First principles investigation of electron mobility enhancement of β-Ga₂O₃ doped with indium

Abstract

β-Ga₂O₃ is one of the new-generation wide-bandgap semiconductor materials that has attracted much attention in recent years. However, the reported room-temperature electron mobility of β-Ga₂O₃ is much lower than GaN and SiC. Alloying Ga₂O₃ is expected to endow the material with superior carrier transport properties. Herein, we mainly investigate the electron mobility of pure Ga₂O₃, In-doped Ga₂O₃, and Al-doped Ga₂O₃ from first-principles considering acoustic deformation potential (ADP) scattering, polar optical phonon (POP) scattering and ionized impurity (IMP) scattering. The structure optimization, electronic band structure, and temperature-dependent and concentration-dependent electron mobility are investigated. The results show that the mobility of In–Ga₂O₃ is always the highest at 105–650 K, and POP scattering is the dominant factor limiting the electron mobility from 150–650 K. The mobility enhancement by In-doping is attributed to the smaller effective mass caused by the In 5s state despite its slightly increased electron–phonon coupling strength. The predicted electron mobilities for Ga₂O₃, Al–Ga₂O₃ and In–Ga₂O₃ at an electron concentration of 1.0 × 10¹⁷ cm⁻³ are 151.5 cm² V⁻¹ s⁻¹, 137.8 cm² V⁻¹ s⁻¹ and 184.9 cm² V⁻¹ s⁻¹ at room temperature, respectively. This work provides an alternative route to enhance the electron mobility of Ga₂O₃ and guides in engineering their electronic transport properties for high-power electronics.

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1. Introduction

Wide bandgap semiconductor materials, represented by gallium oxide (β-Ga₂O₃),¹ aluminum nitride (AlN),² and diamond,³ have become the most promising materials for the new generation of semiconductors. Their high bandgap (>4.5 eV)⁴ and high breakdown voltage (>1800 V)^5,6 significantly expand the depth and breadth of applications in electronic and optical devices. β-Ga₂O₃ has gained market favor due to its high-quality substrates and large-scale substrate production capabilities,⁷ prompting in-depth exploration of performance optimization by the academic and commercial communities. In semiconductor devices, carrier transport properties are crucial as they determine the electrical performance. However, the reported room-temperature electron mobility of β-Ga₂O₃ remains below 200 cm² V⁻¹ s⁻¹,^8–12 which is 1 to 2 orders of magnitude lower than that of third-generation semiconductors like SiC, GaN, etc.,^13,14 hindering the efficiency and application range of Ga₂O₃ devices. Alloying Ga₂O₃ is expected to endow the material with superior carrier transport properties.

Existing research showed that doping could change the carrier mobility of semiconductors. Considering only acoustic phonon (rather than polar phonon) scattering, Zhang found that by doping with Cu and Sn, the electron mobility of Ga₂O₃ was enhanced significantly from 847 cm² V⁻¹ s⁻¹ to 12 409 cm² V⁻¹ s⁻¹ and 4217 cm² V⁻¹ s⁻¹,¹⁵ respectively. Focusing on a simplified model based on deformation potential scattering, Yu found that the electron mobility of Ga₂O₃ was reduced from intrinsic 283.8 cm² V⁻¹ s⁻¹ to 207.7 cm² V⁻¹ s⁻¹, 212.6 cm² V⁻¹ s⁻¹, and 187.3 cm² V⁻¹ s⁻¹ by doping with Si, Ge and Sn.¹⁶ Typically, the structure cells chosen for first principles calculation contain only several dozen atoms, in which even doping with one atom could result in heavy doping, leading to a band structure similar to that of metals or semimetals, when doping with elements that are not isoelectronic with Ga or O. In such cases, using a semiconductor model to calculate its electrical transport properties might not be accurate. Isoelectronic doping does not present this issue. Li calculated the impact of indium doping on the electrical properties of single-layer Ga₂O₃ metal–oxide–semiconductor field-effect transistors (MOSFETs) and discovered that the on-current for high-performance and low-power achieved great improvement at In doping concentrations of 5% and 20%.¹⁷ Using a chemical vapor deposition method, Tian found that In-doping enhanced the mobility of Ga₂O₃ nanobelts compared to pure Ga₂O₃, thereby improving the sensitivity and responsivity of Ga₂O₃ photodetectors,¹⁸ consistent with Ji's experimental results.¹⁹ Both theoretical and experimental studies have shown that the electron mobility of Ga₂O₃ could be improved by doping with indium. However, the scattering mechanisms in In-doped Ga₂O₃ have not been thoroughly studied yet. Moreover, literature has indicated that Al, which belongs to the same group as In, reduced the mobility when doped in Ga₂O₃.^20,21 It is worth noting that these two elements belonging to the same group contribute to completely different effects on Ga₂O₃. Therefore, this research aims to compare various properties of In-doped Ga₂O₃, pure Ga₂O₃ and Al-doped Ga₂O₃, and then to investigate the reasons behind In-doping enhancing the mobility of Ga₂O₃.

In this work, we emphasize the scattering mechanisms determining the electron mobility of Ga₂O₃, In–Ga₂O₃, and Al–Ga₂O₃, taking acoustic deformation potential (ADP) scattering, polar optical phonon (POP) scattering and ionized impurity (IMP) scattering fully into account. We computed the band structures and density of states for three different structures, as well as the necessary parameters for ADP, POP and IMP scattering. Additionally, we calculated the variations in ADP, POP, and IMP scattering between 105 and 650 K and their respective contributions to the total mobility. Lastly, we investigated the mobility enhancement mechanism of In-doped Ga₂O₃.

2. Computational methodology

There are two kinds of unequal Ga atoms in the lattice structure of β-Ga₂O₃, namely Ga_I with tetrahedral coordination and Ga_II with octahedral coordination, as shown in Fig. 1. We have calculated the doping formation energy of In and Al atoms substituting for Ga_I and Ga_II, from which we could conclude that Al atoms preferentially substitute for Ga_II atoms, while In atoms preferentially substitute for Ga_I atoms. All the DFT calculations were performed in the 1 × 2 × 1 monoclinic Ga₂O₃ supercell using the Vienna ab initio simulation package (VASP) code.^22,23 The generalized gradient approximation (GGA) involving the Perdew–Burke–Ernzerhof (PBE)²⁴ functional was employed to describe the exchange correlation potential. The plane wave cutoff energy was set to 400 eV for all calculations. For the optimization and other property calculations, 3 × 6 × 7 and 4 × 7 × 8 Monkhorst–Pack k-points sampling were applied. These calculations were performed with 1 × 10⁻⁸ eV per atom energy tolerance and 0.01 eV Å⁻¹ force tolerance. To counteract the underestimation on the band gaps of GGA, and reproduce the correct band-structure of β-Ga₂O₃, the Heyd–Scuseria–Ernzerhof (HSE06) screened hybrid functional²⁵ was adopted to acquire the band structure as well, and the mixing parameter of 0.39 was set.

Fig. 1 Crystal structure of the Ga₂O₃ computational model. Green spheres represent cation antimony, and red spheres represent anion oxygen.

Based on the momentum relaxation time approximation, the electron mobility, μ_e, can be computed through the linearized Boltzmann transport equation (BTE),^26–29

where α and β denote Cartesian coordinates, n_e is the electron concentration, Ω and Ω_BZ are the volumes of the unit cell and first Brillouin zone, respectively, v_nk,α and v_nk,β are the group velocity weight of band index n and wave vector k, “cb” stands for conduction bands, f⁰_nk is the Fermi–Dirac distribution function, ε_nk is the electron energy, and τ_nk is the electron lifetime. Based on the self-energy relaxation time approximation (SERTA),²⁷ the electron lifetimes can be obtained according towhere τ_nk→mk+q⁻¹ is the scattering rate of the transition from state |nk〉 to state |mk + q〉. Two classes of scattering are implemented in the work: (i) inelastic scattering which occurs via emission or absorption of a phonon and (ii) perfectly elastic scattering in which no energy is gained or lost. In the case of inelastic scattering, the partial decay rate can be written as^30,31where ħ is the reduced Planck constant, n_q is the Bose–Einstein occupation, δ is the Dirac delta function, Δε^nm_k,q = ε_nk − ε_mk+q. |g_nm(k,q)|² represents the matrix element of the interaction, which measures the probability amplitude of the transition from an initial state nk to final state mk + qvia a phonon with wave vector q and frequency ω_q. The −ħω_q and +ħω_q terms correspond to scattering by emission and absorption of a phonon, respectively. The scattering mechanism of POP belongs to inelastic scattering.³²

For elastic scattering, the partial decay rate can be written as

The scattering mechanisms of IMP^33,34 and ADP^35–38 belong to elastic scattering. The matrix elements for ADP, IMP and POP are calculated, respectively. Before the total mobility can be obtained, the basic parameters are calculated firstly. These include the ADP matrix parameters of the fully anisotropic acoustic deformation potential and elastic constants, POP matrix parameters including static (ε_s), high-frequency (ε_∞) dielectric constants and effective phonon frequency (ω_po), etc. Through Fourier interpolation in the first Brillouin zone to compute electron mobility by AMSET,²⁹ we can accurately evaluate the contribution of individual scattering mechanisms on a much-denser mesh of dimensions 13 × 25 × 27 after convergence tests.

3. Results and discussion

3.1. Electronic band structure

The lattice parameters of Ga₂O₃, Al–Ga₂O₃ and In–Ga₂O₃ are listed in Table 1. The calculated Ga₂O₃ lattice parameter results are in good agreement with the experimental data³⁹ (deviation less than 1.8%) and previous calculations,⁴⁰ demonstrating reliability of the employed theoretical methodology. Al–Ga₂O₃ has a slightly smaller lattice parameter, while In–Ga₂O₃ has a slightly larger lattice parameter, as Al atoms are smaller than Ga atoms while In atoms are larger than Ga atoms.

Table 1 Lattice constants of β-Ga₂O₃ compared with literature data

Lattice constant	A	b	c	β
Ga2O3-expt.^39	12.23	3.04	5.80	103.7
Ga2O3-theory^40	12.45	3.08	5.87	103.7
Ga2O3	12.32	3.07	5.70	103.69
Al–Ga2O3	12.30	3.07	5.68	103.64
In–Ga2O3	12.40	3.10	5.73	103.61

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To explore the differences in energy band, electronic occupancy states, and effective electron mass between pure Ga₂O₃ and doped Ga₂O₃, we performed calculations of the band structure, total density of states (TDOS), and partial density of states (PDOS). Fig. 2(a)–(c) present the electronic band structures of Ga₂O₃, Al–Ga₂O₃ and In–Ga₂O₃ with a Brillouin zone path of G–F–Q–Z–G calculated using PBE (orange lines) and HSE (blue lines), respectively. This reveals that the electronic band structures predicted using PBE and HSE demonstrate the same dispersion shape, except for the bandgap values, which means that they have the equivalent effective mass closely related to the curvature of the band at the high symmetry point. The band gap of Ga₂O₃ (Fig. 2a) obtained using HSE and PBE is 4.91 and 2.05 eV respectively. The gap between the bandgap calculated by HSE and the experimental results of 4.86 eV⁴¹ is very small. For Al–Ga₂O₃ (Fig. 2b), the band gap is 5.05 and 2.18 eV, respectively, while for In–Ga₂O₃ (Fig. 2c), the band gap is 4.77 and 1.93 eV, respectively. As HSE requires heavy computational resources, PBE is chosen to predict the carrier transport properties of these three Ga₂O₃ structures.

Fig. 2 Electronic band structures of pure Ga₂O₃, Al–Ga₂O₃, and In–Ga₂O₃ calculated using PBE (orange lines) and HSE (blue lines) functionals.

In this study, we only focus on the electron mobility. From the band diagram, it can be seen that the conduction band minimum (CBM) is located at the gamma point. The electron effective mass is defined as:

where x, y, and z are the directions in the reciprocal space (2π/a), E_n(k⃑) is the dispersion relationship for the n-th electronic band. Using the effective mass calculator (EMC)⁴² and finite difference method, we have calculated the effective mass at CBM, as shown in Table 2. The effective masses of pure Ga₂O₃ along x, y, and z are 0.22, 0.23, and 0.21, which are close to the values reported in the literature of 0.22, 0.22, and 0.21.⁴⁰ The electron effective mass distributions reveal that the electron effective mass is basically isotropic, consistent with the subsequent mobility calculations. The effective masses of Al–Ga₂O₃ are 0.24, 0.24, and 0.23, slightly higher than pure Ga₂O₃, which is unfavorable for carrier transport; meanwhile the effective masses of In–Ga₂O₃ are 0.21, 0.22, and 0.21, slightly lower than pure Ga₂O₃, which is beneficial for carrier transport.

Table 2 Electron effective mass of Ga₂O₃, Al–Ga₂O₃ and In–Ga₂O₃ compared with literature data

Structure	Electron effective mass
Ga2O3^40	0.22	0.22	0.21
Ga2O3	0.22	0.23	0.21
Al–Ga2O3	0.24	0.24	0.23
In–Ga2O3	0.21	0.22	0.21

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The total density of states (TDOS) curves for pure Ga₂O₃, Al–Ga₂O₃, and In–Ga₂O₃ are similar, as shown in Fig. 3. Near the valence band maximum (VBM), the states are mainly contributed by O 2p and Ga 3p states. The O 2p states are localized below VBM, leading to a flat valence band, bringing about large hole effective mass. The DOS near the CBM is very low, resulting in a steep conduction band, meaning Ga₂O₃ has small electron effective mass. Al 3s and Al 3p states have minimal contribution to CBM as shown in Fig. 3(b) and (d). In contrast, In–Ga₂O₃ shows slightly higher In 5s states near the CBM (3.5–5 eV), resulting in a higher TDOS, as shown in Fig. 3(c) and (e). However, in the region that determines the electron effective mass at the CBM (1.93–2.05 eV for In–Ga₂O₃, 2.06–2.15 eV for Ga₂O₃, circled in red), the TDOS of In–Ga₂O₃ is the lowest caused main by the In 5s state, with less localization and a steeper CBM, resulting in a lower electron effective mass. The detailed TDOS of Ga₂O₃ and In–Ga₂O₃ near the CBM is shown in Table S1 (ESI†).

Fig. 3 TDOS of pure Ga₂O₃ (a), Al–Ga₂O₃ (b), and In–Ga₂O₃ (c); PDOS of Al in Al–Ga₂O₃ (d) and PDOS of In in In–Ga₂O₃ (e).

3.2. Electrical transport properties caused by ADP

Based on the electronic band structure, the scattering rate from ADP, POP, and IMP and the electron mobility could be calculated. Piezoelectric scattering was not considered as its proportion is very small. The electron mobility is calculated at moderate electron concentrations (1 × 10¹⁷ cm⁻³). The methodology is introduced firstly for computing the transport properties, which enable us to discuss various factors influencing the electron mobility.

The ADP matrix element is given by²⁹

where D̃_nK is a deformation potential tensor that describes how the electronic states respond to lattice vibrations, Ŝ_l, Ŝ_t1 and Ŝ_t2 represent the unit polarization vectors for the longitudinal acoustic mode (l) and two transverse acoustic modes (t1, and t2), respectively, and ρ represents the density of the material. The amplitude of the strain at any temperature T can be obtained from the potential energy of the acoustic phonon as , where k_B is the Boltzmann constant. 〈mk + q|e^i(q+G)·r|nk〉 is the overlap integral between state |nk〉 and state |mk + q〉, where the factor e^i(q+G)·r accounts for the phase factor associated with the phonon wavevector and reciprocal lattice vector.

The elastic constants (C) of pure Ga₂O₃ calculated in this study are 288.4 GPa, 412.6 GPa, and 401.7 GPa, slightly higher than the experimental values⁴³, as shown in Table 3. The deformation potential (D_A) and elastic constants of In–Ga₂O₃ are the smallest, while those of Al–Ga₂O₃ are the largest, with pure Ga₂O₃ falling in the middle. The value of D_A/C of In–Ga₂O₃ is the smallest, while in Al–Ga₂O₃ it is the largest. According to formula (6), g^ad_nm is proportional to D_A/C, indicating that In–Ga₂O₃ might have the smallest g^ad_nm. Detailed matrix of elastic constants and deformation potential are listed in the ESI.†

Table 3 Elastic constants and deformation potential of Ga₂O₃, Al–Ga₂O₃ and In–Ga₂O₃ compared with literature data

	C (GPa)			D A (eV)
	A	b	c	a	b	c
Ga2O3-expt^43	242.8	343.8	347.4	—	—	—
Ga2O3	288.4	412.6	401.7	3.15	2.16	3.27
Al–Ga2O3	295.8	419.0	406.5	3.29	2.36	3.30
In–Ga2O3	281.8	403.9	385.2	2.88	2.17	2.93

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Fig. 4a shows the temperature-dependent anisotropic mobility caused by only ADP scattering. The calculated mobilities differ by about 30% for the different directions, equivalent to the values observed in ref. 44. Besides, the mobility caused by ADP decreases with increasing temperature. According to formula (6), the scattering caused by deformation potential is positively correlated with temperature; as the temperature rises, deformation potential scattering increases, leading to a decrease in mobility. Fig. 4b shows the isotropically averaged mobility variation of Ga₂O₃, Al–Ga₂O₃, and In–Ga₂O₃ with temperature. The mobility of In–Ga₂O₃ is the highest when only ADP scattering is considered.

Fig. 4 Temperature-dependent electron mobility of Ga₂O₃ along x, y, and z directions (a) and the isotropically averaged mobility variation of Ga₂O₃, Al–Ga₂O₃ and In–Ga₂O₃ (b) when only ADP scattering is considered.

3.3. Electrical transport properties caused by POP

The POP differential scattering rate is given by:where ħω_po represents the frequency of the polar optical phonon mode, n̂ is a unit vector describing the direction of the phonon wavevector, while ε_∞ and ε_s are the high-frequency and static dielectric tensors of the material, respectively. The expression characterizes the difference in dielectric screening between high and low frequencies, influencing the strength of the long-range Coulomb interactions between electrons and phonons in polar materials.

In polar semiconductors, the charge carriers and the surrounding lattice deformation form a so-called polaron,⁴⁵ which determines the nature of carrier transport. Polarons can be classified into small and large ones based on the strength of electron phonon coupling. The lattice deformation in a large polaron is usually moderate and spreads over multiple unit cells, resulting in a larger mobility. The electron–phonon interaction is usually dominated by the coupling of charge carriers to the longitudinal optical phonons, which can be described within the Fröhlich model.⁴⁶ The coupling strength between electrons and polar optical phonons is quantified using the Fröhlich coupling constants^32,47

where is the CBM electron effective mass, and ε₀ is the vacuum permittivity. The above relevant material parameters of Ga₂O₃ are listed in Tables 2 and 5. The calculated α (shown in Table 4) shows the averaged value of 1.16, 1.25 and 1.17 for Ga₂O₃, Al–Ga₂O₃ and In–Ga₂O₃, which falls in the intermediate electron–phonon coupling regime (defined as 0.5 ≤ α ≤ 6). The magnitudes of α for Ga₂O₃ and In–Ga₂O₃ are quite close, suggesting similar electron–phonon interaction strengths. We further estimate the size of polarons in Ga₂O₃, Al–Ga₂O₃ and In–Ga₂O₃, by the Schultz polaron radius (r_f).⁴⁷ The large values of electron polaron radius (which extend over multiple structural units) indicate that the polarons are delocalized in these Ga₂O₃. Al–Ga₂O₃ has the smallest Schultz Polaron radius while Ga₂O₃ has the largest one, suggesting that the Fröhlich coupling strength of In–Ga₂O₃ is slightly larger than Ga₂O₃, and smaller than that of Al–Ga₂O₃.

Table 4 Calculated Fröhlich parameter (α) and Schultz polaron radius (r_f) for electrons in Ga₂O₃, Al–Ga₂O₃, and In–Ga₂O₃ at T = 300 K

	α			r f (Å)
	Ga2O3	Al–Ga2O3	In–Ga2O3	Ga2O3	Al–Ga2O3	In–Ga2O3
Average	1.16	1.25	1.17	145.8	138.1	142.3

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g ^po _nm (k, q) is proportional to the frequency of the polar optical phonon mode, ω_po, and inversely proportional to the electronic high-frequency dielectric constant, ε_∞. As ε_∞ is much smaller than ε_s, the value of is almost equal to .

For purposes of the screening, we assume that the high-frequency dielectric constant is isotropic. Indeed, our DFPT calculations (see Table 5) show that the high-frequency part of the dielectric tensor is nearly isotropic, with values ranging between 3.98 and 4.10 depending on the direction, close to the theoretical value.^11,12 In contrast, the static dielectric tensor (ε_s) shows a stronger directional dependence, which is attributed to the anisotropic ionic contributions to the dielectric tensor in the monoclinic lattice. An averaged value of the computed ε_∞ (4.05) is used for calculating g^po_nm(k, q) of Ga₂O₃. This value is close to the experimental value of ε_∞ = 3.60.⁴⁸ Besides, it is obvious that the ε_∞ of In–Ga₂O₃ is the largest, and the ω_po of In–Ga₂O₃ is the smallest, indicating g^po_nm of In–Ga₂O₃ might be the smallest among these three structures according to formula (7).

Table 5 The parameters of POP scattering of β-Ga₂O₃ compared with literature data

Materials	ε ∞				ε s				ω po (THz)
	a	b	c	Average	a	b	c	Average
Ga2O3^11	4.07	4.14	4.19	4.13	10.99	10.50	13.91	11.80	12.08, 8.46–11.60^49
Ga2O3^12	4.02	4.03	4.15	4.10	11.88	9.22	12.61	11.24
Ga2O3	3.98	4.06	4.10	4.05	10.20	9.55	13.71	11.15	13.12
Al–Ga2O3	3.91	3.99	4.00	3.97	10.09	9.48	13.04	10.87	12.75
In–Ga2O3	4.01	4.10	4.12	4.08	10.32	9.49	13.90	11.24	12.71

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We further analyze the individual contribution of POP scattering, as shown in Fig. 5. The mobility of Ga₂O₃, Al–Ga₂O₃, and In–Ga₂O₃ exhibits similar variation curves with temperature, in which the mobility of In–Ga₂O₃ by POP individual scattering is always the highest. μ_POP reduces monotonically as the temperature increases. Since as temperature further increases, more polar optical phonons are excited which can induce a macroscopic electric field, thus strengthening the electron–phonon coupling.

Fig. 5 Temperature-dependent electron mobility of Ga₂O₃, Al-doped Ga₂O₃ and In-doped Ga₂O₃, where only POP scattering is considered.

3.4. Electrical transport properties caused by IMP

The ionized impurity matrix element is given bywhere n_ii is the ionized impurity density, representing the concentration of the electron that contributes to the interaction. Ze is the charge of the ionized impurity, where Z is the atomic number of the impurity (or the number of charges), and e is the elementary charge. The factor n̂·ε_s·n̂ modulates the strength of the Coulomb interaction due to screening effects. The denominator |q + G|² + β² accounts for the momentum transfer q + G during scattering, with β representing a screening length parameter that modulates the range of the interaction.

As shown in Fig. 6a, the mobility due to IMP slightly increases with rising temperature because, as the temperature increases, the kinetic energy of electrons increases, causing them to move faster and spend less time near recombination centers, thereby reducing the probability of scattering. Besides, our research has shown that the mobility of In–Ga₂O₃ is always higher than that of pure Ga₂O₃ and Al–Ga₂O₃. This might be partly attributed to the larger ε_s of In–Ga₂O₃.

Fig. 6 Temperature-dependent electron mobility of Ga₂O₃ at electron concentrations (1 × 10¹⁷ cm⁻³) where only IMP scattering is considered (a). Temperature-dependent electron mobility of Ga₂O₃ at various electron concentrations where ADP, IMP and POP scattering are considered together (b).

The electron concentration is varied from 10¹³ cm⁻³ to 10¹⁸ cm⁻³ to explore the concentration dependent electron mobility of Ga₂O₃, Al–Ga₂O₃ and In–Ga₂O₃, respectively, as shown in Fig. 6b. With electron concentration increasing from 10¹³ to 10¹⁶ cm⁻³, the electron mobility of In–Ga₂O₃ slightly drops from 228.9 cm² V⁻¹ s⁻¹ to 211.7 cm² V⁻¹ s⁻¹ at 300 K, respectively, which indicates the electron mobility of In–Ga₂O₃ remains essentially stable. After that, a further increase in electron concentration (from 10¹⁶ to 10¹⁸ cm⁻³) rapidly deteriorates electron mobility, from 211.7 cm² V⁻¹ s⁻¹ to 138.4 cm² V⁻¹ s⁻¹ at 300 K. This is caused by the strong scattering of excessive carrier concentration with electrons and phonons. According to formula (9), the scattering intensity is positively correlated with impurity concentration n_ii: the higher the impurity (electron) concentration, the greater the electron scattering intensity, which hinders movement and decreases mobility.

Based on Fig. 4b, 5 and 6a, in the temperature range of 105–150 K, μ_IMP is the lowest, indicating IMP scattering dominates the electron transport. In the temperature range of 150–650 K, μ_ADP and μ_IMP are much higher than μ_POP, indicating that POP scattering dominates the carrier transport of these three Ga₂O₃. Fig. 7 shows the effect of temperature on overall mobility, considering ADP, IMP, and POP together. Following Matthiessen's rule,⁵⁰ the total mobility μ_total is determined from μ_total⁻¹ = μ_ADP⁻¹ + μ_IMP⁻¹ + μ_POP⁻¹. The total mobility is primarily determined by the smallest partial mobility. Overall, ADP has little effect on the total electron mobility. At low temperatures, impurity scattering significantly impacts the total electron mobility; as the temperature increases, optical phonon scattering gradually becomes the dominant scattering mechanism. The magenta curve in Fig. 7 shows the experimentally measured Hall mobility of pure Ga₂O₃,⁹ while the black curve represents the mobility of pure Ga₂O₃ calculated in this study. The high degree of overlap between these two curves indicates the reliability of the calculation method employed in this research. The overall mobility of In–Ga₂O₃ (blue curve) is the highest from 105 to 650 K, and the electron mobilities of Ga₂O₃, Al–Ga₂O₃ and In–Ga₂O₃ are 151.5 cm² V⁻¹ s⁻¹, 137.8 cm² V⁻¹ s⁻¹ and 184.9 cm² V⁻¹ s⁻¹ at 300 K. Since the effective mass of In–Ga₂O₃ is smaller than that of Ga₂O₃ and Al–Ga₂O₃, it still possesses higher mobility despite its slightly larger electron–phonon coupling strength.

Fig. 7 Temperature-dependent electron mobility of Ga₂O₃, Al-doped Ga₂O₃ and In-doped Ga₂O₃, where ADP, IMP and POP scattering are considered together.

4. Conclusions

To summarize, we have investigated the temperature-dependent electron mobility of Ga₂O₃, Al–Ga₂O₃ and In–Ga₂O₃ from first-principles by fully considering the scattering mechanisms of ADP, IMP, and POP. The results show that POP scattering is the dominant factor limiting the electron mobility of Ga₂O₃, Al–Ga₂O₃ and In–Ga₂O₃ between 150 and 650 K. Combining these three scattering mechanisms, the electron mobilities of Ga₂O₃, Al–Ga₂O₃ and In–Ga₂O₃ are 151.5 cm² V⁻¹ s⁻¹, 137.8 cm² V⁻¹ s⁻¹ and 184.9 cm² V⁻¹ s⁻¹ at 300 K. Furthermore, as temperature increases, the electron mobility of three Ga₂O₃ reduces due to more excited optical phonons, thus enhancing the electron–phonon coupling. With electron concentration increasing, the electron mobility of three Ga₂O₃ reduces due to more scattering probabilities. The electron mobility of In–Ga₂O₃ is the highest among these three structures for the decreased effective mass caused by the In 5s state. This work offers an alternative method to boost the electron mobility of Ga₂O₃ and serves as a guide for engineering the electronic transport properties tailored for high-power electronic applications.

Author contributions

L. Z.: conceptualization, software, writing – original draft. J. H.: validation. Y. S.: writing – review & editing. F. L. and P. Z: formal analysis, visualization. D. W.: visualization. K. W.: writing – review & editing, supervision. Y. W.: writing – original draft, supervision.

Data availability

The data that support the findings of this study are available within the article and the ESI.†

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work is financially supported by the National Natural Science Foundation of China (Grant No. 61904071), the QingLan Project of Jiangsu Provincial University of China and the Science Foundation of Jinling Institute of Technology (jit-rcyj-202001). We are thankful for the technology support from the National Supercomputer Center (TianHe-2) in Lvliang.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4cp04220d

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