Modeling the flexoelectric effect in semiconductors via a second-order collocation MFEM

In this paper, the flexoelectric effect in flexoelectric semiconductors (FSs) is studied via a second-order collocation mixed finite element method (MFEM) with the consideration of the strain gradient, electric field gradient, and drift-diffusion motion of carriers. In the present second-order collocation MFEM, the mechanical strain is assumed to be an independent quadratic polynomial function of local coordinates, and the kinematic constraint between the mechanical strain and displacement is approximately satisfied through a collocation method at 9 Gaussian quadrature points in each element. The setup for the electric field and electric potential also follows similar collocation strategy. Thus, except for the fundamental field variables, no additional DOFs are introduced. The accuracy of the proposed second-order collocation MFEM is verified by comparing numerical results with analytical solutions for a cantilever beam example. By performing the numerical experiments using the second-order collocation MFEM, the flexoelectric effect in FSs is simulated. Besides, we propose a novel mechanism for the converse flexoelectric effect in FSs which is induced by the diffusion motion of electrons. Numerical results indicate that by intensifying the diffusion motion of electrons, the converse flexoelectric effect, associated with the electric field gradient, could be enhanced.