A thermodynamically consistent theory for flexoelectronics: Interaction between strain gradient and electric current in flexoelectric semiconductors

This paper presents a continuum theory for flexoelectric semiconductors and analyzes the interaction between electric currents and inhomogeneous deformations, which provides an opportunity for strain gradient engineering. Basic principles for continuum physics, including mass conservation, charge conservation, balance of linear momentum, balance of angular momentum, electrostatics, and thermodynamic laws, are established in the reference configuration for a semiconducting continuum under finite deformation. Then, free-energy imbalance (dissipation inequality) is derived. Based on the dissipation inequality and the Coleman-Noll procedure, thermodynamically consistent constitutive equations are obtained, which account for piezoelectric, flexoelectric, thermoelectric couplings, and drift-diffusion effects for electric currents. The heat conduction equation and Joule heating generation are also derived by combining the energy balance and the second Gibbs relation. Additionally, the principle of virtual work for strain gradient-dependent semiconducting continuum under finite deformation is established. The framework is then geometrically linearized for applications in infinitesimal deformation and small concentration perturbations of free carriers. Based on the reduced linear model, we obtain the exact solutions for the plan-strain problem and then analyze the tuning mechanisms of different mechanical forces on the distribution of free carriers. It is observed that bending and shear deformation would induce the electric polarization and redistribution of free carriers along the thickness direction, whilst extension and thickness-stretch would induce polarization along the axial direction. Furthermore, based on the nonlinear model, we obtain the mechanical effect on the I-V characteristics of p-type flexoelectric semiconductors and flexoelectric PN junctions. Interestingly, mechanical forces can be seen as switches to gate the electric currents in semiconductor devices via flexoelectric polarizations. The theoretical model proposed in this article can guide the design of flexoelectronic devices and can also be used to analyze the flexoelectric effect in piezotronic devices. Since the formulation is based on finite deformation theory, it is also suitable for the analysis and design of flexible electronic devices.