Anisotropy, phonon modes, and free charge carrier parameters in monoclinic $\beta$-gallium oxide single crystals

TL;DR

This study develops a dielectric tensor model for monoclinic materials, applies it to $eta$-Ga$_2$O$_3$ crystals, and determines phonon modes, dielectric constants, and free charge carrier parameters with high accuracy.

Contribution

The paper introduces a novel dielectric tensor model approach for monoclinic materials and applies it to $eta$-Ga$_2$O$_3$, providing detailed phonon and charge carrier parameters.

Findings

Identified all infrared and far-infrared active phonon modes in $eta$-Ga$_2$O$_3$

Derived monoclinic dielectric constants and generalized Lyddane-Sachs-Teller relation

Determined free charge carrier mobility and density parameters

Abstract

We derive a dielectric function tensor model approach to render the optical response of monoclinic and triclinic symmetry materials with multiple uncoupled infrared and farinfrared active modes. We apply our model approach to monoclinic $\beta$-Ga$_2$O$_3$ single crystal samples. Surfaces cut under different angles from a bulk crystal, (010) and ($\bar{2}$01), are investigated by generalized spectroscopic ellipsometry within infrared and farinfrared spectral regions. We determine the frequency dependence of 4 independent $\beta$-Ga$_2$O$_3$ Cartesian dielectric function tensor elements by matching large sets of experimental data using a point by point data inversion approach. From matching our monoclinic model to the obtained 4 dielectric function tensor components, we determine all infared and farinfrared active transverse optic phonon modes with $A_u$ and $B_u$ symmetry, and their eigenvectors within the monoclinic lattice. We find excellent agreement between our model results and results of density functional theory calculations. We derive and discuss the frequencies of longitudinal optical phonons in $\beta$-Ga$_2$O$_3$. We derive and report density and anisotropic mobility parameters of the free charge carriers within the tin doped crystals. We discuss the occurrence of longitudinal phonon plasmon coupled modes in $\beta$-Ga$_2$O$_3$ and provide their frequencies and eigenvectors. We also discuss and present monoclinic dielectric constants for static electric fields and frequencies above the reststrahlen range, and we provide a generalization of the Lyddane-Sachs-Teller relation for monoclinic lattices with infrared and farinfrared active modes. We find that the generalized Lyddane-Sachs-Teller relation is fulfilled excellently for $\beta$-Ga$_2$O$_3$.