Electron and hole mobility of rutile GeO₂ from first principles: An ultrawide-bandgap semiconductor for power electronics

Power electronics are important for the control and conversion of electricity, but inefficiencies cause energy loss during each step in the conversion process, resulting in a combined efficiency of ∼80% or less.¹ In the United States, the existing electricity grid is outdated for modern electricity usage and must be replaced with power-conversion electronics that are able to control the power flow more efficiently. Addressing inefficiencies to improve energy sustainability motivates the ongoing search for new materials for power-electronics devices. Ultrawide-bandgap semiconductors with gaps wider than GaN (3.4 eV) have been the focus of power-electronics materials research.² Important material parameters to consider for power-electronics applications include the possibility of doping (usually n-type, but ambipolar dopability is also desirable for CMOS devices), high carrier mobility $μ$ for fast switching and low Ohmic losses, high thermal conductivity for efficient heat extraction, and a high critical dielectric breakdown field $EC$ and dielectric constant $ε0$ to enable high-voltage operation. The Baliga figure of merit, $BFOM=14ε0μEC3$⁠, quantifies the performance of materials in power-electronic devices.^3,4 The BFOM depends most sensitively on the breakdown field, which increases superlinearly with the increasing bandgap and motivates the search for dopable ultrawide-gap semiconductors.

The β polymorph of gallium oxide (β-Ga₂O₃) has been the recent focus of attention thanks to the availability of native substrates and the n-type dopability with Si or Ge.² While its electron mobility is lower than that of Si, SiC, or GaN, its ultrawide-bandgap of ∼4.5 eV produces a high breakdown field and a BFOM superior to these incumbent technologies.^5–9 However, its low thermal conductivity that prevents heat extraction and the unfeasibility of p-type doping (due to the formation of self-trapped hole polarons¹⁰) limit its applicability.^10–12 To overcome these challenges and advance the frontier of power electronics, new ultrawide-bandgap semiconducting materials must be identified and characterized.

Recently, Chae et al. found that rutile germanium dioxide (r-GeO₂) is a promising ambipolarly dopable semiconductor¹³ with an ultrawide bandgap (4.68 eV). Donors such as Sb_Ge and F_O are shallow (activation energy ∼25 meV), while Al_Ge and Ga_Ge acceptors are deeper with modest ionization energies of ∼0.4–0.5 eV.¹³ However, the co-incorporation of acceptors with hydrogen and subsequent annealing can produce high acceptor concentrations that exceed the Mott-transition limit and enable p-type conduction. Moreover, its measured thermal conductivity of 51 W/m K at 300 K surpasses that of β-Ga₂O₃ and facilitates more efficient thermal management.¹⁴ Rutile GeO₂ displays similar chemical and structural properties to rutile SnO₂, an established n-type transparent conductor.^15,16 However, the wider bandgap of r-GeO₂ is promising for deep-ultraviolet (UV) luminescence and efficient power-electronics applications.^13,17,18 Yet the electrical conductivity of r-GeO₂ and, thus, its viability and efficiency for power-electronics applications remain unexplored.

In this work, we apply predictive atomistic calculations to determine the phonon-limited electron and hole mobilities of r-GeO₂ as a function of temperature and crystallographic orientation. We quantify the intrinsic phonon and carrier-phonon-coupling properties that impact carrier transport. We find that the room-temperature carrier mobility is dominated by the low-frequency polar-optical phonon modes, and the resulting mobility values at 300 K are $μelec,⊥c→=244$ cm² V⁻¹ s⁻¹, $μelec,∥c→=377$ cm² V⁻¹ s⁻¹, $μhole,⊥c→=27$ cm² V⁻¹ s⁻¹, and $μhole,∥c→=29$ cm² V⁻¹ s⁻¹. The electron mobility is higher than that of β-Ga₂O₃, while the hole mobility at high temperature surpasses the values for p-type GaN. Our results demonstrate that r-GeO₂ exhibits a superior BFOM than current semiconductor technologies such as Si, SiC, GaN, and β-Ga₂O₃ and is, therefore, of great interest for high-power and high-temperature electronics applications.

To accurately predict the carrier and phonon properties of r-GeO₂, we use first-principles calculations based on density functional (DFT) and density functional perturbation theories (DFPT) in the local density approximation within Quantum ESPRESSO^19–21 and the iterative Boltzmann transport equation (IBTE) with the EPW code.^22–24 We do not consider spin–orbit coupling. In previous work,¹⁸ we calculated the quasiparticle band structure of r-GeO₂ for the experimental lattice parameters²⁵ using the G₀W₀ method. We find that the effective masses obtained from G₀W₀ are similar to those obtained with a hybrid DFT functional (Table SI in the supplementary material), verifying the reliability of our calculated band parameters. For phonon calculations, the lattice parameters and atomic positions were relaxed to prevent imaginary phonon frequencies, resulting in lattice parameters of $a=4.516 Å$ and $c=2.978 Å$ that differ from experiment²⁵ by +2.5% and +4.1%, respectively. The phonon dispersion, phonon frequencies at Γ, and sound velocities are included in Fig. S1 and Tables SII and SIII in the supplementary material The quasiparticle energies, phonon frequencies, and carrier-phonon coupling matrix elements were calculated on 4 × 4 × 6 Brillouin zone (BZ) sampling grids (using the charge density generated on an 8 × 8 × 12 BZ sampling grid for higher accuracy) and interpolated to fine BZ sampling grids with the EPW code. Carrier velocities were evaluated with the velocity operator,^26,27 and the Fröhlich correction was applied to the carrier-phonon coupling matrix elements $g$⁠.²⁸ The phonon-limited carrier mobilities were calculated over the 100–1000 K temperature range for a carrier concentration of 10¹⁷ cm⁻³.²⁴ The electron and hole mobilities were converged for carrier and phonon BZ sampling grids of 92 × 92 × 138 and 48 × 48 × 72, respectively. We sampled states within energy windows of 530 meV around the carrier Fermi energies, which accounts for energy differences during scattering of up to $ℏωmax+5kBT$ at 1000 K, where $ℏωmax$ is the highest optical phonon energy and $kB$ is the Boltzmann constant. With these calculation parameters, the mobility values at 300 K were converged within 1% for electrons and 2% for holes.

We first discuss the mobility obtained from the iterative solution of the IBTE. The converged carrier mobilities at 300 K are $μelec,⊥c→=244$ cm² V⁻¹ s⁻¹, $μelec,∥c→=377$ cm² V⁻¹ s⁻¹, $μhole,⊥c→=27$ cm² V⁻¹ s⁻¹, and $μhole,∥c→=29$ cm² V⁻¹ s⁻¹. Qualitatively, we expect a lower mobility for transport directions in which carriers have a higher effective mass. Quasiparticle band structure calculations find that the electron mass is anisotropic (⁠$me,⊥c→*=0.36 m0$ and $me,∥c→*=0.21 m0$⁠) and approximately a factor of 2 lighter for the $∥c→$ direction, which is consistent with the approximately 2 times higher mobility along $∥c→$⁠. We also find that the scattering rates for electrons are relatively isotropic (Fig. S2), indicating that the anisotropy of the calculated electron mobility is primarily driven by the anisotropy of the effective mass. On the other hand, the hole mobility is approximately isotropic, which is consistent with the relatively small directional dependence of the effective mass (⁠$mh,⊥c→*=1.29 m0$ and $mh,∥c→*=1.58 m0$⁠) and scattering rates (Fig. S2).

Figure 1 shows the temperature dependence of the mobility. We can accurately fit the temperature dependence with a combination of exponential equations characteristic of electron-phonon scattering by two optical modes,

where $μ1$⁠, $μ2$⁠, $T1$⁠, and $T2$ are fitting parameters to describe the temperature dependence. $μ1$ and $μ2$ are in units of mobility, while $T1$ and $T2$ are in units of temperature. Table I lists the fitted values for each carrier type and direction. The two sets of $μi,Ti$ correspond to the low- and high-energy polar-optical modes that dominate carrier scattering, as we discuss in greater detail below. Our fitted parameters demonstrate that the low-energy polar-optical modes characterized by the $T1$ parameter dominate near room temperature. The high-energy polar-optical modes only become important at high temperatures (higher than 500 K) and to a larger extent for electrons than for holes.

Next, we analyze the carrier-phonon coupling matrix elements and scattering lifetimes to understand how different phonon modes affect carrier scattering in r-GeO₂. We first determine the carrier-phonon coupling matrix elements for the bottom conduction and top valence bands for wave vectors along the Γ—X (⁠$⊥c→$⁠) and Γ—Z (⁠$∥c→$⁠) directions [Figs. 2(a) and 2(d)]. Our results show that the polar-optical modes exhibit the strongest carrier-phonon coupling, as expected in polar materials such as r-GeO₂. However, the higher-frequency modes are not as effective at scattering low-energy carriers at room temperature; they require either high temperatures to enable appreciable phonon occupation numbers and scatter carriers by phonon absorption or high carrier energies to scatter electrons to lower-energy states by phonon emission. Taking the thermal occupation of phonon modes (⁠$nq$⁠) at room temperature $(kBT=26 meV)$ into account, we find the dominant modes for carrier scattering by phonon absorption [$g2nq$⁠, Figs. 2(b) and 2(e)] and phonon emission [$g2nq+1$⁠, Figs. 2(c) and 2(f)]. Electrons and holes most strongly absorb the polar-optical modes with frequencies of 41, 49, 94, and 100 meV. As with phonon absorption, carriers scatter most strongly by phonon emission by the polar-optical phonons, as well as the Raman-active modes at 87 meV and 102 meV. We thus predict that carriers primarily scatter by absorption (and to a lesser extent by emission) of the A_2u and three E_u polar-optical modes. The eigenmodes of these four dominant scattering phonons are shown in Figs. 2(g)–2(j).

Typically, the polar-optical modes limit the room-temperature mobility in oxide materials, and our electron-phonon coupling results combined with the mobility calculations indicate that this is the case in r-GeO₂ as well.²⁹ Further evidence for the role of the polar-optical modes is provided by the mode-resolved carrier scattering rates as a function of carrier energy (Fig. 3). Here, we refer to modes by their numerical index (in order of increasing frequency) since the directional dependence of the LO-TO splitting reorders the mode frequencies. Mode 7 has the strongest effect at low energies and corresponds to the 41 meV E_u mode. In both the valence band Fig. 3(a) and conduction band Fig. 3(b), there is a noticeable step in the magnitude above 40 meV, where phonon emission becomes possible. The same is true for Mode 17 around 100 meV, which corresponds to both the 94 meV A_2u mode and the 100 meV E_u mode, depending on the direction. Modes 9 and 10, which show the next largest contribution to the total electron linewidth, are both (depending on the direction) the 49 meV E_u phonon mode. Finally, Mode 13 is associated with the Raman active 67 meV E_g mode, though its contribution is approximately one order of magnitude smaller than the more dominant polar optical modes.

The carrier mobility is a necessary parameter to understand the performance of a semiconductor in electronic devices. The ultrawide bandgap of r-GeO₂ (4.68 eV)¹⁷ makes it especially suitable for high-power and high-temperature applications.^13,18 Table II lists the material parameters of r-GeO₂ relevant for n-type and p-type power electronics and compares them to incumbent technologies. The breakdown fields of β-Ga₂O₃ (with a gap of 4.5 eV) and r-GeO₂ are evaluated using the breakdown field vs bandgap relation by Higashiwaki et al.⁹ The electron mobility of r-GeO₂ is lower than that of Si, SiC, and GaN by over 70%,^30–32 but higher than that of β-Ga₂O₃. Although the experimental electron mobilities of β-Ga₂O₃ have typically been obtained with Hall measurements, it is the drift mobility that should be applied to evaluate the BFOM. A drift mobility of 80 cm⁻² V⁻¹ s⁻¹ was measured³³ in β-Ga₂O₃ at 300 K, while the highest Hall mobility at 300 K is 184 cm⁻² V⁻¹ s⁻¹.^30,34 However, if the Hall factor at 300 K (⁠$rH=1.68$⁠)³⁵ is applied to convert the Hall to drift mobility (⁠$μHall=μdriftrH$⁠), the highest measured room-temperature drift mobility of Ga₂O₃ is 109 cm⁻² V⁻¹ s⁻¹. Moreover, recent calculations of the phonon-limited carrier mobility of β-Ga₂O₃ from first principles using a similar approach to the present work³⁶ report intrinsic electron and hole drift mobilities at 300 K of 258 cm⁻² V⁻¹ s⁻¹ and 1.2 cm⁻² V⁻¹ s⁻¹, respectively. Overall, r-GeO₂ displays the largest BFOM out of the materials considered here as it exhibits the largest $EC$ value (since it has the widest bandgap), the highest $ε0$⁠, and a higher electron mobility than β-Ga₂O₃.

Though ultrawide-bandgap materials such as nitrides and β-Ga₂O₃ are promising for numerous technological applications, one consistent shortcoming is the relative difficulty of p-type doping and low hole mobilities. This is evident in the differences between electron and hole mobility data in Table II. Our findings of a sizable hole mobility in r-GeO₂, in combination with the results of Chae et al. on the feasibility of ambipolar doping,¹³ demonstrate that it may be a candidate material for high-power CMOS devices, in contrast to β-Ga₂O₃, which is limited by self-trapped hole polarons.¹⁰ Moreover, the hole mobility of r-GeO₂ is comparable to that of GaN at temperatures below ∼500 K, matching or exceeding experimental values and within a factor of two of theoretical values (Fig. S3). At higher temperatures, r-GeO₂ shows higher hole mobility than experimentally reported GaN values despite having a wider bandgap. The combination of a higher $BFOM$ with the prediction of p-type doping and the possibility of efficient hole conduction demonstrate the promise of r-GeO₂ as a superior semiconductor compared to incumbent technologies such as β-Ga₂O₃ and GaN for high-power and high-temperature electronic applications.

In summary, we calculate the phonon-limited electron and hole mobilities of r-GeO₂ as a function of temperature and crystallographic orientation from first principles and provide atomistic insights into the dominant carrier-scattering mechanisms. The combination of its ultrawide bandgap of 4.68 eV with its predicted electron mobility that is higher than that of β-Ga₂O₃ enables a BFOM that surpasses established power-electronics materials such as Si, SiC, GaN, and β-Ga₂O₃. In addition, our results for its sizable hole mobility in combination with the theoretical prediction of its ambipolar dopability¹³ indicate potential applications for p-type and CMOS devices. Our results highlight the advantages of r-GeO₂ compared to incumbent material technologies for power-electronics applications.

See the supplementary material for a comparison of the effective masses calculated with G₀W₀ and the hybrid DFT functional, an analytical discussion of the calculated phonon frequencies and sound velocities, the data for the directionally resolved electron self-energy due to electron-phonon interaction, and a comparison of the hole mobility of r-GeO₂ to available data for GaN.