Demonstration of high mobility and quantum transport in modulation-doped β-(Al_xGa_1-x)₂O₃/Ga₂O₃ heterostructures

Beta-phase gallium oxide (β-Ga₂O₃) is a promising candidate for electronic device applications because of the large bandgap energy (4.7 eV) and the expected high breakdown field of 8 MV/cm.¹ Significantly, β-Ga₂O₃ is the first wide bandgap material that can be grown from the melt, which makes it feasible to achieve large area bulk substrates on a manufacturable scale.^2–6 The expected good transport properties (a mobility of >200 cm²/Vs, and a saturation velocity of ∼2 × 10⁷ cm/s)^1,7–9 make β-Ga₂O₃ very promising for a range of technological applications including high power electronics, detectors, and high frequency transistors. Further bandgap tunability can be realized through the introduction of In and Al into β-Ga₂O₃, leading to β-(In,Ga)₂O₃ and β-(Al,Ga)₂O₃ alloys. β-(Al,Ga)₂O₃ is expected to be stable in the monoclinic phase over a broad range of compositions, and the bandgap can be tuned from the bandgap of β-Ga₂O₃ (4.7 eV) to above 6 eV.^10,11 This enables the realization of several semiconductor heterostructure designs such as modulation-doped electron channels, quantum wells, and superlattices in this semiconductor system.

Recent efforts have led to the demonstration of various device structures with excellent performance, including Schottky diodes,^12,13 metal-oxide-semiconductor field effect transistors (MOSFETs),^1,14–21 and metal-semiconductor field effect transistors (MESFETs).¹ Experimental observations of high breakdown fields above 5 MV/cm for vertical Schottky diodes¹² and 3.8 MV/cm for lateral MOSFET transistors have been reported,¹⁶ which have already surpassed the material limit for GaN and SiC. While excellent device performance has been demonstrated using homoepitaxial β-Ga₂O₃ device structures, the β-(Al_xGa_1-x)₂O₃/Ga₂O₃ heterojunctions have remained under-explored.

Preliminary demonstration of modulation-doped field effect transistors (MODFETs) using β-(Al_xGa_1-x)₂O₃/Ga₂O₃ heterostructures has been reported.^22–24 Sheet charge densities above 5 × 10¹² cm⁻² were measured based on either Si-delta doping²² or Ge doping²³ in the β-(Al_xGa_1-x)₂O₃ layer, but the presence of parallel conduction through the low mobility channel in the (Al_xGa_1-x)₂O₃ layer compromised the transport properties of the two-dimensional electron gas (2DEG). In this study, we show a direct evidence of a quantum confinement of electrons at the β-(Al_xGa_1-x)₂O₃/Ga₂O₃ interface based on temperature-dependent Hall measurements and Shubnikov-de Haas (SdH) oscillations. We demonstrate the room temperature mobility of 180 cm²/Vs and the low temperature mobility of 2790 cm²/Vs achieved using the modulation-doped structure.

Figure 1(a) shows the β-(Al_xGa_1-x)₂O₃/Ga₂O₃ MODFET structure studied in this work. The samples were grown on a (010)-oriented Fe-doped semi-insulating β-Ga₂O₃ substrate using oxygen plasma-assisted molecular beam epitaxy (PA-MBE).^25,26 It consists of an unintentionally doped (UID) β-Ga₂O₃ buffer layer, a 4.5 nm β-(Al_xGa_1-x)₂O₃ spacer, a Si delta-doped layer, and a 22.5 nm β-(Al_xGa_1-x)₂O₃ cap layer. The growth details could be found in the supplementary material. Two samples were compared in this study, with the only difference being the UID β-Ga₂O₃ buffer layer thickness, which is 130 nm and 360 nm for samples A and B, respectively. The (Al_xGa_1-x)₂O₃ layer thickness and the Al composition were estimated to be 27 nm and 18%, respectively, for both samples, based on high resolution XRD measurements of the (020) diffraction [Fig. 1(b)].²⁷ The observed diffraction fringes indicate sharp heterointerfaces. Both samples showed smooth surfaces with a RMS roughness of ∼0.45 nm obtained from AFM measurements as shown in Fig. S1.

The energy band diagram of the MODFET structure was obtained based on a self-consistent solution of the Schrodinger-Poisson equation assuming a conduction band offset (ΔE_C) of 0.4 eV, a surface depletion barrier of 1.4 eV, and a back-depletion due to the Fe-doped semi-insulating substrate (assuming the Fermi level pinned at the midgap), as shown in Fig. 1(c). A 2DEG is expected to form at the (Al_xGa_1-x)₂O₃/Ga₂O₃ interface. When a donor concentration of 4.7 × 10¹² cm⁻² was adopted in the delta-doped layer, the simulated 2DEG density increased from 1.12 × 10¹² cm⁻² to 1.50 × 10¹² cm⁻² as the buffer layer was increased from 130 nm to 360 nm because of a corresponding reduction of backside depletion. It should be noted that the parameters adopted for the band simulations are based on the best estimates we have currently and do provide a self-consistent description of the system. Detailed studies using independent sample series may be necessary to extract these parameters accurately.

To achieve ohmic contact to the channel, the contact regrowth of n⁺⁺ Ga₂O₃ with a high Si doping concentration above 10²⁰ cm⁻³ was carried out using SiO₂ as a regrowth mask. The contact regrowth and device fabrication processes are described in the supplementary material. The highly degenerate doping prevented carrier freeze-out at cryogenic temperatures and ensured the reliability of the low temperature electrical measurements. Ohmic contacts were verified by transfer length measurements, with extracted contact resistances of 9.3 Ω mm for sample A and 4.1 Ω mm for sample B, which were limited by the sidewall contacts between the regrown contacts and the low charge density 2DEG channel.

Temperature-dependent Hall measurements were carried out using a van der Pauw structure with regrown n⁺⁺ Ga₂O₃ contacts, as shown in Fig. 2. Both samples showed a weak temperature dependence in the measured carrier density, which dropped from 1.12 × 10¹² cm⁻² to 1.07 × 10¹² cm⁻² in sample A and from 2.25 × 10¹² cm⁻² to 2.05 × 10¹² cm⁻² in sample B upon lowering the temperature. This is in contrast to carrier freeze-out in bulk-doped β-Ga₂O₃ at low temperatures²⁸ and serves as a direct proof of a degenerate electron gas with no parallel conduction in the (Al_xGa_1-x)₂O₃ barrier layer. Using a higher Si sheet density in the delta-doped layer led to partial freeze-out of charge at low temperatures, which could be attributed to a parallel conduction channel in the barrier layer. The maximum charge density that can be confined in the adopted (Al_0.18Ga_0.82)₂O₃/Ga₂O₃ MODFET structure with a 4.5 nm spacer without introducing a parasitic channel is estimated to be approximately 2 × 10¹² cm⁻². A further increase in the 2DEG charge density requires higher conduction band offset or using a relatively thinner spacer layer.

The room temperature mobility was measured to be 162 cm²/Vs and 180 cm²/Vs for samples A and B, respectively. Both samples showed a sharp increase in the Hall mobility with decreasing temperature, with a peak mobility of 990 cm²/Vs at 60 K for sample A and 2790 cm²/Vs at 50 K for sample B. These are significantly higher than the highest reported mobility values^28,29 obtained in bulk-doped β-Ga₂O₃, an expected benefit of the spatial separation between the impurities and the modulation-doped 2DEG channel. Their mobility values dropped off slightly upon further lowering of the measurement temperature. A similar phenomenon has been observed in early works on modulation-doped AlGaAs/GaAs transistors³⁰ and was attributed to impurity scattering. To understand the scattering mechanisms that cause differences between the samples, the temperature dependence of electron mobility was analyzed by considering various scattering mechanisms (discussed in the supplementary material).

The measured and calculated mobility results are shown in Fig. 2. At low temperature, the remote impurity scattering limited mobility is estimated to be significantly above the measured values, indicating that remote impurity scattering is not a limiting mechanism for the studied MODFET structure. To fit the measured data, the interface roughness and the background impurity density were adjusted in the calculations. The vertical/lateral displacement of the interface was assumed to be 0.45 nm/4.7 nm for both samples for the best fittings of the measured data. The extracted effective charged impurity density was estimated to be 1.2 × 10¹⁸ cm⁻³ in sample A, while it is 1.5 × 10¹⁷ cm⁻³ in sample B. The reduction of the background impurity density in sample B contributed to the notable mobility increase at low temperatures, and it is attributed to less impurity (such as Fe) diffusion from the substrate surface due to a thicker buffer layer growth. The residual background charge could also have contributions from native defects formed during the MBE growth,⁸ such as Ga vacancies (V_Ga), which are expected to be deep acceptors.³¹ We note here that these estimates are based on monovalent charged impurities. Since Ga vacancies can be charged up to the 3+ state, the true background defect density (as opposed to effective defect density) could be lower than 1.5 × 10¹⁷ cm⁻³.

Even though an apparent increase in the low temperature mobility was achieved by increasing the buffer layer thickness, the electron scattering is dominated by polar optical phonon scattering in the high temperature range,^8,9,32 leading to similar mobility values at room temperature for both samples. While longitudinal optical-plasmon coupling could lead to better screening of the optical phonon scattering and therefore higher electron mobilities at room temperature,^8,9 this effect is not significant in the low carrier density range below 2 × 10¹² cm⁻² and was therefore not considered in the mobility calculations. Increasing the 2DEG density is necessary to take advance of the screening effects.

The high channel mobility at low temperature allowed for the measurement of Shubnikov-de Haas oscillations of the transverse magnetoresistance (R_xx) with varied magnetic fields perpendicular to the sample surface. Both samples showed negative magnetoresistance at low magnetic fields (Fig. S2), attributed to weak localization,^33–35 with SdH oscillations developing above four Tesla. Only one period of oscillation was observed for sample A in the magnetic field range below 14 T (Fig. S2), while multiple oscillations developed below 7 T for sample B at varied measurement temperatures benefitting from the higher channel mobility. The oscillation components of R_xx of sample B, with the background subtracted, are shown in Fig. 3(a) as a function of the reciprocal magnetic field (1/B). The 2DEG concentration can be estimated based on the period of Δ(1/B) through the expression Δ(1/B) = e/πℏn_2D.³³ The determined 2DEG densities from SdH oscillation are 1.15 × 10¹² cm⁻² and 1.96 × 10¹² cm⁻² for samples A and B, respectively, which are consistent with low-field Hall measurements.

The angular dependence of the SdH oscillations for sample B was also measured at 3.5 K by tilting the sample surface normal away from the magnetic field direction as shown in Figs. 3(b) and 3(c). Oscillations were observed at tilt angles lower than 60°, and the extracted oscillation periods of Δ(1/B) fell in a linear dependence of cos(θ), indicating that the oscillations only depend on the perpendicular component of the applied magnetic field. This unique feature reinforces the 2DEG nature of the channel formed at the hetero-interface. Notably, spin splitting was observed at low tilt angles (θ) when the magnetic field was above 10 T. The Hall resistivity (R_xy) also exhibited the onset of plateaus (Fig. S3), which corresponds to the quantization of Landau levels. These features enable the further analysis of the quantum phenomena in Ga₂O₃.

The obtained SdH oscillations in sample B make it possible to analyze both the effective mass of the 2DEG and the quantum scattering time (τ_q) in β-Ga₂O₃. The temperature dependence of the SdH oscillation amplitude (A) at a fixed magnetic field (B) can be used to determine the effective mass of the 2DEG. The oscillation amplitude is related to the measurement temperature (T) by³⁶ 

where k_B is the Boltzmann constant and C₁ is a temperature-independent constant at a fixed magnetic field. The dependence of ln(A/T) on temperature and the fittings for the effective mass are plotted in Fig. 4(a) at three magnetic fields. The electron effective mass is extracted to be m* = 0.313 ± 0.015 m₀. While the electron transport is confined in the (010) plane, near isotropic effective mass was predicted³⁷ for β-Ga₂O₃, and the extracted effective mass is in close agreement with the theoretical calculations,³⁷ band structure measurements,^38,39 and optical Hall measurements.⁴⁰ 

Following the estimation of the effective mass, the quantum scattering time (τ_q) can be evaluated using a Dingle plot.^41,42 At a fixed temperature, the oscillation amplitude is related to the inverse of the magnetic field by^41,42

where $χ=2 π 2 m * k BT/ℏeB$⁠, R₀ is the zero field resistance, and C₂ is a constant that is independent of the magnetic field at a given temperature. Figure 4(b) shows the dependence of ln[(A/4R₀)(sinh(χ)/χ)] on 1/B extracted from the SdH oscillation at 3.5 K. Linear fitting of the experimental data gives a quantum scattering time of 0.33 ps. In comparison, the transport lifetime (τ_t) was estimated to be 0.44 ps from the low field Hall mobility (τ_t = m*μ/e) using the extracted effective mass of 0.313m₀. The ratio between the transport lifetime and quantum scattering time is therefore τ_t/τ_q ∼ 1.3. This is close to unity and indicates that the electron scattering events are dominated by large angle scatterings, such as interface roughness scattering or background impurity scattering,⁴³ in agreement with the mobility calculations shown in Fig. 2(c).

To demonstrate the feasibility of device applications, modulation-doped field effect transistors were fabricated using a Pt/Au (=30/130 nm) metal stack to define the Schottky gate contact for sample B. The output and transfer characteristics of three-terminal transistors are shown in Fig. 5. A maximum drain current of I_DS = 46 mA/mm was obtained at a V_DS of 10 V and a V_GS of 2 V. The transconductance (g_m) showed a peak of 39 mS/mm and dropped off at higher gate bias, which we attribute to the decrease in modulation efficiency due to charge transfer into the barrier layer.⁴⁴ I_DS showed above 9 orders of magnitude rectification, and the subthreshold slope is estimated to be 91 mV/decade. The extracted threshold voltage is 0.5 V, corresponding to normally-off operation under the Pt-gate. High frequency small-signal measurements on this device showed a cutoff frequency of 3.1 GHz and the maximum oscillation frequency of 13.1 GHz at a V_DS of 10 V and a V_GS of +1.5 V.

In summary, the formation of a high mobility 2DEG channel was achieved using modulation doping in a (010)-oriented β-(Al_xGa_1-x)₂O₃/Ga₂O₃ heterostructure with Si delta-doping in the barrier layer. The temperature dependent Hall measurement showed nearly constant charge density in the temperature range of 5 K–300 K. Both the room temperature mobility of 180 cm²/Vs and the low temperature peak mobility of 2790 cm²/Vs exceeded the highest experimental mobility values for bulk β-Ga₂O₃. This is attributed to the spatial separation between ionized impurities and the 2DEG. The high mobility values allowed for the observations of the SdH oscillations at cryogenic temperatures. The effective mass in the (010) plane is estimated to be m* = 0.313 ± 0.015 m₀ based on the SdH oscillations. The demonstration of a high-quality heterojunction and quantum transport in the β-(Al_xGa_1-x)₂O₃/Ga₂O₃ wide bandgap semiconductor system reported here lays the foundation for future investigation of the materials science, physics, and device applications of the monoclinic β-(Al_xGa_1-x)₂O₃ semiconductor system.

See supplementary material for more information about the material growth, device fabrication, mobility calculations, and the original data on SdH oscillations.

We acknowledge funding from the Office of Naval Research under Grant No. N00014-12-1-0976 (EXEDE MURI). The project or effort depicted was or is sponsored by the Department of the Defense, Defense Threat Reduction Agency (Grant No. HDTRA11710034). This material is partially based upon the work supported by the Air Force Office of Scientific Research under Award No. FA9550-18RYCOR098. 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. We acknowledge funding from The Ohio State University, Institute of Materials Research (IMR), Multidisciplinary Team Building Grant. We thank Dr. Susanne Stemmer for helpful discussions.