First-principles scattering-state approach for nonlinear electrical transport in nanostructures

We present an ab initio scattering-state method for calculating the electrical transport properties of nanostructures at a finite bias voltage. A typical system of interest consists of two semi-infinite crystalline metal probes and a nanostructure (e.g., an atomic wire or a molecule) placed between the probes. The two metal probes have different chemical potentials at a finite bias voltage. The electronic structure of the system is described by the Kohn-Sham density functional method, with appropriate boundary conditions imposed on the electronic density and potential inside the metal probes. We expand the electronic wave functions with pseudoatomic orbitals and directly solve the self-consistent Kohn-Sham equation to obtain the transmission probabilities of electrons incident from either of the probes. The current through the molecule or nanostructure is then obtained by integrating the transmission between the two chemical potentials. Our scattering-state method provides stable and efficient algorithms for the complex bands, scattering-state wave functions, and the nonequilibrium steady-state electron density. As illustrations of the method, we present the calculated electrical transport properties of a defective carbon nanotube, a four-carbon atomic chain, and a benzene-dithiol molecular junction.

Figure 1

Two-probe setup for conductance measurement of a nanostructure. As an example, a benzene-dithiol molecule is shown, placed between two gold (111) surfaces.

Figure 2

(10,10) carbon nanotube with pentagon-heptagon-pair defect. (a) Atomic structure. (b) Transmission spectra. The solid line is the result of our present work using a double-zeta-polarization basis set to expand the electronic wave functions. An energy-grid spacing of $1\mathrm{meV}$ is used for the calculation of the spectrum and the Fermi energy is set to zero. Empty dots denote the result of the plane-wave-basis calculation (Refs. 27, 28), plotted for comparison. The dashed line is the transmission spectrum without the defect.

Figure 3

Four-carbon-atomic chain between two aluminum bars. (a) Atomic structure. (b) Current-voltage characteristics. Filled dots denote obtained results and the solid line is a guide. Each dot requires a self-consistent calculation for electron redistribution and change in scattering states at the bias voltage. (c) Transmission spectra at the bias voltages of 0, 0.60, and $1.18\mathrm{V}$.

Figure 4

(Color online) A benzene-dithiol molecule $\left({\mathrm{C}}_6{\mathrm{H}}_4{\mathrm{S}}_2\right)$ contacted with gold probes. (a) Local part $\left(V_{\mathit{loc}}\right)$ of the self-consistent Kohn-Sham effective potential when the bias voltage $\left(V_b\right)$ is $2.0\mathrm{V}$. In the upper panel, the potential is plotted in a color scale (in eV) on the plane containing the molecule, with dark solid contours for $V_{\mathit{loc}}=0,-10,-20,\dots ,-50\mathrm{eV}$. In the lower panel, the potential is plotted along the two dotted lines marked on the upper panel. (b) Local-potential difference, $\delta V_{\mathit{loc}}=V_{\mathit{loc}}(V_b=2\mathrm{V})-V_{\mathit{loc}}(V_b=0\mathrm{V})$, plotted (in eV) at the same positions as in (a). Contours in the upper panel represent $\delta V_{\mathit{loc}}=-0.8,-0.6,-0.4,\dots ,0.8\mathrm{eV}$. (c) Current-voltage characteristics obtained with various ${\mathbf{k}}_{\Vert }$-grid points on the $k_x-k_y$ plane.

Figure 5

(Color online) Transmission spectra through a benzene-dithiol molecule between two Au (111) surfaces at various bias voltages: (a) $0\mathrm{V}$, (b) $1.0\mathrm{V}$, (c) $1.6\mathrm{V}$, and (d) $2.0\mathrm{V}$. Each plot shows the transmission spectrum averaged over an $8\times 8$ ${\mathbf{k}}_{\Vert }$ grid and those for ${\mathbf{k}}_{\Vert }=(0,0)$, $(\frac{\pi }{a},0)$, $(0,\frac{\pi }{b})$, and $(\frac{\pi }{a},\frac{\pi }{b})$, with $a=15.1\mathrm{\AA }$ and $b=8.7\mathrm{\AA }$ being the sizes of the supercell in the $x$ and $y$ directions, respectively. The molecule is placed on the $xz$ plane.