Time-dependent quantum transport far from equilibrium: An exact nonlinear response theory

In this work, we present a theory to calculate the time-dependent current flowing through an arbitrary noninteracting nanoscale phase-coherent device connected to arbitrary noninteracting external leads, in response to sharp step- and square-shaped voltage pulses. Our analysis is based on the Keldysh nonequilibrium Green’s-functions formalism, and provides an exact analytical solution to the transport equations in the far from equilibrium, nonlinear response regime. However, the essential feature of our solution is that it does not rely on the commonly used wideband approximation where the coupling between device scattering region and leads is taken to be independent of energy, and as such provides a way to perform transient transport calculations from first principles on realistic systems, taking into account the detailed electronic structure of the device scattering region and the leads. We then perform a model calculation for a quantum dot with Lorentzian linewidth and show how interesting finite-bandwidth effects arise in the time-dependent current dynamics.

Figure 1

Schematic diagram of a nanoscale device in the LDL configuration.

Figure 2

Time-dependent current $J_L\left(t\right)$ through left lead in response to a downward step pulse for different bandwidths: dashed line, WBL $(W=\infty )$; (i) $W=20\Gamma$, (ii) $W=10\Gamma$, (iii) $W=5\Gamma$, (iv) $W=2.5\Gamma$, and (v) $W=\Gamma$. The current is in units of $e\Gamma ∕\hslash$ and the time is in units of $\hslash ∕\Gamma$ where $\Gamma ={\Gamma }_L^0+{\Gamma }_R^0$ is the total linewidth amplitude. Parameters are taken the same as in Ref. 19. When $W=100\Gamma$, the result was found (not shown) to be indistinguishable from the $W=\infty$ curve.

Figure 3

Time derivative of the current through the left lead $d{J}_L\left(t\right)∕dt$ at $t=0$. The bandwidth $W$ is in units of $\Gamma$ and the time derivative ${\partial }_t{J}_L\left(0\right)$ is in units of $e{\Gamma }^2∕{\hslash }^2$. The inset shows a closeup view of the interval around the critical bandwidths $W_{c1}\sim 4.53\Gamma$ and $W_{c2}\sim 0.49\Gamma$ where the slope changes sign.

Figure 4

$1∕e$ turnoff time $\tau$ (in units of $\hslash ∕\Gamma$) as a function of bandwidth $W$ (in units of $\Gamma$).

Figure 5

$1∕e$ turnoff time $\tau$ as a function of total linewidth $\Gamma$. Energies ($\Gamma$ and $W$) are in units of $10{\beta }^{-1}$ and $\tau$ is in units of $\hslash ∕\left(10{\beta }^{-1}\right)$.

Figure 6

Time-dependent current $J_L\left(t\right)$ through left lead in response to an upward step pulse for different bandwidths: dashed line, WBL $(W=\infty )$; (i) $W=20\Gamma$, (ii) $W=10\Gamma$, (iii) $W=5\Gamma$, (iv) $W=2.5\Gamma$, and (v) $W=\Gamma$. Units and parameters are the same as in Fig. 2. Here again, the $W=100\Gamma$ curve (not shown) is indistinguishable from the $W=\infty$ curve.

Figure 7

Time-dependent current $J_L\left(t\right)$ through left lead in response to a square pulse of duration $s=3\hslash ∕\Gamma$ for different bandwidths: dashed line, WBL $(W=\infty )$; (i) $W=20\Gamma$, (ii) $W=10\Gamma$, (iii) $W=5\Gamma$, (iv) $W=2.5\Gamma$, and (v) $W=\Gamma$. Units and parameters are the same as in Fig. 2. Here again, the $W=100\Gamma$ curve (not shown) is indistinguishable from the $W=\infty$ curve.

Figure 8

Time-dependent current $J_L\left(t\right)$ through left lead in response to a square pulse, for different pulse lengths $s$ in units of $\hslash ∕\Gamma$. Other units are the same as in Fig. 2. Dashed line: current during the pulse; solid line: current after the pulse.