Nanoscale device modeling using a conserving analytic continuation technique

We propose an alternative approach to self-consistency and conservation laws in the theory of nonequilibrium Green's functions (NEGF's), which provides an infinite family of conserving but, generally, non-self-consistent approximations. Within any $\Phi$-derivable approximation the associated Born series for the NEGF is shown to be conserving. Expectation values calculated from the Born series are then used to build a Padé table of approximations, while conservation laws are naturally preserved. We implement this technique for the $\Phi$-derivable self-consistent Born approximation (SCBA), for which we obtain a recursion relation that yields the Born series for the NEGF up to any desired order. The expectation values calculated from the Born series are then postprocessed to build a Padé table of conserving approximations. The calculation of the SCBA photocurrent in a biased molecular junction model provides an example where, in addition to conservation laws, a substantial convergence acceleration relative to standard techniques is achieved. The present reformulation of the SCBA might aid convergence to the fully self-consistent results in a wide variety of problems.

Figure 1

Feynman diagrams corresponding to the SCBA (see text). Full lines are interacting NEGF's, thin lines noninteracting NEGF's, and dashed lines free boson GF's. (a) The SCBA Luttinger-Ward functional $\Phi$. (b) The SCBA self-energy. (c) The SCBA NEGF. (d) $G_1$ calculated from Eq. (2) with $n=1$. (e) Self-consistent second Born approximation, which includes second-order diagrams not accounted for by the SCBA. (f) A third-order diagram not accounted for by SC2BA. In (a)–(d) the rightmost equality shows the Born series to second order.

Figure 2

Photocurrent generation and degradation in a biased model molecular junction (see text). (a) In this model there is no direct tunneling. Electrons injected from the left electrode into the highest energy level are reflected back into the electrode. (b) One-photon processes generate a photocurrent: by emmiting and reabsorbing a photon electrons can make transitions between the highest level (coupled to the left electrode) and the second highest (coupled to the right electrode), thereby generating a photocurrent as tunneling into the right electrode becomes possible. (c) An example of two-photon process: we expect higher order photon processes to degrade the one-photon photocurrent, as the charge transported between L and R is now also involved in multiple transitions up and down the level ladder.

Figure 3

Photocurrent (in $\mu$A) as a function of source-drain bias (in volts). In (a)–(c) we show a comparison between currents calculated from the SCBA and the Born series ($I_{N/0}$), for $N=1$, 3, and 5, and the values of $M$ shown. (a) For $M=2.5$ meV excellent agreement between SCBA and Born series currents are obtained, already for $N=3$. (b),(c) For $M=5$–10 meV, the Born series is already divergent, blowing up badly as $M$ is increased—note the log scale in (c). In (d)–(f) we compare SCBA currents with those obtained from the Padé analytic continuation of the Born series for the current, $I_{1/1}$, $I_{2/2}$, and $I_{3/3}$. The Padé approximant $I_{3/3}$ reproduces the exact SCBA result, and is calculated from the Born series including terms up to sixth order. For $M=10$ meV and high bias, $I_{1/1}/I_{\mathrm{SCBA}}=0.68$, $I_{2/2}/I_{\mathrm{SCBA}}=0.90$, and $I_{3/3}/I_{\mathrm{SCBA}}=0.97$.