Band-to-band transitions, selection rules, effective mass, and excitonic contributions in monoclinic $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$

We employ an eigenpolarization model including the description of direction dependent excitonic effects for rendering critical point structures within the dielectric function tensor of monoclinic $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ yielding a comprehensive analysis of generalized ellipsometry data obtained from 0.75–9 eV. The eigenpolarization model permits complete description of the dielectric response. We obtain, for single-electron and excitonic band-to-band transitions, anisotropic critical point model parameters including their polarization vectors within the monoclinic lattice. We compare our experimental analysis with results from density functional theory calculations performed using the Gaussian-attenuation-Perdew-Burke-Ernzerhof hybrid density functional. We present and discuss the order of the fundamental direct band-to-band transitions and their polarization selection rules, the electron and hole effective mass parameters for the three lowest band-to-band transitions, and their excitonic contributions. We find that the effective masses for holes are highly anisotropic and correlate with the selection rules for the fundamental band-to-band transitions. The observed transitions are polarized close to the direction of the lowest hole effective mass for the valence band participating in the transition.

Figure 1

(a) Unit cell of $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ detailing crystallographic and laboratory coordinate systems. (b) Description of the orthogonal laboratory coordinate system within the monoclinic $\mathbf{a}\text{−}\mathbf{c}$ plane. Vector ${\mathbf{c}}^{*}$ is chosen for convenience parallel to laboratory axis $y$, and orthogonal to both a and b. (c) and (d) Ellipsometry plane of incidence for the (010) and ($\overline{2}01$) surfaces of $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ single crystals used in this work, respectively. Samples are rotated stepwise around their surface normal to different measurement positions (see Sec. 3a). In (c), axis b is parallel to the plane of incidence, and in (d) parallel to the sample surface regardless of sample rotation.

Figure 2

The primitive cell and corresponding Brillouin zone used for plotting the band structure of $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$. ${\mathbf{p}}_1$, ${\mathbf{p}}_2$, and ${\mathbf{p}}_3$ denote vectors of the primitive cell (${\mathbf{p}}_3$ not labeled for better clarity of the diagram); ${\mathbf{k}}_1$, ${\mathbf{k}}_2$, ${\mathbf{k}}_3$ denote vectors of the first Brillouin zone in the reciprocal space. Labeling of high-symmetry points as proposed by Setyawan and Curtarolo [61].

Figure 3

Experimental (dotted, green lines) and best match model (solid, red lines) Mueller matrix data obtained from $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ (010) surface at three different azimuthal orientations [P1: $\phi =38.5{\left(1\right)}^{\circ }$, P2: $\phi =77.4{\left(1\right)}^{\circ }$, P3: $\phi =130.42{\left(1\right)}^{\circ }]$. Data were taken at three angles of incidence (${\mathrm{\Phi }}_a={50}^{\circ }$, ${60}^{\circ }$, ${70}^{\circ }$). Vertical lines indicate energies at which CP transitions were suggested by the lineshape analysis. For color code and line styles of vertical lines, refer to Fig. 11. Euler angle parameters $\theta =-0.04{\left(1\right)}^{\circ }$ and $\psi =0.0{\left(1\right)}^{\circ }$ are consistent with the crystallographic orientation of the (010) surface. Note that angles of incidence in element ${\mathrm{M}}_{22}$ are not labeled as they are indistinguishable from each other.

Figure 4

Same as Fig. 3 except for $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ ($\overline{2}01$) surface [P1: $\phi =184.3{\left(1\right)}^{\circ }$, P2: $\phi =228.9{\left(1\right)}^{\circ }$, P3: $\phi =266.7{\left(1\right)}^{\circ }]$. Euler angle parameters $\theta =89.97{\left(1\right)}^{\circ }$ and $\psi =-52.9{\left(1\right)}^{\circ }$ are consistent with the crystallographic orientation of the ($\overline{2}01$) surface. Note that angles of incidences are labeled wherever they are distinguishable.

Figure 5

(a) Dielectric function tensor element ${\varepsilon }_{xx}$, approximately along axis a in our coordinate system, obtained from wavelength-by-wavelength (polyfit) analysis (green dotted lines) and best match MDF analysis (red solid lines). Vertical lines indicate CP transition energy model parameters obtained from MDF analysis. Data from analysis by Sturm et al. are included for comparison (Ref. [9]; open symbols). (b) Imaginary part of the individual CP contributions to the MDF used in this work are shown. For color code and line styles refer to Fig. 11.

Figure 6

Same as Fig. 5 for ${\varepsilon }_{yy}$ approximately along axis c${}^{*}$.

Figure 7

Same as Fig. 5 for ${\varepsilon }_{xy}$ within the $\mathbf{a}\text{−}\mathbf{c}$ plane.

Figure 8

Same as Fig. 5 for ${\varepsilon }_{zz}$ approximately along axis $\mathbf{b}$.

Figure 9

Band structure of $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$. (a) At the GGA-DFT (PBE) level and (b) at the hybrid HF-DFT (Gau-PBE) level.

Figure 10

Band structure in the vicinity of the $\mathrm{\Gamma }$ point. Arrows indicate two lowest vertical band-to-band transitions polarized along the b axis (red dashed arrows) and within the a-c plane (black solid arrows). Note that the reciprocal space direction $\mathrm{\Gamma }\text{−}X$ corresponds to the real space directions parallel to the crystallographic vector b, and the reciprocal space direction $\mathrm{\Gamma }\text{−}Y$ corresponds to the real space direction lying within the a-c plane, inclined at a small angle from the crystallographic vector a.

Figure 11

Transition energies determined by CP-MDF analysis and calculated by DFT in our work, in comparison with data reported by Sturm et al. (Ref. [8]). Short-dashed (Sturm et al.) and dashed lines (this work): excitonic contributions; solid lines: near-band-gap band-to-band transitions; dash-dotted lines: above-band-gap transitions; dotted lines: higher energy transitions. For respective CP-MDF contributions, see Sec. 2c. DFT levels all refer to band-to-band transitions (solid lines). Color code for DFT a-c plane data are intended to match with order of energy levels identified in GSE CP-MDF analysis.

Figure 12

Colors and lines styles as in Fig. 11. (a) MDF-CP transition vectors, $\widehat{\mathbf{j}}=cos{\alpha }_j\widehat{\mathbf{x}}+sin{\alpha }_j\widehat{\mathbf{y}}$, multiplied with their respective CP transition amplitude parameter, $A$. The amplitudes of the transitions have been normalized to the amplitude of the first transition, and the CP-MDF unit vectors have been rotated by $\approx {17}^{\circ }$. Small transition amplitude parameters are multiplied for convenience, as indicated. Labels as given in Table 3. (b) DFT calculated transition matrix elements presented as vectors, $|{\mathcal{M}}_{cv}{|}_a^2\widehat{\mathbf{x}}+{\left|{\mathcal{M}}_{cv}\right|}_{c^{★}}^2\widehat{\mathbf{y}}$. Labels as given in Table 5.