Recent advances in experimental techniques used for probing and controlling nanofabricated devices have stimulated a strong interest in theoretically studying their nonequilibrium properties. In thermal equilibrium, electrons follow the Fermi–Dirac distribution, whereas under nonequilibrium conditions, the electron distribution generally deviates from the Fermi–Dirac form. A typical example of this is a nanowire connected to two electrodes having different electrochemical potentials. The electron distribution in the nanowire exhibits a characteristic “two-step” structure, which can be regarded as a superposition of two Fermi–Dirac distributions shifted by the applied bias voltage.

 Such nonequilibrium electron distributions are known to play a crucial role in a variety of quantum phenomena, such as controlling the critical currents in Josephson junctions, formation of π-junctions, and emergence of unconventional superconducting states. Therefore, developing a theoretical framework to determine nonequilibrium electron distributions is essential for exploring quantum many-body phenomena beyond the equilibrium paradigm.

In this study, we developed a theoretical framework based on the nonequilibrium Green’s function, which provided a unified description of the electron distributions in nanowires connected at both ends to electrodes having different electrochemical potentials. By incorporating both elastic impurity scattering and inelastic phonon scattering within the wire, we systematically analyzed the evolution of the electron distributions across different transport regimes.

 In the ballistic regime, where scattering is negligible, the electron distribution exhibits a spatially uniform two-step structure. In the diffusive regime, which is characterized by multiple impurity scattering, the distribution retains its two-step character, but becomes position-dependent. When inelastic scattering is significant, energy relaxation leads to a gradual crossover toward a local thermal electron distribution that is well approximated by the Fermi–Dirac distribution function.  

 A key achievement of this study is that the three distinct transport regimes are captured within a single theoretical framework. In addition, the microscopic theory used in this study enables us to incorporate effects that are difficult to treat in conventional coarse-grained approaches such as contact resistance and interference arising from reflections at interfaces.

 The versatility of the nonequilibrium Green’s function method suggests that the present framework can be broadly applied to other driven quantum systems such as superconducting heterostructures and periodically driven systems. Thus, this study provides a solid foundation for exploring a wide range of nonequilibrium quantum many-body phenomena.  

(Written by Taira Kawamura on behalf of all the authors)

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