Low-energy Ga₂O₃ polymorphs with low electron effective masses

Abstract

We predict three Ga₂O₃ polymorphs with P2₁/c or Pnma symmetry. The formation energies of P2₁/c Ga₂O₃, Pnma-I Ga₂O₃, and Pnma-II Ga₂O₃ are 57 meV per atom, 51 meV per atom, and 23 meV per atom higher than that of β-Ga₂O₃, respectively. All the polymorphs are shown to be dynamically and mechanically stable. P2₁/c Ga₂O₃ is a quasi-direct wide band gap semiconductor (3.83 eV), while Pnma-I Ga₂O₃ and Pnma-II Ga₂O₃ are direct wide band gap semiconductors (3.60 eV and 3.70 eV, respectively). Simulated X-ray diffraction patterns are provided for experimental confirmation of the predicted structures. The polymorphs turn out to provide low electron effective masses, which is of great benefit to high-power electronic devices.

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Introduction

Wide band gap semiconductors are important for devices requiring high frequency, temperature, and/or power.¹ Many commercial applications use the third generation semiconductors GaN and SiC. However, those suffer from high manufacturing costs and limitations of the achievable performance.^2,3

Gallium oxide, especially thermodynamically stable β-Ga₂O₃, attracts more and more attention, because its high breakdown field leads to a higher Baliga figure of merit for application in high-power electronic devices than provided by SiC and GaN.⁴ The most famous polymorphs of Ga₂O₃ are the α-, β-, γ-, δ-, and ε-phases,^5,6 which are similar in properties such as the band gap, elastic modulus, and electron effective mass.⁷ The hexagonal structure of ε-Ga₂O₃ with space group P6₃mc enables hetero-epitaxial growth on GaN, AlN, ZnO, and Al₂O₃.⁸ While theoretical studies focus on the orthorhombic Pna2₁ phase,^7,9–12 experiments show that ε-Ga₂O₃ is pyroelectric with large polarization and piezoelectric coupling, being able to host a high density electron gas. Various theoretical studies address the physical properties of the β-phase and experimental studies the preparation of non-β-phases.^12–25

In the present work, we predict three Ga₂O₃ polymorphs, which we call P2₁/c Ga₂O₃, Pnma-I Ga₂O₃, and Pnma-II Ga₂O₃ according to their space groups. The formation energy of Pnma-II Ga₂O₃ is found to be 3 meV per atom lower than that of α-Ga₂O₃. A systematic investigation of the predicted polymorphs is performed, including the stability, electronic states, and carrier effective mass.

Computational details

Based on density functional theory, all calculations are performed with the Cambridge Serial Total Energy Package.²⁶ We use the generalized gradient approximation of Perdew–Burke–Ernzerhof for structure optimization and the Heyd–Scuseria–Ernzerhof hybrid functional for calculating electronic properties. Ultrasoft pseudopotentials²⁷ and the Broyden–Fletcher–Goldfarb–Shanno scheme are adopted. Brillouin zone integrations are performed on Monkhorst–Pack 8 × 5 × 14, 6 × 12 × 4, and 7 × 13 × 3 k-grids for P2₁/c Ga₂O₃, Pnma-I Ga₂O₃, and Pnma-II Ga₂O₃, respectively. Elastic constants are calculated by the strain-stress method. Phonon spectra are obtained by density functional perturbation theory.²⁸

Results and discussion

The crystal structures of the predicted Ga₂O₃ polymorphs are shown in Fig. 1. In each case the unit cell contains 8 Ga and 12 O atoms. In P2₁/c Ga₂O₃ and Pnma-I Ga₂O₃ the Ga atoms are 4- or 5-coordinated and the O atoms are 3-coordinated, whereas in Pnma-II Ga₂O₃ the Ga atoms are 4- or 6-coordinated and the O atoms are 3-coordinated (α-Ga₂O₃: O atoms 4-coordinated and Ga atoms 6-coordinated; β-Ga₂O₃: O atoms 3- or 4-coordinated and Ga atoms 4- or 6-coordinated). The lattice parameters of the predicted polymorphs are compared to those of α-Ga₂O₃ and β-Ga₂O₃ in Table 1. The theoretical values of α-Ga₂O₃ and β-Ga₂O₃ show excellent agreement with the experiment (deviation less than 1.8%), demonstrating reliability of the employed theoretical methodology. The densities of the predicted polymorphs are smaller than those of α-Ga₂O₃ and β-Ga₂O₃.

Fig. 1 Crystal structures. Red spheres represent O atoms and green spheres represent Ga atoms.

Table 1 Lattice parameters and densities

Ga2O3		a (Å)	b (Å)	c (Å)	β (°)	ρ (g cm^−3)
α	Theory	4.988		13.625		6.46
	Exp.^29	4.983		13.433		6.47
β	Theory	12.452	3.083	5.876	103.7	5.68
	Exp.^30	12.230	3.040	5.800	103.7	5.94
P21/c	Theory	9.770	8.762	5.810	149.6	4.95
Pnma-I	Theory	7.073	3.311	9.835		5.41
Pnma-II	Theory	5.900	3.093	12.126		5.63

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The phonon spectra of the predicted polymorphs are shown in Fig. 2. Absence of imaginary frequencies indicates dynamical stability. Fig. 2(d) gives the formation energies of selected polymorphs with respect to that of β-Ga₂O₃ (minimum). The values of the predicted polymorphs are much lower than those of polymorphs from the Materials Project (mp). We calculate for mp-13134 a value of 280 meV per atom in agreement with the mp-result of 284 meV per atom. The formation energies of P2₁/c Ga₂O₃, Pnma-I Ga₂O₃, and Pnma-II Ga₂O₃ are only moderately higher (57 meV per atom, 51 meV per atom, and 23 meV per atom, respectively) than that of β-Ga₂O₃. The formation energy of Pnma-II Ga₂O₃ is even 3 meV per atom lower than that of α-Ga₂O₃. To verify the stability of the predicted polymorphs at operating temperature, ab initio molecular dynamics simulations are performed at 1000 K (5 ps with a time step of 1 fs). The results in Fig. 3 show no indication of structural instability.

Fig. 2 (a–c) Phonon spectra. (d) Formation energies with respect to β-Ga₂O₃.

Fig. 3 Ab initio molecular dynamics simulations at 1000 K: total energy and final structures.

The elastic matrix of monoclinic P2₁/c Ga₂O₃ has 13 independent constants and the elastic matrices of orthorhombic Pnma-I and Pnma-II Ga₂O₃ have 9 independent constants. The obtained elastic matrices and the Born mechanical stability conditions are given in the ESI† material. The elastic constants in Table 2 show that the Born mechanical stability criteria are fulfilled for all the predicted polymorphs. The bulk modulus B, shear modulus G, and Young's modulus E of Pnma-II Ga₂O₃ are found to resemble the values of β-Ga₂O₃ (Table 2).

Table 2 Elastic constants (GPa) and moduli (GPa)

Ga2O3	C11	C22	C33	C44	C55	C66	C12	C13	C23	B	G	E
α	365	365	327	75	75	108	149	106	106	197	96	248
β	199	312	298	39	77	95	112	125	62	155	69	180
P21/c	175	253	204	45	59	47	62	66	88	116	53	138
Pnma-I	210	269	257	53	73	51	90	109	122	152	62	164
Pnma-II	311	300	193	81	96	41	59	116	120	154	71	185

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The obtained electronic band structures in Fig. 4 show a quasi-direct band gap for P2₁/c Ga₂O₃ (3.83 eV) and direct band gaps for Pnma-I Ga₂O₃ (3.60 eV) and Pnma-II Ga₂O₃ (3.70 eV). Although these band gaps are smaller than that of β-Ga₂O₃ (4.60 eV),³¹ they exceed those of SiC (2.40 eV), GaN (3.20 eV), and ZnO (3.44 eV).³² Note that the calculated band gap of β-Ga₂O₃ is 4.60 eV, in perfect agreement with the experimental result of ref. 31.

Fig. 4 Electronic band structures. Red, green, cyan, and blue colors represent Ga s, Ga p, O s, and O p contributions, respectively. The high symmetry points are C (0.0, 0.5, 0.5), Y (0.0, 0.5, 0.0), G (0.0, 0.0, 0.0), B (0.5, 0.0, 0.0), A (0.5, 0.5, 0.0), E (0.5, 0.5, 0.5), and Z (0.0, 0.0, 0.5) for P2₁/c symmetry and R (−0.5, 0.5, 0.5), T (−0.5, 0.0, 0.5), Z (0.0, 0.0, 0.5), G (0.0, 0.0, 0.0), X (0.0, 0.5, 0.0), S (−0.5, 0.5, 0.0), and Y (−0.5, 0.0, 0.0) for Pnma symmetry.

We present the calculated electron and hole effective masses along the a-, b-, and c-axes in Table 3. The electron effective masses of Pnma-I Ga₂O₃ are similar to those of α-Ga₂O₃ and β-Ga₂O₃, those of P2₁/c Ga₂O₃ are a bit larger, and those of Pnma-II Ga₂O₃ are significantly larger by a factor of almost 2.5. We find hardly any anisotropy for the electron effective masses. For P2₁/c Ga₂O₃ the minimum appears not in a special direction, for Pnma-I Ga₂O₃ it appears along the a-axis, and for Pnma-II Ga₂O₃ it appears along the b-axis. The hole effective masses show significant anisotropy. For both P2₁/c Ga₂O₃ and Pnma-II Ga₂O₃ the minimum appears along the b-axis and for Pnma-I Ga₂O₃ it appears along the c-axis. P2₁/c Ga₂O₃ and Pnma-II Ga₂O₃ show heavy fermion behavior in specific directions (Table 3).

Table 3 Electron and hole effective masses (multiples of the free electron mass)

Ga2O3	m a	Electron effective mass
		m b	m c	m max	Direction	m min	Direction
α	0.21	0.21	0.21	0.21	[1, 1, 1]	0.21	[1, 1, 1]
β	0.22	0.22	0.21	0.23	[0.68, −0.73, 0.02]	0.21	[−0.22, −0.22, 0.95]
P21/c	0.26	0.26	0.26	0.26	[0, 1, 0]	0.26	[−0.62, 0, 0.78]
Pnma-I	0.20	0.23	0.23	0.23	[0, 1, 0]	0.20	[1, 0, 0]
Pnma-II	0.51	0.51	0.54	0.54	[0, 0, 1]	0.51	[0, 1, 0]

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Ga2O3	m a	Hole effective mass
		m b	m c	m max	Direction	m min	Direction
α	2.54	2.54	0.80	3.47	[0.67, 0.67, 0.31]	0.71	[0.25, 0.25, 0.93]
β	2.81	3.49	4.54	6.06	[0.25, 0.56, 0.79]	2.57	[−0.85, −0.53, 0.04]
P21/c	2.55	2.26	5.03	>50	[−0.56, 0, 0.83]	2.26	[0, 1, 0]
Pnma-I	2.29	2.29	0.31	2.29	[1, 0, 0]	0.31	[0, 0, 1]
Pnma-II	4.07	0.37	>50	>50	[0, 0, 1]	4.07	[0, 1, 0]

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The simulated X-ray diffraction patterns of α-Ga₂O₃, β-Ga₂O₃, P2₁/c Ga₂O₃, Pnma-I Ga₂O₃, and Pnma-II Ga₂O₃ are displayed in Fig. 5. The (111) and (400) peaks of β-Ga₂O₃ (at 34.7° and 29.5°, respectively) coincide with the mp-results. Though they belong to the same space group, the spectra of Pnma-I Ga₂O₃ and Pnma-II Ga₂O₃ reveal notable differences. The strongest peaks are (22–1) at 27.3° for P2₁/c Ga₂O₃, (111) at 33.5° for Pnma-I Ga₂O₃, and (112) at 35.9° for Pnma-II Ga₂O₃. The X-ray characteristics enable identification of the predicted polymorphs in future experiments.

Fig. 5 X-Ray diffraction patterns (Cu source with a wavelength of 1.5406 Å).

Conclusions

In conclusion, three low-energy Ga₂O₃ polymorphs with wide band gaps are predicted to be dynamically and mechanically stable. P2₁/c Ga₂O₃ and Pnma-I Ga₂O₃ realize a novel 5-coordination of the Ga atoms. The band gap is quasi-direct for P2₁/c Ga₂O₃ and direct for Pnma-I Ga₂O₃ and Pnma-II Ga₂O₃. The electron and hole effective masses show significant differences from those of α-Ga₂O₃ and β-Ga₂O₃. Simulated X-ray diffraction patterns enable experimental confirmation of the predicted structures. Remarkably, chemical vapor deposition can be used to synthesize a large variety of Ga₂O₃ polymorphs, adopting temperatures of 410 °C,³³ 550 °C,³⁴ and 650 °C.³⁵ When aiming for synthesis of the predicted polymorphs by this method, identifying a suitable growth temperature is the main experimental uncertainty. However, since structural stability is maintained beyond 1000 K, it is reasonable to expect that appropriate growth parameters can be established.

Data availability statement

The data that supports the findings of this study are available within the article and its supplementary material.

Conflicts of interest

The authors have no conflicts to disclose.

Acknowledgements

The authors acknowledge generous financial support from the National Natural Science Foundation of China (No. 61804120), China Postdoctoral Science Foundation (No. 2019TQ0243 and 2019M663646), Key Scientific Research Plan of the Education Department of Shaanxi Province (Key Laboratory Project) (No. 20JS066), and Young Talent Fund of the University Association for Science and Technology in Shaanxi, China (No. 20190110). The research reported in this publication was supported by funding from King Abdullah University of Science and Technology (KAUST).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d1cp05271c

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