Subdomain-based exponential integrators for quantum Liouville-type equations

Abstract

In order to describe quantum mechanical effects, the use of the von-Neumann equation is apparent. In this work, we present a unified numerical framework so that the von-Neumann equation in center-of-mass coordinates leads to a Quantum Liouville-type equation when choosing a suitable basis. In particular, the proposed approach can be related to the conventional Wigner equation when a plane wave basis is used. The drawback of the numerical methods is the high computational cost. Our presented approach is extended to allow reducing the dimension of the basis, which leads to a computationally efficient and accurate subdomain approach. Not only the steady-state behavior is of interest, but also the dynamic behavior. In order to solve the time-dependent case, suitable approximation methods for the time-dependent exponential integrator are necessary. For this purpose, we also investigate approximations of the exponential integrator based on Faber polynomials and Krylov methods. In order to evaluate and justify our approach, various test cases, including a resonant tunnel diode as well as a double-gate field-effect transistor, are investigated and validated for the stationary and the dynamic device behavior.

Efficiency Analysis of Discontinuous Galerkin Approaches for the Application Onto Quantum-Liouville Type Equations

“…Furthermore, the effective mass m is considered to be spatially constant [13]. On the basis of these conditions, the LVNE reads as [6]…”

“…Being based on the Finite Element method (FEM), transient calculations rely on matrix-vector multiplication when applying the DG approach, making it specifically suitable for the approximation of transient exponential integrators. To approximate the transient exponential operator by the use of matrixvector multiplications, a variety of algorithms including the Krylov methods [5], Faber polynomial-based algorithms [6], or Runge-Kutta schemes [7][8][9] can be utilized. A further benefit of the scheme is the use of block-diagonal matrices.…”

Application of the Tight Binding Method onto the von-Neumann Equation

“…Hence, the von-Neumann equation in centermass coordinates χ, ξ is transformed into a Quantum Liouville-type equation defined in the phase space k. The reason for this transformation is that inflow and outflow boundary conditions with regard to the χ-direction must be integrated into the model by representing it in phase space. These boundary conditions originate from the Wigner formalism [11]. Moreover, a complex absorbing potential (CAP) is applied in ξ-direction to suppress artificial reflections due to the finitiness of the computational domain [9].…”

Computational perspective on recent advances in quantum electronics: from electron quantum optics to nanoelectronic devices and systems

“…For non-trivial interaction mechanisms a closure problem exists in the Wigner formalism [78, p 98]. Other very recent advances of Wigner function methods for electronics (and photonics, but not further discussed here) can be found in a recent special issue [103] and cover, among others, handling of boundary conditions [104][105][106] and modelling and solution approaches [107][108][109][110][111][112][113].…”