First-principles investigation of transient dynamics of molecular devices

Based on the nonequilibrium Green's function (NEGF) and time-dependent density-functional theory (TDDFT), we propose a formalism to study the time-dependent transport behavior of molecular devices from first principles. While this approach is equivalent to the time-dependent wave-function approach within TDDFT, it has the advantage that the scattering states and bound states are treated on equal footing. Furthermore, it is much easier to implement our approach numerically. Different from the time-dependent wave-function $\left[\psi \right(t,E)$] approach, our formalism is in the time space [${\mathbf{G}}^r(t,t^{\prime })$], making this method superior in the time-dependent transport problem with many subbands in the transverse direction. For the purpose of numerical implementation on molecular devices, a computational tractable numerical scheme is discussed in detail. We have applied our formalism to calculate the transient current of two molecular devices Al-1,4-dimethylbenzene-Al and Al-benzene-Al from first principles. In the calculation, we have gone beyond the wideband limit and used the adiabatic local density approximation that was used within TDDFT. It is known that when the wideband limit is abandoned, the boundary condition of the transport problem is non-Markovian, resulting in a memory term in the effective Hamiltonian of the scattering region. To overcome the computational complexity due to the memory term, we have employed a fast algorithm to speed up the calculation and reduced the CPU time from the scaling $N^3$ to $N^2{\log }_2^2\left(N\right)$ for the steplike pulse, where $N$ is the number of time steps in the time evolution of the Green's function. To ensure the accuracy of our method, we have done a benchmark transient calculation on an atomic junction using a time-dependent wave-function approach within TDDFT in momentum space, which agrees very well with the result from our method.

Figure 1

A sketch illustrating the typical two-terminal molecular device which is sandwiched between two semi-infinite external leads. The central scattering region contains a molecular device and several buffer layers of leads. Left and right leads are assumed to be periodic in the transport $z$ direction. ${\mathbf{H}}_{\alpha 0}$ is the unit-cell matrix and ${\mathbf{H}}_{\alpha 1}$ describes the interaction between the nearest unit cell.

Figure 2

Schematic diagram of two molecular devices Al-1,4-dimethylbenzene-Al in panel (a) and Al-benzene-Al in panel (b). The device consists of 1,4-dimethylbenzene and benzene molecules coupled to perfect aluminium atomic electrodes along the (100) direction which extend to the reservoirs at $\pm \infty$.

Figure 3

The schematic plot of computation procedures (from 1 to 15). Here, blue triangle represents direct calculation and green square represents FFT calculation.

Figure 4

The scaling behavior of our numerical scheme with $m_0=16$. The inset shows CPU time versus ${\log }_2\left(m_0\right)$. Blue upper triangle represents CPU time of FFT calculation. Red lower triangle represents CPU time of direction calculation. Black sphere is the total CPU time.

Figure 5

Time-dependent current $I\left(t\right)$ versus time with $V=0.0001$ a.u. for one-dimensional atomic carbon chain. Blue solid line and red dashed line are time-dependent current calculated by using the orbital-space method and the $K$-space method; green dashed line is the dc current at steady-state limit.

Figure 6

In panels (a) and (b), time-dependent current $I\left(t\right)$ versus time with $V=0.0025$ a.u. for Al-1,4-dimethylbenzene-Al device are plotted [panel (b) is for a short-time scale]. Panel (c) shows the largest contribution to the oscillation from a particular scattering state with $E=-0.03459,k=1.81$. (d) Transmission coefficient when the system is in equilibrium state.

Figure 7

Time-dependent current $I\left(t\right)$ versus time with different bias voltages for Al-benzene-Al device. In panel (a), the dotted and solid lines correspond to $V=0.0025$ a.u. and $V=0.005$ a.u., respectively. Panels (b) and (C) show the early behavior of time-dependent current. Blue solid line and red dashed line are time-dependent current and black straight solid and dashed line are current at steady-state limit.