Dynamics of Berry-phase polarization in time-dependent electric fields

TL;DR

This paper develops a nonadiabatic geometric-phase approach to analyze the time-dependent polarization and current in insulators under electric fields, revealing critical field thresholds and resonance behaviors.

Contribution

It introduces a computational framework for dynamic polarization using a nonadiabatic geometric-phase formula applicable to time-dependent electric fields.

Findings

Dynamic polarization follows a nonadiabatic geometric-phase formula.

Stationary solutions exist below a critical electric field.

Dielectric response shows Franz-Keldysh effect under bias.

Abstract

We consider the flow of polarization current J(t)=dP/dt produced by a homogeneous electric field E(t) or by rapidly varying some other parameter in the Hamiltonian of a solid. For an initially insulating system and a collisionless time evolution, the dynamic polarization P(t) is given by a nonadiabatic version of the King-Smith--Vanderbilt geometric-phase formula. This leads to a computationally convenient form for the Schroedinger equation where the electric field is described by a linear scalar potential handled on a discrete mesh in reciprocal space. Stationary solutions in sufficiently weak static fields are local minima of the energy functional of Nunes and Gonze. Such solutions only exist below a critical field that depends inversely on the density of k points. For higher fields they become long-lived resonances, which can be accessed dynamically by gradually increasing E. As an illustration the dielectric function in the presence of a dc bias field is computed for a tight-binding model from the polarization response to a step-function discontinuity in E(t), displaying the Franz-Keldysh effect.