Epitaxial engineering of polar ε-Ga₂O₃ for tunable two-dimensional electron gas at the heterointerface

Ga₂O₃ is emerging as an attractive semiconductor for high-power switching applications due to its high breakdown field and ultra-wide bandgap.^1–4 Amongst the various polymorphs of Ga₂O₃, the β-phase has received the most attention due to its stable form under ambient conditions and the ease of growing large single crystals.^5–7 Recently, carrier confinement and formation of a two-dimensional electron gas (2DEG) have been experimentally demonstrated at the interface of Ga₂O₃ with a wider bandgap alloy (Al_xGa_1–x)₂O₃ by using modulation doping with silicon,⁸ which enables Ga₂O₃-based devices to simultaneously operate at high-power and high-frequencies.^9–11 However, modulation doping results in a modest 2DEG density of ∼10¹² cm⁻² compared to a 2DEG density of ∼10¹³ cm⁻² at the AlGaN/GaN interface.¹² Furthermore, it lowers the mobility of the 2DEG due to impurity scattering. These have led to a search for alternative ways to generate 2DEG in Ga₂O_3.¹³ 

Recently, ferroelectric hysteresis has been reported in thin films of ε-Ga₂O₃, which is a metastable polymorph.¹⁴ Contrary to 2DEG formation in β-Ga₂O₃, the spontaneous polarization of ε-Ga₂O₃ can open a path to achieve 2DEG with high mobility and, possibly, higher sheet charge density without doping. While there have been numerous attempts to grow ε-Ga₂O₃ on various substrates,^15–17 they have been unsuccessful to grow single-phase thin films that are free of defects.¹⁸ This is primarily due to a lack of understanding of the stability of the competing phases of Ga₂O₃ under epitaxial strain. Very recently, ε-Ga₂O₃ thin films have been stabilized on (001) Al₂O₃ substrates by using tin dopants during growth;¹⁹ however, the formation of a 2DEG was not reported. This is because the formation of a 2DEG at the interface of ε-Ga₂O₃ with its lattice-matched substrate, such that the 2DEG resides in the semiconducting ε-Ga₂O₃, also requires a specific band alignment and the knowledge of its spontaneous and piezoelectric polarization constants, all of which are currently missing.

In this letter, we have investigated the energy landscape of various Ga₂O₃ polymorphs under epitaxial strain by combining coincident-site lattice models (CSL) with first-principles density-functional theory (DFT) calculations. We have identified the lattice parameter of the substrates that minimize the epitaxial strain of ε-Ga₂O₃ with respect to other competing phases and recommend a list of commercially available substrates to grow phase-pure ε-Ga₂O₃ without doping. By using group-theoretical methods, we show that ε-Ga₂O₃ is ferroelectric and the polarity of ε-Ga₂O₃ can be switched with an external electric field. Furthermore, by calculating the band alignment of the various lattice-matched substrates, we identify CaCO₃ to be particularly promising as it allows the formation of a 2DEG in ε-Ga₂O₃ due to polarization-induced charges. Finally, we show that an electric field can be used to switch the spontaneous polarization in ε-Ga₂O₃ to obtain a large charge density of 10¹⁴ cm⁻². Therefore, by stabilizing an ultrawide bandgap semiconducting ferroelectric and an electric-field tunable 2DEG, our work paves a way to achieve a new generation of devices.

DFT calculations were performed using the VASP package²⁰ and projector augmented-wave potentials.²¹ The plane-wave basis set was expanded to a cutoff energy of 520 eV to minimize Pulay stress during the structural optimization. The structural optimization was truncated after the Hellmann-Feynman forces were under 0.001 eV/Å. The Brillouin zone was sampled using the Monkhorst-Pack method with k-points grids of 6 × 6 × 2 for α-, 13 × 4 × 4 for β-, and 6 × 4 × 4 for the transformed cell of the ε-phase under epitaxial strain, respectively.²³ The 3d, 4s, and 4p states of Ga and 2s and 2p states of O are taken as valence states, and the exchange-correlation energy of valence electrons was described using the Perdew, Burke, and Ernzerhof (PBE) functional.²² As PBE is known to overestimate the lattice constants, to maintain consistency, we have used PBE-optimized lattice constants for all the substrate candidates, which are shown in supplementary material Table S3.²⁴ To calculate polarization constants, we used the Berry-phase method with a k-point grid of 6 × 4 with 16-point strings.²⁵ To evaluate the dielectric, piezoelectric, and stiffness constants, we employed density functional perturbation theory with an increased cutoff energy of 700 eV.²⁶ The bandgaps and electron affinities were calculated using the Heyd-Scuseria-Ernzerhof (HSE) hybrid functionals with a mixing parameter of 0.35 and 0.15 to fit the experimental bandgaps of β-Ga₂O₃ (Eg = 4.9 eV) and CaCO₃ (Eg = 6.0 eV), respectively.^27–30 Due to the lack of experimental measurements of the bandgap of high-quality ε-Ga₂O₃, combined with the similar theoretical bandgap of ε-Ga₂O₃ and β-Ga₂O₃ calculated using PBE (0.06 eV difference), we have used α = 0.35 to calculate the bandgap of ε-Ga₂O₃. The k-point grid of substrate candidates was sampled with a density of 2000 k-points per reciprocal atom. The electron affinities were calculated using the electrostatic potential of non-polar CaCO₃ (104) and ε-Ga₂O₃ (010) surfaces with the macroscopic electrostatic potential averaging technique. To simulate the surfaces, we used slabs that were thicker than 25 Å and were separated by 20 Å vacuum.^31,32

ε-Ga₂O₃ belongs to the Pna2₁ space group that is a subgroup of hexagonal P6₃mc. It implies that the orthorhombic lattice can be expressed with a basis transformation from the hexagonal lattice. The calculated a = 5.13 Å and b = 8.81 Å lattice parameters of ε-Ga₂O₃ have a ratio of 1.718 $≈3$ that also corresponds to the ratio of the two diagonals of the rhombohedral lattice. According to CSL theory,³³ the epitaxial interface should be constrained such that a repeating unit is formed where the lattice sites of the film and the substrate coincide. Therefore, the (001) plane is promising as it can satisfy the CSL conditions for epitaxial growth on a hexagonal substrate (supplementary material Fig. S1). On the other hand, β-Ga₂O₃ does not have a coincidence lattice with a hexagonal substrate for small epitaxial strains. The reported preferred orientation of β-Ga₂O₃ on hexagonal substrates is the $2¯01$ plane, which has calculated in-plane lattice vectors of 3.09 Å for the [010] direction and 14.98 Å for the $[101¯]$ direction and needs at least 7% strain to fit hexagonal constraints. Furthermore, the large difference of the two in-plane vectors in β-Ga₂O₃ requires a large CSL leading to a number of dangling bonds. ε-Ga₂O₃ can, instead, form a CSL with a smaller unit cell on a hexagonal substrate. This is beneficial to stabilize metastable ε-Ga₂O₃ over stable β-Ga₂O₃. We have also considered the CSL of α-Ga₂O₃, which is less stable compared to the ε and β phases in the bulk form⁵ but has a hexagonal structure and could be expected to be stabilized on hexagonal substrates. The crystal structure of the α, β, and ε-phases of Ga₂O₃ is shown in Fig. 1(a).

We have calculated the energy of the preferred orientation of α, β, and ε-phases of Ga₂O₃ on hexagonal substrates as a function of varying lattice constants of the substrate, as shown in Fig. 1(b). For epitaxial stabilization of ε-Ga₂O₃, on hexagonal substrates, it should have the lowest energy amongst the three competing phases. Furthermore, the lattice mismatch with the substrate should be small, usually within ±3% in the case of oxides,³⁴ to avoid formation of defects caused by strain relaxation. With these constraints, we find hexagonal substrates matching the smallest CSL with ε-Ga₂O₃ and having a lattice constant between 4.97 and 5.12 Å to be most promising. Based on the calculated phase diagram under epitaxial strain, we find that previously used substrates to grow epitaxial ε-Ga₂O₃ impose strains over 3% or stabilize the α or β phases, which explains the poor quality of the deposited thin films (see supplementary material).¹⁸ Based on the identified region of stability of ε-Ga₂O₃, we searched the Materials Project database³⁵ and suggest promising substrates in Table I. We find that non-polar substrates, such as α-Fe₂O₃, CaCO₃, h-BN, and SiO₂, are also commercially available. As discussed below, we find that CaCO₃ is particularly promising to induce 2DEG in ε-Ga₂O₃ due its large bandgap of 6.0 eV and favorable band alignment.²⁷ 

We now focus on identifying the polar properties of ε-Ga₂O₃ and examine whether it is indeed possible to obtain ferroelectric switching to explain the hysteretic behavior reported in recent experiments.¹⁴ Bulk ε-Ga₂O₃ belongs to the non-centrosymmetric Pna2₁ space group. Using Berry-phase calculations, we find that ε-Ga₂O₃ has a spontaneous polarization (P_SP) of 23 μC/cm² oriented along the c-axis, which is in good agreement with a recent theoretical report.¹³ The calculated P_SP of ε-Ga₂O₃ is ten times larger than that of pyroelectric wide bandgap semiconductor GaN (2.9 μC/cm²).³⁶ To switch the dipole moment in ε-Ga₂O₃ to the opposite direction, a transition through an intermediate centrosymmetric (non-polar) supergroup of the Pna2₁ space group is required. Using group-theoretical techniques, as implemented in the Pseudo and Amplimodes programs in the Bilbao crystallographic server,^37–39 we have identified Pnna, Pccn, Pbcn, and Pnma as the four centrosymmetric supergroups from which Pna2₁ can be obtained with minimal atomic distortion. Amongst them, we find that the transition from Pbcn to Pna2₁ space group involves the smallest displacement of all atoms along the $Γ3−$ polar phonon mode and has the smallest energy barrier (E_b) of 0.95 eV, as shown in Fig. 2 (see supplementary material Fig. S2 for other transition pathways). This is a relatively large activation barrier comparable to that of GaFeO₃ (1.05 eV), which shows a high ferroelectric to paraelectric transition temperature of 1368 K.⁴⁰ Such a high activation barrier is expected to stabilize the polarization against thermally activated random dipole switching even at high temperatures during operation, which makes ε-Ga₂O₃ an attractive ferroelectric semiconductor with an ultrawide bandgap.

In addition to P_SP, the use of an epitaxial strain to stabilize ε-Ga₂O₃ is expected to induce piezoelectric polarization (P_PE). For a non-polar substrate, the termination of polarization at the substrate/ε-Ga₂O₃ interface will induce a charge density (σ) with contributions from both P_SP and P_PE that can be expressed by⁴¹ 

where $e31, e32$ and $e33$ are the piezoelectric constants, $ϵ1 and ϵ2$ are the two in-plane strains, and $ϵ3$ is the out-of-plane strain on ε-Ga₂O₃ due to the substrate. The out-of-plane strain can be obtained using the elastic constants of ε-Ga₂O₃: $ϵ3=−ϵ1c13/c33− ϵ2c23/c33$⁠. The calculated piezoelectric and elastic constants are shown in Table II. We find that the piezoelectric constants are comparable to III–V semiconductors. For instance, $e33$ and $e31$ in GaN are 0.73 and −0.49, respectively.³⁶ Due to the magnitude of e₃₁ and e₃₃, even small epitaxial strains can produce a large P_PE.

We use CaCO₃ as the substrate and calculate the charge density at the interface with ε-Ga₂O₃. CaCO₃ imposes a compressive strain of 1.4%, which leads to P_PE = −49 μC/cm², which has an opposite sign to that of P_SP (23 μC/cm²) and points towards the substrate. While P_PE is fixed by the choice of the substrate, P_SP is switchable by an external electric field. Thereby, depending on the orientation of P_SP in ε-Ga₂O₃, the total polarization can be varied from −26 μC/cm² (P_PE + P_SP) to −72 μC/cm² (P_PE − P_SP). The corresponding sheet charge density (σ) varies between 1.6 $×$ 10¹⁴ cm⁻² and 4.4 $×$ 10¹⁴ cm⁻², respectively, which is higher than the density present at AlGaN/GaN⁴¹ and modulation-doped β-Ga₂O₃/Si:(Al_xGa_1−x)₂O₃ heterojunctions.^9–11 Furthermore, the ferroelectric nature of ε-Ga₂O₃ is expected to allow the modulation of the charge density with an external electric field.

To identify the conditions under which the above calculated interface charges are expected to be mobile as opposed to being fixed, we have analyzed the band alignment and potential shift for different P_SP and thicknesses of ε-Ga₂O₃ films on the CaCO₃ substrate. The polarization in ε-Ga₂O₃ is associated with an internal electric field and potential-shift along the [001] direction. Based on the calculated dipole moment in a unit cell of ε-Ga₂O₃, the potential-shift can be estimated as^42–44 

Here, c is the lattice vector along the [001] direction (9.424 Å) of ε-Ga₂O₃ and $ε$ is its calculated static dielectric constant (13.2). The potential shift $ΔV$ of pristine ε-Ga₂O₃ without any strain (P_PE = 0) is −1.98 V/nm. The potential shift $ΔV$ of strained ε-Ga₂O₃ on the CaCO₃ substrate is 2.23 V/nm for P_PE + P_SP and 6.17 V/nm P_PE − P_SP. Therefore, it depends on the direction of P_SP, which can be controlled with an external electric field. With an optimal band alignment between the two materials, the large potential shift can be exploited such that the electrons from the valence band of CaCO₃ spontaneously ionize and spillover to the conduction band of ε-Ga₂O₃ to form a mobile 2DEG at the interface, as shown in Fig. 3. We have calculated band alignment between ε-Ga₂O₃ and the CaCO₃ substrate based on their bulk bandgap, electron affinity, and potential shift. We find that the two materials form a staggered gap of 2.86 eV at the heterointerface (supplementary material), where the band alignment is determined by the Anderson rule without considering the polarity.⁴⁵ Figure 3 shows the schematic band alignment at the ε-Ga₂O₃(001)/CaCO₃(104) interface and the different spontaneous polarization and thicknesses of ε-Ga₂O₃ under which a mobile 2DEG is expected to form. The direction of the total polarization is always towards the substrate as it is determined by P_PE, regardless of the direction of P_SP. For a thin layer of ε-Ga₂O₃ (<2.7 nm), if P_SP is parallel to P_PE (i.e., P_PE + P_SP), the strong field of 6.17 V/nm drives the conduction band of ε-Ga₂O₃ above the valence band of CaCO₃. This results in ionization of the valence electrons of CaCO₃ and a mobile 2DEG on the ε-Ga₂O₃ side. On the other hand, when the P_SP is switched such that it is antiparallel to P_PE (i.e., P_PE − P_SP), the interface charges are confined to the valence band of ε-Ga₂O₃ and are expected to be immobile. For ε-Ga₂O₃ films with the thickness above 2.7 nm, mobile 2DEG is expected for both the directions of P_SP; however, the sheet charge density can be tuned between 1.6 $×$ 10¹⁴ cm⁻² and 4.4 $×$ 10¹⁴ cm⁻² with an external electric field. We would like to point out that the exact sheet charge density and the critical thickness for the formation of 2DEG will also depend on the quality of the heterointerface, including the presence of defects and intermixing as is observed in the 2DEG formed at the LaAlO₃/SrTiO₃ heterointerface.^46–48 

In conclusion, we have investigated a pathway to stabilize metastable, polar ε-Ga₂O₃ using epitaxial strain and have identified promising substrate candidates. We have also calculated possible switching pathways for ε-Ga₂O₃ and predict it to be a ferroelectric wide bandgap semiconductor. Furthermore, we predict the formation of 2DEG at the interface of ε-Ga₂O₃ with CaCO₃ substrates with a sheet charge density that is two orders of magnitude higher than that obtained using modulation doping in β-Ga₂O₃/(Al_xGa_1−x)₂O₃. Due to the ferroelectric nature of ε-Ga₂O₃, we show that the interface 2DEG density can be modulated using an external electric field, which opens a pathway to design new device architectures. The polarization-induced 2DEG in ε-Ga₂O₃ is also expected to result in devices that can simultaneously operate at high-power and high frequencies.

See supplementary material for the choice of CSL, the effect of PBE functional, and further substrate candidates for the epitaxial growth.

We are thankful to Prof. Sriram Krishnamoorthy of University of Utah for helpful discussions. This work used computational resources of the Extreme Science and Engineering Discovery Environment (XSEDE), which was supported by National Science Foundation Grant No. ACI-1053575.