Electron charges and distribution profiles induced by polarization gradients at the interfaces of pseudomorphic, hexagonal ScxAl1−xN/GaN- and ScxAl1−xN/InN-heterostructures are simulated by using a Schrödinger–Poisson solver across the entire range of random and metal-face ScxAl1−xN-alloys, considering the transition from wurtzite to hexagonal layered crystal structure. In contrast to previous calculations of polarization-induced sheet charges, we use Dryer’s modern theory of polarization, which allows for consideration of the spontaneous polarization measured on ferroelectric ScxAl1−xN-layers. Because the sheet density of the electrons accumulating at the heterostructure interfaces can strongly depend both on the data set of the piezoelectric and structural coefficients and on the alloying region of the ScxAl1−xN-layers in which the transition from the wurtzite to the hexagonal layered crystal structure occurs, we have calculated the charge carrier sheet densities and profiles for three representative data sets and evaluated their relevance for devices. We predict electron sheet densities of ( 2.26 ± 0.20 ) × 10 14 c m 2 and ( 6.25 ± 0.20 ) × 10 14 c m 2 for all three sets of data for Ni/AlN/InN- and Ni/ScN/InN-heterostructures, respectively. We demonstrate that the polarization-induced interface charges of Ni/ScxAl1−xN/InN-heterostructures are always positive, tend to increase with increasing Sc-content, and can cause electron accumulations that lead to flooding of the triangular quantum wells at the semiconductor interface. We identify Ni/ScxAl1−xN/GaN-heterostructures with 0.13 x 0.19 as particularly promising candidates for the processing of energy-efficient high electron mobility transistors due to their missing or low mechanical strain and their large electron sheet densities between ( 4.11 ± 0.20 ) × 10 13 c m 2 and ( 6.37 ± 0.20 ) × 10 13 c m 2. Furthermore, we present simulation results of highly strained Ni/ScxAl1−xN/GaN-heterostructures for 0.81 x 1.0, which point to electron accumulations of up to ( 8.02 ± 0.40 ) × 10 14 c m 2. These heterostructures are not suitable for transistor devices, but they may be of great interest for the implementation of low impedance contacts.

Alloying aluminum-nitride (AlN) with scandium-nitride (ScN) to form high quality crystalline thin films of wide bandgap semiconductors with hexagonal crystal structure has drawn tremendous attention due to the significantly enhanced piezoelectric response, the high pyroelectric polarization, and the ferroelectric behavior it enables.1–3 The high piezoelectric, pyroelectric, and ferroelectric polarizations in wurtzite ScxAl1−xN-crystals enrich the dimension of polarization engineering in group-III-nitride based heterostructures and provide opportunities for the integration of novel functionalities into electronic and optoelectronic devices. The possibility for hexagonal ScxAl1−xN-layers to be grown biaxially compressive as well as biaxially tensile strained on GaN- and InN-buffer layers creates the opportunity to engineer heterostructures with interfaces free of polarization sheet charges, with the benefit to light emitting devices, because of the missing Stark effect4 or to design heterostructures with high positive as well as negative polarization-induced interface charges in order to optimize electron and hole distribution profiles according to the need of electronic devices.5,6 Analogous to GaxAl1−xN and InxAl1−xN, ScxAl1−xN is a promising barrier layer in group-III-nitride based high electron mobility transistors (HEMTs), but offers larger gradients in polarization at the barrier–channel interface and, therefore, higher sheet electron concentrations of the two-dimensional electron gases (2DEGs) confined in triangular quantum wells.7,8 Pseudomorphic, nitrogen-polar Sc0.18Al0.82N/GaN-based heterostructures of hexagonal crystals with sheet electron concentrations confined at a lattice-matched barrier/buffer interface of up to 4.88 × 1013 cm−2 have been demonstrated by Wang et al.9 This sheet charge is comparable to the one of a 2DEG confined at the interface of a pseudomorphic, metal-polar AlN/GaN-heterostructure.10–12 A Sc0.19Al0.81N/GaN-heterostructure, as opposed to an AlN/GaN-heterostructure, does not face high thermal and lattice mismatch and is not limited in barrier thickness, as no strain-relaxation-related formation of dislocations or cracking occurs. Recently, it was demonstrated that thin layers of hexagonal ScxAl1−xN can be grown on GaN by molecular beam epitaxy13–15 as well as by metalorganic chemical vapor deposition16,17 and processed into power electronic and high-frequency transistors. Hardy et al. have developed ScxAl1−xN/GaN-based high electron mobility transistors (HEMTs) that simultaneously achieved high current density (>3 A/mm) and breakdown voltage (>60 V) as well as excellent millimeter wave frequency performance. In addition, Adamski et al.5 predicted a large polarization discontinuity of 1.358 C m 2 at the interface of wurtzite GaN ( 000 1 ¯ ) and rock-salt ScN ( 111 ), which is capable of attracting an electron sheet density of up to 8.5 × 1014 cm−2. This electron sheet charge is about 100 times higher in comparison to the 2DEGs confined in state-of-the-art high-power millimeter wave transistors based on Ga0.75Al0.25N/GaN-heterostructures.

The large range of polarization-induced interface charges simulated as well as the experimental proof of high performance HEMTs motivate our scientific advance to determine the electron sheet densities as well as distribution profiles at interfaces of pseudomorphic, hexagonal, metal face ScxAl1−xN/GaN- and ScxAl1−xN/InN-heterostructures, in order to identify the structures with the highest potential for processing novel optoelectronic and electronic devices. A prerequisite for this is an extensive understanding of the structural, mechanical, and polarization properties of hexagonal ScN- and AlN-crystals as well as their ternary alloys, spanning the entire range of compositions. At ambient temperature and pressure, AlN is known to crystallize in the hexagonal wurtzite structure with the space group P 6 3 m c, the lattice parameters a A l N = 3.110 Å, c A l N = 4.994 Å and an internal cell parameter of u = 0.381 (Ref. 6 and references listed). The aluminum and nitrogen atoms form individual sublattices, which are comparable to hexagonal close-packed structures (Fig. 1).

FIG. 1.

Schematic representation of the hexagonal wurtzite (left hand side), hexagonal layered (middle position), and cubic rock-salt crystal lattice (right hand side) of ScxAl1−xN-alloys.

FIG. 1.

Schematic representation of the hexagonal wurtzite (left hand side), hexagonal layered (middle position), and cubic rock-salt crystal lattice (right hand side) of ScxAl1−xN-alloys.

Close modal

Due to its wurtzite crystal structure, AlN is non-centrosymmetric (i.e., lacks inversion symmetry) and shows piezoelectric and pyroelectric properties, which centrosymmetric crystals lack.18,19 AlN can be forced to crystallize in the zinc blende cubic structure with the space group F 4 ¯ 3 m and NaCl-like rock-salt structure (Fig. 1) with the space group F m 3 ¯ m.20,21 The zinc blende structure of AlN is stable only when it is very thin (1.5–2.0 nm) and it transforms to the wurtzite structure at a larger thickness.22,23 There are experimental measurements showing that AlN transforms from the wurtzite to the rock-salt phase under high pressure.24,25 Upon releasing pressure, the rock-salt phase persists down to ambient conditions.

The hexagonal structure of ScN has been predicted to be a metastable state, which has the non-polar flattened-layer structure with the space group P63mmc. The crystal structure of hexagonal ScN closely relates to the wurtzite structure, corresponding to the internal cell parameter u = 0.5. The lattice parameters of hexagonal ScN predicted to range between a ScN = 3.692 3.723 Å and c ScN = 4.501 4.522 Å.26–28 Important to note here is that polar structures of ScN are energetically unstable. ScN in both non-polar cubic rock-salt and hexagonal phases should be non-piezoelectric along the [001]-direction as well as non-pyroelectric due to its crystal symmetries. The calculated total energy of cubic ScN is lower than that of hexagonal ScN by 0.18 eV/f.u. and wurtzite AlN is more stable in energy compared to the cubic rock-salt AlN by 0.33 eV/f.u.28 Based on these simulated results, a phase transition from the hexagonal to the cubic crystal structure must be expected if, starting from wurtzite AlN, an increasing number of Al-atoms are substituted by Sc-atoms. Zhang et al.26 and Talley et al.29 predict a phase transition of single crystals to occur from wurtzite to rock-salt structure at alloy compositions of x = 0.56 and 0.67, respectively. Density functional theory studies predict an elastic softening of wurtzite ScxAl1−xN-alloys caused by the competition between coordination states of nitrogen in hexagonal ScN and wurtzite AlN.3,30,31 The lower electronegativity of Sc- compared to Al-atoms also affects chemical bonds in a way that generates greater Born effective charges. These findings are thought to be responsible for the increased values of piezoelectric coefficients observed for wurtzite ScxAl1−xN-alloys in comparison to AlN.

In this publication, we focus on the determination of polarization-induced electron sheet charges and distribution profiles located at interfaces of hexagonal, pseudomorphic, metal face ScxAl1−xN/GaN- and ScxAl1−xN/GaN-heterostructures considering the changes in elastic and piezoelectric coefficients as well as in spontaneous polarization caused by an increasing substitution of Al- by Sc-atoms in the top layer. Due to the small dependence of the piezoelectric and elastic coefficients on the biaxial strain calculated by Daoust et al.32 for a given Sc-content, we base our simulations on measured and simulated coefficients for unstrained crystals and layers. We assume that the hexagonal crystal structure of ScxAl1−xN is retained in the heterostructures considered, spanning the entire range of conceivable alloys, meaning that we neglect the transition to cubic phase. Stabilization of the hexagonal structure for high alloy compositions (x > 0.5) might be conceivable by epitaxial processes, which can be implemented far from thermodynamic equilibrium like magnetron sputtering33,34 or pulsed laser ablation.35,36 Embedding thin ScxAl1−xN-layers in superlattices or heterostructures based on InN/ScxAl1−xN/InN-stacks might be another possibility to preserve the hexagonal crystal structure toward higher Sc-contents.37,38 In addition, one should keep in mind that the maximum possible Sc-content of application relevant crystals with hexagonal structure is not necessarily limited by the critical layer thickness (i.e., by the stress-induced formation of dislocations), as practical applications of ScxAl1−xN may tolerate dislocation densities of 106–108 cm−2.39 

To calculate electron sheet densities and distribution profiles induced by gradients in piezoelectric and spontaneous polarization present at ScxAl1−xN/buffer interfaces, we have to determine the piezoelectric and spontaneous polarization of the ternary alloy, keeping in mind that the buffer layers are pyroelectric but assumed to be unstrained and free of piezoelectric polarization. As a first step, we will determine the piezoelectric polarization of ScxAl1−xN for pseudomorphic heterostructures grown along the [0001]-axis. The ScxAl1−xN-layers are forced to adapt to the lattice parameter of the GaN- or InN-buffer crystals ( a buffer = a GaN = 3.1986 Å or a buffer = a InN = 3.6848 Å ).6 The resulting biaxial strain is isotropic in their basal plane ( ε 1 ( x ) = ε 2 ( x ) ) and no force is applied to the ScxAl1−xN-layers in the direction of growth and, as a consequence, they relax freely along the [0001]-axis by adapting ε 3 ( x ). The strain in the basal plane ε 1 buffer ( x ) and in growth direction ε 3 buffer ( x ) for a given buffer layer and alloy composition of x can be determined by
ε 1 buffer ( x ) = a ( x ) a 0 ( x ) a 0 ( x ) = a buffer a 0 ( x ) a 0 ( x ) ,
(1a)
ε 3 buffer ( x ) = c ( x ) c 0 ( x ) c 0 ( x ) = a buffer a 0 ( x ) a 0 ( x ) 2 C 13 ( x ) C 33 ,
(1b)
where a ( x ) and c ( x ) are the lattice parameters of the strained ScxAl1−xN-layers and a 0 ( x ), c 0 ( x ) and C 13 ( x ), C 33 ( x ) are the lattice parameters and stiffness coefficients of the relaxed ScxAl1−xN-crystals, respectively. The lattice parameters a 0 ( x ) and c 0 ( x ) predicted by first-principles calculations of Zhang et al.,26 Furuta et al.,27 Talley et al.,29 Urban et al.,40 and Wang et al.41 as well as measured by x-ray diffraction6,42,43 vs alloy composition x are presented in Fig. 2. These sets of data mainly differ in the range of alloy compositions in which the transition from wurtzite to hexagonal layered structures occurs. The simulated and measured lattice parameters a 0 ( x ) increase across the entire range of alloy composition, if the number of Al-atoms substituted by Sc-atoms rises. The lattice parameters simulated by Zhang et al.26 can be approximated by Vegard’s law for the intervals 0.0 ≤ x ≤ 0.50 and 0.63 ≤ x ≤ 1.0 with slopes of ( 0.51 ± 0.03 ) and ( 0.43 ± 0.03 ) Å, respectively, and are provided in Fig. 2 (solid line) to guide the eye. In the interval 0.5 ≤ x ≤ 0.625, a 0 ( x ) increases in a non-linear fashion from ( 3.369 ± 0.005 ) to ( 3.548 ± 0.005 ) Å and the dependence on alloy composition is somewhat more pronounced in comparison to the other two intervals mentioned.
FIG. 2.

Lattice parameters a 0 ( x ) and c 0 ( x ) of relaxed ScxAl1−xN-layers predicted by first-principles calculations of Zhang et al. (rhombus), Furuta et al. (squares), Talley et al. (circles), Urban et al. (triangles), and Wang et al. (crosses)26,27,29,40,41 (supercell) as well as determined experimentally by Ambacher et al. (black circles), Deng et al. (black triangles) and Teshigahara et al. (black squares)6,42,43 vs alloy composition x. The pointed, solid, and dashed lines are intended to guide the eye along the measured data as well as along the simulated data of Zhang et al.26 and Talley et al.29 

FIG. 2.

Lattice parameters a 0 ( x ) and c 0 ( x ) of relaxed ScxAl1−xN-layers predicted by first-principles calculations of Zhang et al. (rhombus), Furuta et al. (squares), Talley et al. (circles), Urban et al. (triangles), and Wang et al. (crosses)26,27,29,40,41 (supercell) as well as determined experimentally by Ambacher et al. (black circles), Deng et al. (black triangles) and Teshigahara et al. (black squares)6,42,43 vs alloy composition x. The pointed, solid, and dashed lines are intended to guide the eye along the measured data as well as along the simulated data of Zhang et al.26 and Talley et al.29 

Close modal
By looking at the lattice parameters c 0 ( x ) simulated and measured by the authors mentioned above, it can be noticed that in the interval 0.4 x 0.8 the functional behavior depends on the predicted range of x, in which the transition from wurtzite to hexagonal layered structure occurs. Whereas Furuta et al. and Zhang et al. predict the transition, indicated by a reduction in c 0 ( x ) from ( 5.002 ± 0.021 ) to ( 4.470 ± 0.026 Å ) as well as in c 0 ( x ) a 0 ( x ) from ( 1.492 ± 0.003 ) to ( 1.256 ± 0.015 ), to occur in the interval 0.44 ≤ x ≤ 0.67, Talley et al. predict a comparable change in the structural parameters for 0.56 ≤ x ≤ 0.78. Consequently, the interval in which the transition is predicted determines the range in which the strains ε 1 GaN ( x ) and ε 1 InN ( x ) of the ScxAl1−xN-layers grown on GaN- and InN-buffer change in a nonlinear fashion. Our measurements of the lattice parameters and the experimental data from Teshigahara et al. are in somewhat better agreement with the simulation results from Zhang et al. than with those of Talley et al. This suggests a transition from the wurtzite structure to the rock-salt or hexagonal layered structure at about x 0.5.6,42,26,29 Seeing as though the dependency of the lattice parameter a 0 ( x ) on the composition of the ScxAl1−xN-crystals can be approximated by Vegard’s law for low and high alloy compositions, the change in strain ε 1 buffer ( x ) with increasing substitution of the Al-atoms by Sc-atoms can also be described by linear functions dependent on x. For the calculation of the piezoelectric polarization based on the data of Zhang et al., we use the following linear approximations:
ε 1 GaN ( x ) = { ( 0.153 ± 0.005 ) x + ( 0.029 ± 0.003 ) for 0.0 x 0.50 , ( 0.105 ± 0.005 ) x ( 0.034 ± 0.003 ) for 0.63 x 1.0 ,
(2a)
ε 1 InN ( x ) = { ( 0.171 ± 0.004 ) x + ( 0.153 ± 0.003 ) for 0.0 x 0.50 , ( 0.118 ± 0.004 ) x ( 0.083 ± 0.003 ) for 0.63 x 1.0 ,
(2b)
which are shown in Fig. 3.
FIG. 3.

Strains ε 1 buffer ( x ) and ε 3 buffer ( x ) of hexagonal ScxAl1−xN-layers grown pseudomorphically on relaxed GaN- and InN-buffer layer vs alloy composition x. The strains ε 1 GaN ( x ) and ε 1 InN ( x ) are calculated from lattice parameters predicted by Zhang et al. (rhombus), Furuta et al. (squares), Talley et al. (circles), Urban et al. (triangles), and Wang et al. (crosses)26,27,29,40,41 (supercell) as well as determined experimentally by Ambacher et al. (black circles), Deng et al. (black triangles), and Teshigahara et al. (black squares).6,42,43 The solid and dashed lines are intended to guide the eye along the simulated data of Zhang et al.26 and Talley et al.29 

FIG. 3.

Strains ε 1 buffer ( x ) and ε 3 buffer ( x ) of hexagonal ScxAl1−xN-layers grown pseudomorphically on relaxed GaN- and InN-buffer layer vs alloy composition x. The strains ε 1 GaN ( x ) and ε 1 InN ( x ) are calculated from lattice parameters predicted by Zhang et al. (rhombus), Furuta et al. (squares), Talley et al. (circles), Urban et al. (triangles), and Wang et al. (crosses)26,27,29,40,41 (supercell) as well as determined experimentally by Ambacher et al. (black circles), Deng et al. (black triangles), and Teshigahara et al. (black squares).6,42,43 The solid and dashed lines are intended to guide the eye along the simulated data of Zhang et al.26 and Talley et al.29 

Close modal

The strain ε 3 buffer ( x ) of ScxAl1−xN-layers along the [0001]-direction is related to the strain in the basal plane ε 1 buffer ( x ) by the ratio of stiffness coefficients C 13 ( x ) C 33 ( x ) [see Eq. (1b)], which is also included in the calculation of the piezoelectric polarization. This ratio, determined for the simulated stiffness coefficients of Caro et al.,3 Talley et al.,29 Urban et al.,40 and Zhang et al.,44 is increasing, if starting with AlN, the number of Al-atoms substituted by Sc-atoms rises, reaching its maximum in the interval of transition from wurtzite to the hexagonal layered structure (Fig. 4). Because this phase transition is predicted for higher alloy compositions by Talley et al. in comparison to Zhang et al. and Urban et al., the alloy compositions at which this ratio C 13 ( x ) C 33 ( x ) reaches its maximum differ. But, in any modeling of structural parameters, the ratio of stiffness coefficients decreases with increasing alloy composition in the interval 0.64 ≤ x ≤ 1.0. Talley et al. calculated very similar stiffness coefficients C 13 and C 33 of about 90 and 365 GPa for wurtzite AlN and hexagonal layered ScN, respectively. Comparing the stiffness coefficients of the binary crystals with ScxAl1−xN in the transition region from wurtzite to hexagonal layered crystal structure, the value of C 13 has increased by 40% while the value of C 33 has decreased by 70%, indicating a pronounced softening of the alloys along the c-axis. Compared to the changes in C 13 ( x ), the variation in the stiffness coefficient C 33 ( x ) is much more pronounced and, therefore, dominates the functional course of the ratio C 13 ( x ) C 33 ( x ) as well as the dependence of strain ε 3 buffer ( x ) on alloy composition.

FIG. 4.

Ratio of stiffness coefficients C 13 ( x ) C 33 ( x ) vs alloy composition x, determined based on the simulated stiffness coefficients of Caro et al. (squares), Talley et al. (circles), Urban et al. (triangles), and Zhang et al. (rhombus).3,29,39,44 The solid and dashed lines are intended to guide the eye along sets of data, which mainly differ in the range of alloy compositions in which the phase transition from wurtzite to hexagonal layered structures occurs.

FIG. 4.

Ratio of stiffness coefficients C 13 ( x ) C 33 ( x ) vs alloy composition x, determined based on the simulated stiffness coefficients of Caro et al. (squares), Talley et al. (circles), Urban et al. (triangles), and Zhang et al. (rhombus).3,29,39,44 The solid and dashed lines are intended to guide the eye along sets of data, which mainly differ in the range of alloy compositions in which the phase transition from wurtzite to hexagonal layered structures occurs.

Close modal
It can be seen from Fig. 3 that the Sc0.19Al0.81N/GaN- and Sc0.73Al0.27N/InN-heterostructure can be epitaxially grown lattice-matched and, as a result, the ScxAl1−xN-barrier is unstressed [ ε 1 GaN ( x ) = ε 3 GaN ( x ) = 0 , for x = 0.19 ; ε 1 InN ( x ) = ε 3 InN ( x ) = 0, for x = 0.73 ] and free from piezoelectric polarization. For all other combinations of buffer layer and alloy composition, the ScxAl1−xN-layers are biaxially strained. The signs of the strains are related to the orientation of the piezoelectric polarization vector. For ε 1 buffer ( x ) 0 , ε 3 buffer ( x ) 0 , the ScxAl1−xN-layers are under biaxial tensile strain, and the vector of the piezoelectric polarization points from the surface of the heterostructure toward its interface ( P PE buffer ( x ) 0), if the crystals are metal-polar. The gradient in piezoelectric polarization causes a positive polarization-induced interface charge (as discussed in more detail in Sec. III C). If ε 1 buffer ( x ) 0 , ε 3 buffer ( x ) 0, the layers are under biaxial compressive strain, and the piezoelectric polarization vector points toward the heterostructure surface ( P PE buffer ( x ) 0 ) and the interface sheet charge induced by the gradient of piezoelectric polarization is negative. For biaxially strained ScxAl1−xN-layers, the piezoelectric polarization oriented parallel to the [0001]-axis can be calculated by6 
P PE buffer ( x ) = 2 ε 1 buffer ( x ) ( e 31 ( x ) e 33 ( x ) C 13 ( x ) C 33 ( x ) ) ,
(3a)
where
e 31 ( x ) e 33 ( x ) C 13 ( x ) C 33 ( x ) 0.
(3b)

The functional relationship between the piezoelectric coefficients e 31 ( x ) and e 33 ( x ) and the alloy composition for hexagonal ScxAl1−xN-layers as simulated are visualized in Fig. 5.

FIG. 5.

Simulated functional relationships between the piezoelectric coefficients e 31 ( x ), e 33 ( x ) , and the alloy composition x for hexagonal ScxAl1−xN-layers [Caro et al. (squares),3 Zhang et al. (rhombus),26 Momida et al. (crosses),28 Talley et al. (circles),29 and Urban et al. (triangles)].40 The pointed, solid, and dashed lines are intended to guide the eye along sets of data,40,26,29 which mainly differ in the range of alloy compositions in which the phase transition from wurtzite to hexagonal layered structures occurs.

FIG. 5.

Simulated functional relationships between the piezoelectric coefficients e 31 ( x ), e 33 ( x ) , and the alloy composition x for hexagonal ScxAl1−xN-layers [Caro et al. (squares),3 Zhang et al. (rhombus),26 Momida et al. (crosses),28 Talley et al. (circles),29 and Urban et al. (triangles)].40 The pointed, solid, and dashed lines are intended to guide the eye along sets of data,40,26,29 which mainly differ in the range of alloy compositions in which the phase transition from wurtzite to hexagonal layered structures occurs.

Close modal
Besides the fact that the values of e 31 ( x ) are negative and approximately a factor of 3–6 less compared to e 33 ( x ), the functional dependence of these two piezoelectric coefficients on alloy composition is similar. If, starting with AlN, the number of Al-atoms substituted by Sc-atoms rises, the values of the piezoelectric coefficients | e 31 ( x ) | and | e 33 ( x ) | increase in a non-linear fashion from ( 0.50 ± 0.09 ) C m 2 to ( 1.26 ± 0.05 ) C m 2 and ( 1.60 ± 0.12 ) C m 2 to ( 6.10 ± 0.16 ) C m 2, respectively. The alloy composition at which both coefficients reach their maximum values is dependent on the interval in which the transition from wurtzite to hexagonal layered structure is predicted. For x 0.75, a significant decrease in the piezoelectric coefficients is determined by all models, as the crystal structures approximate the non-polar hexagonal layered ScN. The drop in the values of the piezoelectric coefficients becomes particularly clear if they are plotted in relation to the ratio of the lattice parameters (Fig. 6). When the lattice parameter ratio is slightly reduced from (1.240 ± 0.005) to ( 1.228 ± 0.005 ), the values of piezoelectric coefficients | e 31 | and | e 33 | are reduced from ( 1.21 ± 0.02 ) C m 2 and ( 6.26 ± 0.02 ) C m 2, respectively, to almost zero. For the interval ( 1.250 ± 0.005 ) c 0 ( x ) a 0 ( x ) ( 1.610 ± 0.005 ), the correlation between the piezoelectric coefficients and the ratio of lattice parameters as simulated by Zhang et al.26 and Talley et al.29 can be described by linear approximations,
e 31 ( c 0 ( x ) a 0 ( x ) ) = ( ( 1.667 ± 0.040 ) c 0 ( x ) a 0 ( x ) ( 3.262 ± 0.040 ) ) C m 2 ,
(4a)
e 33 ( c 0 ( x ) a 0 ( x ) ) = ( ( 12.86 ± 0.04 ) c 0 ( x ) a 0 ( x ) + ( 22.20 ± 0.07 ) ) C m 2 .
(4b)
FIG. 6.

Piezoelectric coefficients e 31 and e 33 vs the ratio of lattice parameter c 0 ( x ) a 0 ( x ) for hexagonal ScxAl1−xN-layers [Zhang et al. (rhombus),26 Talley et al. (circles),29 and Urban et al. (triangles)40].

FIG. 6.

Piezoelectric coefficients e 31 and e 33 vs the ratio of lattice parameter c 0 ( x ) a 0 ( x ) for hexagonal ScxAl1−xN-layers [Zhang et al. (rhombus),26 Talley et al. (circles),29 and Urban et al. (triangles)40].

Close modal

We also observed good agreement of the simulated data from Urban et al.40 with the linear approximations. However, the slope of the function e 33 ( c 0 ( x ) a 0 ( x ) ) is twice as large compared to Eq. (4b).

Taking advantage of the elastic and piezoelectric coefficients and Eq. (3a), the piezoelectric polarizations P PE buffer ( x ) of the ScxAl1−xN-layers grown pseudomorphically on GaN- and InN-buffer layers are calculated and presented in Fig. 7. The values of piezoelectric polarization of AlN-barriers on metal-polar GaN- and InN-buffer layers are negative ( P PE GaN ( x = 0 ) = ( 0.048 ± 0.005 ) C m 2 , P PE InN ( x = 0 ) = ( 0.305 ± 0.014 ) C m 2 ) and larger for an InN-buffer in comparison to a GaN-buffer layer because of the more severe mismatch. Taking the data sets of Zhang et al.26 and Talley et al.29 to calculate the piezoelectric polarization of ScxAl1−xN on GaN-buffer layer, a nonlinear enhancement is observed, causing the piezoelectric polarization to vanish for x = ( 0.19 ± 0.01 ), and reaching maximum values of ( 1.17 ± 0.02 ) C m 2 and ( 1.77 ± 0.02 ) C m 2, respectively, at an alloy composition of x = ( 0.72 ± 0.03 ).

FIG. 7.

Piezoelectric polarizations P PE buffer ( x ) caused by biaxial strains within barriers of pseudomorphic, ScxAl1−xN/GaN- and ScxAl1−xN/InN-heterostructures vs alloy composition x. The solid, dashed, and dotted lines visualize the piezoelectric polarization calculated based on the piezoelectric and stiffness coefficients provided by Zhang et al. (rhombus),26 Talley et al. (circles),29 and Urban et al. (triangles).40 

FIG. 7.

Piezoelectric polarizations P PE buffer ( x ) caused by biaxial strains within barriers of pseudomorphic, ScxAl1−xN/GaN- and ScxAl1−xN/InN-heterostructures vs alloy composition x. The solid, dashed, and dotted lines visualize the piezoelectric polarization calculated based on the piezoelectric and stiffness coefficients provided by Zhang et al. (rhombus),26 Talley et al. (circles),29 and Urban et al. (triangles).40 

Close modal

The piezoelectric polarization of ScxAl1−xN on InN-buffer becomes more negative if an increasing number of Sc-atoms is added to the crystal lattice. At x = ( 0.53 ± 0.03 ), a minimal strain induced polarization of ( 0.45 ± 0.03 ) C m 2 is determined. In the range of alloy composition in which the lattice is strongly changed by the transition from wurtzite to the hexagonal layered structure, the piezoelectric polarization P PE InN ( x ) is significantly enhanced to a positive value of ( 1.17 ± 0.02 ) C m 2. For x > 0.83, the positive piezoelectric polarization of ScxAl1−xN on GaN- and on InN-buffer layers decreases and vanishes as the crystal structure approaches the non-polar hexagonal layered ScN-structure.

At this point, we have achieved a description of the piezoelectric polarization of the hexagonal ScxAl1−xN-layers and its gradient in comparison to the unstrained GaN- and InN-buffer layers over the entire alloy range. Because sheet charges are induced by gradients in piezoelectric and spontaneous polarization at the interfaces of the heterostructures under consideration, the next step toward the determination of polarization-induced electron sheet densities and distribution profiles is a precise determination of the spontaneous polarization of the ternary ScxAl1−xN-crystals.

Unstrained AlN, which has a wurtzite crystal structure, is distorted from the tetrahedral structure in an equilibrium state and exhibits a spontaneous polarization (PSP) in the c-axis.19,45,46 AlN shows pyroelectric and piezoelectric properties, but no ferroelectric behavior has been observed at room temperature. In addition to the polarization being non-linear in dependence of the electric field strength (E), pyroelectric and ferroelectric materials demonstrate a spontaneous non-zero polarization even when the electric field applied is zero. The distinguishing feature of ferroelectrics is that the spontaneous polarization can be reversed by a suitably strong electric field applied in the opposite direction.47,48 The polarization is, therefore, dependent not only on the current electric field but also on its history, yielding a hysteresis loop P(E). PSP equals the maximum retentive polarization value (Pr) that can be obtained from a hysteresis loop, if the volume of the ferroelectric investigated is free of charge and behave like a single crystal domain.

In 2019, Fichtner et al.2 observed the ferroelectricity of ScxAl1−xN at Sc-concentrations in the range of 0.27 ≤ x ≤ 0.43 experimentally for the first time. The range of alloy composition in which ferroelectricity was demonstrated at room temperature was confirmed and extended by Yasuoka et al. (0.10 ≤ x ≤ 0.34),49 Drury et al. (x = 0.4),50 Gund et al. (x = 0.2, 0.3),51 and Wang et al. (0.14 ≤ x ≤ 0.36).9, Figure 8 shows a polarization-electric field strength loop P(E) for a thin film ( d = 320 nm ) of Sc0.3Al0.7N deposited by co-sputtering on a Mo/Si(111) substrate, partially covered by a Ti/Pt/Au top contact.52 The retentive polarization P r ( E = 0 ) measured equals, for the sample investigated, its spontaneous polarization P SP ( x = 0.3 ) = ( 1.17 ± 0.03 ) C m 2. The value of spontaneous polarization is confirmed by the reversal charge q ( E ) measured as a current over time during the switch in polarization at the coercive field strength of E c ( x = 0.3 ) = ( 4.0 ± 0.3 ) MV cm (shown as open squares in Fig. 8). The spontaneous polarizations P SP ( x ) determined experimentally from P r ( x ) or the charge transfer during polarization switching52 combined with the data provided in Refs. 2, 27, 41, 45, 49, 50, and 52 are compiled in Fig. 9 and compared to the spontaneous polarizations P SP GaN and P SP InN of the buffer layers as predicted by Dreyer et al.45 

FIG. 8.

Experimentally determined polarization P ( E ) (open circles) and reversal charge q ( E ) (open squares) vs applied electric field strength E for a thin film ( d = 320 nm ) of Sc0.3Al0.7N deposited by co-sputtering on a Mo/Si(111) substrate.2,52 The star indicates the retentive polarization P r ( E = 0 ) which equals for the sample investigated, its spontaneous polarization P S P ( x = 0.3 ).

FIG. 8.

Experimentally determined polarization P ( E ) (open circles) and reversal charge q ( E ) (open squares) vs applied electric field strength E for a thin film ( d = 320 nm ) of Sc0.3Al0.7N deposited by co-sputtering on a Mo/Si(111) substrate.2,52 The star indicates the retentive polarization P r ( E = 0 ) which equals for the sample investigated, its spontaneous polarization P S P ( x = 0.3 ).

Close modal
FIG. 9.

Predicted [gray circles,27 gray squares (supercells),41 white squares (VCA),41 cross with shading45] and experimentally determined (black squares,2 black circles,49 black triangle,50 black rhombuses52) spontaneous polarization P S P ( x ) vs alloy composition x for random ScxAl1−xN-alloys with hexagonal crystal structure. The horizontal dashed lines indicate the spontaneous polarizations of the GaN- and InN-buffer layer.45 

FIG. 9.

Predicted [gray circles,27 gray squares (supercells),41 white squares (VCA),41 cross with shading45] and experimentally determined (black squares,2 black circles,49 black triangle,50 black rhombuses52) spontaneous polarization P S P ( x ) vs alloy composition x for random ScxAl1−xN-alloys with hexagonal crystal structure. The horizontal dashed lines indicate the spontaneous polarizations of the GaN- and InN-buffer layer.45 

Close modal
The report of Wang et al.41 on first-principles calculations aimed at elucidating the microscopic mechanisms underlying the enhancement of the piezoelectric response and the potential for ferroelectric switching in ScxAl1−xN-alloys up to x = 0.5. The calculations are based on density functional theory (DFT), and alloys are treated in two distinct ways: by performing explicit supercell calculations and by using the virtual crystal approximation (VCA). The authors find that, as the Sc-concentration increases, the spontaneous polarization of the alloy decreases, and so does the energy barrier for switching the direction of polarization. In addition, the results of first-principles calculations performed by Furuta et al.27 are provided in Fig. 9 to support a systematic understanding of the dependence of the spontaneous polarization P S P ( x ) on the Sc-concentration, spanning the entire range of alloys with hexagonal crystal structure. The fit to the data of Furuta et al. (solid line in Fig. 9) has been used for the determination of the total polarization P T buffer ( x ) of the biaxial strained ScxAl1−xN-barriers. The total polarization is the sum of the piezoelectric and spontaneous polarization,
P T buffer ( x ) = P S P ( x ) + P PE buffer ( x ) ,
(5)
which is shown in Fig. 10 for pseudomorphic ScxAl1−xN/GaN- and ScxAl1−xN/InN-heterostructures over the entire range of ternary alloys. By introducing Sc into the upper layer of a heavily strained AlN/InN-heterostructure, the total polarization P T InN ( 0 ) = ( 1.04 ± 0.04 ) C m 2 decreases continuously, and reaches a value of ( 0.15 ± 0.01 ) C m 2 or ( 0.15 ± 0.05 ) C m 2, in the lattice-matched Sc(0.73±0.03)Al(0.27±0.03)N/InN-heterostructure, as calculated from the data sets of Zhang et al. and Talley et al., respectively.
FIG. 10.

Calculated total polarization P T buffer ( x ) = P PE buffer ( x ) + P SP ( x ) within the ternary alloys of pseudomorphic, metal-polar ScxAl1−xN/GaN- and ScxAl1−xN/InN-heterostructures vs alloy composition x. The solid, dashed, and dotted lines visualize the total polarization calculated based on the piezoelectric and stiffness coefficients provided by Zhang et al. (rhombus),26 Talley et al. (circles),29 and Urban et al. (triangles).40 

FIG. 10.

Calculated total polarization P T buffer ( x ) = P PE buffer ( x ) + P SP ( x ) within the ternary alloys of pseudomorphic, metal-polar ScxAl1−xN/GaN- and ScxAl1−xN/InN-heterostructures vs alloy composition x. The solid, dashed, and dotted lines visualize the total polarization calculated based on the piezoelectric and stiffness coefficients provided by Zhang et al. (rhombus),26 Talley et al. (circles),29 and Urban et al. (triangles).40 

Close modal

Spontaneous polarization dominates the total polarization of the hexagonal ScxAl1−xN-layers deposited on InN up to alloy compositions of x 0.5, exceeding the value of piezoelectric polarization by more than a factor of two. Mainly due to the higher piezoelectric coefficient e 33 ( x ) (Fig. 5) the piezoelectric and total polarization determined from the data set of Urban et al. decreases more rapidly in the interval 0 x 0.5 compared to the values discussed above, reaching values of P PE InN ( 0.5 ) = ( 0.79 ± 0.03 ) C m 2 and P T InN ( 0.5 ) = ( 0.11 ± 0.03 ) C m 2, respectively. For ScxAl1−xN/InN-heterostructures with x ( 0.82 ± 0.02 ) ( P T InN ( 0.82 ) = ( 0.11 ± 0.03 ) C m 2 ), the ternary alloys exceed the non-polar hexagonal layered structure, recognizable by a disappearing total polarization. This is also the case with pseudomorphic ScxAl1−xN/GaN-heterostructures. However, the drop in total polarization is much more pronounced for these structures. This is due to the very high positive piezoelectric polarization caused by enormous strains in the ScxAl1−xN-layers with high Sc-contents. Maximum total polarizations of ( 1.92 ± 0.04 ) C m 2 and ( 1.30 ± 0.03 ) C m 2 are calculated at a Sc concentration of x = ( 0.68 ± 0.02 ) based on the data sets of Zhang et al. and Talley et al., respectively. Its change as a function of Sc-content is relatively small before the increase in total polarization in the interval 0.56 x 0.68. For AlN/GaN-heterostructures, a total polarization of ( 1.28 ± 0.01 ) C m 2 is calculated. This decreases only slightly to ( 1.22 ± 0.02 ) C m 2 for ScxAl1−xN/GaN-heterostructures with alloy compositions of up to x = ( 0.44 ± 0.02 ). It should be emphasized that the total polarization of the ternary alloy in the lattice-matched Sc(0.19±0.01)Al(0.81±0.01)N/GaN- ( P T GaN ( 0.19 ) = ( 1.24 ± 0.03 ) C m 2 ) differs significantly from the spontaneous polarization of the GaN-buffer layer ( P SP GaN = 1.312 C m 2 )44 and can cause gradients of polarization at the ScxAl1−xN/buffer interfaces. This is also the case for the lattice-matched Sc(0.73±0.03)Al(0.27±0.03)N/InN-heterostructures in which the spontaneous polarization of the InN-buffer ( P SP InN = 1.026 C m 2 )44 exceeds the total polarization of Sc0.73Al0.27N significantly, causing a gradient of polarization at the heterostructure interface.

A polarization-induced charge distribution is always associated with a gradient of polarization in space. As a consequence, localized polarization-induced sheet charges ( σ buffer ( x ) ) are generated at interfaces of ScxAl1−xN/buffer heterostructures, where the piezoelectric and spontaneous polarization can change abruptly. These can be calculated by
σ buffer ( x ) = Δ P SP buffer ( x ) 2 ε 1 buffer ( x ) ( e 31 ( x ) e 33 ( x ) C 13 ( x ) C 33 ( x ) P S P ( x ) ) ,
(6)
where Δ P SP buffer ( x ) = P SP buffer P SP ( x ) is the gradient in spontaneous polarization.45 The polarization-induced bound interface sheet charges of metal polar heterostructures, which are quantified by the number of elementary charges per area ( 1 e σ buffer ( x ) , e = 1.602 × 10 19 C ), are determined through division by the electron charge and presented in Fig. 11. Whenever these sheet charges are positive, they can attract mobile electrons via Coulomb interaction and can cause an accumulation of these charge carriers at the heterostructure interface. For pseudomorphic-, metal polar AlN/GaN-heterostructures, the polarization-induced interface sheet charge is positive ( 1 e σ GaN ( 0 ) = ( 6.75 ± 0.68 ) × 10 13 c m 2 ) causing the formation of a two-dimensional electron gas (2DEG). If Al-atoms of the AlN-barrier are replaced by Sc-atoms, the gradients in piezoelectric and spontaneous polarization at the interface to GaN decrease, causing a reduction of the polarization-induced interface charge.
FIG. 11.

Polarization-induced interface sheet charge 1 e σ buffer ( x ) of pseudomorphic, metal-polar ScxAl1−xN/GaN- and ScxAl1−xN/InN-heterostructures vs alloy composition x. The sheet charge predicted for the ( 000 1 ¯ ) GaN / ( 111 ) ScN-interface is shown for comparison (open square5).

FIG. 11.

Polarization-induced interface sheet charge 1 e σ buffer ( x ) of pseudomorphic, metal-polar ScxAl1−xN/GaN- and ScxAl1−xN/InN-heterostructures vs alloy composition x. The sheet charge predicted for the ( 000 1 ¯ ) GaN / ( 111 ) ScN-interface is shown for comparison (open square5).

Close modal

The alloy compositions of the ScxAl1−xN-layers at which the polarization-induced sheet charges become minimal or begin to bear a negative sign depend very much on the data set on which the calculations are based. Using the fits to the data set simulated by Talley et al., we determine a positive sheet charge over the whole range of alloy compositions. The simulated structural, elastic, and piezoelectric coefficients from Urban et al. and Zhang et al. lead to values of zero for the sheet charge and a corresponding limitation in alloy compositions allowing the formation of polarization-induced 2DEGs at x = ( 0.25 ± 0.01 ) and x = ( 0.42 ± 0.01 ), respectively. Interfaces between hexagonal group-III nitrides with a zero-polarization gradient are not only noteworthy as a possible limit for the electron accumulations at ScxAl1−xN/buffer interfaces, but they also have a potential of interest to the realization of optoelectronic devices. If it can be confirmed experimentally that interfaces between ScxAl1−xN and GaN can be created that are free of polarization-induced charges, a corresponding ScxAl1−xN/GaN/ScxAl1−xN-structure would form a quantum well that is free of a Stark effect and, therefore, suitable for processing energy-efficient UV light-emitting diodes.53 

Let us return to the consideration of such interfaces for the accumulation of electrons. A second interval of alloy compositions can be identified, in which the sheet charges are positive, based on the data set of Zhang et al. We calculate a sharp increase in positive interface sheet charges starting at alloy compositions of x = ( 0.82 ± 0.01 ), which extends up to the ScN/GaN-heterostructure, where the maximum value of 1 e σ GaN ( 1.0 ) = ( 8.21 ± 0.80 ) × 10 14 c m 2 occurs. Although the hexagonal layered ScN has neither a spontaneous nor a piezoelectric polarization, the polarization gradient at the interface to the wurtzite GaN-buffer is very high due to its spontaneous polarization. It should be noted that the maximum interfacial charge found is very close to that which Adamski et al. predicted for a ( 000 1 ¯ ) GaN / ( 111 ) ScN-heterostructure ( 1 e σ GaN ( 1.0 ) = 8.5 × 10 14 c m 2 ) .5 

Hexagonal ScxAl1−xN/InN-heterostructures show high gradients of polarization for all barrier alloys and positive polarization-induced bound interface charges if they are metal polar. The polarization-induced sheet charges of pseudomorphic AlN/InN-heterostructures are determined to be 1 e σ InN ( 0 ) = ( 2.23 ± 0.20 ) × 10 14 c m 2 increasing to 1 e σ InN ( 1.0 ) = ( 6.40 ± 0.60 ) × 10 14 c m 2 if all Al-atoms are substituted by Sc-atoms. It is interesting to note that the positive interface charge of the lattice-matched heterostructure based on InN-buffer layers is 1 e σ InN ( 0.73 ) = ( 6.56 ± 0.66 ) × 10 14 c m 2 and, thus, very close to the maximum value found for all ScxAl1−xN/InN-heterostructures based on the data set of Talley et al. Based on the comparison on the slightly smaller sheet charge of the lattice-matched Sc0.73Al0.27N/InN-heterostructure, calculated based on Zhang's set of data ( 1 e σ InN ( 0.73 ) = ( 5.50 ± 0.50 ) × 10 14 c m 2 ), it is still a factor of 15 larger in comparison to the sheet charge of the lattice-matched Sc0.19Al0.81N/GaN-heterostructure ( 1 e σ GaN ( 0.19 ) = ( 3.85 ± 1.02 ) × 10 13 c m 2 ). In general, lattice-matched heterostructures with large positive polarization-induced interface charges are of interest for the development of robust and energy-efficient electronic devices, as they allow for the accumulation of high electron densities without a distortion of the crystal lattice. For this reason, the electron sheet densities n s buffer ( x ) accumulated at interfaces of pseudomorphic, hexagonal ScxAl1−xN/buffer heterostructures are simulated as a function of the alloy composition of the ScxAl1−xN-layers in the following, utilizing the Schrödinger–Poisson solver nextnano and the interface sheet charges ( 1 e σ buffer ( x ) ) calculated prior.54 We will determine the profile of the potential energy of the conduction band edge and the electron states near the interface for representative heterostructures in addition to their electron distribution profiles. In order to ensure charge neutrality between localized and mobile charges across each heterostructure, the polarization-induced interface charges are compensated by localized polarization-induced surface charges and charges localized at the buffer/substrate interface. The negative charge associated with the accumulation of mobile electrons at the ScxAl1−xN/buffer interface is balanced by localized, ionized surface donors. To determine n s buffer ( x ) for low as well as for very high polarization-induced interface charges, the ground state as well as higher energy levels of the one-dimensional quantum well, formed at the ScxAl1−xN/buffer interface, are considered. To calculate the conduction band edge profile of the ScxAl1−xN/buffer heterostructures, the relative permittivity coefficients and bandgaps of the ScxAl1−xN- and buffer layers are required. Figure 12 shows the relative permittivity coefficients of the ScxAl1−xN-layers ( ε 33 ( x ) ) measured by Yanagitani and Suzuki,55 Wingqvist et al.,56 and Baeumler et al.57 for 0 ≤ x ≤ 0.63, and the buffer layers ( ε 33 GaN = 10.28 , ε 33 InN = 14.61 ),19 as well as a value for ScN with rock-salt structure ( ε 33 r s S c N ) provided by Adamski et al.5 

FIG. 12.

Measured (symbols: black squares,55 black triangles,56 and black circles57) and fitted relative permittivity ε 33 ScAlN ( x ) of ScxAl1−xN vs alloy composition x. The permittivity ε 33 ScN for ( 111 ) ScN as predicted by Adamski et al. is shown in addition (open square5). The relative permittivities ε 33 GaN and ε 33 InN of the buffer layers are provided for comparison.19 

FIG. 12.

Measured (symbols: black squares,55 black triangles,56 and black circles57) and fitted relative permittivity ε 33 ScAlN ( x ) of ScxAl1−xN vs alloy composition x. The permittivity ε 33 ScN for ( 111 ) ScN as predicted by Adamski et al. is shown in addition (open square5). The relative permittivities ε 33 GaN and ε 33 InN of the buffer layers are provided for comparison.19 

Close modal

Due to the lack of measured or simulated values for ε 33 h - ScN and because of the comparable values calculated for the relative permittivities of wurtzite and zinc blende AlN, GaN or InN,19 the fitted curve to the relative permittivity values ε 33 ( x ) measured by Yanagitani and Suzuki55 (solid line in Fig. 12) was created assuming ε 33 h - ScN ε 33 rs - ScN = 25. We find the best agreement with the experimental data when the permittivity increases linearly up to a Sc content of x = 0.4 and reaches a value that is almost twice the permittivity of AlN. For higher Sc contents, the permittivity increases stronger and reaches a value of about 55 at x = 0.63. We interpret the strong increase in permittivity as a consequence of the transition from the ferroelectric wurtzite structure to the paraelectric hexagonal layered structure of ScxAl1−xN, as it occurs, e.g., in BaTi2O5.58 However, this also means that after the structural transition and at very high Sc contents ( x > 0.63 ), the permittivity of ScxAl1−xN must decrease again. We took this effect into account in our fitted curve and let the permittivity drop to the value of ε 33 rs - ScN = 25 for x = 1.0.

In order to describe the bandgap E g ( x ) of hexagonal ScxAl1−xN-crystals as a function of the percentage of Al-atoms substituted by Sc-atoms, a fit to the data from Zhang et al.26 was created,
E g ( x ) = { ( 2.25 x 2 3.72 x + 6.28 ) eV for 0 x < 0.5 , ( 15.68 x + 11.75 ) eV for 0.5 x < 0.63 , ( 0.88 x 2 2.86 x + 3.47 ) eV for 0.63 x < 1.0.
(7)

From AlN to Sc0.5Al0.5N, we approximate the decrease in bandgap from 6.28 to 3.99 eV by a quadratic equation with a bowing parameter of 2.25 eV. In the interval of transition from wurtzite to the hexagonal layered structure (0.5 ≤ x ≤ 0.63), the slightly stronger decrease in the bandgap with an increase in the Sc content is approximated by a linear function. For ScxAl1−xN-crystals with x 0.63, the further decrease in the bandgap is again approximated by a quadratic function with a bowing parameter of 0.88 eV, which equals a value of 1.49 eV for hexagonal layered ScN.26,59 For alloy compositions of up to x = 0.41, the approximated data are found to be in good agreement with the experimental results of Baeumler et al.57, Figure 13 presents the simulated and measured data for E g ( x ) combined with the bandgaps of the GaN- and InN-buffer layer, as well as data for the conduction band offsets Δ E CB GaN ( x ) experimentally determined by Jin et al.60 The experimental data of the conduction band discontinuities, which are available up to a Sc content of x = 0.2, tend to exceed the functional curve used for the calculation of the electron sheet charges, as will be explained later in more detail. For ScxAl1−xN/GaN-heterostructures with 0 ≤ x < 0.51, the bandgap of wurtzite ScxAl1−xN is predicted to be larger in comparison to the bandgap of the buffer layer ( E g GaN = 3.42 eV ). For higher alloy compositions, the bandgap of GaN starts to exceed the bandgap of ScxAl1−xN. As a consequence, the center of the polarization-induced electron distribution profile is shifted into the ScxAl1−xN-layer. It should be noted that electron transport through ScxAl1−xN (except for x = 1), compared to GaN and InN, will face a strong alloy scattering, reducing the mobility and drift velocity of the electrons. In case of the ScxAl1−xN/InN-heterostructures, the bandgaps of the barriers are larger in comparison to the bandgap of the buffer layer ( E g InN = 0.65 eV ) across the entire range of alloy compositions, with the consequence that the centers of the electron distribution profiles are always located in the InN-buffer layer.

FIG. 13.

Measured (black circles and black rhombuses57) and calculated (gray rhombuses26) bandgap E g ( x ) of ScxAl1−xN with hexagonal crystal structure vs alloy composition x. The bandgap E g GaN and E g InN of the buffer layers are indicated by horizontal dashed lines for comparison. Conduction band offsets Δ E CB GaN ( x ) of pseudomorphic ScxAl1−xN/GaN-heterostuctures determined by Jin et al.60 experimentally for alloy compositions up to x = 0.2 are shown as black squares. The conduction band offsets used as an approximation in the calculation of the electron sheet densities by nextnano are indicated by the dotted line.

FIG. 13.

Measured (black circles and black rhombuses57) and calculated (gray rhombuses26) bandgap E g ( x ) of ScxAl1−xN with hexagonal crystal structure vs alloy composition x. The bandgap E g GaN and E g InN of the buffer layers are indicated by horizontal dashed lines for comparison. Conduction band offsets Δ E CB GaN ( x ) of pseudomorphic ScxAl1−xN/GaN-heterostuctures determined by Jin et al.60 experimentally for alloy compositions up to x = 0.2 are shown as black squares. The conduction band offsets used as an approximation in the calculation of the electron sheet densities by nextnano are indicated by the dotted line.

Close modal
Analogous to the earlier calculations of electron sheet charges and distributions in GaxAl1−xN/GaN-heterostructures, the boundary and interface conditions for solutions of the Schrödinger–Poisson solver are determined by the potential of the conduction band edge ( E CB ) at the surface of the ScxAl1−xN-layer, at the ScxAl1−xN/buffer interface, and at the buffer/substrate interface. At the surface of the heterostructure, the potential is determined by the barrier height of a Ni Schottky contact ( e ϕ ( x ) ).61–63 At the interface, the potential of the conduction band edge changes abruptly. This discontinuity of the conduction band edge ( Δ E CB buffer ( x ) ) is approximated by two-thirds of the bandgap difference between the ScxAl1−xN and buffer layer.64,65 Since the buffer should be electrically insulating for transistor devices, the potential of the Fermi level at the buffer/substrate interface ( E F buffer ) is placed in the middle of the bandgap of the buffer ( E g buffer ). The input of the boundary and interface conditions into the Schrödinger–Poisson solver is approximated as follows:
E F buffer = 1 2 E g buffer ,
(8)
e ϕ ( x ) = 1 3 E g ( x ) ,
(9)
Δ E C buffer ( x ) = { 2 3 ( E g ( x ) E g buffer ) for E g ( x ) E g buffer , 2 3 ( E g buffer E g ( x ) ) for E g buffer > E g ( x ) .
(10)
The number of electrons occupying energy states within the triangular quantum wells located near the heterostructure interfaces can be estimated using the effective carrier mass in the channel material ( m channel ( x ) ).66–68 To calculate the ScxAl1−xN/InN-heterostructures, we can use the effective electron mass of the InN-channel, m InN = 0.055 m e ( m e = 9.1094 × 10 31 kg )69 across the entire range of alloy compositions of the ScxAl1−xN-barriers. Regarding the ScxAl1−xN/GaN-heterostructures with barrier alloy compositions of 0.0 x 0.5, we use m GaN = 0.222 m e.70 Equivalent to the conduction band offsets, we have to consider that for high Sc-contents the bandgap of the ScxAl1−xN-layer becomes smaller in comparison to the GaN-buffer layer. The center of distribution of accumulated electrons is shifted from GaN into the ScxAl1−xN-layer, when the Sc content is increased from x = 0.50 to 0.56. To take into account the shift in the center of electron distribution and the associated increase in the effective electron mass ( m channel ( x ) ) when calculating the electron sheet density of ScxAl1−xN/GaN-heterostructures in which E g GaN > E g ( x ), we use a linear approximation between the effective electron masses of wurtzite AlN and rock-salt ScN.71,72 We justify the use of the effective electron mass of rock-salt ScN to approximate the value of the hexagonal layered structure by the very similar effective electron masses of cubic and hexagonal InN, GaN, and AlN crystals determined,73 
m channel ( x ) = { m GaN = 0.222 m e for E g GaN E g ( x ) , m ( x ) = ( 0.10 x + 0.30 ) m e for E g GaN > E g ( x ) .
(11)
In order to calculate the electron sheet densities of pseudomorphic, metal-polar Ni/ScxAl1−xN/buffer heterostructures with relevance to HEMTs, we restrict our simulations to barrier and buffer thicknesses of d ScAlN = 25 nm and d buffer = 4000 nm, respectively. Consequently, the electron sheet charge can be calculated using the integral of the electron density ( n e ( d ) ) over the depth,
n s = 0 4025 nm n e ( d ) Δ d .
(12)

In Fig. 14, the electron sheet densities n s buffer ( x ) are presented for Ni/ScxAl1−xN/GaN- and Ni/ScxAl1−xN/InN-heterostructures vs alloy composition. We observe a good agreement for the electron sheet densities n s GaN ( x = 0 ) = 5.48 × 10 13, 5.87 × 10 13, and 6.50 × 10 13 c m 2 calculated based on the data sets of Urban et al., Zhang et al., and Talley et al., respectively. Seeing as though the positive polarization-induced sheet charge decreases if Al-atoms are substituted by Sc-atoms in the ScxAl1−xN-barrier, the electron sheet density is reduced as well. In order to provide representative values, we calculated the electron sheet density for an alloy composition of 0.22 for the same data sets and obtained n s GaN ( 0.22 ) = 1.60 × 10 13, 3.98 × 10 13, and 3.88 × 10 13 c m 2.

FIG. 14.

Polarization-induced electron sheet densities n s buffer ( x ) of pseudomorphic, Ni/ScxAl1−xN/GaN- and Ni/ScxAl1−xN/InN-heterostructures with barrier and buffer thicknesses of 25 and 4000 nm, respectively, vs alloy composition x. The solid, dashed, and dotted lines are fits to the calculated results based on the data sets provided by Zhang et al. (rhombus),26 Talley et al. (circles),29 and Urban et al. (triangles).40 Black and dark gray symbols represent electron sheet densities n s GaN ( x ) and n s InN ( x )confined in one dimensional (triangular) quantum wells. Open and light gray symbols represent the electron sheet densities accumulated at interfaces of Ni/ScxAl1−xN/InN- and Ni/ScxAl1−xN/GaN-heterostructures exceeding the confined sheet densities. The stars mark alloy compositions and sheet densities for which the electron distribution and conduction band edge are presented in Figs. 1517.

FIG. 14.

Polarization-induced electron sheet densities n s buffer ( x ) of pseudomorphic, Ni/ScxAl1−xN/GaN- and Ni/ScxAl1−xN/InN-heterostructures with barrier and buffer thicknesses of 25 and 4000 nm, respectively, vs alloy composition x. The solid, dashed, and dotted lines are fits to the calculated results based on the data sets provided by Zhang et al. (rhombus),26 Talley et al. (circles),29 and Urban et al. (triangles).40 Black and dark gray symbols represent electron sheet densities n s GaN ( x ) and n s InN ( x )confined in one dimensional (triangular) quantum wells. Open and light gray symbols represent the electron sheet densities accumulated at interfaces of Ni/ScxAl1−xN/InN- and Ni/ScxAl1−xN/GaN-heterostructures exceeding the confined sheet densities. The stars mark alloy compositions and sheet densities for which the electron distribution and conduction band edge are presented in Figs. 1517.

Close modal

The electron sheet charges determined based on the data of Urban et al. decrease faster by raising the Sc-content of the ScxAl1−xN-barrier in comparison to the equivalent values that result using the data provided by Zhang et al. and Talley et al. A consequence of this strong decrease is that electron accumulations are only observed up to an alloy composition of x = 0.24, while for the data set of Zhang et al. the formation of 2DEGs is possible up to a Sc-content of x = 0.36. If we use the data of Talley et al., the formation of electron accumulations in ScxAl1−xN/GaN-heterostructures is possible across the entire alloy range. Here, we observe only a minimal electron sheet density of ( 1.39 ± 0.30 ) × 10 13 c m 2 at an alloy composition of x = 0.39. By increasing the Sc-content further, the electron sheet density increases in a non-linear fashion and the center of the electron distribution is shifted from the GaN-buffer into the ScxAl1−xN-layer. For alloy compositions x > 0.84, the huge number of electrons attracted by the positive polarization-induced sheet charges cannot be confined in the one-dimensional quantum wells that form at the ScxAl1−xN/GaN-interfaces. The Fermi level ( E F ) no longer intersects the conduction band edge at the level of the band discontinuity, but at a smaller depth and, thus, within the ScxAl1−xN-layer. The quantum wells of the heterostructures are “flooded” by mobile carriers and only part of the electrons accumulated are confined in the one-dimensional quantum well. We find a good agreement between the electron sheet densities determined based on the data sets of Zhang et al. and Talley et al. for x > 0.87. For both set of data, we calculate the highest electron sheet density of ( 8.04 ± 0.20 ) × 10 14 c m 2, of which ( 2.63 ± 0.20 ) × 10 14 c m 2 are confined in the quantum well, for a pseudomorphic, hexagonal ScN/GaN-heterostructure. We observe significant differences in the electron sheet densities caused by the higher positive piezoelectric polarization of the biaxial compressive strained ScxAl1−xN-layers calculated based on the data set of Zhang et al. in comparison to Talley et al. While we determined electron accumulation across the entire range of ScxAl1−xN-alloys based on the data from Talley et al., the negative interface charges derived from the data provided by Zhang et al. result in no electron accumulation in the interval 0.31 x 0.81. In this case, we observe an extremely steep increase in the electron sheet density starting at x = 0.81 and reaching a value of ( 8.96 ± 0.20 ) × 10 13 c m 2 at a Sc content of x = 0.84. The dependence of the electron sheet density ( n s InN ( x ) ) accumulated at interfaces of Ni/ScxAl1−xN/InN-heterostructures on the alloy of the ternary crystal is not as complex compared to Ni/ScxAl1−xN/GaN-heterostructures and the deviations resulting from the different data sets for the piezoelectric and elastic coefficients are less pronounced. Very similar values of electron sheet densities of ( 2.26 ± 0.20 ) × 10 14 c m 2 and ( 6.25 ± 0.20 ) × 10 14 c m 2 are determined from all three sets of data for Ni/AlN/InN- and Ni/ScN/InN-heterostructures. The electron sheet densities obtained from the data sets do not deviate by more than 10% in the intervals of alloy composition 0.0 x 0.35 and 0.81 x 1.0. Between these two intervals, the electron sheet density also increases as the Sc content of the ScxAl1−xN-layers increases, but the absolute values obtained from Urbans data set are significantly larger than those of Talley et al., and these in turn are higher than those which belong to the set of Zhang et al. (Fig. 14). A representative example for the differences observed is the electron sheet concentration n s InN ( x = 0.5 ), which is calculated to be ( 6.11 ± 0.20 ) × 10 14, ( 4.24 ± 0.20 ) × 10 14, and ( 3.90 ± 0.20 ) × 10 14 c m 2 for the three sets of data. Different negative values for the piezoelectric polarization of the ScxAl1−xN-layer are decisive for this, which are partly caused by the different Sc-contents for which the transition from the wurtzite to the hexagonal layered crystal structure is predicted.

In opposition to Ni/ScxAl1−xN/GaN-, none of the pseudomorphic Ni/ScxAl1−xN/InN-heterostructures are able to confine all accumulated electrons within the one-dimensional quantum well. Mainly due to the reduction in conduction band discontinuity, the confined part of the electron sheet density is reduced from ( 1.59 ± 0.08 ) × 10 14 to ( 1.27 ± 0.15 ) × 10 14 c m 2 if half of the Al-atoms in the barrier are substituted by Sc-atoms. With Sc contents x > 0.56, the potential energy of the ground state ( E 0 ) of the quantum well shifts above the upper edge of the band discontinuity, resulting in the charge carrier being significantly broadened. In general, this means a deterioration in the electronic transport properties due to greater alloy scattering, which compensates at least part of the reduction in sheet resistance caused by the higher electron sheet density.

For the following calculations of the electron distribution and conduction band edge profiles of representative and device-relevant heterostructures, we focus on the simulation results, achieved based on the sets from Zhang et al. We justify this approach with the argument that Zhang's simulation results show the best agreement with our experimental data regarding the structural properties of the ScxAl1−xN-layers. Figure 15 shows the electron distribution and conduction band edge profile ( E CB ( d ) ) of the lattice-matched Sc0.19Al0.81N/GaN-heterostructure viewed from the surface. Due to the polarization-induced electric field in the Sc0.19Al0.81N-barrier, which is partly screened by ionized surface donors and mobile electrons, the potential energy of the conduction band edge decreases with increasing depth ( d ). The band discontinuity of the conduction band edge is visible at the interface of the heterostructure, which results from the larger bandgap of the Sc0.19Al0.81N-barrier compared to the GaN-buffer layer. The triangular quantum well, which forms near the interface, has two quantized states below the Fermi level. These states are occupied by electrons that form the two-dimensional electron gas, as indicated by the narrow half-width of the electron distribution ( F W H M n e = 1.4 nm ) and a high maximum electron density of n e ( d = 25.5 nm ) = ( 2.37 ± 0.30 ) × 10 20 c m 3 . The majority of the electrons, which are part of the electron sheet charge of n s GaN ( 0.19 ) = ( 4.11 ± 0.40 ) × 10 13 c m 2, occupy the ground state ( E 0 ) of the quantum well ( n s ( E 0 ) = ( 3.36 ± 0.4 ) × 10 13 c m 2 ). This has a positive effect on the electron transport properties in the direction of the basal plane, which, in combination with the high electron sheet density, makes the lattice-matched Sc0.19Al0.81N/GaN-heterostructure a highly interesting candidate for processing robust and energy-efficient HEMTs. The question is, whether the output power of transistors can be enhanced, if the total sheet charge can be increased even further, assuming that the drift velocity of the electrons is not significantly reduced by their higher number. As shown in Fig. 14, the total electron sheet charge for polarization-induced, positive interface charges increases in the interval 0 x 0.44, by decreasing the Sc-content of the barrier. As a consequence, by reducing the Sc-content from x = 0.19 0.13, the electron sheet density that is confined in the triangular quantum well can be increased further by 55 % without the critical layer thickness falling below a barrier thickness of 25 nm.43 This indicates that besides the lattice-matched heterostructure the range of barrier alloys 0.13 x 0.19 is of high potential for the processing and evaluation of Ni/ScxAl1−xN/GaN-based power electronic devices.

FIG. 15.

Electron density vs depth n e ( d ) (solid gray line) and conduction band edge profile E C B ( d ) (solid black line) for a Sc0.19Al0.81N/GaN-heterostructure with Sc0.19Al0.81N- and GaN-buffer thicknesses of 25 and 4000 nm, respectively. The Fermi level ( E F ) and the potential energies of the quantized states of the triangular quantum well ( E x ) are shown as horizontal gray-dashed lines. The calculated electron sheet density ( n s GaN ( 0.19 ) ) and that of the ground state ( n s ( E 0 ) ), which are confined in the triangular quantum well, are given as numerical values in the figure, along with the full-width at half-maximum of the electron distribution ( F W H M n e ).

FIG. 15.

Electron density vs depth n e ( d ) (solid gray line) and conduction band edge profile E C B ( d ) (solid black line) for a Sc0.19Al0.81N/GaN-heterostructure with Sc0.19Al0.81N- and GaN-buffer thicknesses of 25 and 4000 nm, respectively. The Fermi level ( E F ) and the potential energies of the quantized states of the triangular quantum well ( E x ) are shown as horizontal gray-dashed lines. The calculated electron sheet density ( n s GaN ( 0.19 ) ) and that of the ground state ( n s ( E 0 ) ), which are confined in the triangular quantum well, are given as numerical values in the figure, along with the full-width at half-maximum of the electron distribution ( F W H M n e ).

Close modal

Additional electrons are predicted to be confined in a metal-polar Sc0.84Al0.16N/GaN-heterostructure, where the maximum and the charge center of the 2DEG are located on the side of the Sc0.84Al0.16N-layer. A maximum electron concentration and a sheet charge of n e ( d = 24.3 nm ) = ( 3.31 ± 0.30 ) × 10 20 c m 3 and n s GaN ( 0.84 ) = ( 8.96 ± 0.40 ) × 10 13 c m 2 are determined, respectively (Fig. 16). The electrons are distributed across at minimum over four energy levels in the quantum well, with only about half of the charge carriers occupying the ground state. The simulated heterostructure has two major disadvantages for processing high electron mobility transistors: First, the average distance of the electrons to the Ni-contact is less than the thickness of the Sc0.84Al0.16N-layer, which can lead to increased gate leakage currents, and second, its strain is too high ( ε 1 GaN ( 0.8 ) = ( 0.11 ± 0.01 ) ) to realize voltage-resistant and robust devices. As Adamski et al. pointed out before,5 these disadvantages can be circumvented if GaN-barriers with a [0001]-oriented wurtzite structure are epitaxially grown on non-polar ScxAl1−xN-buffers ( x > 0.8 ) with a [111] rock-salt structure.

FIG. 16.

Electron density vs depth n e ( d ) (solid gray line) and conduction band edge profile E C B ( d ) (solid black line) for a Sc0.84Al0.16N/GaN-heterostructure with Sc0.84Al0.16N- and GaN-buffer thicknesses of 25 and 4000 nm, respectively. The Fermi level ( E F ) and the potential energies of the quantized states of the triangular quantum well ( E x ) are shown as horizontal gray-dashed lines. The calculated electron sheet density ( n s GaN ( 0.84 ) ) and that of the ground state ( n s ( E 0 ) ), which are confined in the triangular quantum well, are given as numerical values in the figure, along with the half-width at half-maximum of the electron distribution ( F W H M n e ).

FIG. 16.

Electron density vs depth n e ( d ) (solid gray line) and conduction band edge profile E C B ( d ) (solid black line) for a Sc0.84Al0.16N/GaN-heterostructure with Sc0.84Al0.16N- and GaN-buffer thicknesses of 25 and 4000 nm, respectively. The Fermi level ( E F ) and the potential energies of the quantized states of the triangular quantum well ( E x ) are shown as horizontal gray-dashed lines. The calculated electron sheet density ( n s GaN ( 0.84 ) ) and that of the ground state ( n s ( E 0 ) ), which are confined in the triangular quantum well, are given as numerical values in the figure, along with the half-width at half-maximum of the electron distribution ( F W H M n e ).

Close modal

Despite very high polarization gradients at their interfaces, these structures have little or no strain. However, even in these heterostructures, a significant reduction in the electron transport properties compared to heterostructures with GaN- or InN-channel layers is to be expected due to alloy scattering and an overflow of electrons from the quantum well. An overflow of electrons from the quantum wells also limits the usefulness of Ni/ScxAl1−xN/InN-heterostructures. As a representative example, the conduction band edge and electron distribution profile of the lattice-matched Ni/Sc0.73Al0.27N/InN-heterostructures are presented in Fig. 17. An enormously high sheet charge of n s InN ( 0.73 ) = 5.46 × 10 14 c m 2 is predicted, composed of charge carriers occupying more than six different energy levels. The half-width of the electron distribution indicates that they will no longer have all the good transport properties of a 2DEG. Furthermore, estimates allow the conclusion that the dielectric strength of the Sc0.73Al0.27N-layer will not be sufficient to completely deplete the electron accumulation via an applied gate voltage. This heterostructure is, therefore, not suitable for the processing of transistors, despite the high level of electrical conductivity that is to be expected. However, it can very well be useful for horizontal electrical connections in GaN-based integrated circuits or between devices, as well as for realizing low-impedance contacts.

FIG. 17.

Electron density vs depth n e ( d ) (solid gray line) and conduction band edge profile E C B ( d ) (solid black line) for a lattice-matched Sc0.73Al0.27N/InN-heterostructure with a barrier and buffer thicknesses of 25 and 4000 nm, respectively. The Fermi level ( E F ) and part of the potential energies of the quantized states of the “flooded” quantum well ( E x ) are shown as horizontal gray-dashed lines. The calculated electron sheet density ( n s InN ( 0.73 ) ) and that of the ground state ( n s ( E 0 ) ) are given as numerical values in the figure, along with the half-width at half-maximum of the electron distribution ( F W H M n e ).

FIG. 17.

Electron density vs depth n e ( d ) (solid gray line) and conduction band edge profile E C B ( d ) (solid black line) for a lattice-matched Sc0.73Al0.27N/InN-heterostructure with a barrier and buffer thicknesses of 25 and 4000 nm, respectively. The Fermi level ( E F ) and part of the potential energies of the quantized states of the “flooded” quantum well ( E x ) are shown as horizontal gray-dashed lines. The calculated electron sheet density ( n s InN ( 0.73 ) ) and that of the ground state ( n s ( E 0 ) ) are given as numerical values in the figure, along with the half-width at half-maximum of the electron distribution ( F W H M n e ).

Close modal

For the first time, we have calculated the polarization-gradient-induced electron sheet charges and electron distribution profiles in pseudomorphic, hexagonal ScxAl1−xN/GaN- and ScxAl1−xN/InN-heterostructures across the entire range of possible alloys of the ternary ScxAl1−xN-layers. Because the sheet density of the electrons accumulating at the heterostructure interfaces can strongly depend both on the data set of the piezoelectric and structural coefficients and on the alloying region of the ScxAl1−xN-layers in which the transition from the wurtzite to the hexagonal layered crystal structure takes place, we have calculated the carrier densities and profiles for three representative data sets and evaluated their relevance for devices. In contrast to earlier work, the simulation method of Bernardini et al. for determining gradients of the spontaneous polarization is replaced by the modern method of polarization calculation invented by Dreyer et al.45 This theory allows the use of spontaneous polarization values as they are obtained experimentally from polarization hysteresis of ferroelectric ScxAl1−xN-layers and thereby increases the reliability of the simulated results. Based on the gradients in spontaneous and piezoelectric polarization determined, Schrödinger–Poisson simulations were carried out in order to evaluate the application potential of Ni/ScxAl1−xN/GaN- and Ni/ScxAl1−xN/InN-heterostructures for processing energy efficient high electron mobility transistors. The lattice-matched Ni/Sc0.19Al0.81N/GaN-heterostructure deserves special attention due to its high electron sheet density n s GaN ( 0.19 ) = 4.11 × 10 13 c m 2. The majority of the electrons occupy the ground state of the triangular quantum well ( n s ( E 0 ) = 3.36 × 10 13 c m 2 ). By reducing the Sc-content of the ScxAl1−xN-barrier from x = 0.19 to 0.13, the total electron sheet charge confined in the one-dimensional quantum well can be increased further by 55 % without the critical layer thickness falling below a barrier thickness of 25 nm. This indicates that beside the lattice-matched heterostructure the entire range of barrier alloys 0.13 x 0.19 is of high potential for the processing of Ni/ScxAl1−xN/GaN-based power electronic devices outperforming conventional Ni/Ga0.25Al0.75N/GaN-based devices by a factor of up to 7 in terms of electron sheet density. An even larger electron sheet density of 8.04 × 10 14 c m 2 is calculated for hexagonal Ni/ScN/GaN-heterostructures. However, in these heterostructures, the focus of electron accumulation is in the upper, extremely strained ScN-layer. For this reason, Ni/ScN/GaN-heterostructures will only allow extremely thin ScN-layer thicknesses and favor high gate leakage currents. In combination with the results of Adamski et al.,5 our simulations indicate that alternative Ni/GaN(0001)/ScN(111)-heterostructures, in which the GaN-layer forms the barrier, can be realized nearly lattice-matched and additionally enable an enormously huge electron sheet density. However, these electron densities are so high that they cannot be completely depleted by an applied negative gate voltage due to the limited breakdown field strength of GaN. We come to a comparable conclusion when evaluating pseudomorphic Ni/ScxAl1−xN/InN-heterostructures in which the electron sheet density is always greater than ( 2.26 ± 0.20 ) × 10 14 c m 2 and the limiting factor for a complete depletion of the electron accumulation is the breakdown field strength of the ScxAl1−xN-barrier. For alloy compositions of x 0.57, the potential energy of the ground state shifts above the band discontinuity of the ScxAl1−xN/InN-interface, accompanied by an “overflow” of the triangular quantum wells due to the high density of polarization-induced electrons. These heterostructures are not suitable for transistors, but they could be of great interest for the realization of low-impedance contacts. In opposition to ScxAl1−xN/InN-heterostructures, we have identified ScxAl1−xN/GaN-interfaces based on the data set of Zhang et al., which can be free of polarization-induced charges. For two alloy compositions of x = ( 0.37 ± 0.01 ) and ( 0.81 ± 0.01 ), the gradient in total polarization vanishes, creating the chance of field-free GaN-based quantum wells or quantum barriers. Sc0.37Al0.73N/GaN/ Sc0.37Al0.73N-heterostructures could be of great importance for the development of light-emitting diodes, as they can be used to generate UV-light efficiently since the electron hole recombination is not reduced by the Stark effect.

This work was partially supported by the German Science Foundation (DFG) under Project No. AM 105/50-1. In addition, the work was partially supported by the Gips-Schüle-Stiftung and the Carl-Zeiss-Stiftung (project “SCHARF”). The electrical characterization of the ScxAl1−xN/GaN-heterostructures is supported by the Federal Ministry of Education and Research (BMBF) within the project “EdgeLimit.”

The authors have no conflicts to disclose.

O. Ambacher: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). S. Mihalic: Data curation (equal); Investigation (equal); Resources (equal). B. Christian: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal).

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

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