Trigonal αGa2O3 is an ultrawideband semiconductor with potential applications in power electronics and ultraviolet opto-electronic devices. In this Letter, we calculate the low field electron mobility in αGa2O3 from first principles calculations. The effect of all the 30 phonon modes is taken into account for the transport calculation. The phonon dispersion and the Raman and Infrared spectra are calculated under the density functional perturbation theory formalism and compared with experiments. The electron–phonon interaction (EPI) elements on a dense reciprocal space grid are obtained using the Wannier function interpolation. The full energy dispersion of the phonons is included in both the polar and nonpolar EPI calculations. The electron mobility is then evaluated incorporating the effects of the polar, nonpolar, and ionized impurity scattering using Rode's iterative method. At room temperature, the low field isotropic average electron mobility is estimated to be ∼220 cm2/Vs predominantly limited by the polar optical phonon scattering at a doping density of 1.0×1015cm3. The anisotropy in the mobility arising from the phonon scattering is also evaluated. Temperature and dopant concentration variation of mobility is also studied, which can help in optimization of the growth for transport measurements.

Gallium oxide (Ga2O3) has emerged as a promising candidate for power and radio frequency (RF) electronics and ultraviolet (UV) optoelectronic applications owing to its large bandgap (4.7–5.3 eV).1–4 Its high breakdown field strength and the associated high Baliga Figure of Merit (BFoM)5,6 make it an attractive choice for high-voltage power electronics devices.

Ga2O3 is known to exist in six polymorphs of α, β, γ, δ, ε,7 and κ.8 Of the known polymorphs, the β phase is the most thermodynamically stable phase,9 and hence, it is also the most studied phase. Extensive theoretical and experimental research for the β phase has resulted in the well-developed bulk and epitaxial substrate growth and in successful fabrication of a lateral transistor with breakdown voltages of 8 kV10 and Schottky barrier diodes,11,12 to name a few. In addition, extensive transport studies13–16 have been carried out identifying high polar optical phonon (POP) scattering17 as the limiting mechanism for the mobility.

On the contrary, αGa2O3 has received comparatively less attention as it is thermodynamically semi-stable, making its synthesis challenging, which generally requires high growth temperatures and pressures.18 However, recently, thin films were successfully grown under low temperature conditions on the sapphire substrate using atomic layer deposition18 and the HVPE technique with low background densities,19 paving the way for low-cost and high-quality substrates. αGa2O3 can be used in a wide array of applications by alloying it with structurally similar materials such as corundum αAl2O3 to fabricate tunable bandgap heterostructures and with materials such as Cr2O3 and Fe2O3 for magnetoelectric devices as well. It has a higher bandgap of about 5.3 eV1 compared to the β phase, and hence, a better performance in terms of breakdown voltage is expected for high power device applications, making it a material of significant interest.

In contrast to βgallia, very few reports exist on the fundamental electron transport mechanism in αGa2O3. With the maturity in the growth of the α phase, it is important to investigate the electron transport phenomena to understand the limiting mechanisms and also provide guidance to growth studies. Although there are a few reports on the electronic structure and the optical properties of the α phase,2,9 they do not explore electron transport.

αGa2O3 is a polar semiconductor similar to β-gallia, and hence, it is anticipated that at room temperature, the polar optical phonon scattering (POP) mechanism will dominate the low-field electron mobility compared to nonpolar deformation potential (DP) and ionized impurity (II) scattering. However, the higher crystal symmetry (R3¯c) of αGa2O3 will have fewer IR active phonon modes, which could potentially lead to higher mobility than βGa2O3. In order to comprehensively understand the low-field electron transport mechanism in αGa2O3, all the phonon modes were included in the calculations taking into account full dispersion on a dense grid. We have then resolved POP scattering mechanism modewise to identify the dominant scattering modes. The effect of temperature on the electron mobility for different donor concentrations is also presented in this work.

We first begin with the standard density functional theory (DFT) calculation using Quantum Espresso,20,21 a plane wave pseudopotential-based open source package. The exchange–correlation energy in our calculations is obtained under the local density approximation (LDA). We have used a Γ-centered 6×6×6k-point grid with a plane wave energy cutoff of 1088 eV to obtain the converged charge density. The gallium 3d orbital was explicitly included as part of the valence band in our pseudopotential in order to account for the correct phase stability of different Ga2O3 polymorphs.22 All the calculations were performed using a 10 atom rhombohedral unit cell shown in Fig. 1 realized using VESTA.23 The structural parameters including the atomic positions are relaxed to minimize the forces to be lower than 2.5 meV/Å on each atom. The converged lattice and atomic position parameters obtained match closely with the experimentally obtained results shown in Table I.

FIG. 1.

(a) The 20-atom conventional cell with equivalent Gallium atoms surrounded by polyhedra. (b) The 10-atom primitive rhombohedral unit cell with α=β=γ=55.87°. The six oxygen atoms are represented by red spheres.

FIG. 1.

(a) The 20-atom conventional cell with equivalent Gallium atoms surrounded by polyhedra. (b) The 10-atom primitive rhombohedral unit cell with α=β=γ=55.87°. The six oxygen atoms are represented by red spheres.

Close modal
TABLE I.

Calculated and experimentally observed lattice parameters and electronic bandgap corresponding to different pseudopotentials.

PropertyThis workExperiment24 Experiment1 Experiment2 
a (Å) 4.987 4.983 5.04 … 
c (Å) 13.359 13.433 13.56 … 
Fractional coordinates 
zGa (12c) 0.3565 0.3554 … … 
xO (18e) 0.3018 0.3049 … … 
Bandgap (eV)     
LDA 2.74 … … … 
HSE 5.24 … 5.3 5.61 
PropertyThis workExperiment24 Experiment1 Experiment2 
a (Å) 4.987 4.983 5.04 … 
c (Å) 13.359 13.433 13.56 … 
Fractional coordinates 
zGa (12c) 0.3565 0.3554 … … 
xO (18e) 0.3018 0.3049 … … 
Bandgap (eV)     
LDA 2.74 … … … 
HSE 5.24 … 5.3 5.61 

Next, we calculate the electronic structure under the LDA on a coarse 6×6×6 Γ centered grid. The converged coarse grid Hamiltonian is then interpolated onto a fine grid along the high symmetry Brillouin zone direction using the technique of Wannier interpolation implemented in the Wannier90 code.25 For the low-field electron transport studies, we interpolate the lowest conduction band contributed mainly by the Ga 4s orbital. Using the hybrid functional (HSE) incorporating 35% exact exchange gave a better estimate of the bandgap (see Table I); however, we have used the LDA electronic structure for our calculations of scattering rates. The calculated band structure is provided in the supplementary material, which shows close match with the previously obtained results.9 From the band structure, we obtain an isotropic electron effective mass of 0.25m0 which is comparable to the previously obtained value,9 where m0 represents the rest mass of electrons. The effective mass calculated using HSE is 0.26m0, which is similar to that obtained in the LDA.

We then calculate the lattice dynamics under the density functional perturbation theory (DFPT26) formalism as implemented in Quantum Espresso. The phonon dispersion is first obtained on a coarse Γ-centered 6×6×6 grid, and then the dynamical matrix along the symmetrical irreducible Brillouin zone directions is obtained using standard Fourier interpolation. The calculated phonon dispersion is shown in Fig. 2. The long-range correction to the dynamical matrix arising due to the macroscopic electric field in polar materials is explicitly included during interpolation.

FIG. 2.

Calculated phonon dispersion from DFPT where the long-range interaction results in the LO–TO split represented by the discontinuities at the Γ point toward the Z (the A2u modes) directions represented as the red scattered plot. The other IR (the Eu) modes are represented by green dots, the Raman modes by orange dots, acoustic by black dots, and the neither IR nor Raman active modes (non-IR-Raman) by blue dots.

FIG. 2.

Calculated phonon dispersion from DFPT where the long-range interaction results in the LO–TO split represented by the discontinuities at the Γ point toward the Z (the A2u modes) directions represented as the red scattered plot. The other IR (the Eu) modes are represented by green dots, the Raman modes by orange dots, acoustic by black dots, and the neither IR nor Raman active modes (non-IR-Raman) by blue dots.

Close modal

As seen in Fig. 2, the 10-atom unit cell results in 30 phonon modes, of which 10 are IR active with the representation 2A2u+4Eu and 12 modes are Raman active with the representation 2A1g+5Eg. Of the remaining eight modes, three are acoustic and 5 (2A1u+3A2g) are neither IR nor Raman active (silent modes). The calculated IR and Raman spectra match closely with the experimental values shown in Table II. The two A2u IR modes are polarized along the z-axis, while the four (doubly degenerate) Eu modes are polarized along the Cartesian x and y directions. The polarization directions for the IR modes are provided in the supplementary material. The well-known phenomena of longitudinal optical–transverse optical (LO–TO) split due to the coupling of the TO phonon modes with the macroscopic polarization are also shown in Fig. 2 along the ΓZ direction. Only the two A2u modes (z-polarized) become LO and split to a higher energy (red scattered plot in Fig. 2).

TABLE II.

Comparison of the calculated IR and the Raman spectrum with the experimental observation; for the IR spectrum, the degenerate modes Eu are polarized in the x and y Cartesian direction and A2u modes in the z direction. The modes with * are doubly degenerate.

Mode symmetryActivityωcm1 (meV)Experiment2 (cm1)Experiment28 (cm1)
A1g (1) Raman 210.3 (25.9) … 218.2 
Eu (TO1)* IR 220.8 (27.2) NA … 
Eg (1) Raman 236.6 (29.2) … 240.7 
A2u (TO1) IR 270.6 (33.4) 280.0 … 
Eg (2)* Raman 282.6 (34.8) … 285.3 
Eg (3)* Raman 315.3 (38.9) … 328.8 
Eu (TO2)* IR 328.8 (40.6) 333.4 … 
Eg (4)* Raman 426.7 (52.7) … 430.7 
Eu (TO3)* IR 462.2 (57.0) 469.9 … 
A2u (TO2) IR 535.1 (66.0) 544.0 … 
Eu (TO4)* IR 553.6 (68.3) 562.7 … 
A1g (2) Raman 555.1 (68.5) … 569.7 
Eg (5)* Raman 669.1 (82.6) … 656.7 
Mode symmetryActivityωcm1 (meV)Experiment2 (cm1)Experiment28 (cm1)
A1g (1) Raman 210.3 (25.9) … 218.2 
Eu (TO1)* IR 220.8 (27.2) NA … 
Eg (1) Raman 236.6 (29.2) … 240.7 
A2u (TO1) IR 270.6 (33.4) 280.0 … 
Eg (2)* Raman 282.6 (34.8) … 285.3 
Eg (3)* Raman 315.3 (38.9) … 328.8 
Eu (TO2)* IR 328.8 (40.6) 333.4 … 
Eg (4)* Raman 426.7 (52.7) … 430.7 
Eu (TO3)* IR 462.2 (57.0) 469.9 … 
A2u (TO2) IR 535.1 (66.0) 544.0 … 
Eu (TO4)* IR 553.6 (68.3) 562.7 … 
A1g (2) Raman 555.1 (68.5) … 569.7 
Eg (5)* Raman 669.1 (82.6) … 656.7 

The Born effective charge and the high frequency dielectric tensor are also calculated at the Γ point as the part of the DFPT calculation. The LO and TO energies are tabulated in Table III, which are used to calculate the DC dielectric constants along the three Cartesian directions using the Lyddane–Sachs–Teller27 relation. Strong anisotropy is seen in the DC dielectric constant due to the direction-dependent LO–TO split.

TABLE III.

Calculated LO–TO split of the IR active modes. Direction-dependent dielectric constant arising from LO–TO split.

X directionY directionZ direction
Eu Eu A2u 
TO (meV) LO (meV) TO (meV) LO (meV) TO (meV) LO (meV) 
27.26 27.27 27.26 27.27 33.41 53.97 
40.60 46.31 40.60 46.31   
57.06 67.72 57.06 67.72 66.06 83.60 
68.35 84.58 68.35 84.58   
ε = 4.62  ε = 4.62  ε = 4.46  
εDC = 12.98  εDC = 12.98  εDC = 18.67  
X directionY directionZ direction
Eu Eu A2u 
TO (meV) LO (meV) TO (meV) LO (meV) TO (meV) LO (meV) 
27.26 27.27 27.26 27.27 33.41 53.97 
40.60 46.31 40.60 46.31   
57.06 67.72 57.06 67.72 66.06 83.60 
68.35 84.58 68.35 84.58   
ε = 4.62  ε = 4.62  ε = 4.46  
εDC = 12.98  εDC = 12.98  εDC = 18.67  

The electron–phonon interaction (EPI) is calculated as gelph=ψk+q|Vscfq|ψk,29 where ψ is the electronic wavefunction and Vscf/q is the perturbation in the self-consistent potential experienced by electrons due to lattice motion. The EPI elements are first calculated on a coarse mesh and then interpolated onto a fine mesh using Wannier–Fourier interpolation as implemented in the EPW30–32 package. The short-range EPI elements are interpolated using the standard Wannier technique using EPW package, whereas the long-range elements, which result in the polar optical interactions, are calculated on a fine mesh separately using the method described by Verdi and Giustino,33 which is a generalization of the Fröhlich interaction for materials having multiple optical modes. As stated earlier, we use Fourier interpolation to obtain the dynamical matrix on a dense grid and then the corresponding eigenvectors and eigenvalues of the matrix are used for the calculation of the dipole moment and, thus, the polar EPI elements. The convergence test for different k and q grid sizes is provided in the supplementary material. For the calculation of the EPI elements, we have used a commensurate grid of size 30×30×30q points and a k-grid size of 60×60×60 for both the long- and the short-range interactions with an energy cutoff of 0.4 eV from the bottom of the conduction band minimum.

Next, we calculate the POP and the nonpolar DP scattering rates using Fermi's Golden rule. The reciprocal space integration is carried out numerically using the technique of Gaussian smearing of 5 meV for the energy conservation δ function to account for the anisotropy in the EPI elements. The ionized impurity (II) scattering rate is calculated using the Brooks Herring model34 (not ab initio) with Debye screening.

To calculate the scattering rates shown in Fig. 3, we have used the n-type donor concentration of 1.0×1017cm3. We take into account partial dopant ionization with an activation energy of 31 meV17 assuming silicon as a potential shallow donor. The POP scattering is the most dominant mechanism at 300 K as shown in Fig. 3, similar to the β phase. A very important feature of the POP scattering rate shown in Fig. 3 is the presence of two strong emission peaks. We have resolved the POP scattering rates into mode-wise contribution in order to identify the dominant modes as discussed in the following paragraphs.

FIG. 3.

The polar optical, ionized impurity, and nonpolar scattering rate calculated for a dopant concentration of 1×1017cm3 at 300 K with two strong emission peaks marked as emssion1 and emssion2, respectively. The empirical fit35 of the first principles POP scattering gives a single LO energy of 73 meV shown in the inset.

FIG. 3.

The polar optical, ionized impurity, and nonpolar scattering rate calculated for a dopant concentration of 1×1017cm3 at 300 K with two strong emission peaks marked as emssion1 and emssion2, respectively. The empirical fit35 of the first principles POP scattering gives a single LO energy of 73 meV shown in the inset.

Close modal

The IR active modes of the phonon spectrum have a nonzero dipole moment and, thus, are responsible for the POP scattering. The spectrum of the IR modes is presented in Table II with their TO energies. The LO–TO split in polar materials results in the LO modes having higher energy than the corresponding TO mode. The list of the TO modes and their corresponding LO energies for the α-phase are presented in Table III. As seen, the LO–TO split is direction dependent due to the different polarization direction of each mode. It is also proportional to the dipole strength, and so modes having high dipole strength should contribute most to the POP scattering rate. From Table III, it is found that the A2u (TO2) mode with the TO energy of 66.06 meV shifts to the LO energy of 83.60 meV [A2u(LO2)], indicating a very large dipole strength, and, thus, is the most dominant POP scattering mode. The dipole strength of this mode is large enough to offset the low phonon occupancy at 300 K. Also from Table III, it is found that the degenerate Eu (TO4) modes with a TO energy of 68.35 meV shift to an LO energy of 84.58 meV, which is very close to the A2u (LO2) energy. This large LO–TO shift again indicates the presence of a strong dipole, also resulting in a high scattering rate. These two modes are merged as one peak in Fig. 3 as they are very close in energy. The third most dominating mode, which is the Eu (TO2) mode, has a TO energy of 40.6 meV and the corresponding LO energy of 46.31 meV, again indicating a very strong dipole.

The first emission peak in Fig. 3 is due to the Eu mode with a TO energy of 40.6 meV (LO = 46.31 meV), and the next emission peak is the result of two strong modes, A2u and Eu, with TO energies of 66.06 meV (LO = 83.60 meV) and 68.35 meV (LO = 84.58 meV), respectively. We have also fitted the ab initio calculated POP scattering with the popular Frölich-derived scattering rate with a fitted LO energy of 73 meV, as shown in the inset of Fig. 3. The isotropic average scattering rates of the top five most dominant modes are shown in Fig. 4 where the emission peaks represent the corresponding LO energies for each mode. Also from Fig. 4, we see a finite contribution from a non-IR mode [Raman mode (Eg)], because it has a nonzero dipole moment away from the Γ point. However, the most dominant modes still correspond to the IR active modes due to their high dipole strengths and 1/q dependence of the Fröhlich interaction. The contribution of all the 30 phonon modes to the POP scattering is included in the supplementary material.

FIG. 4.

The five most dominant modes contributing to the POP scattering taking full dispersion into account. The top four modes correspond to IR active modes having LO energies of 83.60 meV, 84.58 meV, 46.31 meV, and 57.0 meV, respectively. The curve in green represents the non-IR mode also having low but finite contribution to the POP scattering.

FIG. 4.

The five most dominant modes contributing to the POP scattering taking full dispersion into account. The top four modes correspond to IR active modes having LO energies of 83.60 meV, 84.58 meV, 46.31 meV, and 57.0 meV, respectively. The curve in green represents the non-IR mode also having low but finite contribution to the POP scattering.

Close modal

We then use Rode's iterative method36 to calculate the perturbation in the electron distribution [g(E)] and the low field electron mobility. We use a 60×60×60k-point and 30×30×30q-point commensurate grid for these calculations. Both temperature- and doping density-dependent low field mobility (isotropic average) is calculated. As seen in Figs. 5(a) and 5(b), the behavior is typical of semiconductors where the transport mechanism is dominated by the II scattering at low temperatures and as the temperature is increased, the electron mobility increases due to the reduction in the II scattering until the POP scattering mechanism starts dominating, resulting in the decrease in the electron mobility. For a low doping density of 1.0×1015cm3, the calculated electron mobility is 220 cm2/Vs at 300 K. Although compared to the β phase, the α phase had fewer IR active modes and a lower electron effective mass, the high dipole strengths of the IR active modes result in a mobility that is similar to the β phase. With a bandgap interpolated critical field strength of 9 MV/cm, the BFoM is 3706 times that of Si, making it an attractive choice for power devices. The BFoM is, in fact, larger than βGa2O3 primarily due to its larger critical field strength. However, as seen in Figs. 5(a) and 5(b), the experimentally measured mobilities are low, which could be attributed to the high density of impurity elements and dislocation density in the mist-CVD-grown films.37 The calculated anisotropy (RTA mobility) at room temperature is shown in the inset in Fig. 5(c); the mobility is lowest in the Cartesian z-direction, which can be attributed to the strong A2u (TO2) POP mode.

FIG. 5.

(a) and (b) The variation of calculated mobility (isotropic average) with temperature for different dopant concentrations between 10 K and 500 K. (c) Room temperature isotropic average mobility as a function of doping. The inset shows the calculated RTA mobility in different Cartesian directions.

FIG. 5.

(a) and (b) The variation of calculated mobility (isotropic average) with temperature for different dopant concentrations between 10 K and 500 K. (c) Room temperature isotropic average mobility as a function of doping. The inset shows the calculated RTA mobility in different Cartesian directions.

Close modal

In conclusion, we have theoretically investigated the electron–phonon coupling in αGa2O3 from first principles. The low field electron mobility was then calculated for a varying range of temperatures and dopant concentrations. It is seen that the POP is the dominant scattering mechanism. The mode-resolved scattering rate for the dominant mechanism is also presented, which shows two TO modes, with energies of 66.06 meV [A2u (TO2)] and 68.35 meV [Eu (TO4)] being the dominant ones. Also upon inclusion of full phonon dispersion, we see that even the non-IR modes contribute finitely to the POP scattering rate. The theoretically calculated mobilities are comparable to βGa2O3, giving a BFoM of 3706 times that of Si. The anisotroy in the POP scattering leads to aniostropy in the mobility.

See the supplementary material for the bands using LDA and HSE approximations, convergence studies, and mode-wise scattering rate.

The authors acknowledge the support from the Air Force Office of Scientific Research under Award No. FA9550-18-1-0479 (Program Manager: Ali Sayir), from NSF under Award Nos. ECCS 1607833 and ECCS–1809077, and the Center for Computational Research (CCR) at University at Buffalo. We thank Dr. Krishnendu Ghosh for helpful discussions.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Supplementary Material