Donors and deep acceptors in β-Ga₂O₃

Excellent performance improvements in Ga₂O₃ power electronic transistors have been made since the first demonstration of Ga₂O₃ MESFETs¹ and MOSFETs.² Breakdown voltages for Ga₂O₃ Schottky diodes have reached 1.1 kV (Ref. 3) and 1.6 kV,⁴ while breakdown voltages for MOSFETs are as high as 740 V.⁵ Lateral device electric fields of at least 3.8 MV/cm (Ref. 6) and vertical device electric fields of 5.1 MV/cm (Ref. 3) have been demonstrated, along with on-currents of 1.5 A/mm.^7,8 Radio frequency operation of Ga₂O₃ MOSFETs, with f_t and f_max as high as 3.3 GHz and 12.9 GHz, respectively, has also been reported.⁹ The rapid pace of Ga₂O₃ device development can be attributed in part to the effective and controllable n-type doping of Ga₂O₃, which has been achieved using tin (Sn),^1–17 silicon (Si),^17–24 and, more recently, germanium (Ge),²⁵ consistent with the results from DFT calculations.²⁶ Some previous studies have examined the transport properties of shallow donors in Ga₂O₃; however, there remain some discrepancies regarding the energies of the shallow donors. Estimates of the donor energies range from 7.4 meV to 60 meV for Sn^2,27,28 and from 16 meV to 50 meV for Si,^29–35 and we have previously reported a 17.5 meV donor energy for Ge.³⁶ Recent EPR studies report that Si may also exhibit a DX⁻ state at an energy of 49 meV;³⁴ however, other groups have reported no evidence for a DX⁻ state.³⁷ In addition to the shallow donor impurities, some impurities have been reported to induce insulating behavior in Ga₂O₃, acting as deep acceptors, including magnesium (Mg)^38–41 and iron (Fe).^42,43

Given the wide range of donor energies reported in the literature for shallow donors and the limited data available on the deep acceptors, we have undertaken a study to understand the transport properties of Si, Ge, Fe, and Mg doped Ga₂O₃. In the case of shallow donors Si and Ge, by simultaneously and self-consistently fitting both the temperature dependent mobility and temperature dependent carrier density, we are able to accurately determine the donor energy for both Si and Ge to be 30 meV. Additionally, our transport measurements are consistent with Si and Ge acting as typical shallow donors, rather than shallow DX centers. We also report that EFG grown Fe doped Ga₂O₃ remains weakly n-type, as determined by the high temperature Hall effect measurement, with an acceptor energy for Fe of 0.86 eV relative to the Ga₂O₃ conduction band. Finally, we report that the conductivity of CZ grown Ga₂O₃ pulled from a melt with 0.1 mol. % MgO shows an activation energy of 1.1 eV, consistent with the previous report.³⁹ 

To study the properties of shallow donors in Ga₂O₃, we have measured and analyzed the temperature dependent carrier density and mobility of several samples determined by van der Pauw and Hall effect measurements in a four terminal configuration. To accurately estimate the donor energies, the carrier density was fit using the charge neutrality equation⁴⁴ and the mobility fit using the solution to the Boltzmann transport equation in the relaxation time approximation,⁴⁵ including the Hall factor,⁴⁶ where ionized impurity,⁴⁷ neutral impurity,⁴⁸ and polar optical phonon⁴⁹ scattering mechanisms were included. An effective mass m_∗ = 0.3m_o,^50–52 a low frequency relative dielectric constant κ_S = 10,^53,54 a high frequency relative dielectric constant κ_∞ = 3.5,^54–56 an effective phonon energy ℏω = 44 meV,³² and an effective number of phonon modes M = 1.5 were used in the calculation. A detailed summary of the relevant equations from the cited references can be found in the supplementary material. Iteration was used to simultaneously and self-consistently fit both the carrier density and mobility. Uncertainty in the compensating acceptor concentration can propagate to the estimated donor energy when fitting the temperature dependent carrier density alone in moderately doped samples. With our approach, we are able to independently determine the compensating acceptor concentration from the ionized impurity limited mobility, allowing for more accurate estimation of the donor energies by avoiding said propagation. Others have used parts of this approach studying the properties of Si doped Ga₂O₃.^30–32 

The Si doped samples include a bulk CZ sample from Northrop Grumman SYNOPTICS (Sample 3), a bulk EFG sample from Tamura Corporation (Sample 4), and two epitaxial films grown by LPCVD on c-sapphire substrates with 3.5° (Sample 5) and 6° (Sample 6) offcuts.⁵⁷ Glow discharge mass spectrometry (GDMS) and secondary ion mass spectrometry (SIMS) analysis of EFG Ga₂O₃ samples similar to Sample 4 and SIMS analysis of a CZ grown sample similar to Sample 3 confirm that Si is the dominant unintentional donor in the bulk melt-grown Ga₂O₃ used in this study. Our result is consistent with a previous study which determined that the unintentional Si doping comes from the Ga₂O₃ powder used as the source material for bulk melt-growth.¹⁹ The LPCVD films were intentionally doped using SiCl₄. To make ohmic contacts, four 150 nm Ti/500 nm Au contacts were deposited on the sample edges and annealed in a tube furnace under Ar gas flow up to 450 °C. Sample 1 and Sample 2 are Ge doped MBE grown Ga₂O₃ epitaxial films on semi-insulating substrates whose fabrication and growth details are published elsewhere.^25,36 Figures 1 and 2 show the experimentally measured Hall carrier density and Hall mobility for the samples, along with the temperature dependent fittings. The mobility due to individual scattering mechanisms is shown only for Sample 3. Parameters of the temperature dependent fittings for the samples are shown in Table I. The donor energies for the samples, along with several samples from the literature, are summarized in Fig. 3. As the figure shows, the donor energies for samples seem to converge to a value of 30 meV as the donor concentration approaches 1 × 10¹⁷ cm⁻³ for both Si and Ge donors. However, as the donor concentration increases above 4 × 10¹⁷ cm⁻³, the donor energy begins to decrease, as is expected for highly doped semiconductors when an impurity band begins to form.^58,59 Consistent with this hypothesis, the decrease in the donor energy occurs as the donor density approaches N_dn = (0.2/α)³ = 1.46 × 10¹⁸ cm⁻³, where α is the effective Bohr radius for gallium oxide, which is the estimated density at which a Mott metal-insulator transition would occur for the donor level.^58,59

The above analysis assumes that the Si and Ge donors behave as typical shallow donors in Ga₂O₃, but recent EPR measurements have suggested that Si may behave as a shallow DX center.³⁴ By contrast, others have reported that they do not see evidence of DX center behavior in Ga₂O₃.³⁷ To address this open question in light of our Hall effect measurements, let us consider the charge neutrality equation for a DX center⁶⁰ 

where N_c is the conduction band effective density of states, N_ac the compensating acceptor concentration, N_dn the donor concentration, E_dn the donor energy, $F$_1/2 the normalized Fermi-Dirac integral of order one half, E_f the Fermi level, and U the interaction energy for two electrons on a single donor. The interaction energy, U, is the energy difference between two electrons on a single donor (DX⁻ state) and two non-interacting electrons on two different donors (neutral state). U is negative, which indicates that the DX⁻ state is the lower energy state, usually due to lattice distortion. For a normal shallow donor, U is a large positive number due to coulomb repulsion of the electrons, and the conventional charge neutrality equation is recovered. Using Eq. (1) and the parameters listed in the first two entries of Table II, a comparison of the temperature dependent carrier density and ionized impurity density for the typical normal shallow donor model and the shallow DX donor model is plotted in Fig. 4. For the models in Fig. 4, values for the normal donor model were chosen to be representative of the experimentally characterized samples listed in Table I, while the values for the DX donor model were chosen to match the temperature dependent carrier density of the normal donor model, with U = −20 meV as suggested by the recent EPR study.³⁴ As shown, it is possible to find a set of parameters for the DX donor model which almost match the temperature dependent carrier density of the normal donor model; however, as the dashed lines show, the density of ionized impurities at low temperatures is about a factor of five larger for the DX donor model. Because the density of ionized impurities is so much larger for the DX donor model, it underestimates the experimentally measured mobility of our samples at low temperatures where ionized impurity scattering dominates, as shown in Fig. 5, with fitting parameters shown in Table II. This fact means that the DX donor model cannot be used to fit our measured Hall mobility and carrier density data. This analysis assumes that the spatial distribution of donors in the negatively charged DX⁻ state and those in the typical positively charged ionized state is uncorrelated, so that all donors act as point-charge-like scattering centers. However, there is some evidence that the distributions of DX⁻ donors and positively charged ionized donors can become correlated, acting as weaker dipole-like scattering centers.⁶¹ While such a correlation may partially account for the discrepancy between our measured mobility data and the DX donor mobility models presented in Fig. 5, it does not fully account for the factor of 4 to 5 discrepancy in low temperature mobility for Si doped Sample 3 and Sample 4. Therefore, our measured Hall data for these Si doped samples are consistent with a normal shallow donor and are inconsistent with a shallow DX center. This observation agrees with Irmscher,³⁷ who also reports no evidence of DX behavior. We note, however, that we are only able to rule out DX center behavior in our samples because they have a compensation ratio N_ac/N_dn less than one third. For compensation ratios of one third or greater in the normal donor model, it is always possible to find a set of parameters for the DX donor model which match both the temperature dependent carrier density and the low temperature ionized impurity density. This fact means that a normal shallow donor and shallow DX donor can only be distinguished using simultaneous, self-consistent carrier density and mobility fitting for compensation ratios less than one third, as is the case here. Considering the remarkable consistency in experimental results across a wide variety of growth methods, including EFG, CZ, MBE, and LPCVD, along with the fact that Si and Ge are both group IV elements on the periodic table, we can conclude that Si and Ge act as typical shallow donors in all of the samples presented here.

In addition to measurements on Si and Ge donor doped samples, we have characterized Fe and Mg acceptor doped samples as well using the same four terminal van der Pauw and Hall effect measurement techniques. Due to the semi-insulating nature of these samples, high temperature Hall effect measurements were necessary to thermally activate carriers to enable Hall effect measurements. Measurements were performed under N₂ gas flow in a tube furnace with an external silicon carbide heater and an electromagnet. Figure 6 shows the Hall carrier density for an Fe doped, EFG grown semi-insulating substrate from Tamura Corporation (Sample 7). The Hall effect measurement was possible for temperatures above 400 K, with the sign of the Hall effect indicating that the β-Ga₂O₃ remains weakly n-type at elevated temperatures even with Fe doping. Glow discharge mass spectrometry analysis (GDMS) performed on a similar Fe doped sample indicates a concentration of about 1 × 10¹⁷ cm⁻³ for the unintentional Si donor and 8 × 10¹⁷ cm⁻³ for the intentional Fe acceptor. Therefore, we ascribe the observed n-type behavior to thermal activation of electrons on ionized Fe acceptors into the conduction band. Fitting with the charge neutrality equation yields an acceptor energy E_c − E_ac of 860 meV. Note that this acceptor energy is referenced to the conduction band edge due to the n-type behavior. This acceptor energy near the conduction band is consistent with preliminary DFT calculations of Fe doped Ga₂O₃ which indicate that Fe induces midgap states closer to the conduction band.⁴³ The acceptor energy is somewhat higher than that recently observed via DLTS for Fe impurities in n-type bulk substrates;⁶² however, this difference can be explained by the broadening of the Fe acceptor energy level at the much higher Fe concentration in this semi-insulating substrate. Because the carrier density does not saturate at higher temperatures, it is not possible to estimate the absolute value of the Fe and Si concentrations in this sample. However, the ratio of Fe acceptors to Si donors, N_ac/N_dn, is uniquely determined to be 1.65 by fitting the temperature dependent carrier density using the charge neutrality equation. We note that this ratio is smaller than one would expect based on the GDMS results, which could suggest that not all of the Fe dopants in the sample are electrically active. Finally, the inset of Fig. 6 shows the temperature dependent conductivity measured by the van der Pauw method for an Mg doped sample grown by Northrop Grumman SYNOPTICS using the CZ method (Sample 8). The melt from which the sample was pulled contained 0.1 mol. % of MgO. The Hall effect measurement was attempted, but it was not possible to resolve the Hall voltage due to a low signal to noise ratio as a result of the very low conductivity for the sample. Least squares fitting of the temperature dependent conductivity indicates an empirical activation energy of 1.1 eV, which is consistent with a previous report on the activation energy of highly Mg doped Ga₂O₃.³⁹ Because Si contamination is present in these CZ grown samples, it is expected that the activation energy of 1.1 eV is approximately equal to the acceptor energy for Mg. However, without the Hall effect measurement, it is not possible to determine the carrier type of the Mg doped sample. This fact means that we are unable to determine if this 1.1 eV activation energy is referenced to the conduction band or valence band edge of Ga₂O₃ from experiment, although recent DFT studies indicate that the Mg acceptor level is closer to the valence band.⁶³ 

In conclusion, simultaneous, self-consistent fitting of the temperature dependent carrier density and mobility of n-type β-Ga₂O₃, as measured by the van der Pauw and Hall effect methods, indicates a donor ionization energy of 30 meV for Si and Ge shallow donors. Accurate determination of the donor energy is enabled by reliable estimation of the compensating acceptor concentration through fitting of the low temperature ionzied impurity limited mobility. Additionally, comparison of our Hall effect data to appropriate models indicates that Si and Ge act as typical shallow donors in the β-Ga₂O₃ samples presented here, as opposed to shallow DX centers. Finally, Fe doped β-Ga₂O₃ is shown to remain weakly n-type by high temperature Hall effect measurements, with an acceptor energy of 860 meV relative to the Ga₂O₃ conduction band.

See supplementary material for a summary of the relevant equations from the cited references used to fit the temperature dependent carrier density and mobility data.

This material is partially based upon the work supported by the Air Force Office of Scientific Research under Award No. FA9550-18RYCOR098. J.S.S. was also supported by the Defense Threat Reduction Agency through program HDTRA-17-1-0034. The content of the information does not necessarily reflect the position or the policy of the federal government, and no official endorsement should be inferred.