Anisotropy, phonon modes, and free charge carrier parameters in monoclinicβ-gallium oxide single crystals

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 far-infrared active modes. We apply our model approach to monoclinic β-Ga 2 O 3 single-crystal samples. Surfaces cut under different angles from a bulk crystal, (010) and (201), are investigated by generalized spectroscopic ellipsometry within infrared and far-infrared spectral regions. We determine the frequency dependence of 4 independent β-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 infrared and far-infrared 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 β-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 β-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 far-infrared active modes. We find that the generalized Lyddane-Sachs-Teller relation is fulfilled excellently for β-Ga 2 O 3 .

Coordinate-Invariant Lyddane-Sachs-Teller Relationship for Polar Vibrations in Materials with Monoclinic and Triclinic Crystal Systems

“…[30], where the application of the LST relation was not discussed. A generalized ellipsometry analysis [9] of phonon modes and free charge carrier properties in β-Ga 2 O 3 was reported very recently, and an alternative approach to the LST relation was suggested for materials with monoclinic symmetry [31]. Virtually no information is available on triclinic materials, which appears as a widely uncharted field of condensed matter physics.…”

“…The latter can be obtained, for example, from density-functional perturbation theory calculations [44], and from matching model equations using Eqs. (5)- (6) to experimental data [31]. The displacement vectors of all TO and LO modes is thereby determined, and the index l unambiguously identifies mode and vector via Eq.…”

“…Pavinich and Belousov suggested use of the subdeterminant of the dielectric tensor within the a-c plane for long wavelength phonon mode analysis of monoclinic LiAlSi 2 O 6 [24]. A derivation of the Pavinich and Belousov equation for monoclinic crystal systems is shown by Schubert et al [31] and applied to β-Ga 2 O 3 . Setting ω ¼ 0 in Eq.…”

“…For mononclinic and triclinic systems, the CC-LST relations become problematic, because, in general, for such systems an α direction cannot be found anymore along which the displacement directions of multiple TO and LO modes line up. For example, the displacement vectors of all TO and LO modes within the monoclinic plane of β-Ga 2 O 3 each possess a different direction, which was obtained from experiment and verified by density-functional theory calculations [31]. Furthermore, the ratios equaled in Eq.…”

“…The dielectric function tensor in Eq. (6) has six independent complex-valued parameters (spectra), which can be obtained by experiment, for example, using generalized spectroscopic ellipsometry [9,31,42]. Two characteristic sets of N optical modes, transverse optical (TO; ω TO;l ), and longitudinal optical (LO; ω LO;l ), can be obtained, respectively, from the roots of the determinants of ε −1 , and ε,…”