The electron effective masses of ScxAl1xN and AlxGa1xN, two of the most promising wide bandgap materials for power and RF electronic applications, have been calculated using the predictions of the density functional theory (DFT). More specifically, the unfolding technique has been adopted to extract the effective band structure of the two alloys under investigation. It has been found that the AlGaN effective masses m approximately follow the Vegard law. On the contrary, due to the larger amount of disorder inside the crystal, the ScAlN shows a non-monotonic change of m as a function of the Sc concentration, which requires the DFT calculations to be consistently performed for an accurate prediction. The ScAlN effective masses as a function of Sc content have been reported in the range 0x0.25 for the first time.

Ultrawide-bandgap nitride materials are nowadays extensively used to produce electronic devices with high power density and efficiency when compared to their silicon counterparts. Actually, high electron mobility transistors (HEMTs) based on AlGaN/GaN heterostructures are widely studied because of their very promising electron concentration in the 2D electron gas (2DEG) (12×1013cm2) and very high electron mobility (10.000 cm2V/s).1 The advantageous high 2DEG concentrations are due to the spontaneous and piezoelectric polarization in the AlGaN/GaN heterostructure. In the last few years, ScAlN, a scandium-based alloy, has been proposed as a substitute of the AlGaN for the next HEMT generation due to its superior material properties, such as the 3× higher spontaneous polarization.2 As shown by Hardy et al.,3 this results in 2DEG concentrations of 3.4×1013cm2. Moreover, ScAlN-based HEMT prototypes have been recently manufactured confirming theoretical predictions.4,5 The phenomena that dominate on the charge transport in such devices can be accurately described by performing TCAD simulations, relying on the solution of the drift–diffusion model in a discrete domain and accounting for the main features of heterostructures and polarization. To this purpose, the transport model needs to be specialized also for the ScAlN material using a specific set of physical models and parameters. As a starting point, the ScAlN energy structure is needed to correctly account for the energy gap (Eg) and the density of states through the effective masses (m), adopted when assuming a parabolic dispersion relation E(k) close to the conduction band minimum. The parabolic band is expressed as

E(k)=2k22m,
(1)

where is the reduced Plank constant and k is the wave vector. From Eq. (1), the density of states (DOS) can be analytically calculated.6 Such quantity provides the amount of free charges in the bulk and their temperature dependency. When treating alloys, such as AlGaN and ScAlN, supercells (SCs) are needed to properly take into account the random distribution of atoms inside the lattice. However, by performing density functional theory (DFT) simulations using SCs, the number of energy eigenvalues becomes proportional to the SC size, leading to a much more complex band structure, which needs to be treated with appropriate techniques. To this purpose, the unfolding technique has been developed to map the energy eigenvalues obtained from SC calculations into an effective band structure (EBS) recovering an approximate dispersion relation E(k), which collects the main features of the alloy properties.7–9 Recently, it has been adopted to extract the electron effective mass of the lowest-lying conduction band close to the Γ point of Al0.5Ga0.5N.10 Despite DFT simulations of ScAlN SCs were recently performed by Zhang et al.,11 the effects of the Sc incorporation on the low energy transport properties have not been clarified yet. In this work, for the first time, the electron effective masses of ScAlN and AlGaN have been extracted for different values of the mole fraction x using the EBS. The analysis confirms that the ScAlN alloy shows a much more complex structure with respect to AlGaN due to the rock-salt configuration of the ScN that mixes with the AlN wurtzite structure. The reported results are fundamental in order to extend the future TCAD analysis to such kind of new material.

The electronic properties of ScAlN and AlGaN have been calculated using the density functional theory (DFT) via the Quantum ESPRESSO (QE) package.12 Norm-conserving (NC) pseudo-potential has been employed with the energy cut-off fixed to 100Ry. A 8×8×8 Monkhorst–Pack grid has been used to sample the Brillouin zone of the wurtzite AlN (w-AlN) and wurtzite GaN (w-GaN) primitive cells (PCs). For the SC self-consistent calculations, a 4×4×4 k-point grid has been used. The general gradient approximation (GGA) as parameterized by Perdew, Burke, and Ernzerhof (PBE)13 has been adopted for the variable-cell relaxation. The lowest energy configuration has been achieved with all forces smaller than 0.001 Ry/aB and the pressure below 0.5 kbar.

In order to emulate the random position of Sc and Al atoms in the alloy, the special quasi-random structure (SQS) technique has been employed. More specifically, the Monte Carlo algorithm implemented in the Alloy Theoretic Automatic Toolkit (ATAT)14 has been used to generate 2×2×2 supercells containing a total number of 32 atoms, 16 of nitrogen and 16 of alloy atoms mixed with different concentrations.

Starting from the self-consistent solutions of the Kohn–Sham equations, the unfolding technique has been used to construct the EBS of both ScxAl1xN and AlxGa1xN for different values of x. To ensure the quality of the unfolding procedure around the Γ point, the values of the spectral weights have been checked. They are defined as8 

PKm(ki)=n|Km|kin|2,
(2)

where m and n correspond to band indices, while k and K are wave vectors of the PC and the SC, respectively. This quantity represents the amount of Bloch character ki preserved in |Km at En=Em. If the SC is obtained by repeating identical PCs, the spectral weights reduce to

PKm(ki)=gn(ki)δ(EmEn),
(3)

where gn(ki) is the bulk degeneracy of the PC state |Kin at En. In w-AlN and w-GaN supercells, PKm(ki)=1 at each eigenvalue En(ki) of the PC, 0 otherwise. When considering pseudo-binary alloys, the spectral weights are no more δ functions. However, if the perturbation to the crystal symmetry is small, the spectral weights corresponding to the lowest-lying conduction band around the Γ point are still close to 1. Under such assumptions, the effective masses can be calculated by fitting the DFT bands around the minimum of the conduction band against the Kane model for non-parabolic spherical bands, which gives more accurate results when compared to Eq. (1). It reads

ε(1+αε)=2k22m.
(4)

As the energetic minimum is characterized by an ellipsoidal shape, the electron effective mass can be calculated by analyzing the band structure near Γ along the A direction (m, parallel to the c direction of the crystal) and along the M and K directions (m, perpendicular to the c direction). It has been confirmed that the electron effective mass in the M and K directions is equal.15 Thus, m=(mmm)1/3 is the corresponding DOS effective mass.6 The ScAlN DOS has been calculated integrating the Brillouin zone on an 8×8×8 k-point grid adopting the tetrahedron method.16 

DFT simulations of bulk GaN and AlN have been performed to choose the most suitable approximations for the exchange-correlation potential. Lattice parameters of w-AlN and w-GaN, resulting from structural optimization, are in nice agreement with previous works based on GGA-PBE simulations and experiments as reported in Table I. For the sake of comparison, starting from the relaxed structure obtained from PBE, the following exchange correlation potential has been tested to calculate the electronic properties of the materials: the GGA-PBE potential used for relaxation, the modified Becke–Johnson (mBJ) with default parameters (which, in the QE package, is implemented only for norm-conserving pseudo-potential), and the hybrid pseudo-potential as defined by Heyd–Scuseria–Ernzerhof (HSE)17 with a 4×4×4 q-point grid. In the latter case, the Wannier90 code18 has been used to efficiently calculate the band structure. The energy gap and the effective masses have been reported in Table II and compared with previous works based on DFT simulations and experiments. As expected, calculations performed with the PBE approximations provide a strong underestimation of the energy gap Eg. On the contrary, a good agreement has been found between mBJ-PBE and HSE, which, in terms of Eg, are closer to the corresponding experimental values.11 As far as the calculation of the effective masses is concerned, it has been found that the HSE approach provides the most accurate results when compared with experimental data,19 while mBJ meta-GGA overestimates them along both the monitored crystalline orientations. In contrast, PBE provides good results for w-AlN while slightly underestimates m for w-GaN.20 The PBE potential has been then chosen as the most suitable approximation of the exchange-correlation term in the Kohn–Sham equations for the following reasons: (i) it is a full ab initio method since there are no fitting parameters, (ii) it has a lower computational cost when compared to mBJ and, especially, to HSE, and (iii) the error on the effective masses is similar or even lower then the mBJ one.

TABLE I.

Lattice constants of w-GaN and w-AlN obtained from variable-cell relaxation.

a (Å)c (Å)u (Å)
w-GaN PBE (this work) 3.2129 5.2332 0.3770 
 PBE (Ref. 113.244 5.276 0.377 
 PBE (Ref. 193.210 5.232 0.377 
 Exp. (Ref. 193.19 5.200 0.377 
w-AlN PBE (this work) 3.1269 5.0129 0.3817 
 PBE (Ref. 113.125 5.010 0.382 
 PBE (Ref. 193.13 5.039 0.381 
 Exp. (Ref. 193.11 4.97 0.382 
a (Å)c (Å)u (Å)
w-GaN PBE (this work) 3.2129 5.2332 0.3770 
 PBE (Ref. 113.244 5.276 0.377 
 PBE (Ref. 193.210 5.232 0.377 
 Exp. (Ref. 193.19 5.200 0.377 
w-AlN PBE (this work) 3.1269 5.0129 0.3817 
 PBE (Ref. 113.125 5.010 0.382 
 PBE (Ref. 193.13 5.039 0.381 
 Exp. (Ref. 193.11 4.97 0.382 
TABLE II.

Calculated energy gap and electron effective masses of w-GaN and w-AlN.

Eg (eV)m (m0)m (m0)
w-GaN PBE (this work) 1.76 0.17 0.15 
 PBE 2 × 2 × 2 SC (this work) 1.76 0.17 0.15 
 PBE (Ref. 111.58 n.a. n.a. 
 PBE (Ref. 20n.a. 0.17 0.16 
 PBE (Ref. 191.69 n.a. n.a. 
 mBJ (this work) 2.93 0.26 0.24 
 mBJ (Ref. 112.88 n.a n.a 
 HSE (this work α = 0.25) 2.80 0.21 0.17 
 HSE (this work α = 0.30) 3.03 0.21 0.17 
 HSE (Ref. 15α = 0.31) 3.48 0.22 0.19 
 Exp. (Ref. 213.50 0.18–0.29 0.18–0.29 
w-AlN PBE (this work) 4.08 0.30 0.28 
 PBE 2 × 2 × 2 SC (this work) 4.08 0.30 0.28 
 PBE (Ref. 114.09 n.a. n.a. 
 PBE (Ref. 194.02 n.a. n.a. 
 mBJ (this work) 5.51 0.40 0.37 
 mBJ (Ref. 115.57 n.a n.a. 
 HSE (this work α = 0.25) 5.41 0.34 0.33 
 HSE (this work α = 0.30) 5.68 0.31 0.31 
 HSE (Ref. 15α = 0.31) 6.04 0.32 0.30 
 Exp. (Ref. 216.28 0.29–0.45 0.29–0.45 
Eg (eV)m (m0)m (m0)
w-GaN PBE (this work) 1.76 0.17 0.15 
 PBE 2 × 2 × 2 SC (this work) 1.76 0.17 0.15 
 PBE (Ref. 111.58 n.a. n.a. 
 PBE (Ref. 20n.a. 0.17 0.16 
 PBE (Ref. 191.69 n.a. n.a. 
 mBJ (this work) 2.93 0.26 0.24 
 mBJ (Ref. 112.88 n.a n.a 
 HSE (this work α = 0.25) 2.80 0.21 0.17 
 HSE (this work α = 0.30) 3.03 0.21 0.17 
 HSE (Ref. 15α = 0.31) 3.48 0.22 0.19 
 Exp. (Ref. 213.50 0.18–0.29 0.18–0.29 
w-AlN PBE (this work) 4.08 0.30 0.28 
 PBE 2 × 2 × 2 SC (this work) 4.08 0.30 0.28 
 PBE (Ref. 114.09 n.a. n.a. 
 PBE (Ref. 194.02 n.a. n.a. 
 mBJ (this work) 5.51 0.40 0.37 
 mBJ (Ref. 115.57 n.a n.a. 
 HSE (this work α = 0.25) 5.41 0.34 0.33 
 HSE (this work α = 0.30) 5.68 0.31 0.31 
 HSE (Ref. 15α = 0.31) 6.04 0.32 0.30 
 Exp. (Ref. 216.28 0.29–0.45 0.29–0.45 

In order to check the consistency of the simulation setup, the fully relaxed band structure of 2×2×2 SCs of w-GaN and w-AlN has been calculated and compared with the PC calculations reported above. As expected, no differences in the energy gap and effective masses have been found. Moreover, the observed differences on the lattice constants can be considered negligible. Thus, simulations of the two alloys under investigation have been carried out. The EBS of AlxGa1xN and ScxAl1xN for four different values of x is reported in Figs. 1 and 2, respectively. Since both AlN and GaN are stable in the wurtzite phase, the increase of disorder in AlxGa1xN is small in the whole composition range; thus, the unfolding procedure can be successfully applied. The corresponding lattice parameters as a function of x are reported in Table III showing an approximately linear decrease with x (Vegard’s law). The same considerations apply to the electron effective masses and to the energy gap as shown in Fig. 3. On the contrary, ScN is stable in the rock-salt phase. For this reason, in ScxAl1xN, the amount of disorder strongly increases with x making useless the concept of EBS for high Sc concentrations. The DFT calculations of ScxAl1xN correspond to w-AlN for x=0 and to layered-hexagonal ScN (lh-ScN) for x=1 with the internal parameter u [that characterizes the distance between the (0001) planes containing metal atoms and N atoms] fixed to u=0.5. The latter structure is unstable, and the equilibrium crystal structure of ScN is cubic rock-salt. However, the Sc–N bond lengths in lh-ScN are very close to the one of cubic rock-salt ScN.22 The calculated lattice parameters are reported in Table III. In Fig. 4, the ScAlN lattice parameters are compared with those of Ref. 11 showing good agreement. The Vegard law can be applied on a for 0x0.50,11 while it is not valid for c since it is a non-monotonic function of x. However, if x0.25, the conduction band minimum is still located in the Γ-point and the corresponding value of the spectral weights7–9 are close to 1. Therefore, it is still possible to extract m around the Γ point. It is worth nothing that, for RF and power electronic applications, the adopted Sc concentration ranges between x=10% and x=26%.4,23,24 Thus, the present analysis is sufficient to guarantee an accurate estimation of the electron effective masses for TCAD simulations of Sc-based HEMTs. In Fig. 5, the energy gap and the electron effective masses are reported as functions of x. As expected, the energy gap decreases with x, while an anomalous behavior of both masses m and m has been found. Along the ΓM direction, m slightly decreases in the range 0x0.1875, while it significantly increases at x=0.25. The opposite is true along the ΓA direction for m. The electron effective masses of AlGaN and ScAlN are mostly affected by two factors: the variation of the chemical composition and the consequent alteration of the crystal structure. In AlGaN, both these phenomena result in an increase of m with x, leading to the Vegard law. In order to gain a physical insight on the role played by the Sc concentration on m, two different sets of simulations have been performed: (i) calculation of the ScAlN band structure without performing relaxation (with lattice constants and atomic positions of w-AlN) and (ii) calculation of the AlN band structure using lattice constants and atomic positions from the variable-cell relaxation of the ScxAl1xN. The electron effective masses along both the ΓM and ΓA directions have been calculated and reported in Fig. 6. It has been found that the presence of Sc atoms leads to an increase of m in both directions. Vice versa, the increase of the lattice constants a and c along with the internal parameter u reduces m. The superposition of these two phenomena leads to the observed dependence of the effective mass on x in Fig. 5.

FIG. 1.

EBS of AlxGa1xN for different Al concentrations (x = 0, 0.0625, 0.125, 0.25).

FIG. 1.

EBS of AlxGa1xN for different Al concentrations (x = 0, 0.0625, 0.125, 0.25).

Close modal
FIG. 2.

EBS of ScxAl1xN for different Sc concentrations (x = 0, 0.0625, 0.125, 0.25).

FIG. 2.

EBS of ScxAl1xN for different Sc concentrations (x = 0, 0.0625, 0.125, 0.25).

Close modal
FIG. 3.

Electron effective masses and the energy gap of AlxGa1xN as a function of the Al concentration.

FIG. 3.

Electron effective masses and the energy gap of AlxGa1xN as a function of the Al concentration.

Close modal
FIG. 4.

Lattice constants of ScxAl1xN as a function of the Sc content compared with the previous work in the literature. The black vertical dashed line indicates the transition from wurtzite to an unstable layered-hexagonal phase.

FIG. 4.

Lattice constants of ScxAl1xN as a function of the Sc content compared with the previous work in the literature. The black vertical dashed line indicates the transition from wurtzite to an unstable layered-hexagonal phase.

Close modal
FIG. 5.

Electron effective masses and energy gap of ScxAl1xN as a function of the Sc concentration.

FIG. 5.

Electron effective masses and energy gap of ScxAl1xN as a function of the Sc concentration.

Close modal
FIG. 6.

Solid lines: electron effective masses of ScAlN as a function of x calculated without performing variable-cell relaxation. Dashed lines: electron effective masses extracted from simulation of 2×2×2 AlN SC performed using lattice constants and atomic positions from variable-cell relaxation of ScAlN.

FIG. 6.

Solid lines: electron effective masses of ScAlN as a function of x calculated without performing variable-cell relaxation. Dashed lines: electron effective masses extracted from simulation of 2×2×2 AlN SC performed using lattice constants and atomic positions from variable-cell relaxation of ScAlN.

Close modal
TABLE III.

Calculated lattice parameter a and c of ScxAl1−xN and AlxGa1−xN. 0 (PC) corresponds to calculations on a single cell with x = 0, and 0 (SC) corresponds to calculations on the corresponding supercell.

ScxAl1−xNAlxGa1−xN
xa (Å)c (Å)a (Å)c (Å)
0 (PC) 3.1269 5.0129 3.2129 5.2332 
0 (SC) 3.1267 5.0132 3.2123 5.2343 
0.0625 3.1534 5.0356 3.2072 5.2221 
0.125 3.1797 5.0547 3.2013 5.2090 
0.1875 3.1994 5.0900 n.a. n.a. 
0.25 3.2409 5.0875 3.1894 5.1836 
0.50 n.a. n.a. 3.1675 5.1311 
0.75 n.a. n.a. 3.1469 5.0741 
n.a. n.a. 3.1267 5.0132 
ScxAl1−xNAlxGa1−xN
xa (Å)c (Å)a (Å)c (Å)
0 (PC) 3.1269 5.0129 3.2129 5.2332 
0 (SC) 3.1267 5.0132 3.2123 5.2343 
0.0625 3.1534 5.0356 3.2072 5.2221 
0.125 3.1797 5.0547 3.2013 5.2090 
0.1875 3.1994 5.0900 n.a. n.a. 
0.25 3.2409 5.0875 3.1894 5.1836 
0.50 n.a. n.a. 3.1675 5.1311 
0.75 n.a. n.a. 3.1469 5.0741 
n.a. n.a. 3.1267 5.0132 

Differently from w-AlN, where the second lowest-lying minimum of the conduction band is located along the M–L direction, in ScAlN, it is found to be in the K-point, and as x increases, it moves toward lower energies (Fig. 2). This is clearly correlated with the transition from a direct to an indirect gap (ΓK). For x=1 (lh-ScN), the conduction band minimum is located in the K-point of the Brillouin and can be mostly ascribed to d-orbitals of Sc atoms.25 By increasing x, the number of states provided by such orbitals increase, and due to lattice deformation toward a layered-hexagonal crystal structure, the K-point minimum moves toward lower energies. As shown in Fig. 7, the DOS corresponding to different values of x is significantly influenced by the change of the local minimum in the K-point, leading to a shift of the corresponding DOS peak, which might influence the high-energy transport properties.

FIG. 7.

Density of states of ScxAl1xN for different values of the Sc concentration (x = 0, 0.0625, 0.125, 0.25). Dots show the value of the energy corresponding to the minimum of the conduction band in the K-point extracted from the EBS. Black square: value of the energy corresponding to the minimum of the conduction band along the M–L direction extracted from the w-AlN EBS.

FIG. 7.

Density of states of ScxAl1xN for different values of the Sc concentration (x = 0, 0.0625, 0.125, 0.25). Dots show the value of the energy corresponding to the minimum of the conduction band in the K-point extracted from the EBS. Black square: value of the energy corresponding to the minimum of the conduction band along the M–L direction extracted from the w-AlN EBS.

Close modal

In this paper, first-principles calculations of the electron effective masses of AlxGa1xN and ScxAl1xN have been performed using the unfolding technique. It has been confirmed that in AlxGa1xN, the wurtzite crystal structure is approximately preserved independently on x. For this reason, both Eg and m linearly increase with the Al concentration. On the contrary, in ScxAl1xN, the amount of disorder strongly increases with x; thus, the effective band structure can be extracted only for 0x0.25. In this range, the lattice constants c and a are non-monotonic functions of x. In addition, the minimum in the K valley moves toward lower energies, leading to new peaks in the density of states. To conclude, the present analysis gives an accurate estimation of the electron effective mass of AlxGa1xN and ScxAl1xN as functions of x in the most interesting ranges of values for future applications. In the former case, the Vegard law can be adopted to nicely predict the linear change of m. In the latter one, DFT simulations are required to calculate m, which is a fundamental ingredient to perform accurate TCAD analyses of HEMTs for power and RF applications at the design stage.

This research was partially funded by the EC project GAN4AP (G.A. 101007310) via the IUNET Consortium.

The authors have no conflicts to disclose.

Luigi Balestra: Conceptualization (equal); Data curation (lead); Methodology (equal); Validation (lead); Visualization (lead); Writing – original draft (lead). Elena Gnani: Conceptualization (equal); Methodology (equal); Validation (supporting); Writing – review & editing (supporting). Susanna Reggiani: Funding acquisition (lead); Methodology (equal); Supervision (lead); Validation (supporting); Writing – review & editing (supporting).

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

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