Technical Papers and Presentations

INTRODUCTION The coupling of the Schrödinger and Maxwell equations allows to predict the self-generated electromagnetic (EM) field for a quantum charged particle moving in a domain. Such problem is really challenging, and it must be investigated in order to simulate the behavior of a particle in an EM field. The COMSOL Multiphysics® simulation software provides a useful platform for the solution of such a problem thanks to its high versatility and the potentiality to solve custom problems. THEORY The Hamiltonian of a particle moving in an electromagnetic field is provided by (1) H=(p-qA)^2/2m+qϕ (1) where A and ϕ are the vector and scalar potentials respectively, derived using the Lorentz Gauge as in (2). ∇^2 A-1/c^2 ∂t^2 A=-μJ ∇^2 ϕ-1/c^2 ∂t^2 ϕ=-ρ/ε (2) From the time-dependent Schrödinger equation it is possible to calculate the quantum charge distribution (ρ) and current density (J) defined as follows in (3a) and (3b). J=q[ℏ/2im (ψ^* ∇ψ-ψ∇ψ^* )-q/m |ψ|^2 A ] (3a) ρ=q|ψ|^2 (3b) IMPLEMENTATION The coupling is realized in COMSOL Multiphysics® using the Coefficient Form PDE interface. An absorbing region with low reflections necessary to simulate the transport without undesired reflections. That can be easily included in the module using a 2D source region with an imaginary potential defined by an analytical function. The computed quantum current density is exported as an interpolation function. Using the Electromagnetic Waves, Transient interface present in the RF Module is than possible to solve for the EM generated by a surface current density in a 3D domain. RESULTS The feasibility of calculating the EM field generated by a propagating quantum wave packet has been demonstrated. The behavior of such a wave packet has also been simulated considering a constant magnetic field of B_0=0.5 T along the y-axis as shown in Fig. (2b). Such field rotate the direction of the wave packet as expected from classical electrodynamics of a charged particle due to the Lorentz force. CONCLUSION This work presents a quantum–electromagnetic coupling in time domain simulations. This coupling could be of great interest in the simulation of the coherent quantum electric transport especially for ballistic device design and optimization. In future work we expect the potentiality of directly coupling an electromagnetic field in the same COMSOL® model It can be used to optimize device geometry and investigate the behavior of coherent quantum transport.