Dipole analysis of the dielectric function of color dispersive materials: Application to monoclinic ${\mathrm{Ga}}_2{\mathrm{O}}_3$

We apply a generalized model for the determination and analysis of the dielectric function of optically anisotropic materials with color dispersion to phonon modes and show that it can also be generalized to excitonic polarizabilities and electronic band-band transitions. We take into account that the tensor components of the dielectric function within the Cartesian coordinate system are not independent of each other but are rather projections of the polarization of dipoles oscillating along directions defined by the, non-Cartesian, crystal symmetry and polarizability. The dielectric function is then composed of a series of oscillators pointing in different directions. The application of this model is exemplarily demonstrated for monoclinic ($\beta$-phase) ${\mathrm{Ga}}_2{\mathrm{O}}_3$ bulk single crystals. Using this model we are able to relate electronic transitions observed in the dielectric function to atomic bond directions and orbitals in the real space crystal structure. For a thin film revealing rotational domains we show that the optical biaxiality is reduced to uniaxial optical response.

Figure 1

Relation between the laboratory coordinate system ($x$, $y$, and $z$) and the system of the excitation ($x^{\prime }$, $y^{\prime }$, and $z^{\prime }$). The blue double arrow denotes the direction of the dipole moment of the excitation as discussed in the text.

Figure 2

Schematic representation of the unit cell of $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$. The tetrahedrally coordinated Ga atoms are shown in blue, whereas the octahedrally coordinated ones are shown in green. The oxygen atoms are marked in red. The $\left(\overline{2}01\right)$ plane is highlighted by the gray rectangle. (Image created by VESTA [23].)

Figure 3

Experimental (symbols) and calculated (lines) infrared spectra of the MM elements of a $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ bulk single crystal for an angle of incidence of ${70}^{\circ }$. The corresponding orientation of the crystal is given by the Euler angles on top of each column in the $yzx$ notation.

Figure 4

Experimental (symbols) and calculated (lines) spectra of the MM elements of a $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ bulk single crystal for an angle of incidence of ${70}^{\circ }$. The corresponding orientation of the crystal is given by the Euler angles on top of each column in the $yzx$ notation.

Figure 5

Dielectric function (black solid line) of a $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ bulk single crystal in the infrared and UV spectral range. The red to blue colored solid lines represent the contribution in the investigated UV spectral range, whereas the red dashed lines represent the contribution of the high-energy contributions. The arrows in the insets depict the orientation of the corresponding dipole moments of the phonons and the excitonic transitions in the $x\text{−}z$ plane ($a\text{−}c$ plane) and their relative amplitude ratio.

Figure 6

(a) and (b) Schematic representation of projections of the crystal structure of $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ into the $x\text{−}y$ plane ($a\text{−}b$ plane) (a) and $x\text{−}z$ plane ($a\text{−}c$ plane) (b). The unit cell is indicated by the black framed box. Bonds are indicated by lines between the atoms. The tetrahedrally coordinated Ga(I) atoms are shown in blue and the octahedrally coordinated Ga(II) are shown in green. The oxygen atoms are marked in red. (c) Sublayers of the $x\text{−}z$ plane ($a\text{−}c$ plane) (left: sublayer 1, right: sublayer 2) as indicated in (a). Here the oxygen atoms at different lattice sites are highlighted by colors as O(I) red, O(II) orange, and O(III) yellow (see also text). The dashed colored arrows relate the dipole directions of the transitions $X_1\cdots X_4$ to atomic bonds within the crystal structure. Please note that only one example is shown for each transition. (Images created by VESTA [23].)

Figure 7

Real (a) and imaginary (b) part of the thin film's pseudodielectric function for angles of incidence ${60}^{\circ }$ and ${70}^{\circ }$. The experimental and calculated data are shown as symbols and red solid lines, respectively.

Figure 8

Real (a) and imaginary (b) part of the tensor components of the ${\mathrm{Ga}}_2{\mathrm{O}}_3$ thin film (solid lines). For comparison, the components calculated by Eq. (11) using the single crystal values are shown as dashed lines.