{\text{d}}t}} = \left[ {H_{{\text{s}}} - i\eta } \right]\left( {\left| {\left| {S_{{\text{s}}} } \right\rangle } \right\rangle } \right) + \left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle . \\ \end{aligned} $$

(D1)

Now, from (D1),

$$ \begin{aligned} & \Rightarrow i\hbar \frac{{{\text{d}}\left( {\left| {S_{{\text{s}}} } \right\rangle } \right)}}{{{\text{d}}t}} = \left[ {H_{{\text{s}}} - i\eta } \right]\left( {\left| {S_{{\text{s}}} } \right\rangle } \right) + \left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle \\ & \Rightarrow - i\hbar \frac{{{\text{d}}\left( {\left\langle {S_{{\text{s}}} } \right|} \right)}}{{{\text{d}}t}} = \left( {\left\langle {S_{{\text{s}}} } \right|} \right)\left[ {H_{{\text{s}}} - i\eta } \right]^{ + } + \left\langle \psi \right|\left[ {\tau_{{\text{s}}} } \right]. \\ \end{aligned} $$

(D2)

Now, the time derivative of the term $\left| {S_{{\text{s}}} } \right\rangle \left\langle {S_{{\text{s}}} } \right|$ is given as

$$ \begin{aligned} & \frac{{{\text{d}}\left( {\left| {S_{{\text{s}}} } \right\rangle \left\langle {S_{{\text{s}}} } \right|} \right)}}{{{\text{d}}t}} = \frac{{{\text{d}}\left( {\left| {S_{{\text{s}}} } \right\rangle } \right)}}{{{\text{d}}t}}\left\langle {S_{{\text{s}}} } \right| + \left| {S_{{\text{s}}} } \right\rangle \frac{{{\text{d}}\left( {\left\langle {S_{{\text{s}}} } \right|} \right)}}{{{\text{d}}t}} \\ & \quad \Rightarrow i\hbar \frac{{{\text{d}}\left( {\left| {S_{{\text{s}}} } \right\rangle \left\langle {S_{{\text{s}}} } \right|} \right)}}{{{\text{d}}t}} = \left( {i\hbar \frac{{{\text{d}}\left( {\left| {S_{{\text{s}}} } \right\rangle } \right)}}{{{\text{d}}t}}} \right)\left\langle {S_{{\text{s}}} } \right| + \left| {S_{{\text{s}}} } \right\rangle \left( {i\hbar \frac{{{\text{d}}\left( {\left\langle {S_{{\text{s}}} } \right|} \right)}}{{{\text{d}}t}}} \right) \\ & \quad \Rightarrow i\hbar \frac{{{\text{d}}\left( {\left| {S_{{\text{s}}} } \right\rangle \left\langle {S_{{\text{s}}} } \right|} \right)}}{{{\text{d}}t}} = \left[ {H_{{\text{s}}} - i\eta } \right]\left( {\left| {S_{{\text{s}}} } \right\rangle \left\langle {S_{{\text{s}}} } \right|} \right) \\ & \quad \quad + \left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle \left\langle {S_{{\text{s}}} } \right| - \left| {S_{{\text{s}}} } \right\rangle \left\langle {S_{{\text{s}}} } \right|\left[ {H_{{\text{s}}} - i\eta } \right]^{ + } \\ & \quad \quad - \left| {S_{{\text{s}}} } \right\rangle \left\langle \psi \right|\left[ {\tau_{{\text{s}}} } \right] \\ & \quad \Rightarrow i\hbar \frac{{{\text{d}}\left( {\left| {S_{{\text{s}}} } \right\rangle \left\langle {S_{{\text{s}}} } \right|} \right)}}{{{\text{d}}t}} = \left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle \left\langle {S_{{\text{s}}} } \right| - \left| {S_{{\text{s}}} } \right\rangle \left\langle \psi \right|\left[ {\tau_{{\text{s}}} } \right] \\ & \quad \Rightarrow \frac{{{\text{d}}\left( {\left| {S_{{\text{s}}} } \right\rangle \left\langle {S_{{\text{s}}} } \right|} \right)}}{{{\text{d}}t}} = \frac{{\left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle \left\langle {S_{{\text{s}}} } \right|}}{i\hbar } - \frac{{\left| {S_{{\text{s}}} } \right\rangle \left\langle \psi \right|\left[ {\tau_{{\text{s}}} } \right]}}{i\hbar }. \\ \end{aligned} $$

(D3)

Now, by taking the trace of the time-derivative of the term $\left| {S_{{\text{s}}} } \right\rangle \left\langle {S_{{\text{s}}} } \right|$, the Eq. (D3) becomes

$$ \begin{aligned} & {\text{Trace}}\left[ {\frac{{{\text{d}}\left( {\left| {S_{{\text{s}}} } \right\rangle \left\langle {S_{{\text{s}}} } \right|} \right)}}{{{\text{d}}t}}} \right] = \frac{{{\text{Trace}}\left[ {\left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle \left\langle {S_{{\text{s}}} } \right|} \right]}}{i\hbar } \\ & \quad \quad - \frac{{{\text{Trace}}\left[ {\left| {S_{{\text{s}}} } \right\rangle \left\langle \psi \right|\left[ {\tau_{{\text{s}}} } \right]} \right]}}{i\hbar } \\ & \quad \Rightarrow {\text{Trace}}\left[ {\frac{{{\text{d}}\left( {\left\langle {{S_{{\text{s}}} }} \mathrel{\left | {\vphantom {{S_{{\text{s}}} } {S_{{\text{s}}} }}} \right. \kern-0pt} {{S_{{\text{s}}} }} \right\rangle } \right)}}{{{\text{d}}t}}} \right] \\ & \quad = \frac{{{\text{Trace}}\left[ {\left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle \left\langle {\Phi_{{\text{s}}} } \right| + \left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle \left\langle {\chi_{{\text{s}}} } \right|} \right]}}{i\hbar } \\ & \quad \quad - \frac{{{\text{Trace}}\left[ {\left| {\Phi_{{\text{s}}} } \right\rangle \left\langle \psi \right|\left[ {\tau_{{\text{s}}} } \right] + \left| {\chi_{{\text{s}}} } \right\rangle \left\langle \psi \right|\left[ {\tau_{{\text{s}}} } \right]} \right]}}{i\hbar } \\ & \quad \Rightarrow \frac{{{\text{d}}\left( {\left\langle {{S_{{\text{s}}} }} \mathrel{\left | {\vphantom {{S_{{\text{s}}} } {S_{{\text{s}}} }}} \right. \kern-0pt} {{S_{{\text{s}}} }} \right\rangle } \right)}}{{{\text{d}}t}} = \frac{{{\text{Trace}}\left[ {\left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle \left\langle {\Phi_{{\text{s}}} } \right|} \right]}}{i\hbar } \\ & \quad \quad + \frac{{{\text{Trace}}\left[ {\left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle \left\langle {\chi_{{\text{s}}} } \right|} \right]}}{i\hbar } - \frac{{{\text{Trace}}\left[ {\left| {\Phi_{{\text{s}}} } \right\rangle \left\langle \psi \right|\left[ {\tau_{{\text{s}}} } \right]} \right]}}{i\hbar } \\ & \quad \quad - \frac{{{\text{Trace}}\left[ {\left| {\chi_{{\text{s}}} } \right\rangle \left\langle \psi \right|\left[ {\tau_{{\text{s}}} } \right]} \right]}}{i\hbar } \\ & \quad \Rightarrow \frac{{{\text{d}}\left( {\left\langle {{S_{{\text{s}}} }} \mathrel{\left | {\vphantom {{S_{{\text{s}}} } {S_{{\text{s}}} }}} \right. \kern-0pt} {{S_{{\text{s}}} }} \right\rangle } \right)}}{{{\text{d}}t}} = \frac{{{\text{Trace}}\left[ {\left\langle \psi \right|\left[ {\tau_{{\text{s}}} } \right]\left| {\Phi_{{\text{s}}} } \right\rangle } \right]}}{i\hbar } \\ & \quad \quad + \frac{{{\text{Trace}}\left[ {\left\langle \psi \right|\left[ {\tau_{{\text{s}}} } \right]\left| {\chi_{{\text{s}}} } \right\rangle } \right]}}{i\hbar } - \frac{{{\text{Trace}}\left[ {\left\langle {\Phi_{{\text{s}}} } \right|\left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle } \right]}}{i\hbar } \\ & \quad \quad - \frac{{{\text{Trace}}\left[ {\left\langle {\chi_{{\text{s}}} } \right|\left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle } \right]}}{i\hbar } \\ & \quad \Rightarrow I_{{\text{s}}} = \frac{{{\text{Trace}}\left[ {\left\langle \psi \right|\left[ {\tau_{{\text{s}}} } \right]\left| {\Phi_{{\text{s}}} } \right\rangle - \left\langle {\Phi_{{\text{s}}} } \right|\left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle } \right]}}{i\hbar } \\ & \quad \quad - \frac{{{\text{Trace}}\left[ {\left\langle {\chi_{{\text{s}}} } \right|\left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle - \left\langle \psi \right|\left[ {\tau_{{\text{s}}} } \right]\left| {\chi_{{\text{s}}} } \right\rangle } \right]}}{i\hbar }. \\ \end{aligned} $$

(D4)

Appendix E: Evaluation of the terminal currents (part 2)

Now, the expression of $I_{{\text{s}}}$ becomes_,

$$ \begin{aligned} I_{{\text{s}}} & = \frac{{{\text{Trace}}\left[ {\left\langle \psi \right|\left[ {\tau_{{\text{s}}} } \right]\left| {\Phi_{{\text{s}}} } \right\rangle - \left\langle {\Phi_{{\text{s}}} } \right|\left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle } \right]}}{i\hbar } \\ & \quad - \frac{{{\text{Trace}}\left[ {\left\langle {\chi_{{\text{s}}} } \right|\left[ {\tau_{{\text{s}}}^{ + } } \right]\left| \psi \right\rangle - \left\langle \psi \right|\left[ {\tau_{{\text{s}}} } \right]\left| {\chi_{{\text{s}}} } \right\rangle } \right]}}{i\hbar } \\ & = \frac{{{\text{Trace}}\left[ {\left\{ S \right\}^{ + } G_{{\text{D}}}^{{\text{A}}} \left\{ {S_{{\text{s}}} } \right\} - \left\{ {S_{{\text{s}}} } \right\}^{ + } G_{{\text{D}}}^{{\text{R}}} \left\{ S \right\}} \right]}}{i\hbar } \\ & \quad - \frac{{{\text{Trace}}\left[ {\left\{ \psi \right\}^{ + } \left[ {\tau_{{\text{s}}} } \right]g_{{\text{s}}}^{{\text{A}}} \left[ {\tau_{{\text{s}}}^{ + } } \right]\left\{ \psi \right\} - \left\{ \psi \right\}^{ + } \left[ {\tau_{{\text{s}}} } \right]g_{{\text{s}}}^{{\text{R}}} \left[ {\tau_{{\text{s}}}^{ + } } \right]\left\{ \psi \right\}} \right]}}{i\hbar } \\ & = \frac{{{\text{Trace}}\left[ {\left\{ {S_{{\text{s}}} } \right\}^{ + } G_{{\text{D}}}^{{\text{A}}} \left\{ {S_{{\text{s}}} } \right\} - \left\{ {S_{{\text{s}}} } \right\}^{ + } G_{{\text{D}}}^{{\text{R}}} \left\{ {S_{{\text{s}}} } \right\}} \right]}}{i\hbar } \\ & \quad - \frac{{{\text{Trace}}\left[ {\left\{ \psi \right\}^{ + } \left[ {\tau_{{\text{s}}} } \right]g_{{\text{s}}}^{{\text{A}}} \left[ {\tau_{{\text{s}}}^{ + } } \right]\left\{ \psi \right\} - \left\{ \psi \right\}^{ + } \left[ {\tau_{{\text{s}}} } \right]g_{{\text{s}}}^{{\text{R}}} \left[ {\tau_{{\text{s}}}^{ + } } \right]\left\{ \psi \right\}} \right]}}{i\hbar } \\ & = \frac{{{\text{Trace}}\left[ {S_{{\text{s}}}^{ + } AS_{{\text{s}}} } \right]}}{\hbar } - \frac{{{\text{Trace}}\left[ {\left\{ \psi \right\}^{ + } \Gamma_{{\text{s}}} \left\{ \psi \right\}} \right]}}{\hbar } \\ & = \frac{{{\text{Trace}}\left[ {S_{{\text{s}}} S_{{\text{s}}}^{ + } A} \right]}}{\hbar } - \frac{{{\text{Trace}}\left[ {\Gamma_{{\text{s}}} \left\{ \psi \right\}\left\{ \psi \right\}^{ + } } \right]}}{\hbar } \\ & = \frac{1}{\hbar }\int {\frac{{{\text{d}}E}}{2\pi }} \left( {{\text{Trace}}\left[ {\Gamma_{{\text{s}}} A} \right]f - {\text{Trace}}\left[ {\Gamma_{{\text{s}}} G^{n} } \right]} \right). \\ \end{aligned} $$

(E1)