Microscopic analysis of high harmonic generation in semiconductors with degenerate bands

Abstract

Based on the multiband semiconductor Bloch equations a microscopic approach to high-harmonic generation in crystalline solids which is able to properly describe degenerate bands and band crossings is presented and analyzed. It is well-known that numerical band structure calculations typically provide electronic wave functions with an undetermined k-dependent phase which results in matrix elements which contain arbitrary k-dependent phases. In addition, such approaches usually mix degenerate bands and bands with an energy difference smaller than the numerical precision in an arbitrary way for each point in k-space. These ambiguities are problematic if one considers the dynamics induced by electric fields since the matrix elements of the position operator involve a derivative of the wave functions with respect to k. When the light-matter interaction is described in the length gauge, the problem of arbitrary phases and degenerate subspace mixing of Bloch states is solved by adopting a smooth gauge transformation along the field direction. The results obtained within this method are validated by comparing with calculations in the velocity gauge. Although we obtain in both gauges the same overall result, the length gauge is advantageous since it converges with a smaller number of bands and thus requires significantly less numerical effort than the velocity gauge. Also an unique distinction between inter-and intraband contributions and thus an instructive physical interpretation is possible in the length gauge whereas in the velocity gauge this is unclear. The computed polarization-direction-dependent high-harmonic spectra agree well with experimental data reported for GaAs. Furthermore, it is demonstrated that, under proper conditions, the Berry curvature is largely responsible for the even-order harmonics which are polarized perpendicular to the driving field.

Probing Intervalence Band Coupling via High‐Harmonic Generation in Binary Zinc‐Blende Semiconductors

“…Multiple recent studies shaped the understanding of the contributions of the Berry connection and the gauge-invariant Berry curvature to HHG. [25][26][27][28] For excitation with linear-polarized pulses, the Berry connection and curvature were shown to predominantly contribute to perpendicular-polarized HHG, whereas they are less important for parallel-polarized HH emission. This is especially true for the above bandgap high harmonic emission.…”

“…Important features are quantum interference effects in many-band configurations, [20,23] Berry curvature and the closely associated gaugedependent Berry connection as well as the phase of the complex valued transition dipoles. [24][25][26][27][28] Quantum interference, i.e., the nonperturbative interference of direct and indirect excitation pathways, was shown to explain the emission of parallel polarized even-order harmonics and its subcycle time structure, [20] the Berry curvature dominates the emission of perpendicular-polarized HHG and the response to very low excitation frequencies. [25][26][27] Herein, we focus on the mechanism of quantum interference and provide a theoretical analysis for the resulting intensity ratio between even and odd harmonic emission induced by strong-field THz excitation in semiconductors.…”

“…[24][25][26][27][28] Quantum interference, i.e., the nonperturbative interference of direct and indirect excitation pathways, was shown to explain the emission of parallel polarized even-order harmonics and its subcycle time structure, [20] the Berry curvature dominates the emission of perpendicular-polarized HHG and the response to very low excitation frequencies. [25][26][27] Herein, we focus on the mechanism of quantum interference and provide a theoretical analysis for the resulting intensity ratio between even and odd harmonic emission induced by strong-field THz excitation in semiconductors. To determine the dependency of the emission intensity, a systematic analysis and comparison of HHG in the III-V semiconductors, GaAs, InAs, and InP, are presented.…”

Non-Degenerate Two-Photon Absorption in ZnSe: Experiment and Theory

“…A major difficulty in the combination of dynamical equations in the length gauge with rather realistic band-structure models is the random gauge of the numerically obtained eigenfunctions/matrix elements at each k-point. We build on earlier works [24][25][26][27], to reconstruct a globally smooth gauge in the modeled region of k-space. This is achieved by applying a parallel transport gauge (PTG), where additionally the lines in k-space are connected via a common starting point.…”

An explicit formula for high-order sideband polarization by extreme tailoring of Feynman path integrals

“…More details on the scattering effects have also been investigated through the coupling between the density matrix elements and four-point correlations [30,31,33]. In the simplest case, the SBEs are solved in the singleelectron limit, where the Coulomb interaction between the charge carriers are neglected [3,4,11,14,19,20,[36][37][38][39][40][41][42][44][45][46][47][48][49][50][51][52][53][54]. Another important aspect is the global gauge symmetry, which has long been ignored in the study of HHG and is paid attention to only recently [21,[46][47][48][49][50][51][52][53][54].…”