Theory of quantum transport in disordered systems driven by voltage pulse

Predicting time-dependent quantum transport in the transient regime is important for understanding the intrinsic dynamic response of a nanodevice and for predicting the limit of how such a device can switch on or off a current. Theoretically, this problem becomes quite difficult to solve when the nanodevice contains disorder because the calculated transient current must be averaged over many disorder configurations. In this work, we present a theoretical formalism to calculate the configuration averaged time-dependent current flowing through a phase coherent device containing disorder sites where the transient current is driven by sharply turning on and off the external bias voltage. Our theory is based on the Keldysh nonequilibrium Green's function (NEGF) formalism and is applicable in the far from equilibrium nonlinear response quantum transport regime. The effects of disorder scattering are dealt with by the coherent potential approximation (CPA) extended in the time domain. We show that after approximations such as CPA and vertex corrections for calculating the multiple impurity scattering in the transient regime, the derived NEGFs perfectly satisfy a Ward identity. The theory is quantitatively verified by comparing its predictions to the exact solution for a tight-binding model of a disordered two-probe transport junction.

Figure 1

Schematic plot of a two-probe quantum transport junction under voltage pulses denoted by ${\mathrm{\Delta }}_{L,R}\left(t\right)$. The central channel region is sandwiched between two semi-infinite leads. A particular disorder configuration of the central region is shown in the plot, where the black dots represent atomic defects.

Figure 2

Diagrammatic expansion of ${\mathbf{G}}^r$ [see Eq. (27)]. A cross represents a scattering site.

Figure 3

Diagrammatic representation of a particular term in the expansion of $\left[{\mathbf{G}}^r\left({\omega }_1\right)\mathbf{\Gamma }{}_{\alpha }{\mathbf{G}}^a\left({\omega }_2\right)\right]$.

Figure 4

Diagrammatic representation of (a) the ladder approximation (29) and (30), (b) the recursive relation (32), and (c) the relation between the dressed vertex, the bare vertex, and the vertex correction.

Figure 5

Numerical calculation of configuration-averaged current flowing through the left lead, in response to upward (upper panels) or downward (lower panels) step-shaped voltage pulses. Numerical results for $x=0.3$ are obtained from the brute-force calculation or with CPA methods (with or without VC). Results for $x=0$ and 1 are also plotted for reference. The current is in units of $e{\mathrm{\Gamma }}_0/\hslash$ and time in units of $\hslash /{\mathrm{\Gamma }}_0$. Other parameters are $p=1{\mathrm{\Gamma }}_0,{\varepsilon }_A^0=0{\mathrm{\Gamma }}_0,{\varepsilon }_B^0=0.5{\mathrm{\Gamma }}_0$. A legend is shown in the upper left panel. Insets: short time behavior of the current.

Figure 6

Numerical calculation of configuration-averaged current flowing through a disordered tight-binding chain of six sites, in response to an upward step-shaped voltage pulse with ${\mathrm{\Delta }}_L=3{\mathrm{\Gamma }}_0,{\mathrm{\Delta }}_R=0$. The current is in units of $e{\mathrm{\Gamma }}_0/\hslash$ and time in units of $\hslash /{\mathrm{\Gamma }}_0$. Other parameters are $p=1{\mathrm{\Gamma }}_0,{\varepsilon }_A^0=0{\mathrm{\Gamma }}_0,{\varepsilon }_B^0=1{\mathrm{\Gamma }}_0$. Inset: short time behavior of the current.

Figure 7

Time $\tau$ at maximum current $J_{\max }$ versus $J_{\max }$. The current is in units of $e{\mathrm{\Gamma }}_0/\hslash$ and time in units of $\hslash /{\mathrm{\Gamma }}_0$. The system is a disordered tight-binding chain of six sites with different $x$ values, driven by an upward step-shaped voltage pulse with ${\mathrm{\Delta }}_L=3{\mathrm{\Gamma }}_0,{\mathrm{\Delta }}_R=0$.

Figure 8

Time-dependent “local density of states” $\left[\mathrm{Im}{\langle {-\mathbf{A}}_L(\varepsilon ,t)\rangle }_{11}\right]$ of the leftmost site at impurity concentrations (a) $x=0$ and (b) $x=0.3$, with ${\varepsilon }_A^0=0$ and ${\varepsilon }_B^0=1{\mathrm{\Gamma }}_0$. Time is in units of $\hslash /{\mathrm{\Gamma }}_0$ and energy in units of ${\mathrm{\Gamma }}_0$. The system is a disordered tight-binding chain of six sites, driven by an upward step-shaped voltage pulse with ${\mathrm{\Delta }}_L=3{\mathrm{\Gamma }}_0,{\mathrm{\Delta }}_R=0$.

Figure 9

The (a) real and (b) imaginary parts of the vertex correction ${\left[{\overline{\mathbf{G}}}^r\left({\omega }_1\right){\sum }_n{\mathbf{\Lambda }}_n{\overline{\mathbf{G}}}^a\left({\omega }_2\right)\right]}_{11}$ on the leftmost site of a disordered tight-binding chain of six sites. ${\omega }_{1/2}$ is in units of ${\mathrm{\Gamma }}_0$. The disorder related parameters are $x_A=0.7,x_B=0.3,{\varepsilon }_A^0=0$, and ${\varepsilon }_B^0=1{\mathrm{\Gamma }}_0$.