Numerical determination of a non-equilibrium many-body statistical operator for quasi-bound electrons in a gated nanowire system

We present a numerical approach to construct a non-equilibrium many-body statistical operator $\hat{\rho}_\mathrm{rel}$ for an adaptive subspace of relevant quasi-bound electronic states in a semiconductor nanowire-based field-effect transistor (NWFET). As a constraint for $\hat{\rho}_\mathrm{rel}$, we assume that the single-particle density matrix $\rho_1$ is a given quantity, resulting from a non-equilibrium Green's function (NEGF) calculation for the NWFET for a given set of applied voltages. Two different orthonormal (ON) eigenbases for $\hat{\rho}_\mathrm{rel}$ are considered: (A) a Slater determinant basis of natural orbitals (eigenstates of $\rho_1$) and (B) the eigenbasis of the projected many-body Hamiltonian $\hat{H}_\mathrm{rel}$ within a relevant Fock subspace of the system. As for the eigenvalues $w_n$ of $\hat{\rho}_\mathrm{rel}$, we furthermore assume that $w_n$ have a generalized Boltzmann form, parameterized by effective electrochemical potentials of natural orbitals and a given temperature. From the determined $\hat{\rho}_\mathrm{rel}$, in turn, one can calculate expectation values for any many-body observable within the relevant subspace. As an example, we analyze the electron density and the covariance of the density-density correlation function for representative electronic preparations of the NWFET.