Vibrational and electron-phonon coupling properties of \b{eta}-Ga2O3 from first-principles calculations: Impact on the mobility and breakdown field

The wide band gap semiconductor \b{eta}-Ga2O3 shows promise for applications in high-power and high-temperature electronics. The phonons of \b{eta}-Ga2O3 play a crucial role in determining its important material characteristics for these applications such as its thermal transport, carrier mobility, and breakdown voltage. In this work, we apply predictive calculations based on density functional theory and density functional perturbation theory to understand the vibrational properties, phonon-phonon interactions, and electron-phonon coupling of \b{eta}-Ga2O3. We calculate the directionally dependent phonon dispersion, including the effects of LO-TO splitting and isotope substitution, and quantify the frequencies of the infrared and Raman-active modes, the sound velocities, and the heat capacity of the material. Our calculated optical-mode Gr\"uneisen parameters reflect the anharmonicity of the monoclinic crystal structure of \b{eta}-Ga2O3 and help explain its low thermal conductivity. We also evaluate the electron-phonon coupling matrix elements for the lowest conduction band to determine the phonon mode that limits the mobility at room temperature, which we identified as a polar-optical mode with a phonon energy of 29 meV. We further apply these matrix elements to estimate the breakdown field of \b{eta}-Ga2O3. Our theoretical characterization of the vibrational properties of \b{eta}-Ga2O3 highlights its viability for high-power electronic applications and provides a path for experimental development of materials for improved performance in devices.


I. INTRODUCTION
An increasing amount of recent experimental and theoretical research has focused on the β phase of gallium oxide (β-Ga 2 O 3 ), with a primary focus on its applications in high-power electronic devices. 1 Because of its wider band gap and, correspondingly, its larger estimated breakdown voltage, β-Ga 2 O 3 has been identified as a promising alternative for power electronics compared to other wide-gap semiconductors such as GaN and SiC. [2][3][4][5][6][7] The band gap of β-Ga 2 O 3 is an especially important property to consider, with various reports placing it within a broad energy range of 4.4-5.0 eV. 2,3,8−10 Reasons for this uncertainty could include experimental growth conditions, sample type (i.e. film, bulk, etc.), sample quality, and light polarization.
However, recent detailed experimental and theoretical absorption onset results point toward a fundamental band gap near 4.5 eV, i.e. at the lower limit of the reported range. 2,11,12 Although the band gap of Ga 2 O 3 is lower than estimated in earlier reports, it is still much larger than other a) Author to whom correspondence should be addressed. Electronic mail: kioup@umich.edu. power-electronics materials, such as Si (1.1 eV), GaAs (1.4 eV), 4H-SiC (3.3 eV), or GaN (3.4 eV). 3 Since the breakdown field of a material increases strongly with increasing band-gap value, a reinvestigation of the breakdown field estimate is needed. This is achievable by studying the phonon and electron-phonon coupling properties. From the phonon dispersion, one can also obtain important information such as sound velocities and longitudinal optical (LO)-transverse optical (TO) splitting. Phonon-phonon interactions can provide information on the anharmonicity of different phonon modes, which can be used to explain finite thermal conductivity. Limited carrier mobility and breakdown field can both be explained by the electron-phonon coupling in a material. Each of these properties is important for high-power electronic applications.
In addition to the requirement of a wide band gap and large breakdown field, materials used in power electronics need to also exhibit large electron mobility to minimize Joule heating, as well as a high thermal conductivity to facilitate heat extraction. While the band gap and estimated breakdown field of β-Ga 2 O 3 make it promising for power electronics, its electron mobility and thermal conductivity are lower than desired for such applications. An initial estimate placed the electron mobility of β-Ga 2 O 3 around 300 cm 2 V -1 s -1 . 3 However, both experimental and theoretical subsequent reports find much lower values. Various scattering mechanisms including acoustic deformation potential, ionized impurity, neutral impurity, and polar optical (PO) phonon scattering were analyzed alongside Hall-effect measurements by Ma et al. The results showed that the dominant mechanism limiting electron mobility in β-Ga 2 O 3 is PO phonon scattering, which limits the room temperature mobility to < 200 cm 2 V -1 s -1 (for doping densities less than ~10 18 cm -3 ). 13 Zhang et al. recently grew β-(Al x Ga 1-x ) 2 O 3 /Ga 2 O 3 heterostructures with modulation doping, resulting in a high mobility 2D electron gas at the interface. They measured the highest experimental room temperature mobility in bulk β-Ga 2 O 3 to date: 180 cm 2 V -1 s -1 . 14 Two separate ab initio calculations using the Boltzmann transport equation (BTE) take into account scattering by PO phonons and impurities as well, resulting in room-temperature mobility values around 115 cm 2 V -1 s -1 for n = 1.1×10 17 cm -3 (Ghosh and Singisetti) 15 and 155 m 2 V -1 s -1 for n = 10 17 cm -3 (Kang et al.). 16 On the other hand, the measured thermal conductivity depends on the crystallographic direction due to the anisotropic monoclinic crystal structure 17 , but the highest value is only 27 W m -1 K -1 along the [010] direction, and the lowest is 10 W m -1 K -1 along the [100] direction. First-principles calculations estimated the thermal conductivity along several directions (both cross-plane and in-plane) for a range of film thicknesses and found the highest value to be 21 W m -1 K -1 along the [010] direction. 18 To fully realize the promise of Ga 2 O 3 as a high-power electronic material, the properties of phonons and their interactions with electrons and each other must be quantified to elucidate the atomistic origins of the inherent limits to its electronic and thermal transport properties.
In this work, we apply first-principles calculations based on density functional theory (DFT) and density functional perturbation theory (DFPT) to calculate the fundamental vibrational properties of β-Ga 2 O 3 , such as its phonon dispersion, phonon-phonon interactions, and electron-phonon coupling properties, and compare to available experimental data. Our results show that the monoclinic crystal structure strongly influences the phonon modes by causing strong anharmonicities, which are detrimental to the thermal conductivity. Moreover, our evaluated electron-phonon coupling matrix elements point to a specific low-frequency polar optical phonon mode that limits the electron mobility at room temperature. These matrix elements are also used to estimate a value of 6.8 MV/cm for the breakdown field of β-Ga 2 O 3 .
Our results highlight the viability of β-Ga 2 O 3 for applications in high-power electronics and propose a path for experimental development for improved performance in devices.
This manuscript is organized as follows. In Section II we describe the computational methodology. Section III discusses the vibrational properties, including phonon frequencies, isotope effects, sound velocities, and heat capacity. Phonon-phonon interactions and their implications for thermal transport are described in Section IV, followed by a description of how the strong electron-phonon coupling in this material impacts the mobility and breakdown field in Section V. Section VI summarizes the key results of this work.

II. COMPUTATIONAL METHODS
Our computational methods are based on DFT and DFPT. The Quantum ESPRESSO software package was used for the calculations within the plane-wave pseudopotential formalism and the local density approximation (LDA) for the exchange-correlation potential. 19 Brillouin-zone sampling grid of 4×8×4 were determined to converge the total energy of the system to within 1 mRy/atom. The experimental lattice parameters (a = 12.214 Å, b = 3.0371 Å, and c = 5.7981 Å) 23 and atomic positions for β-Ga 2 O 3 were converted to the primitive cell and used as starting points for the structural relaxation calculation. For reference, the primitive-cell lattice and the reciprocal lattice vectors are referred in this paper as ! = −1.53, −1.46, 5.96 , ! = 3.06, 0, 0 , ! = 0, 5.82, 0 , and ! = 0, 0, 0.17 , ! = 0.33, 0, 0.08 , and ! = 0, 0.17, 0.04 , respectively. The structure was relaxed from the experimental lattice parameters and atomic positions until the total force on all atoms was less than 2×10 -5 Ry/a 0 and the total stress was less than 4×10 -8 Ry/a 0 3 along each direction.
Phonon frequencies were determined with DFPT for a 4×8×4 grid of phonon wave vectors using a stricter plane-wave cutoff energy of 170 Ry. The acoustic sum rule was applied to the phonon modes at Γ to ensure that the acoustic modes satisfy → 0 as → 0. Using the dynamical matrices produced during the phonon calculations, the interatomic force constants were generated via a Fourier interpolation and applied to determine the phonon dispersion with where is the total mass of the atoms in the unit cell, is the frequency of phonon mode with wave vector and index , ! is the derivative of the self-consistent potential resulting from ionic displacement by phonon , and ! and ! ! are electronic wave functions at bands and with wave vectors and + . All ! ,! ! were determined for phonon wave vectors along the reciprocal lattice directions by evaluating the potential derivative ! for all phonon modes with DFPT and calculating the matrix element in (1) using the conduction-band electron wave functions at Γ and at . This method for the calculation of electron-phonon coupling matrix elements yields accurate values even for the polar-optical modes for ≠ 0. 24

A. Phonon frequencies
The calculated phonon dispersion of β-Ga 2 O 3 is shown in Fig. 1 We also report the LO frequencies for each IR-active mode as calculated along the three conventional crystal axes for better comparison with experiment. As expected from the crystallographic anisotropy, the LO-TO splitting depends on the direction, and some directions show larger splitting than others for a given phonon mode. The LO frequency for IR-active mode #5 agrees extraordinarily well across all reports (theoretical and experimental). A coincidental finding is that our LO frequencies agree better with the reported experimental TO frequencies     25 and by Machon et al. 29

C. Sound velocities and heat capacity
The phonon dispersion is also crucial in determining other properties of a material such as the sound velocity and heat capacity ! , which are important in the evaluation of the lattice thermal conductivity. Table III  comparison, the lowest sound velocity in GaN is ~4 km/s, and the largest is ~8 km/s. 35 Since the sound velocities of Ga 2 O 3 are only slightly lower than GaN, they cannot account for its much lower thermal conductivity, the origin of which will be examined later. The heat capacity at constant volume of β-Ga 2 O 3 is calculated from the Helmholtz free energy according to . First, the phonon density of states (DOS) was calculated with DFPT in Quantum ESPRESSO. From the phonon DOS, the Helmholtz free energy was calculated as a function of temperature, and the second derivative with respect to temperature was taken using the finite-difference method to obtain the heat capacity. Fig. 3 shows the heat capacity at constant volume ! over a temperature range from 0-1000 K. At high temperatures, the heat capacity reaches the Dulong-Petit limit of 3 !"#$ ! = 4.14×10 -22 J/K per primitive cell. Also shown in Fig. 3

A. Grüneisen parameters
The lattice thermal conductivity of β-Ga 2 O 3 at room temperature is strongly influenced by anharmonic phonon-phonon interactions, which are quantified by the Grüneisen parameters.
We calculated the Grüneisen parameters for the acoustic and optical phonon modes to understand the origin of the low thermal conductivity of β-Ga 2 O 3 . The average Grüneisen parameter for the acoustic modes is obtained by fitting the energy versus volume curve for hydrostatic deformation to the Murnaghan equation of state 37 , given by: where is the total energy of the system, is the bulk modulus, ′ is the pressure derivative of the bulk modulus, is the system volume, and ! is the equilibrium volume. Volume values spanning a range of ± 20% around the equilibrium were used for the fit.
where is the linear thermal expansion coefficient and ! is the molar volume. For comparison, applying the same methods to wurtzite GaN yields a bulk modulus of 192 GPa, pressure derivative of 4.44, and Grüneisen parameter equal to 1.39. Since the acoustic-mode for GaN is larger than that for β-Ga 2 O 3 , yet GaN exhibits a higher thermal conductivity, we conclude that the optical modes of β-Ga 2 O 3 must have a strong impact on limiting thermal conductivity.
For the optical modes, the mode Grüneisen parameters ! were calculated according to their definition: where ! is the frequency of phonon mode at wave vector . The unit-cell volume was hydrostatically varied within ±1% around its equilibrium value and the atomic positions were relaxed to minimize forces on the atoms. The resulting structural parameters were subsequently used to calculate phonon frequencies.    18 These thermal-conductivity values are much lower than those of other power electronic materials such as GaN (~130 W m -1 K -1 ) and SiC (~500 W m -1 K -1 ). 42 We attribute the low thermal conductivity of β-Ga 2 O 3 to the large values for ! (e.g., for IR-active modes #5, #9, #7, #11, and #12 and Raman-active modes #7, #8, and #12) and, in particular, to IR-active mode #5 ( = 303.17 cm !! ) with the largest mode Grüneisen parameter of 2.32.

A. Electron-phonon coupling matrix elements
We studied electron-phonon interactions in β-Ga 2 O 3 to gain a fundamental understanding of how they impact the electron transport properties relevant for power electronics. We evaluated the electron-phonon coupling matrix elements between states at Γ and at several points along each reciprocal lattice direction in increments of 0.1 . Fig. 5 shows these electron-phonon coupling matrix elements for the bottom conduction band as a function of the phonon wave vector. Fig. 5 also displays the magnitude of the phonon-absorption ( ! ) and phononemission ( ! + 1 ) contributions of each phonon mode to the mobility, where are the phonon occupation numbers given by the Bose-Einstein distribution: evaluated at 300 K ( ! = 25.85 meV). For clarity, the plots in Fig. 5 only include the modes that have the top three largest values for either the matrix element squared, phonon-absorption, or phonon-emission terms along each direction. Note that the phonon frequencies given in Fig. 5 correspond to the frequencies of the modes at the calculated wave vectors closest to, rather than at, Γ since these are the specific wave vectors used in Eqs. (6) and (7).
As in polar materials in general, the phonon modes that dominate electron-phonon coupling in β-Ga 2 O 3 are the polar LO phonon modes 43 and show a diverging behavior as → 0.
The interaction of electrons with the polar LO phonons is described by the Fröhlich expression of the general form: for each phonon mode and wave vector along each reciprocal lattice vector direction . We determined the numerators from the calculated matrix elements for the wave vectors closest to Γ along each direction. Table VI

B. Origin of mobility limit
Our results for the electron-phonon coupling matrix elements enable us to identify the limiting factors to the room-temperature mobility and interpret experimental results. The modes that couple strongest to electrons along each direction are the highest-frequency ones (772 cm -1 along ! , 756 cm -1 along ! , and 745 cm -1 along ! ). Those modes dominate phonon emission and, as discussed in Section III.C, play an important role in determining the dielectric-breakdown properties. For low-field transport, however, the thermal energy of electrons (~! ) is insufficient to cause electron scattering by the emission of those high-frequency phonons. Hence we turn our attention to those phonon modes that couple strongly to electrons yet have an energy that is sufficiently low (on the order of ! or less) that allows them to be emitted by thermal electrons. The low-energy modes also have large phonon occupation numbers that facilitate electron scattering by phonon absorption. Of all the modes that dominate electron-phonon coupling, the mode with frequency 235 cm -1 has a phonon energy (29 meV)  Our results agree with previously reported values for the frequency of the PO phonon mode that limits the room-temperature mobility (21 meV and 44 meV). 13,15 Hall-effect measurements by Ma et al. found that, for doping densities < 10 18 cm -3 , the electron mobility is intrinsically limited by phonons to < 200 cm 2 V -1 s -1 at 300 K. 13 The authors also estimated the dimensional Fröhlich coupling constant (α F ) to be almost three times stronger than that of GaN.
The atomistic origin behind the stronger electron-phonon scattering in Ga 2 O 3 is the low symmetry of the crystal structure, which hosts low-energy polar optical modes. The lack of such low-frequency modes leads to a higher intrinsic electron mobility in GaN than β-Ga 2 O 3 .

C. Dielectric breakdown
We applied our calculated electron-phonon coupling matrix elements to estimate the intrinsic breakdown electric field of β-Ga 2 O 3 following to the first-principles methodology developed by Sun et al. 44 The breakdown-field value !" is estimated according to: for !"# ≤ ≤ !"# + ! , where ! is the experimental band gap of β-Ga 2 O 3 (4.5 eV), is the electron mass, is the electron charge, is the electron relaxation time, and is the net rate of energy loss. is given by Equation (3)  represents the time it takes for a single electron to independently relax from a certain energy to another. is given by Equation (6) in Ref. 44 and is calculated similarly to but accounts for energy exchanges between electrons and phonons either through phonon absorption or emission. Complete details for this method are described in the work by Sun et al. 44 !" is calculated in an energy range equal to the energy of the band gap, starting from the bottom of the CBM to ! higher in the conduction band(s). This limit is set by assuming that once an electron reaches energies higher than the band gap, impact ionization will occur at a rate of 100%. This assumption holds for a material sample with infinite size. Practical devices, however, have a finite channel length, and thus not all electrons that reach this energy impact ionize within the time that they cross the device. Therefore, the breakdown field in devices can be higher than our estimate for the bulk intrinsic material, and its analysis requires impact ionization coefficients of electrons such as those calculated by Ghosh and Singisetti. 45 To determine the breakdown field, we applied our previously calculated GW band structure 12 and the fitted Fröhlich parameters (see Table VI) for the electron-phonon coupling matrix elements to evaluate the scattering time and energy loss rate as a function of the electron energy above the conduction band minimum, similar to Ref. 46. We assumed screening by a carrier concentration of 10 16 cm -3 and a temperature of 300 K. We considered the contributions by the dominant phonon modes, which altogether account for approximately 80% of the total magnitude of the electron-phonon interaction.
The argument of the max function on the right-hand side of Eq. (8)  to Si, respectively. 3 Our new estimate of 6.8 MV/cm reduces the BFOM for β-Ga 2 O 3 to ~2,115 (~1.6 times lower) at low frequencies. Additionally, the value used for mobility in the previous report was 300 cm 2 V -1 s -1 , which is actually an extrapolated experimental value. 5 However, recent reports in the literature place it in a lower range around ~200 cm 2 V -1 s -1 . 13,50−53 Assuming a mobility of 200 cm 2 V -1 s -1 and a breakdown field of 6.8 MV/cm, the low-frequency BFOM of Ga 2 O 3 is ~1,410 relative to Si, which is still 1.6 times greater than that of GaN. Despite the lower breakdown-field estimate (6.8 MV/cm) in combination with the lower mobilities recently measured, β-Ga 2 O 3 still shows superior performance for power electronics compared to GaN.
Increasing the band gap even slightly by, e.g., alloying or strain, is a promising method to increase the breakdown field and the BFOM. From Fig. 6, for an energy equal to that of the conduction band minimum plus 4.5 eV (the experimental ! ), the average breakdown field is estimated to be 5.4 MV/cm, with values of 4.8, 5.5, and 5.9 MV/cm along the , , and directions, respectively. Increasing the band gap to 4.7 eV causes a 20% increase in the average breakdown field estimate (6.5 MV/cm), with direction-dependent values of 6.1, 6.5, and 6.8 MV/cm along the , , and directions. Accounting for the remaining 20% of the electronphonon interaction would raise this value to ~8.1 MV/cm, bringing the breakdown field estimate to the original estimate of 8 MV/cm. Possible methods to realize the larger band gap include strain engineering or alloying with aluminum. The latter of which has already been demonstrated, and a band gap range from 5.2-7.1 eV was measured when using 24%-100% Al. 54 For applications in high-power electronics, future work should be done to improve the breakdown field of β-Ga 2 O 3 by increasing the band gap as well as increasing the electron mobility and thermal conductivity.

VI. CONCLUSIONS
In summary, we investigated the phonon properties, phonon-phonon interactions, and electron-phonon scattering of β-Ga 2 O 3 in the anisotropic monoclinic crystal. We derived the directionally dependent phonon dispersion curves, LO-TO splittings, sound velocities and found good agreement with experiment. Oxygen substitution by O 15 has the largest isotopic effects on the phonon frequencies. Our calculated Grüneisen parameters indicate that the optical modes show stronger anharmonicities than the acoustic ones and suppress the thermal conductivity of β-Ga 2 O 3 . We also determined that the low-symmetry crystal structure gives rise to a low-frequency polar-optical mode with a phonon energy of 29 meV that dominates electron scattering at room temperature and limits the mobility. Our value for the breakdown field of β-Ga 2 O 3 is 6.8 MV/cm, which is in good agreement with empirical estimates that use the revised band-gap value of 4.5 eV. We validate that β-Ga 2 O 3 has a higher Baliga FOM compared to GaN. Our results identify the microscopic origins of the thermal and electron transport limits in β-Ga 2 O 3 and propose strategies to increase the breakdown field to 8 MV/cm by increasing the band gap either by growth under strain or by alloying with Al 2 O 3 .