Time-dependent density functional theory with ultrasoft pseudopotentials: Real-time electron propagation across a molecular junction

A practical computational scheme based on time-dependent density functional theory (TDDFT) and ultrasoft pseudopotentials (USPP) is developed to study electron dynamics in real time. A modified Crank-Nicolson time-stepping algorithm is adopted, under plane-wave basis. The scheme is validated by calculating the optical absorption spectra for a sodium dimer and a benzene molecule. As an application of this USPP-TDDFT formalism, we compute the time evolution of a test electron packet at the Fermi energy of the left metallic lead crossing a benzene-(1,4)-dithiolate junction. A transmission probability of 5–7%, corresponding to a conductance of $4.0–5.6\mathrm{\mu }\mathrm{S}$, is obtained. These results are consistent with complex band structure estimates and Green’s function calculation results at small bias voltages.

Figure 1

(Color online) Optical absorption spectra of ${\mathrm{Na}}_2$ cluster obtained from direct time-stepping TDLDA calculation using norm-conserving TM pseudopotentials. The results should be compared with Fig. 1 of Marques et al. (Ref. 63).

Figure 2

(Color online) Optical absorption spectrum of benzene $\left({\mathrm{C}}_6{\mathrm{H}}_6\right)$ molecule. The results should be compared with Fig. 2 of Marques et al. (Ref. 29).

Figure 3

Illustration of the Landauer transmission formalism.

Figure 4

(Color online) Atomistic configurations of our USPP-TDDFT simulations. Au: yellow (light gray), S: magenta (large dark gray), C: black (small dark gray), and H: white. (a) A 12-atom Au chain. Bond length: $\mathrm{Au}\text{−}\mathrm{Au}$ $2.88\mathrm{\AA }$. (b) BDT $\left(\text{−}\mathrm{S}\text{−}{\mathrm{C}}_6{\mathrm{H}}_4\text{−}\mathrm{S}\text{−}\right)$ junction connected to Au chain contacts. Bond lengths: $\mathrm{Au}\text{−}\mathrm{Au}$ $2.88\mathrm{\AA }$, $\mathrm{Au}\text{−}\mathrm{S}$ $2.41\mathrm{\AA }$, $\mathrm{S}\text{−}\mathrm{C}$ $1.83\mathrm{\AA }$, $\mathrm{C}\text{−}\mathrm{C}$ $1.39\mathrm{\AA }$, and $\mathrm{C}\text{−}\mathrm{H}$ $1.1\mathrm{\AA }$.

Figure 5

Band structure of a one-atom Au chain with 64 Monkhorst-Pack (Ref. 70) $\mathbf{k}$ sampling in the chain direction. The Fermi level, located at $-6.65\mathrm{eV}$, is marked as the dashed line.

Figure 6

(Color online) Evolution of modulated Fermi electron density in time along the chain direction. The electron density, in the unit of ${\mathrm{\AA }}^{-1}$, is an integral over the perpendicular $yz$ plane and normalized along the $x$ direction, which is then color coded.

Figure 7

(Color online) Projected density of states (PDOS) of the 12-atom Au chain.

Figure 8

(Color online) Evolution of filtered wave package density in time along the chain direction. The electron density, in the unit of ${\mathrm{\AA }}^{-1}$, is a sum over the perpendicular $yz$ plane and normalized along the $x$ direction. The normalized electron density is color coded by the absolute value.

Figure 9

(Color online) $R(x^{\prime },t)$ versus time plot. Curves are measured in ten different regions with different $x^{\prime }$ positions, which equally divide the region from the right S atom to the boundary on the right-hand side.