  An accurate description of 2-D quantum transport in a double-gate metal oxide
semiconductor filed effect transistor (dgMOSFET) requires a high-resolution
solver to a coupled system of the 4-D Wigner equation and 2-D Poisson equation.
In this paper, we propose an operator splitting spectral method to evolve such
Wigner-Poisson system in 4-D phase space with high accuracy. After an operator
splitting of the Wigner equation, the resulting two sub-equations can be solved
analytically with spectral approximation in phase space. Meanwhile, we adopt a
Chebyshev spectral method to solve the Poisson equation. Spectral convergence
in phase space and a fourth-order accuracy in time are both numerically
verified. Finally, we apply the proposed solver into simulating dgMOSFET,
develop the steady states from long-time simulations and obtain numerically
converged current-voltage (I-V) curves.
