Inelastic scattering in nanoscale devices: One-shot current-conserving lowest-order approximation

We describe a lowest-order approximation (LOA) to the nonequilibrium Green's function in the presence of interactions, and generally address how one can build $\Phi$-derivable one-shot approximations that satisfy the continuity equation. These approximations produce conserved electronic currents in one shot, requiring only one self-energy evaluation and, when applicable, they are as accurate as but much faster than the corresponding self-consistent approximation. This challenges the currently adopted view that heavy self-consistent calculations are necessary to get a satisfactory prediction of transport in nanoscale structures. We illustrate this with the case of electron-phonon scattering expressed within the self-consistent Born approximation (SCBA). In the LOA, the SCBA is further approximated by accounting only for one-phonon processes. LOA and SCBA are compared in one-dimensional wire where electrons interact with one optical phonon mode at room temperature. The LOA is found to provide a considerable reduction in computational time. Its limitations and extensions to include two-phonon processes are discussed.

Figure 1

In all the figures, thick solid lines represent the self-consistent electron GF $G$, thin solid lines represent the non-self-consistent electron GF $g_0$, while dashed lines represent the noninteracting phonon propagator, $D$. Left panel: (a) the self-energy diagram in the SCBA; (b) to find $\Phi$ from $\Sigma$, close $\Sigma$ with a $G$ and divide by the resulting number of $G$'s. By $\Phi =1/2DGG$, we mean $\Phi =\frac{1}{2}\int d3d4G(4,3)G(3,4)D(3,4)$; (c) when $\Phi$ is built as in (b), then $\Sigma$ can be obtained back from $\Phi$ as shown. Right panel: writing the Dyson equation as $G=G_0+\delta G$ in (d), we show the diagrams for the exact $\delta G$; in (e) we show $\delta G$ in the LOA; in (f) we show an extension of the LOA that includes two-phonon processes (2LOA).

Figure 2

Current conservation in the LOA. For the model described in the text, the LOA is numerically shown to exactly satisfy current conservation $I_L=-I_R$. The graph has been obtained plotting $I_L$ vs $-I_R$ for different electron-phonon couplings ${10}^{-4}<M<{10}^{-3}$ eV${}^2$ and applied bias voltages $0<V<0.4$ V.

Figure 3

Top panel: For $M=2\times {10}^{-4}$ eV${}^2$: (a) the current-voltage characteristics; (b) spectral resolution of the current density in the left interface, $J_L\left(\omega \right)$, calculated for $V=0.2$ V; (c) the same as in (b) but for the right interface. Middle panel: (d), (e), and (f) are the same as (a), (b), and (c) but calculated for $M=4\times {10}^{-4}$ eV${}^2$. Bottom panel: As above but for $M=8\times {10}^{-4}$ eV${}^2$. For the one-phonon mode with $M=8\times {10}^{-4}$ eV${}^2$ and $V=0.2$ V, the phonon-induced on-current degradation is about 11$\%$ in the LOA compared to 8$\%$ in the SCBA.