We report the strain and stress relationships for the three lowest energy direct band to band transitions at the Brillouin zone center in monoclinic β-Ga2O3. These relationships augment four linear perturbation parameters for situations, which maintain the monoclinic symmetry, which are reported here as numerical values obtained from density functional theory calculations. With knowledge of these perturbation parameters, the shift of each of the three lowest band to band transition energies can be predicted from the knowledge of the specific state of strain or stress, thus providing a useful tool for modeling performance of power electronic devices and rational strain engineering in heteroepitaxy.

Fundamental physical properties, such as energy levels in solid state matter, are characteristically affected by the state of strain and its induced stress, which cause the lattice parameters to shift from their equilibrium values. Inherent strain and stress are expected in heteroepitaxial systems, and hence, the relationships for the deviation of electronic states and phonon modes are of particular interest. For certain lattice parameter deviations, the strain/stress induced change to the electronic states can be described as first-order (linear) perturbations of the quantum states of the crystal in equilibrium. Deformation potentials (defined by group theory and the theory of invariants for each specific crystal symmetry)1 can be used to model the strain/stress induced changes. Using these deformation potentials, the shifts of energy levels, for example, bandgap energies and phonon mode frequencies, can be calculated. In materials with high crystal symmetry, strain and stress relationships have been extensively investigated, by theory and experiment, e.g., for phonon modes,2–29 and electronic band structures.30 This includes symmetry-conserving strain and stress relationships for the zinc blende structure GaAs31–34 and wurtzite structure GaN and AlN.35–39 

Low-symmetry materials studies of strain effects are scarce. An example of recent interest is the monoclinic β phase of Ga2O3, a new ultrawide bandgap material, which has attracted significant research attention due to its high potential for high-power switching devices for a sustainable energy economy.40 In addition, β-Ga2O3 is very promising for harsh environment electronics and for optoelectronic applications in the deep ultraviolet spectral range.41–43 Strain effects onto electronic properties are very important for device architectures based on β-Ga2O3, which involve heterostructures with its alloys Al2O3 and In2O3 for band engineering, envisioned as key components of next generation electronic devices. In our recent work,44 we presented the linear perturbation theory strain and stress relationships for infrared and Raman active phonon modes in crystals with monoclinic (C2h) symmetry for strain and stress situations, which maintain the monoclinic symmetry of the crystal. We demonstrated that for each phonon mode, one needs four linear deformation potentials to describe the energy shift of the mode vs all four independent tensor elements of strain, ϵxx,ϵxy,ϵyy, and ϵzz, and similarly four linear deformation potentials to describe the energy shift caused by all four independent stress tensor elements, σxx,σxy,σyy, and σzz. Using β-Ga2O3 as an example, we determined the strain and stress deformation potentials for Raman-active phonon modes and infrared-active transversal optical (TO) phonon modes using density-functional perturbation theory (DFPT) calculations.

Because the concept for the linear strain and stress relationships is general, it is expected to be applicable to all energy levels of a given lattice system, not just its lattice phonon frequencies. Specifically, optical transitions between energy bands from the symmetry point of view are similar to infrared-active phonon modes, i.e., dipole transitions between energy levels, one of which in monoclinic crystals of C2h symmetry is classified as either Au (transition dipole oriented parallel to the symmetry axis of the crystal) or Bu (transition dipole normal to the symmetry axis). The purpose of this Letter is to validate the conclusions of our previous paper,44 i.e., the strain–stress relationships for phonon modes in monoclinic symmetry crystals, to electronic energy levels of monoclinic β-Ga2O3. Specifically, we show that similarly to phonon modes, four linear deformation potential parameters are sufficient to describe the energy shift of selected energy bands in the Brillouin zone center in the presence of strain or stress, i.e., according to the general formula for the dependence on strain,44 

Eη(ϵ)=Eη(0)+Pη,xxϵxx+Pη,xyϵxy+Pη,yyϵyy+Pη,zzϵzz
(1)

and similarly for the dependence on stress

Eη(σ)=Eη(0)+P̃η,xxσxx+P̃η,xyσxy+P̃η,yyσyy+P̃η,zzσzz,
(2)

where Eη(ϵ) and Eη(σ) are energy positions of a band η in the presence of strain and stress and Pη,… and P̃η, are band-specific strain and stress linear deformation potentials, respectively. Furthermore, we propose a simple model for the energies of the direct band to band transitions connecting the conduction band bottom and the top valence band levels at the Brillouin zone center in β-Ga2O3 as a function of strain and stress. We explore the strain/stress relationships of the electronic energy levels by the consistent application of the density functional theory (DFT), which we use to determine the numerical values of the linear deformation potentials Pη,… and P̃η, for selected energy bands. We note that stress and strain tensors are related by the elastic tensor C, σ=Cϵ, and elements of C are given in, for example, Refs. 44 and 45.

In the current Letter, we rely on the same set of structures that we used in our previous study to identify the phonon deformation potential parameters.44 Therefore, we only describe the calculations very briefly here as all the details are available in the previous paper.44 The entire dataset comprises of 64 structures representing different deformation patterns, although not all of them were used for the determination of deformation potentials. All density functional theory calculations were performed using the plane wave code Quantum Espresso.46 The generalized-gradient-approximation (GGA) density functional of Perdew, Burke, and Ernzerhof (PBE)47 was used in combination with norm-conserving Troullier–Martins pseudopotentials generated using FHI98PP48,49 code and available in the Quantum Espresso pseudopotentials library. The pseudopotential for gallium did not include the semicore 3d states in the valence configuration. All calculations were performed with a very high electronic wavefunction cutoff of 400 Ry, and a dense shifted 8×8×8 Monkhorst-Pack50 grid for sampling of the Brillouin zone. A convergence threshold of 1×1012 Ry was used to reach self-consistency. The unit cells in all the calculations were aligned with a Cartesian reference system, where the symmetry axis of the monoclinic cell (the crystallographic vector b) is aligned with the Cartesian axis z, and the monoclinic plane stays in the xy plane, with the crystallographic vector a parallel to x. Thus, all the physical properties described by second-rank tensors have four non-zero components with the shear component xy (for a sketch of the unit cell with the reference system used, see Fig. S1 in the supplementary material).

We considered a range of different deformation scenarios: hydrostatic pressure (with equal diagonal components of the stress tensor, obtained by requesting the target pressure during the variable-cell structural relaxation, as implemented in the software), uniaxial stress (with a single non-zero component of the stress tensor, obtained by a constrained structural relaxation in which a specific lattice parameter was kept constant, as described in detail in the supplementary material), and uniaxial strain (with a single non-zero component of the strain tensor, obtained by the application of a predefined amount of strain to the crystal lattice along one Cartesian direction at a time, followed by the structural relaxation of the ionic positions). In all scenarios, we ensured that the symmetry of the monoclinic cell was not further reduced to triclinic, i.e., all deformation scenarios studied did not induce shear-stresses and/or shear-strains involving the monoclinic axis b. All structures were relaxed with tight convergence thresholds of 1×106 Ry for energy and 1×105 Ry/bohr for forces. We obtained the band eigenvalues at the Brillouin zone center, including the lowest unoccupied band, by performing additional non-self-consistent calculations for all the structures. The raw data, i.e., the lattice parameters and the energies of energy bands included in the present study for all the structures, are provided in the supplementary material file.

Another essential ingredient for the present work, helping us decide on which bands we should focus our attention, is recent theoretical and combined spectroscopic ellipsometry and theoretical studies of optical properties and band to band transitions in β-Ga2O3.45,51–54 The analysis of the dielectric tensor measured using spectroscopic ellipsometry yields four critical points corresponding to band to band transitions near the onset of optical absorption51,53 (each of them except one in the model of Mock et al.51 with an excitonic component not relevant here). The lowest energy transition is polarized within the monoclinic plane, roughly in the direction of crystallographic vector c. The second-lowest transition, also polarized within the monoclinic plane, is polarized roughly in the direction of crystallographic vector a, up to an angle of 25°. The third- and fourth-lowest transitions are polarized along the material symmetry axis (crystallographic vector b).

This structure of the critical points polarized in different directions explains what was nicknamed “the anisotropy of the absorption edge,” and what electronically likely involves different band pairs producing optical transitions at different energies and directions. This picture is remarkably consistent with results of first principles calculations, which show, through the analysis of optical matrix elements between the valence and conduction bands,51,53 or the symmetry classification of valence bands,52 which of the valence bands can produce optical transitions to the lowest conduction band. Because ellipsometry only measures direct transitions and because the conduction band minimum in β-Ga2O3 is without doubt at the center of the Brillouin zone, we focus onto the Γ point bands. We label the band to band transitions using the scheme Γcv, used by Mock et al.,51 where c is the index of the conduction band (here, c =1 for all the lowest transitions) and v is the index of the valence band (both indices running from the bandgap). The two top valence bands belong to the irreducible representation Bu,52 and thus, the two lowest energy transitions, Γ11 and Γ12, are polarized within the monoclinic plane. Additionally, the computed optical matrix elements show that Γ11 is polarized close to the crystallographic vector c, and Γ12 is polarized close to the crystallographic vector a, just like the corresponding two critical points in the experimental dielectric function spectra for the tensor elements corresponding to the monoclinic ac plane. The third and fourth transitions, Γ14 and Γ16, occur along the symmetry axis b from valence bands belonging to the irreducible representation Au. The transition energies and relative differences between energy bands for the three lowest band to band transitions, one in each major directions, which, hence, define the optical bandgap of β-Ga2O3, are collected in Table I.

TABLE I.

DFT-calculated and experimental energies of selected near-bandgap energy bands at the Brillouin zone center (upper part), and energies of the corresponding band to band transitions (lower part). For transitions polarized within the monoclinic plane, the orientation of the transition dipole with respect to the crystallographic vector a is given in brackets, where available. The band energies have been shifted, so that the energy of the top valence band (v =1) is zero.

Band index/TransitionPBE This workGau-PBE Ref. 51 QSGWa Ref. 52 GWb Ref. 53 Exp. Ref. 53 Exp. Ref. 51 
c =2.349 4.740 4.933 5.05 5.15 5.04 
v =0.0 0.0 0.0 0.0 0.0 0.0 
v =−0.270 −0.229 −0.133 −0.24 −0.22 −0.36 
v=4c −0.709 −0.610 −0.669 −0.57 −0.53 −0.60 
Γ11 2.349 (111°) 4.740 (113°) 4.933 5.05 (111°) 5.15 (110°) 5.04 (115°) 
Γ12 2.619 (15°) 4.969 (20°) 5.066 5.29 (17.7°) 5.37 (17°) 5.40 (25.2°) 
Γ14* 3.058 5.350 5.602 5.62 5.68 5.64 
Band index/TransitionPBE This workGau-PBE Ref. 51 QSGWa Ref. 52 GWb Ref. 53 Exp. Ref. 53 Exp. Ref. 51 
c =2.349 4.740 4.933 5.05 5.15 5.04 
v =0.0 0.0 0.0 0.0 0.0 0.0 
v =−0.270 −0.229 −0.133 −0.24 −0.22 −0.36 
v=4c −0.709 −0.610 −0.669 −0.57 −0.53 −0.60 
Γ11 2.349 (111°) 4.740 (113°) 4.933 5.05 (111°) 5.15 (110°) 5.04 (115°) 
Γ12 2.619 (15°) 4.969 (20°) 5.066 5.29 (17.7°) 5.37 (17°) 5.40 (25.2°) 
Γ14* 3.058 5.350 5.602 5.62 5.68 5.64 
a

QSGW - quasiparticle self-consistent GW.

b

GW refers to Hedin's GW approach.

c

Fifth valence band in the results from Ref. 52, fourth otherwise.

In the current study, the calculations of the strain effects were performed using relatively simple DFT at the non-hybrid GGA level. Hence, the value of the energy gap between the valence and the conduction bands is greatly underestimated, as expected. Nevertheless, the transition energies all involve the lowest conduction band only and depend primarily on the structure of the valence bands, which are obtained here at a reasonably accurate level of theory. It is then appropriate to correct the underestimation of the bandgap energy by an arbitrary offset.

As postulated, we use four linear deformation potentials to model the strain/stress energy shifts of the energy bands. Figure 1 shows the DFT-computed (squares) and best-match linear strain potential parameters calculated (crosses) band energies vs strain tensor elements ϵxx,ϵxy,ϵyy, and ϵzz (left column) and vs stress tensor elements σxx,σxy,σyy, and σzz (right column). For each energy band, a four-dimensional dataset is obtained with either strain or stress tensor elements as a base. Hence, each figure contains four panels, where the frequencies are plotted vs one of the strain tensor elements. Note that in order to present the four-dimensional dataset, for every data point {Ei,(xx,xy,yy,zz)}, the same energy is plotted four times, once vs each of its strain coordinates. The dataset comprises DFT calculations with various scenarios of different hydrostatic stress, uniaxial stress, and uniaxial strain. In order to avoid possible non-linearities, in our best-match model analysis, we limited the permissible strains (stresses) to maximum of ±0.0035 (±12.5 kbar), which resulted in data points for 35 structures being used for the determination of strain deformation potentials and 37 structures being used for the determination of stress deformation potentials. Within the permissible strain/stress values, the DFT-calculated band energies show a linear shift, and the four deformation potential parameters reproduce all DFT calculated data with great accuracy. The resulting deformation potentials are listed in Table II.

FIG. 1.

DFT-derived (squares) and best-match linear strain deformation potentials calculated (crosses) energies of selected energy bands vs strain tensor elements ϵxx,ϵxy,ϵyy, and ϵzz (left column) and stress tensor elements σxx,σxy,σyy, and σzz (right column), under various deformations (see the text).

FIG. 1.

DFT-derived (squares) and best-match linear strain deformation potentials calculated (crosses) energies of selected energy bands vs strain tensor elements ϵxx,ϵxy,ϵyy, and ϵzz (left column) and stress tensor elements σxx,σxy,σyy, and σzz (right column), under various deformations (see the text).

Close modal
TABLE II.

Linear strain (Pη,…) and stress (P̃η,) potentials for energies of selected near-bandgap energy bands in β-Ga2O3 in units of eV/(unit strain) and meV/kbar, respectively. The permitted maximum strain was limited to ±0.0035. The maximum permitted stress was limited to ±12.5 kbar.

ModePη,xxPη,xyPη,yyPη,zzP̃η,xxP̃η,xyP̃η,yyP̃η,zz
c =−18.78 5.130 −20.04 −18.02 5.200 0.063 84 3.642 2.975 
v =−10.40 4.784 −14.11 −8.462 2.184 −1.972 3.332 1.224 
v =−11.03 2.845 −11.55 −10.28 3.202 0.2970 2.037 1.619 
v =−10.35 1.793 −8.179 −11.69 3.220 0.5091 0.8561 2.272 
ModePη,xxPη,xyPη,yyPη,zzP̃η,xxP̃η,xyP̃η,yyP̃η,zz
c =−18.78 5.130 −20.04 −18.02 5.200 0.063 84 3.642 2.975 
v =−10.40 4.784 −14.11 −8.462 2.184 −1.972 3.332 1.224 
v =−11.03 2.845 −11.55 −10.28 3.202 0.2970 2.037 1.619 
v =−10.35 1.793 −8.179 −11.69 3.220 0.5091 0.8561 2.272 

Knowledge of the deformation potentials for each energy band allows us to propose a simple model for the strain/stress dependence of measurable optical transition as differences of respective deformation potential parameters of the bands involved in the transition. For the base (strain-free) energies of the transitions, we use the experimental values from Ref. 51. Hence, the lowest Γ11 transition as a function of strain can be expressed as

EΓ11=5.048.38ϵxx+0.346ϵxy5.93ϵyy9.56ϵzzeV.
(3)

Considering the polarization of the Γ11 transition is close to the [001] direction, it may be used as an approximate expression of the direct bandgap in that direction. Similarly, for [100] and [010] directions, we can use

EΓ12=5.407.75ϵxx+2.29ϵxy8.49ϵyy7.74ϵzzeV
(4)

and

EΓ14=5.648.43ϵxx+3.34ϵxy11.9ϵyy6.33ϵzzeV,
(5)

respectively. Obtaining analogous formulas for the stress dependence of the energy of each of these transitions is rather straightforward.

In conclusion, we have validated the group-theoretical approach to modeling effects of symmetry-conserving lattice deformations in linear approximation onto the band energies in β-Ga2O3, and we have determined the strain and stress deformation potentials of selected energy bands for β-Ga2O3 using density-functional theory calculations. It allowed us to obtain a simple model of the bandgap energy vs strain and stress, and we anticipate its use for modeling effects of strain and stress in heterostructures for future electronic materials based on β-Ga2O3 and related alloys. The same approach is expected to be valid for all monoclinic crystals as well.

See the supplementary material file for details on the Cartesian reference system, the derivation of the deformation potentials, DFT calculations of the structures under uniaxial stress, and lattice parameters and band energies for the structures included in the study.

This work was supported in part by the National Science Foundation (NSF) under Award Nos. NSF DMR 1808715 and NSF/EPSCoR RII Track-1, Emergent Quantum Materials and Technologies (EQUATE) under Award No. OIA-2044049, and by the Air Force Office of Scientific Research under Award Nos. FA9550-18-1-0360, FA9550-19-S-0003, and FA9550-21-1-0259. This work was also supported by the Knut and Alice Wallenbergs Foundation award “Wide-bandgap semiconductors for next generation quantum components.” M.S. acknowledges the University of Nebraska Foundation and the J. A. Woollam Foundation for support. This work was also supported in part by the Swedish Research Council under VR Award No. 2016-00889, the Swedish Foundation for Strategic Research under Grant Nos. RIF14-055 and EM16-0024, by the Swedish Governmental Agency for Innovation Systems VINNOVA under the Competence Center Program Grant No. 2016-05190, and by the Swedish Government Strategic Research Area in Materials Science on Functional Materials at Linköping University under Faculty Grant SFO Mat LiU No. 2009-00971. DFT calculations were in part performed at the Holland Computing Center of the University of Nebraska, which receives support from the Nebraska Research Initiative.

The authors declare no conflict of interest.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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