To make complementary GaN electronics a desirable technology, it is essential to understand the low mobility of 2D hole gases in III-Nitride heterostructures. This work derives both the acoustic and optical phonon spectra present in one of the most prominent p-channel heterostructures (the all-binary GaN/AlN stack) and computes the interactions of these spectra with the 2D hole gas, capturing the temperature dependence of its intrinsic mobility. Finally, the effects of strain on the electronic structure of the confined 2D hole gas are examined and a means is proposed to engineer the strain to improve the 2D hole mobility for enhanced p-channel device performance, with the goal of enabling wide-bandgap CMOS.

Decades after the celebrated invention of Mg p-doping^{1} in Gallium Nitride (GaN) and the subsequent development of GaN-based light emitting diodes, the manipulation of holes in GaN remains a fundamental challenge. Consequently, despite the expected dominance of GaN High Electron Mobility Transistors (HEMTs) in the coming generation of power electronics^{2} and communications systems,^{3} there is no complementary p-channel device which can be readily integrated. This incompleteness restricts the possible circuit topologies and system designs achievable in GaN electronics but arises quite straightforwardly from the physics of the GaN valence band. These bands, both heavy and deep in energy, have proven difficult to contact with typical metal workfunctions,^{4} difficult to dope with high efficiency,^{5} and difficult to flow current through with high conductivity. Nonetheless, the commercial interest in generating complementary GaN-based circuits^{6} and scientific interest in studying highly degenerate hole physics^{7} have prompted great recent progress in p-channel devices.^{8}

Of the various structures^{6,9–16} which have been proposed as a platform for p-channel III-Nitride electronics, the single GaN/AlN heterojunction field-effect transistor has received recent attention for its high sheet conductance^{7} and excellent device performance.^{8} In this structure, depicted in Fig. 1(a), the all-binary materials provide a straightforwardly repeatable growth with no possible parasitic electron channels, a tremendous hole-inducing polarization-charge for low sheet-resistance, and maximal bandgaps for extreme voltage-handling capability. Given the recent reports of temperature-dependent transport studies in this heterostructure^{7} and recent first-principles suggestions of possible enhancements to hole mobility in bulk p-GaN,^{17} this work presents a model to explain the measured mobility of the 2D hole gas (2DHG) at the GaN/AlN interface and evaluates the potential of strain-engineering approaches to alter the band structure in a favorable way. First, the valence band structure including confinement and multiband mixing effects is computed. Then, the spectra of both acoustic and optical phonons in the heterostructure are determined and the mobility limitation due to these mechanisms is derived. Finally, the effects of strain on the band structure and mobility are presented.

The multiband k.p (MBKP) approach, based on Burt Exact Envelope Function Theory,^{18,19} describes the electronic states of heterostructures wherein multiple subbands may be intermixed by nonuniform potentials and material interfaces. MBKP has been extended to wurtzite heterostructures by various authors^{20,21} and is available in certain commercial packages.^{22} The PyNitride software package^{23} employed here self-consistently^{24} solves the 6 × 6 MBKP equation of the wurtzite valence band (see the supplementary material) simultaneously with the Poisson equation, accounting for the large fixed interface polarization charge. Figure 1(a) depicts a highly confined 2D hole gas at the GaN/AlN interface. The hole gas represents contributions mainly from the first quantized subband of both the heavy hole (HH) and light hole (LH) bands (including spin, this is four subbands). The transverse dispersion is shown in Fig. 1(b). By density, as clear from Fig. 1(c), the HH band dominates, though as shown in Fig. 1(d), the LH band has lighter in-plane mass near the Fermi energy, and so it contributes significantly to transport. To evaluate transport, we proceed to describe the phonons.

The Dielectric Continuum model^{25} describes polar optical phonons (POPs) in arbitrary heterostructures. As POP scattering is the main limitation on electron mobility in quality GaN, numerous authors have invested significant theoretical effort into the elaboration of POP spectra in various wurtzite heterostructures.^{26–30} A structure as simple as this, in fact, can be solved analytically. For a uniaxial crystal, the effect of polar optical phonons can be incorporated into two frequency (*ω*) dependent, directional dielectric functions, $\epsilon \u22a5$ and $\epsilon \u2225$,

where those longitudinal (*ω _{LO}*) and transverse (

*ω*) POP frequencies and high-frequency dielectric constant $\u03f5\u221e$ can be determined experimentally for the materials in play (see tabulation in Komirenko

_{TO}*et al.*

^{29}). For a mode with in-plane wavevector

*q*, the POP problem reduces to solving a frequency-dependent Poisson eigenvalue equation

for the potential *ϕ*. At every characteristic frequency appearing in Eq. (1), a dielectric constant changes sign, which changes the character of the involved modes (locally decaying vs oscillating). Altogether, there are three classes of modes depending on the energy range: (1) oscillating in GaN, decaying in AlN, (2) oscillating in AlN, decaying in GaN, and (3) decaying bidirectionally from a GaN/AlN interface. For each class, both transverse and longitudinal polarizations are possible. The spectrum and example modes are depicted in Fig. 2, and the solution is elaborated in the supplementary material.

The Elastic Continuum model^{25} aptly describes acoustic phonons in arbitrary heterostructures. Given the centrality of “optical” phonons in electron-based devices, the literature on acoustic phonons in wurtzite heterostructures^{31–34} is significantly less comprehensive. The basic approach is to link a continuum Newton's law with a material stress-strain relation

where *u _{i}* is the local displacement vector,

*ρ*is the density,

*T*is the stress tensor,

_{ij}*c*is the stiffness tensor, and

_{ijkl}*ϵ*is the strain tensor $\u03f5ij=12(\u2202rjui+\u2202riuj)$. Due to symmetry constraints on the stiffness tensor, the in-plane shear component is uncoupled (“Y” modes), while longitudinal and out-of-plane components constitute two coupled differential equations (“XZ” modes). These are solved by the Finite Element Method. For both types of modes, there are two characters possible depending on the energy range. At energies above $\u210fvtq$, where

_{ijkl}*v*is the relevant AlN sound velocity, modes are able to propagate in the AlN and are thus of a continuum sort. Below this energy, any modes which exist must decay into the AlN, so are GaN-confined modes. The spectrum and example modes are revealed in Fig. 3, and the solution methodology is elaborated in the supplementary material.

_{t}We now turn to the interactions of the phonons and carriers. First, both types of phonon modes are normalized^{25} by the quantization condition $\u222bd3r\u2192\u2009\rho (r\u2192)|u\u2192(r)|2=\u210f2\omega $. Once the oscillations are normalized, the Hamiltonian term for each mode can be generated. For optical phonons, this is just *H _{pop}* = −

*eϕ*(

*r*). For acoustic phonons, the deformation potential

*H*=

_{adp}*D*(

*ϵ*) is a six-by-six matrix function of position: at each position, it recruits the valence deformation matrix of the local material and also the local strain from a given acoustic mode. There is also a (coherently combined) piezo term

*H*= −

_{pz}*eϕ*(

*r*) where

*ϕ*is found

^{32}by solving the Poisson equation given the piezoelectric charge induced by the mode (see the supplementary material). Finally, to account for the low-temperature mobility, which is around 200 cm

^{2}/Vs in these structures to date, some extrinsic limitation must be included. The exact cause (e.g

*.,*interface roughness, dislocation, impurity, etc) is irrelevant to this study, since all these elastic mechanisms are temperature-independent and have similar dependence on effective masses. So the exact cause is not deduced here, but rather a generic scatterer with a constant scalar matrix element is applied to set the low-temperature mobility to 200 cm

^{2}/Vs. This matrix element is the only “tuning parameter.”

To calculate the perturbed carrier distribution, we employ the linearized Boltzmann Transport Equation (LBTE)

where $E\u2192$ is the in-plane electric field, *f* ^{0} is the equilibrium occupation function, *f ^{A}* is a small change in occupation to be solved for, and the transition rate

*R*is given by

where *α* = ±1 represents the absorption/emission, respectively, *ψ _{km}* is the state with in-plane wavevector $k\u2192$, of subband

*m*, $Hl(q)$ is the perturbation due to mode

*l*with wavevector $q\u2192=k\u2192\u2032\u2212k\u2192$, and

*E*/

*ε*are the electronic/phonon energies. Once this is discretized on a k-space mesh, the LBTE is solved as a linear matrix equation for the change in occupation $\u2202fA/\u2202E$ under an applied field, from which the mobility is extracted. The results, Fig. 4, compare agreeably to experiment over a wide temperature range and verify Poncé's prediction

^{17}that acoustic phonons dominate scattering at room temperature.

Now, we discuss what can be done to improve the mobility. In a recent first-principles study of the mobility of bulk p-type GaN, Poncé *et al.*^{17} suggested the application of significant tensile in-plane strain (or compressive c-axis strain) to raise the split-off band (SO) above the heavy-hole and light-hole bands. The lighter mass of the split-off band would allow for a drastically increased hole mobility. Such a proposal, while potentially revolutionary for bulk p-GaN, is difficult to apply to the particular heterostructure discussed here-in, since the GaN-pseudomorphic-to-AlN is already compressively strained by 2.4% as-grown and that large strain would have to be overcome first before applying further tensile strain. Moreover, applying tensile strain dramatically changes the interface between GaN and AlN, significantly reducing the valence band offset (VBO) and thus the confining potential. (The role of strain in the asymmetry of the GaN/AlN and AlN/GaN VBO is well-known.)^{35}

This raises the question of whether reasonable strain can improve the mobility of holes in this heterostructure, without access to the deep SO band. Based on the theory of Suzuki and Uenoyama^{36} in the context of lasers, two converse techniques may be suggested employing strain along only one in-plane axis. Dasgupta *et al.*^{37} considered the application of compressive strain along the direction of current flow. Conversely, Gupta *et al.*^{38} tested the application of tensile strain perpendicular to current flow. Figures 5 and 6 consider what effects these two proposals have, not on bulk p-GaN as with previous authors but rather on the heterojunction band structure. To take full advantage of the changing band structure near the band edge, the charge density is lowered (by thinning the well layer to 8 nm) and kept constant at 1 × 10^{13}/cm^{2} by variable applied bias even as the strain is changed. These adjustments to thickness and charge match closely the actual 2DHG environment in p-channel FETs.^{8} In parts (b) and (c) of these figures, uniaxial in-plane strain is seen to split the topmost two-bands. Depending on the sign and orientation of the strain, the topmost band may become light or heavy in the current flow direction (*x*).

In the case of tensile *x*-strain, Fig. 5(c), or compressive *y*-strain, Fig. 6(b), the topmost band is heavy along *x*, and so even though the available scattering DOS is reduced by the band-splitting, the net mobility remains low. However, in the case of compressive *x*-strain, Fig. 5(b), or tensile *y* strain, Fig. 6(c), the topmost band is light along *x* and interband scattering is diminished. As such, these strain conditions result in an enhanced mobility. The mobility improvement due to *x*-compressive strain is more pronounced than that due to *y*-tensile strain because of the aforementioned effect which tensile strain has on the VBO. However, these two enhancement mechanisms are not mutually exclusive: a fin relaxation and a compressive regrowth would complement each other via the Poisson effect to reduce the total amount of stress which each mechanism would have to apply.

Of course, any enhancements will still be limited by whatever extrinsic mechanisms are at play, and so Figs. 5 and 6 include two mobility curves: (1) a (solid blue) prediction which assumes the extrinsic scattering element is similar to the present data and (2) a (dashed blue) intrinsic calculation using only phonon mechanisms, corresponding to space for further improvements in growth.

In conclusion, this work has combined solutions for both the optical and acoustic phonon spectra in a GaN/AlN heterostructure with a multiband description of the hole gas to model the hole mobility. The model qualitatively matches experimental observations and can be used to estimate what sort of intrinsic mobility enhancements are possible by strain engineering, suggesting that a unixial compression is the best approach for optimizing the GaN/AlN hole gas. A high-mobility 2DHG could join the celebrated GaN 2DEG to realize the future of energy-efficient complementary GaN-based circuits.

See the supplementary material for mathematical and numerical simulation details.

This work was supported by Intel Corp, by AFOSR Grant No. FA9550-17-1-0048, and by NSF Grant Nos. 1710298/1534303. The authors appreciate discussions with Samuel Poncé, Feliciano Giustino, Tomás Palacios, Nadim Chowdhury, Paul Fischer, and Sansaptak Dasgupta.

## References

_{x}Ga

_{1−x}N quantum wells