Rutile germanium dioxide (r-GeO_{2}) is a recently predicted ultrawide-bandgap semiconductor with potential applications in high-power electronic devices, for which the carrier mobility is an important material parameter that controls the device efficiency. We apply first-principles calculations based on density functional and density functional perturbation theory to investigate carrier-phonon coupling in r-GeO_{2} and predict its phonon-limited electron and hole mobilities as a function of temperature and crystallographic orientation. The calculated carrier mobilities at 300 K are $\mu elec,\u22a5c\u2192=244$ cm^{2} V^{−1} s^{−1}, $\mu elec,\u2225c\u2192=377$ cm^{2} V^{−1} s^{−1}, $\mu hole,\u22a5c\u2192=27$ cm^{2} V^{−1} s^{−1}, and $\mu hole,\u2225c\u2192=29$ cm^{2} V^{−1} s^{−1}. At room temperature, carrier scattering is dominated by the low-frequency polar-optical phonon modes. The predicted Baliga figure of merit of *n*-type r-GeO_{2} surpasses several incumbent semiconductors such as Si, SiC, GaN, and *β*-Ga_{2}O_{3}, demonstrating its superior performance in high-power electronic devices.

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process, resulting in a combined efficiency of ∼80% or less.^{1} In the United States, the existing electricity grid is outdated for modern electricity usage and must be replaced with power-conversion electronics that are able to control the power flow more efficiently.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 $\mu $ 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 $\epsilon 0$ to enable high-voltage operation. The Baliga figure of merit, $BFOM=14\epsilon 0\mu EC3$, 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*-Ga_{2}O_{3}) 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 polarons^{10}) 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-GeO_{2}) is a promising ambipolarly dopable semiconductor^{13} 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-Ga_{2}O_{3} and facilitates more efficient thermal management.^{14} Rutile GeO_{2} displays similar chemical and structural properties to rutile SnO_{2}, an established *n*-type transparent conductor.^{15,16} However, the wider bandgap of r-GeO_{2} is promising for deep-ultraviolet (UV) luminescence and efficient power-electronics applications.^{13,17,18} Yet the electrical conductivity of r-GeO_{2} 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-GeO_{2} 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 $\mu elec,\u22a5c\u2192=244$ cm^{2} V^{−1} s^{−1}, $\mu elec,\u2225c\u2192=377$ cm^{2} V^{−1} s^{−1}, $\mu hole,\u22a5c\u2192=27$ cm^{2} V^{−1} s^{−1}, and $\mu hole,\u2225c\u2192=29$ cm^{2} V^{−1} s^{−1}. The electron mobility is higher than that of *β*-Ga_{2}O_{3}, while the hole mobility at high temperature surpasses the values for p-type GaN. Our results demonstrate that r-GeO_{2} exhibits a superior BFOM than current semiconductor technologies such as Si, SiC, GaN, and *β*-Ga_{2}O_{3} and is, therefore, of great interest for high-power and high-temperature electronics applications.

To accurately predict the carrier and phonon properties of r-GeO_{2}, 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-GeO_{2} for the experimental lattice parameters^{25} using the G_{0}W_{0} method. We find that the effective masses obtained from G_{0}W_{0} 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 $a=4.516\u2009\xc5$ and $c=2.978\u2009\xc5$ that differ from experiment^{25} 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 10^{17} 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 $\u210f\omega max+5kBT$ at 1000 K, where $\u210f\omega max$ 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 $\mu elec,\u22a5c\u2192=244$ cm^{2} V^{−1} s^{−1}, $\mu elec,\u2225c\u2192=377$ cm^{2} V^{−1} s^{−1}, $\mu hole,\u22a5c\u2192=27$ cm^{2} V^{−1} s^{−1}, and $\mu hole,\u2225c\u2192=29$ cm^{2} 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 ($me,\u22a5c\u2192*=0.36\u2009m0$ and $me,\u2225c\u2192*=0.21\u2009m0$) and approximately a factor of 2 lighter for the $\u2225c\u2192$ direction, which is consistent with the approximately 2 times higher mobility along $\u2225c\u2192$. 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 ($mh,\u22a5c\u2192*=1.29\u2009m0$ and $mh,\u2225c\u2192*=1.58\u2009m0$) 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 $\mu 1$, $\mu 2$, $T1$, and $T2$ are fitting parameters to describe the temperature dependence. $\mu 1$ and $\mu 2$ 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 $\mu i,Ti$ 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.

Parameters . | Electron, $\u22a5c\u2192$ . | Electron, $\u2225c\u2192$ . | Hole, $\u22a5c\u2192$ . | Hole, $\u2225c\u2192$ . |
---|---|---|---|---|

$\mu 1$ (cm^{2} V^{-1} s^{−1}) | 43.8 | 85.9 | 2.08 | 2.27 |

$T1$ (K) | 576 | 525 | 765 | 761 |

$\mu 2$ (cm^{2} V^{−1} s^{−1}) | 21.1 | 22.2 | 0.669 | 0.603 |

$T2$ (K) | 1221 | 1268 | 2691 | 2395 |

Parameters . | Electron, $\u22a5c\u2192$ . | Electron, $\u2225c\u2192$ . | Hole, $\u22a5c\u2192$ . | Hole, $\u2225c\u2192$ . |
---|---|---|---|---|

$\mu 1$ (cm^{2} V^{-1} s^{−1}) | 43.8 | 85.9 | 2.08 | 2.27 |

$T1$ (K) | 576 | 525 | 765 | 761 |

$\mu 2$ (cm^{2} V^{−1} s^{−1}) | 21.1 | 22.2 | 0.669 | 0.603 |

$T2$ (K) | 1221 | 1268 | 2691 | 2395 |

Next, we analyze the carrier-phonon coupling matrix elements and scattering lifetimes to understand how different phonon modes affect carrier scattering in r-GeO_{2}. We first determine the carrier-phonon coupling matrix elements for the bottom conduction and top valence bands for wave vectors along the Γ—X ($\u22a5c\u2192$) and Γ—Z ($\u2225c\u2192$) 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-GeO_{2}. 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 $(kBT=26\u2009meV)$ 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-GeO_{2} 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-GeO_{2} (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-GeO_{2} relevant for *n*-type and *p*-type power electronics and compares them to incumbent technologies. The breakdown fields of *β*-Ga_{2}O_{3} (with a gap of 4.5 eV) and r-GeO_{2} are evaluated using the breakdown field vs bandgap relation by Higashiwaki *et al.*^{9} The electron mobility of r-GeO_{2} is lower than that of Si, SiC, and GaN by over 70%,^{30–32} but higher than that of *β*-Ga_{2}O_{3}. Although the experimental electron mobilities of *β*-Ga_{2}O_{3} 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 measured^{33} in *β*-Ga_{2}O_{3} 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 ($\mu Hall=\mu driftrH$), the highest measured room-temperature drift mobility of Ga_{2}O_{3} is 109 cm^{−2} V^{−1} s^{−1}. Moreover, recent calculations of the phonon-limited carrier mobility of *β*-Ga_{2}O_{3} from first principles using a similar approach to the present work^{36} 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-GeO_{2} 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 $\epsilon 0$, and a higher electron mobility than *β*-Ga_{2}O_{3}.

Material . | Static dielectric constant, $\epsilon 0$ . | Electron mobility, $\mu $ (cm^{2} V^{−1} s^{−1})
. | Hole mobility, $\mu $ (cm^{2} V^{−1} s^{−1})
. | Dielectric breakdown field, $EC$ (MV cm^{−1})
. | n-BFOM (10^{6} V^{2} Ω^{−1} cm^{−2})
. | p-BFOM (10^{6} V^{2} Ω^{−1} cm^{−2})
. |
---|---|---|---|---|---|---|

Si | 11.9^{37} | 1240 (drift)^{31} | 0.3^{9} | 8.8 | ||

4H-SiC | 9.7^{31} | 980 (drift)^{31} | 2.5^{9} | 3300 | ||

GaN | 10.4 ($\u2225c\u2192$)^{38} | 1000 (drift)^{32} | 31 (Hall)^{39} | 3.3^{9} | 8300 | 260 |

β-Ga_{2}O_{3} | 10.0^{40} | 184; 180 (Hall)^{30,34} | … | 6.4^{9} | 11 000; 10 000 6300; 4600 | |

109; 80 (drift)^{33,34} | ||||||

258 (drift)^{36} | 1.2 (drift)^{36} | 5.8^{36} | 11 000 | 52 | ||

r-GeO_{2} | 14.5 ($\u22a5c\u2192$)^{41} | 244 (drift, $\u22a5c\u2192$) | 27 (drift, $\u22a5c\u2192$) | 7.0 | 27 000 | 3000 |

12.2 ($\u2225c\u2192$)^{41} | 377 (drift, $\u2225c\u2192$) | 29 (drift, $\u2225c\u2192$) | 7.0 | 35 000 | 2700 |

Material . | Static dielectric constant, $\epsilon 0$ . | Electron mobility, $\mu $ (cm^{2} V^{−1} s^{−1})
. | Hole mobility, $\mu $ (cm^{2} V^{−1} s^{−1})
. | Dielectric breakdown field, $EC$ (MV cm^{−1})
. | n-BFOM (10^{6} V^{2} Ω^{−1} cm^{−2})
. | p-BFOM (10^{6} V^{2} Ω^{−1} cm^{−2})
. |
---|---|---|---|---|---|---|

Si | 11.9^{37} | 1240 (drift)^{31} | 0.3^{9} | 8.8 | ||

4H-SiC | 9.7^{31} | 980 (drift)^{31} | 2.5^{9} | 3300 | ||

GaN | 10.4 ($\u2225c\u2192$)^{38} | 1000 (drift)^{32} | 31 (Hall)^{39} | 3.3^{9} | 8300 | 260 |

β-Ga_{2}O_{3} | 10.0^{40} | 184; 180 (Hall)^{30,34} | … | 6.4^{9} | 11 000; 10 000 6300; 4600 | |

109; 80 (drift)^{33,34} | ||||||

258 (drift)^{36} | 1.2 (drift)^{36} | 5.8^{36} | 11 000 | 52 | ||

r-GeO_{2} | 14.5 ($\u22a5c\u2192$)^{41} | 244 (drift, $\u22a5c\u2192$) | 27 (drift, $\u22a5c\u2192$) | 7.0 | 27 000 | 3000 |

12.2 ($\u2225c\u2192$)^{41} | 377 (drift, $\u2225c\u2192$) | 29 (drift, $\u2225c\u2192$) | 7.0 | 35 000 | 2700 |

Though ultrawide-bandgap materials such as nitrides and *β*-Ga_{2}O_{3} 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-GeO_{2}, 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*-Ga_{2}O_{3}, which is limited by self-trapped hole polarons.^{10} Moreover, the hole mobility of r-GeO_{2} 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-GeO_{2} 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-GeO_{2} as a superior semiconductor compared to incumbent technologies such as *β*-Ga_{2}O_{3} and GaN for high-power and high-temperature electronic applications.

In summary, we calculate the phonon-limited electron and hole mobilities of r-GeO_{2} 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*-Ga_{2}O_{3} enables a BFOM that surpasses established power-electronics materials such as Si, SiC, GaN, and *β*-Ga_{2}O_{3}. In addition, our results for its sizable hole mobility in combination with the theoretical prediction of its ambipolar dopability^{13} indicate potential applications for p-type and CMOS devices. Our results highlight the advantages of r-GeO_{2} compared to incumbent material technologies for power-electronics applications.

See the supplementary material for a comparison of the effective masses calculated with G_{0}W_{0} 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-GeO_{2} to available data for GaN.

## AUTHORS' CONTRIBUTIONS

K.B. and K.A.M. contributed equally to this work.

This work was supported as part of the Computational Materials Sciences Program funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0020129. It used resources of the National Energy Research Scientific Computing (NERSC) Center, a DOE Office of Science User Facility supported under Contract No. DE-AC02–05CH11231. K.A.M. acknowledges the support from the National Science Foundation Graduate Research Fellowship Program through Grant No. DGE 1256260. K.B. acknowledges the support of the DOE Computational Science Graduate Fellowship Program through Grant No. DE-SC0020347.

## DATA AVAILABILITY

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