Time-dependent quantum transport: A practical scheme using density functional theory

We present a computationally tractable scheme of time-dependent transport phenomena within open-boundary time-dependent density functional theory. Within this approach all the response properties of a system are determined from the time propagation of the set of “occupied” Kohn-Sham orbitals under the influence of the external bias. This central idea is combined with an open-boundary description of the geometry of the system that is divided into three regions: left∕right leads and the device region (“real simulation region”). We have derived a general scheme to extract the set of initial states in the device region that will be propagated in time with proper transparent boundary-condition at the device∕lead interface. This is possible due to a new modified Crank-Nicholson algorithm that allows an efficient time-propagation of open quantum systems. We illustrate the method in one-dimensional model systems as a first step towards a full first-principles implementation. In particular we show how a stationary current develops in the system independent of the transient-current history upon application of the bias. The present work is ideally suited to study ac transport and photon-induced charge-injection. Although the implementation has been done assuming clamped ions, we discuss how it can be extended to include dissipation due to electron-phonon coupling through the combined simulation of the electron-ion dynamics as well as electron-electron correlations.

Figure 1

The schematic sketch of an electrode-junction-electrode system with semiperiodic electrodes.

Figure 2

The continuum states of the square potential barrier at different energies with leads at zero potential. Upper panel: eigenstates for $\epsilon =0.45\mathrm{a.u.}$, just below the barrier height of $0.5\mathrm{a.u.}$. Lower panel: eigenstates at $\epsilon =0.6\mathrm{a.u.}$.

Figure 3

The time evolution of a Gaussian wave packet with initial width $1.0\mathrm{a.u.}$ and initial momentum $0.5\mathrm{a.u.}$ for various propagation times. The transparent boundary conditions allow the wave packet to pass the propagation region without spurious reflections at the boundaries.

Figure 4

The time evolution of the current for a system where initially the potential is zero in the leads and the propagation region. At $t=0$, a constant bias with opposite sign in the left and right leads is switched on, $U=U_L=-U_R$ (values in atomic units). The propagation region extends from $x=-6$ to $x=+6\mathrm{a.u.}$ The Fermi energy of the initial state is ${\epsilon }_F=0.3\mathrm{a.u.}$ The current in the center of the propagation region is shown.

Figure 5

Time evolution of the current through a double square potential barrier in response to an applied constant bias (given in atomic units) in the left lead. The potential is given by $V\left(x\right)=0.5\mathrm{a.u.}$ for $5⩽\mid x\mid ⩽6\mathrm{a.u.}$ and zero otherwise; the propagation region extends from $x=-6$ to $x=+6\mathrm{a.u.}$ The Fermi energy of the initial state is ${\epsilon }_F=0.3\mathrm{a.u.}$ The current in the center of the structure is shown.

Figure 6

The time evolution of the total number of electrons in the region $\mid x\mid ⩽6$ for the double square potential barrier of Fig. 5.

Figure 7

The time evolution of the current for a double square potential barrier when the bias is switched on in two different manners: in one case, the bias $U_0$ is suddenly switched on at $t=0$ while in the other case the same bias is achieved with a smooth switching $U\left(t\right)=U_0{\sin }^2\left(\omega t\right)$ for $0<t<\pi ∕\left(2\omega \right)$. The parameters for the double barrier and the other numerical parameters are the same as the ones used in Fig. 5. $U_0$ and $\omega$ given in atomic units.

Figure 8

The time evolution of the current for a square potential barrier in response to a time-dependent, harmonic bias in the left lead, $U_L\left(t\right)=U_0\sin \left(\omega t\right)$ for different amplitudes $U_0$ (values in a.u.) and frequency $\omega =1.0\mathrm{a.u.}$ The potential is given by $V\left(x\right)=0.6\mathrm{a.u.}$ for $\mid x\mid ⩽6.0\mathrm{a.u.}$ and zero otherwise. The propagation region extends from $x=-6$ to $x=+6\mathrm{a.u.}$ The Fermi energy of the initial state is ${\epsilon }_F=0.5\mathrm{a.u.}$ The current at $x=0$ is shown.