Nonequilibrium electron distribution function in a voltage-biased nanowire: A nonequilibrium Green's function approach

Abstract

We develop a theoretical framework to determine distribution functions in nonequilibrium systems coupled to equilibrium reservoirs, by using the nonequilibrium Green's function technique. As a paradigmatic example, we consider the nonequilibrium distribution function in a nanowire under a bias voltage. We model the system as a tight-binding chain connected to reservoirs with different electrochemical potentials at both ends. For electron scattering processes in the wire, we consider both elastic scattering from impurities and inelastic scattering from phonons within the self-consistent Born approximation. We demonstrate that the nonequilibrium distribution functions, as well as the electrostatic potential profiles, in various scattering regimes are well described within our framework. This scheme will contribute to advancing our understanding of quantum many-body phenomena driven by nonequilibrium distribution functions that have different functional forms from the equilibrium ones.

Figures (16)

• Figure 1: A nanowire connected between two electrodes with different electrochemical potentials due to the bias voltage. These electrodes can be approximated as isolated systems in thermal equilibrium, where electrons follow the Fermi-Dirac distribution function. On the other hand, electrons at position $x$ in the wire follow a nonequilibrium distribution function $f^{\rm neq}_x(\omega)$, which in general has a different functional form from the Fermi-Dirac distribution function. The form of the nonequilibrium distribution function $f^{\rm neq}_x(\omega)$ depends on scattering processes experienced by electrons as they traverse the wire Pothier1997GueronThesisAnthore2003Huard2005HuardThesisPierreThesisAnneThesisTikhonov2020Franceschi2002Chen2009Bronn2013. When the wire length is shorter than the electron inelastic mean free path, $f^{\rm neq}_x(\omega)$ exhibits the two-step structure at low temperatures, reflecting the different electrochemical potentials in the electrodes.
• Figure 2: Schematic picture of our model. Two free-fermion $\alpha$ ($={\rm L}, {\rm R}$) reservoirs are connected to both ends of the tight-binding chain with $N$ sites. The $\alpha$ reservoir is in the thermal equilibrium state characterized by the electrochemical potential $\mu_\alpha$ and the temperature $T_{\rm env}$. The potential difference $\mu_{\rm L}-\mu_{\rm R}$ equals the applied bias voltage $eV$ across the wire. Within the chain, electrons scatter from randomly distributed impurities with the potential $U_{{\rm imp}, j}$. The electrons also interact with local phonons of frequency $\Omega_{\rm ph}$, where $g_{\rm ph}$ represents the electron-phonon coupling constant.
• Figure 3: Calculated electron distribution function $f_j^{\rm neq}(\omega)$ in a nanowire under bias voltage. The position $x_j$ along the wire is defined by Eq. \ref{['eq.xj']}. We show results for different values of impurity scattering strength $\gamma_{\rm imp}$ and electron-phonon coupling strength $\gamma_{\rm ph}$. We set $N=201$, $eV/t = 0.4$, $\gamma_{\rm lead}/t=1$, $\Omega_0/t=0.05$ and $T_{\rm env}/t=0.02$. These values are used in the following figures unless otherwise mentioned.
• Figure 4: (a) Position $x_j$ dependence of the weight function $w_j(\omega)$ in Eq. \ref{['eq.wj']}. We show the results for $\omega/t=0$, $0.1$, and $0.2$ in the ballistic limit ($\gamma_{\rm imp}=0$). (b) Impurity scattering strength $\gamma_{\rm imp}$ dependence of the weight function $w_j(\omega)$. As a typical example, we take $\omega/t=0.2$. Circles show $w_j$ calculated directly from the definition in Eq. \ref{['eq.wj']}. For comparison, solid and dashed lines show $w_j$ obtained by fitting the calculated distribution function $f^{\rm neq}_j(\omega)$ using Eq. \ref{['eq.fneq.ballistic']} for $\gamma_{\rm imp}/t=\sqrt{0.005}$ and $\sqrt{0.03}$, respectively. As $\gamma_{\rm imp}$ increases, the deviation between the two becomes more pronounced, reflecting the breakdown of Eq. \ref{['eq.fneq.ballistic']} in the diffusive regime.
• Figure 5: Calculated distribution function $f_j^{\rm neq}(\omega)$ for (a) $\gamma_{\rm imp}/t=\sqrt{0.01}$, (b) $\gamma_{\rm imp}/t=\sqrt{0.05}$, and (c) $\gamma_{\rm imp}/t=\sqrt{0.3}$. We show the distribution function at $x_j=0.1$ (near the left reservoir), $x_j=0.5$ (in the middle of the wire), and $x_j=0.9$ (near the right reservoir). We set $\gamma_{\rm ph}=0$ for all panels.