3.9.3 Current Density

To derive an equation for the current density one uses the conservation law of quantum mechanical variables [93]. The starting point is the subtraction of equation (3.62) from (3.63)

$$\begin{array}{l}\displaystyle i\hbar \left(\partial_{t_1}+\p... ...athrm{<}(13)\ \Sigma^\mathrm{a}(32) \ \right] \ , \end{array}$$ (3.85)

where $H_0(1)=-{\hbar^2}/{2m}\ensuremath{{\mathbf{\nabla}}}^2_1+U(1)$ has been assumed. By taking the limit $1\rightarrow 2$ ( ${\bf r_2}\rightarrow {\bf r_1}$ and $t_2\rightarrow t_1$) and assuming that the right-hand-side of (3.85) approaches zero in this limit, one obtains

$$\begin{array}{l}\displaystyle i\hbar \lim_{t_2\rightarrow t_... ...abla}}}_{\bf r_2}) G^{<}(12) \right) \ = \ 0 \ . \end{array}$$ (3.86)

By multiplying both sides by $-\ensuremath {\mathrm{q}}$ and recalling the definition of the charge density, one recovers the continuity equation

$$\begin{array}{l}\displaystyle \partial_{t_1} \varrho({\bf r_... ...f{\nabla}}}\cdot {\bf J}({\bf r_1},t_1) \ = 0 \ , \end{array}$$ (3.87)

where the current density is defined as

$$\begin{array}{l}\displaystyle {\bf J}({\bf r_1},t_1) \ = -\f... ...2}\right) G^{<}({\bf r_1},t_1;{\bf r_2},t_1) \ . \end{array}$$ (3.88)

Under steady-state condition the current density takes the form [60]

$$\begin{array}{l}\displaystyle {\bf J}({\bf r_1}) \ \displays... ..._{\bf r_2}\right) G^{<}({\bf r_1},{\bf r_2},E) \ . \end{array}$$ (3.89)

M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors