Asymptotic self-consistency in quantum transport calculations

Ab initio simulations of quantum transport commonly focus on a central region which is considered to be connected to infinite leads through which the current flows. The electronic structure of these distant leads is normally obtained from an equilibrium calculation, ignoring the self-consistent response of the leads to the current. We examine the consequences of this, and show that the electrostatic potential $\Delta \varphi$ is effectively being approximated by the difference between electrochemical potentials $\Delta \mu$, and that this approximation is incompatible with asymptotic charge neutrality. In a test calculation for a simple metal-vacuum-metal junction, we find significant errors in the nonequilibrium properties calculated with this approximation, in the limit of small vacuum gaps. We provide a scheme by which these errors may be corrected.

Figure 1

(Color online) The figure shows the difference between electrochemical potentials $\Delta \mu$ as a function of the drop in the electrostatic potential $\Delta \varphi$, in units of the equilibrium Fermi energy, for different values of the electrode-electrode distance $L$. $L=0.25r_s$ (solid line), $0.5r_s$ (dashed), $1r_s$ (short dashes), $1.5r_s$ (dots), and $2r_s$ (dot-dashed). $\Delta \mu \approx \Delta \varphi$ only for large electrode-electrode spacings. For reference $E_F=3.131\mathrm{eV}$.

Figure 2

(Color online) (a) Resistivity dipoles calculated for different values of $\Delta \varphi$ with $L=1.5r_s$, using $\Delta \varphi =\Delta \mu$. The arrows indicate the presence of unphysical charges at the edges of the numerical grid. (b) Same as in (a) but calculated using our neutrality scheme. Solid line $\Delta \varphi ∕E_F=0.075$, dashed $\Delta \varphi ∕E_F=0.05$, dotted $\Delta \varphi ∕E_F=0.025$. The vertical lines indicate the positions of the edges of both jellium surfaces.

Figure 3

(Color online) $J\text{−}\Delta V$ characteristics for different electrode-electrode spacings. $L=$(a) $1r_s$; (b) $1.5r_s$; (c) $3r_s$. For the solid lines $\Delta V=\Delta \varphi =\Delta \mu$, for the dotted lines $\Delta V=\Delta \varphi$, and for the dashed lines $\Delta V=\Delta \mu$.