*β*-Ga_{2}O_{3} is a transparent conducting oxide that, due to its large bandgap of 4.8 eV, exhibits transparency into the UV. However, the free carriers that enable the conductivity can absorb light. We study the effect of free carriers on the properties of Ga_{2}O_{3} using hybrid density functional theory. The presence of free carriers leads to sub-band-gap absorption and a Burstein-Moss shift in the onset of absorption. We find that for a concentration of 10^{20} carriers, the Fermi level is located 0.23 eV above the conduction-band minimum. This leads to an increase in the electron effective mass from 0.27–0.28 m_{e} to 0.35–0.37 m_{e} and a sub-band-gap absorption band with a peak value of 0.6 × 10^{3} cm^{–1} at 3.37 eV for light polarized along the *x* or *z* direction. Both across-the-gap and free-carrier absorption depend strongly on the polarization of the incoming light. We also provide parametrizations of the conduction-band shape and the effective mass as a function of the Fermi level.

*β*-Ga_{2}O_{3} exhibits very good electronic conductivity in spite of its bandgap of 4.8 eV; this large gap makes it transparent into the UV.^{1–4} This combination allows for applications in devices such as deep-UV blind detectors and contacts for solar cells.^{5,6} It also enables high-power devices, such as high-voltage metal-semiconductor field-effect transistors and Schottky barrier diodes.^{7–9}

*β*-Ga_{2}O_{3}, which has a monoclinic crystal structure (*C*2/*m*), is the most stable structure of Ga_{2}O_{3}.^{10} It has two inequivalent Ga sites and three inequivalent O sites,^{10} with 10 atoms in the primitive unit cell and 20 atoms in the conventional unit cell; the latter is shown in Fig. 1(a). The bandgap is indirect, but the difference with the direct bandgap (at Γ) is very small [less than 0.05 eV (Refs. 11 and 12)] so that for all intents and purposes Ga_{2}O_{3} can be considered a direct-band-gap material.^{12,13} The large bandgap [∼4.8 eV (Refs. 1–4)] is a necessary but not sufficient condition to allow for transparency; indeed, since the material needs to be conducting, electrons are present in the conduction band, and those free carriers can absorb photons by exciting carriers to higher-energy states in the conduction band. Free-carrier absorption thus places a fundamental limit on the transparency of the material, and addressing this limit will be one focus of the present work.

Si and Sn are the most commonly used dopants in Ga_{2}O_{3}. Typical carrier concentrations range from 10^{16} to 10^{19} cm^{−3},^{14–17} but concentrations of up to 9.1 × 10^{19} have been reported.^{18} Room-temperature mobilities of over 100 cm V^{−1 }s^{−1} have been observed^{15,19} and have theoretically been shown to potentially exceed 160 cm V^{−1 }s^{−1}, even when ionized impurity scattering is taken into account.^{20} The presence of these free carriers can lead to light absorption and other changes in the material properties. In a semiconductor with a small effective mass such as Ga_{2}O_{3}, the density of states of the conduction band is quite small and the Fermi level will rise rapidly above the conduction-band minimum (CBM) when the electron concentration is increased. This gives rise to an increase in the onset for optical absorption (the Burstein-Moss shift^{21,22}) as well as a possible change in the effective mass of the electrons, which affects the mobility. At the same time, the filling of the conduction band with free carriers leads to additional sub-band-gap absorption. In this letter, we report a detailed hybrid density functional theory (DFT) study of all these effects, in particular the sub-band-gap absorption in the visible and UV region of the spectrum.

Our DFT calculations use projector augmented wave potentials^{23} in a plane-wave basis set with an energy cutoff of 400 eV, within the Vienna *Ab initio* Simulation Package (VASP).^{24} We use the hybrid functional of Heyd, Scuseria, and Ernzerhof (HSE)^{25} with a mixing parameter of 35%, which has been shown to accurately describe the structure and the electronic properties of Ga_{2}O_{3.}^{12} We calculate the absorption coefficient based on the complex dielectric function. The frequency-dependent imaginary part of the dielectric function [*ε*_{2}(*ω*)] is evaluated in the independent-particle approach as

where *V* is the unit cell volume, $Nk$ is the number of **k** points, $Mv,c,k$ is the optical transition matrix element, and *ϵ* is the band energy. The summation runs over the valence bands *v*, the conduction bands *c*, and the **k** points. The real part of the dielectric function is obtained using the Kramers-Kronig relation. Both real and imaginary parts are used to calculate the absorption coefficient.

To converge the sum in Eq. (1), a very fine grid of **k** points in the Brillouin zone is required. In addition, the Kramers-Kronig transformation requires a large number of conduction bands. Calculating the band energies and optical transition matrix elements for such a large number of **k** points and conduction bands is very computationally demanding. We enabled this by using 80 maximally localized Wannier functions,^{26} as implemented in the Wannier90 package,^{27} to perform these calculations on an interpolated 100 × 100 × 100 **k**-point grid and for bands up to an energy of 25 eV above the CBM. The same **k**-point grid is used to calculate the density of states. As starting projections for the Wannier functions, we use *s* and *p* projections on the Ga and O atoms, supplemented by random projections. The disentanglement is performed until the difference of the spread between steps is smaller than 10^{−10}. Subsequently, 10 000 spread-minimization steps are performed, resulting in spread differences between steps better than 10^{−7}.

Because of the monoclinic symmetry of Ga_{2}O_{3}, the Brillouin zone has many high-symmetry points and lines.^{12} In Fig. 1(b), we focus on two segments of the band structure, Γ-X and Γ-M; a full band structure can be found in Ref. 12. The CBM is located at Γ. We fit the lowest conduction band to a hyperbolic equation

where $m\Gamma *$ is the electron effective mass at Γ, *ϵ* and *k* are the band energy and momentum, and *α* is the non-parabolicity parameter. This allows us to quantify both the effective mass at Γ and the non-parabolicity. Including the latter provides a much better fit to the first-principles band structure than a parabolic fit [which corresponds to *α *= 0 in Eq. (2)], as shown in Fig. 1(b).

The lowest conduction band of Ga_{2}O_{3} is only slightly anisotropic, and we find the effective mass to be between 0.27 and 0.28 m_{e}, depending on the direction. Values along several high-symmetry directions are listed in Table I; they were obtained by fitting to a **k**-point path of length 0.15 bohr^{−1}. The non-parabolicity parameter is also quite isotropic. The effective mass values are in good agreement with experimental measurements^{11,28} and previous HSE calculations.^{12,20}

Direction . | $m\Gamma *\u2009(me)$ . | α (eV^{−1})
. |
---|---|---|

Γ → N | 0.27 | 0.23 |

Γ → Y | 0.28 | 0.22 |

Γ → M | 0.27 | 0.20 |

Γ → X | 0.27 | 0.22 |

Γ → Z | 0.27 | 0.23 |

Direction . | $m\Gamma *\u2009(me)$ . | α (eV^{−1})
. |
---|---|---|

Γ → N | 0.27 | 0.23 |

Γ → Y | 0.28 | 0.22 |

Γ → M | 0.27 | 0.20 |

Γ → X | 0.27 | 0.22 |

Γ → Z | 0.27 | 0.23 |

Accurately calculating the Fermi integrals requires the density of states on a very fine mesh, which we obtained by a Wannier function interpolation. For *n* larger than 1.1 × 10^{19} cm^{−3}, the Fermi level (*E _{F}*) will be above the CBM (indicated by the horizontal dashed line in Fig. 2), and the effective mass will change. This change can be calculated by calculating the second derivative of Eq. (2), giving

The effective masses as a function of the carrier concentration are shown in Fig. 2 (right axes). Due to the small anisotropy of the non-parabolicity parameter, the masses away from Γ will be direction-dependent. The range of values is indicated by the two curves in Fig. 2, where the lower curve corresponds to the Γ-M and the upper curve to the Γ-N direction.

For *n* = 10^{20} cm^{−3}, *E _{F}* is 0.23 eV above the CBM and the effective mass increases to 0.35–0.37

*m*, depending on the direction; if an electron concentration of 10

_{e}^{21}cm

^{−3}could be achieved,

*E*would be 1.06 eV above the CBM and the effective mass would be between 0.78 and 0.89

_{F}*m*.

_{e}Now, we investigate direct absorption, first in undoped Ga_{2}O_{3}. Because of the low symmetry, the across-the-gap absorption depends strongly on the direction of the incoming light, as can be seen in Fig. 3. The onset of absorption occurs at the bandgap for light polarized along Z. The onset occurs at higher energies for the *x* and *y* directions, because the transitions from the highest valence band to the lowest conduction band are dipole forbidden for these directions; only transitions from states lower in the valence band are allowed. This anisotropy is in agreement with experimental absorption measurements^{2–4} and previous calculations of the dielectric function^{13,29,30} and the absorption coefficient.^{31} Our calculations do not include excitonic effects, which are known to lead to a redshift of the absorption edge and a redistribution of the dipole weights, leading to an increase in absorption in the energy range reported here.^{30,32}

Now, we examine the effect of adding free carriers, which also leads to strongly anisotropic absorption. We first focus on the *z* direction, which is the direction with the lowest onset of across-the-gap absorption (Fig. 3). Figure 4 shows the calculated absorption coefficient for 10^{19} cm^{−3}, 10^{20} cm^{−3}, and 10^{21} cm^{−3} carriers, compared with the undoped results (*n *= 0). The presence of free carriers shifts the onset of across-the-gap absorption to higher energies, as transitions can take place only to unoccupied states and electrons occupy the lowest conduction-band states. This Burstein-Moss shift has been observed experimentally in Si-doped Ga_{2}O_{3}.^{33} At the same time, transitions from occupied states in the lowest conduction band to higher-lying conduction bands become possible, including at energies within the bandgap of Ga_{2}O_{3}.

For *n* = 10^{19} cm^{−3}, this induces an absorption peak at 3.45 eV, corresponding to transitions from the bottom of the lowest conduction band to the second conduction band. The magnitude of this peak is rather small (25 cm^{−1}). The small shoulder right before the onset of the strong across-the-gap absorption corresponds to transitions from the bottom of the lowest conduction band to the third conduction band.

For *n* = 10^{20} cm^{−3}, the sub-band-gap absorption peaks are larger in magnitude and occur at 3.37 eV (peak value of 0.6 × 10^{3} cm^{−1}) and 4.67 eV (peak value of 1.3 × 10^{3} cm^{−1}). These values correspond to transitions to the second conduction band and the fourth conduction band, taking into account that the Fermi level is located 0.23 eV above the CBM. The shift of the peak position towards lower energies can be explained by the larger filling of the lowest conduction band, which enables transitions with lower energy. The shoulder around 4.43 eV corresponds to transitions to the third conduction band.

For *n* = 10^{21} cm^{−3}, the sub-band-gap absorption becomes continuous with an onset at around 2.3 eV, due to overlapping peaks at 2.91 eV, 4.13 eV, and 5.31 eV, and an absorption coefficient larger than 10^{3} cm^{−1}. At this high free-carrier concentration, the Fermi level is 1.06 eV above the CBM, resulting in a shift towards lower energies of the absorption. The free-carrier absorption bands that were identified at lower electron concentrations are still present, but they are significantly broadened and also shifted to lower energy, because initial (occupied) states with higher energy in the conduction band are now available. In addition, sub-band-gap transitions now appear at energies above 5 eV, arising from the excitation of electrons to even higher-lying conduction-band states. In lightly doped materials, these transitions would lie above the across-the-gap absorption edge, but the opening of the gap due to the Burstein-Moss shift makes these absorption bands emerge within the gap.

The polarization dependence of the free-carrier absorption is examined in Fig. 5, which shows absorption for light polarized along the three Cartesian directions for *n* = 10^{20} cm^{−3}. The sub-band-gap absorption at 3.37 eV is very weak for the *y* direction (5 cm^{−1}), but light polarized along the *x* direction exhibits very strong absorption (1.8 × 10^{3} cm^{−1}), even stronger than the *z*-direction polarization (0.6 × 10^{3} cm^{−1}). In all cases, this corresponds to transitions to the second conduction band. The differences in absorption strength are related to the magnitude of the optical transition matrix elements, i.e., the degree to which the transitions are dipole allowed.

Similar differences, but for different polarization directions, are observed for the peak at 4.67 eV (which is related to transitions to the fourth conduction band), with polarization along *z* now leading to stronger absorption. For light polarized along *x* or *y*, a smaller peak is evident at 4.43 eV, which arises from transitions to the third conduction band; for *z*-polarized light, this absorption is visible as a shoulder.

We display the combined effects of free-carrier concentration and light polarization in Fig. 6, where the absorption coefficient (color scale) is shown for light polarized along the three Cartesian directions and for carrier concentrations between 10^{18} cm^{−3} and 10^{21} cm^{−3}. The Burstein-Moss shift of the edge of the across-the-gap absorption is clearly visible. For low carrier concentrations, the onset occurs at different energies for light polarized along different directions (as shown in Fig. 3), which is caused by the anisotropy of the crystal. Similarly, the sub-band-gap absorption depends on the light polarization (see Fig. 5). With the increasing free-carrier concentration, the sub-band-gap absorption broadens towards smaller energies, since an increased filling of the conduction band allows additional transitions with smaller energies. This effect is also illustrated in Fig. 4 for select free-carrier concentrations.

The absorption coefficients reported here for sub-band-gap absorption are substantial and at least as important as absorption induced by defect-to-band transitions. Typical photoionization cross sections are around 10^{18} cm^{2}.^{34} Even a fairly high defect concentration of 10^{20} cm^{−3} would then lead to an absorption coefficient an order of magnitude lower than the free-carrier absorption at *n* = 10^{20} cm^{−3}. We note that experimental measurements on Si-doped Ga_{2}O_{3} have already produced evidence for sub-band-gap absorption at high Si concentrations.^{33}

Our results show that for device applications that rely on transparency in the UV region, free-carrier concentrations above 1 × 10^{19} cm^{−3} could induce strong below-band-gap absorption. Increasing the conductivity of the material will come at the expense of reduced transparency. Our results presented in Figs. 4, 5, and 6 provide researchers with the means to evaluate this tradeoff. At lower carrier concentrations, sub-band-gap absorption is much less of a concern: for *n* = 10^{19} cm^{−3}, the absorption coefficient (for *x*-polarized light) at 3.37 eV decreases to about 60 cm^{−1}, and for *n* = 10^{18} cm^{−3}, this is further decreased to less than 2 cm^{−1}.

In conclusion, we have performed an in-depth investigation of the effects of doping on *β*-Ga_{2}O_{3}. Undoped Ga_{2}O_{3} has a nearly isotropic effective mass of 0.27–0.28 m_{e}, and the lowest conduction band can be described well with a hyperbolic equation, with a small (0.21 eV^{−1}) non-parabolicity parameter that is also fairly isotropic. We calculated the position of the Fermi level as a function of carrier concentration and provided an analytical formula and parameters for the increase in effective mass when the Fermi level rises above the CBM. This increase can be substantial: for *n* = 10^{20} cm^{−3}, the effective mass increases to 0.35–0.37 m_{e}. At carrier concentrations above 10^{19} cm^{−3}, the onset of across-the-gap absorption shifts to higher energies (the Burstein-Moss shift) and substantial sub-band-gap absorption occurs due to free-carrier transitions to higher-lying conduction bands. We provided quantitative information about both types of absorption as a function of electron concentration and showed that they depend strongly on the polarization of the incoming light.

This work was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) under Award No. DE-SC0010689. Computing resources were provided by the Center for Scientific Computing at the CNSI and MRL: an NSF MRSEC (DMR-1121053) and NSF CNS-0960316, and by the Extreme Science and Engineering Discovery Environment (XSEDE), which was supported by NSF Grant No. ACI-1548562.