Wurtzite phonons and the mobility of a GaN/AlN2D hole gas

Cite as: Appl. Phys. Lett. 114, 253501 (2019);doi: 10.1063/1.5099957Submitted: 13 April 2019 . Accepted: 1 June 2019 .Published Online: 24 June 2019

Samuel James Bader,1,a) Reet Chaudhuri,2 Martin F. Schubert,3 Han Wui Then,4 Huili Grace Xing,2,5,6

and Debdeep Jena2,5,6

AFFILIATIONS1School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA2School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14853, USA3X Development LLC, 100 Mayfield Ave., Mountain View, California 94043, USA4Intel Corporation, 2501 NE Century Blvd., Hillsboro, Oregon 97124, USA5Department of Materials Science and Engineering, Cornell University, Ithaca, New York 14853, USA6Kavli Institute at Cornell, Ithaca, New York 14853, USA

a)sjb353@cornell.edu. URL: http://sambader.net

ABSTRACT

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

Published under license by AIP Publishing. https://doi.org/10.1063/1.5099957

Decades after the celebrated invention of Mg p-doping1 inGallium Nitride (GaN) and the subsequent development of GaN-basedlight emitting diodes, the manipulation of holes in GaN remains a fun-damental challenge. Consequently, despite the expected dominance ofGaN High Electron Mobility Transistors (HEMTs) in the coming gen-eration of power electronics2 and communications systems,3 there isno complementary p-channel device which can be readily integrated.This incompleteness restricts the possible circuit topologies and systemdesigns achievable in GaN electronics but arises quite straightforwardlyfrom the physics of the GaN valence band. These bands, both heavyand deep in energy, have proven difficult to contact with typical metalworkfunctions,4 difficult to dope with high efficiency,5 and difficult toflow current through with high conductivity. Nonetheless, the com-mercial interest in generating complementary GaN-based circuits6 andscientific interest in studying highly degenerate hole physics7 haveprompted great recent progress in p-channel devices.8

Of the various structures6,9–16 which have been proposed as aplatform for p-channel III-Nitride electronics, the single GaN/AlNheterojunction field-effect transistor has received recent attention forits high sheet conductance7 and excellent device performance.8 In this

structure, depicted in Fig. 1(a), the all-binary materials provide astraightforwardly repeatable growth with no possible parasitic electronchannels, a tremendous hole-inducing polarization-charge for lowsheet-resistance, and maximal bandgaps for extreme voltage-handlingcapability. Given the recent reports of temperature-dependent trans-port studies in this heterostructure7 and recent first-principles sugges-tions of possible enhancements to hole mobility in bulk p-GaN,17 thiswork presents a model to explain the measured mobility of the 2Dhole gas (2DHG) at the GaN/AlN interface and evaluates the poten-tial of strain-engineering approaches to alter the band structure in afavorable way. First, the valence band structure including confine-ment and multiband mixing effects is computed. Then, the spectraof both acoustic and optical phonons in the heterostructure aredetermined and the mobility limitation due to these mechanisms isderived. Finally, the effects of strain on the band structure andmobility are presented.

The multiband k.p (MBKP) approach, based on Burt ExactEnvelope Function Theory,18,19 describes the electronic states of heter-ostructures wherein multiple subbands may be intermixed by nonuni-form potentials and material interfaces. MBKP has been extended to

Appl. Phys. Lett. 114, 253501 (2019); doi: 10.1063/1.5099957 114, 253501-1

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wurtzite heterostructures by various authors20,21 and is available incertain commercial packages.22 The PyNitride software package23

employed here self-consistently24 solves the 6! 6 MBKP equation ofthe wurtzite valence band (see the supplementary material) simulta-neously with the Poisson equation, accounting for the large fixed inter-face polarization charge. Figure 1(a) depicts a highly confined 2D holegas at the GaN/AlN interface. The hole gas represents contributionsmainly from the first quantized subband of both the heavy hole (HH)and light hole (LH) bands (including spin, this is four subbands). Thetransverse dispersion is shown in Fig. 1(b). By density, as clear fromFig. 1(c), the HH band dominates, though as shown in Fig. 1(d), theLH band has lighter in-plane mass near the Fermi energy, and soit contributes significantly to transport. To evaluate transport, weproceed to describe the phonons.

The Dielectric Continuum model25 describes polar optical pho-nons (POPs) in arbitrary heterostructures. As POP scattering is themain limitation on electron mobility in quality GaN, numerousauthors have invested significant theoretical effort into the elaborationof POP spectra in various wurtzite heterostructures.26–30 A structureas simple as this, in fact, can be solved analytically. For a uniaxial crys-tal, the effect of polar optical phonons can be incorporated into twofrequency (x) dependent, directional dielectric functions, e?and ek,

e?¼ e1x2

LO?# x2

x2TO?# x2 and ek ¼ e1

x2LOk # x2

x2TOk # x2 ; (1)

where those longitudinal (xLO) and transverse (xTO) POP frequenciesand high-frequency dielectric constant !1 can be determined

experimentally for the materials in play (see tabulation in Komirenkoet al.29). For a mode with in-plane wavevector q, the POP problemreduces to solving a frequency-dependent Poisson eigenvalue equation

@z!k@z/ ¼ q2!?/ (2)

for the potential /. At every characteristic frequency appearing inEq. (1), a dielectric constant changes sign, which changes the characterof 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 inGaN, and (3) decaying bidirectionally from a GaN/AlN interface. Foreach class, both transverse and longitudinal polarizations are possible.The spectrum and example modes are depicted in Fig. 2, and the solu-tion is elaborated in the supplementary material.

The Elastic Continuummodel25 aptly describes acoustic phononsin arbitrary heterostructures. Given the centrality of “optical” phononsin electron-based devices, the literature on acoustic phonons in wurt-zite heterostructures31–34 is significantly less comprehensive. The basicapproach is to link a continuum Newton’s law with a material stress-strain relation

q@2ui@t2¼@Tij

@rj; Tij ¼ cijkl!kl; (3)

where ui is the local displacement vector, q is the density, Tij is thestress tensor, cijkl is the stiffness tensor, and !ijkl is the strain tensor!ij ¼ 1

2 ð@rj ui þ @ri ujÞ. Due to symmetry constraints on the stiffnesstensor, the in-plane shear component is uncoupled (“Y” modes), whilelongitudinal and out-of-plane components constitute two coupled dif-ferential equations (“XZ” modes). These are solved by the FiniteElement Method. For both types of modes, there are two characters

FIG. 1. (a) A 13 nm GaN layer on top of a thick AlN buffer induces a two-dimensionalhole gas of density '4.4! 1013/cm2, which is confined at the interface by strongpolarization fields. (b) The relevant bands are the first spin-degenerate HH subband-pair (blue) and LH subband-pair (red). (c) The DOS of the two bands is indicated bysolid lines and the occupation of the bands by filled shapes. (d) The effective massesvs position in k-space, capped by the occupation probability as a function of k. Theenergy axes of (b) and (c) align, and the k-axes of (b) and (d) align. Dashed linesthat show the effective masses near the Fermi energy are guides to the eye.

FIG. 2. Polar optical phonons. (a) Spectrum of the various extra-ordinary POPmodes, with the character of each mode labeled. The bottom three bands (withinred frequency labels) are predominantly transverse (TO), and the top three bands(within blue frequency labels) are predominantly longitudinal (LO). Examples of (b)modes confined to GaN, (c) to the interface, and (d) to AlN. For each, the potentialof both possible polarizations is shown. As expected, the TO phonons contributeweaker potential.

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possible depending on the energy range. At energies above !hvtq, wherevt is the relevant AlN sound velocity, modes are able to propagate inthe AlN and are thus of a continuum sort. Below this energy, anymodes which exist must decay into the AlN, so are GaN-confinedmodes. The spectrum and example modes are revealed in Fig. 3, andthe solution methodology is elaborated in the supplementary material.

We now turn to the interactions of the phonons and carriers.First, both types of phonon modes are normalized25 by the quantiza-tion condition

Ðd3~r qð~rÞj~uðrÞj2 ¼ !h

2x. Once the oscillations are nor-malized, the Hamiltonian term for each mode can be generated. Foroptical phonons, this is just Hpop¼ #e/(r). For acoustic phonons, thedeformation potential Hadp ¼ D (!) is a six-by-six matrix function ofposition: at each position, it recruits the valence deformation matrix ofthe local material and also the local strain from a given acoustic mode.There is also a (coherently combined) piezo termHpz¼ #e/(r) where/ is found32 by solving the Poisson equation given the piezoelectriccharge induced by the mode (see the supplementary material). Finally,to account for the low-temperature mobility, which is around200 cm2/Vs in these structures to date, some extrinsic limitation mustbe included. The exact cause (e.g., interface roughness, dislocation,impurity, etc) is irrelevant to this study, since all these elastic mecha-nisms are temperature-independent and have similar dependence oneffective masses. So the exact cause is not deduced here, but rather ageneric scatterer with a constant scalar matrix element is applied to setthe low-temperature mobility to 200 cm2/Vs. This matrix element isthe only “tuning parameter.”

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

q~E!h(rkf 0mðkÞ ¼

X

k0m0Rk0m0km f Am ðkÞ # Rkm

k0m0 fAm0ðk

0Þ; (4)

where ~E is the in-plane electric field, f 0 is the equilibrium occupationfunction, f A is a small change in occupation to be solved for, and thetransition rate R is given by

Rk0m0km ¼

2p!h

X

al

jhw k0m0 jHlðqÞjw kmij2

! N þ 12

1# að Þ þ af 0m0ðk0Þ

" #

! dðEk0m0 # Ekm # aelðqÞÞ; (5)

where a ¼61 represents the absorption/emission, respectively, w km isthe state with in-plane wavevector~k, of subband m, HlðqÞ is the per-turbation due to mode l with wavevector~q ¼~k

0#~k, and E/e are the

electronic/phonon energies. Once this is discretized on a k-spacemesh, the LBTE is solved as a linear matrix equation for the change inoccupation @f A=@E under an applied field, from which the mobility isextracted. The results, Fig. 4, compare agreeably to experiment over awide temperature range and verify Ponc"e’s prediction17 that acousticphonons dominate scattering at room temperature.

Now, we discuss what can be done to improve the mobility. In arecent first-principles study of the mobility of bulk p-type GaN, Ponc"eet 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

FIG. 3. Acoustic phonons. Spectrum of the (a) XZ polarized and (b) Y polarizedmodes. In each, there is a characteristic velocity dividing discrete modes and con-tinuum modes:

ffiffiffiffiffiffiffiffiffiffiffiffiC44=q

pand

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðC11 # C12Þ=2q

p, respectively. (c) An example of a

continuum Y mode: oscillations are more concentrated in the GaN where the higherdensity leads to lower sound velocity. The lower sound velocity of GaN relative toAlN also allows for confinement, such as in (d), which depicts a confined XZ mode.

FIG. 4. (a) The hole occupation of the HH band at equilibrium and (b) the antisym-metric change in occupation per applied electric field in the þx direction, obtainedby direct solution of the linearized Boltzmann Transport Equation. (The hole popula-tion is enhanced at negative kx, which, given the negative group velocity for holes,implies a rightward current.) (c) Model mobility vs Hall measurements reported byChaudhuri et al.7Dashed curves are also shown for the various scattering mecha-nisms alone (polar optical, acoustic, and extrinsic). Note: the model is obtained bya full solution incorporating all mechanisms simultaneously, not by a Matthiessenapproximation of the component limitations.

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heavy-hole and light-hole bands. The lighter mass of the split-off bandwould allow for a drastically increased hole mobility. Such a proposal,while potentially revolutionary for bulk p-GaN, is difficult to applyto 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 beforeapplying further tensile strain. Moreover, applying tensile strain dra-matically changes the interface between GaN and AlN, significantlyreducing the valence band offset (VBO) and thus the confining poten-tial. (The role of strain in the asymmetry of the GaN/AlN andAlN/GaN VBO is well-known.)35

This raises the question of whether reasonable strain canimprove the mobility of holes in this heterostructure, without accessto the deep SO band. Based on the theory of Suzuki and Uenoyama36

in the context of lasers, two converse techniques may be suggestedemploying strain along only one in-plane axis. Dasgupta et al.37 con-sidered the application of compressive strain along the direction ofcurrent flow. Conversely, Gupta et al.38 tested the application oftensile strain perpendicular to current flow. Figures 5 and 6 considerwhat effects these two proposals have, not on bulk p-GaN as with pre-vious authors but rather on the heterojunction band structure. Totake full advantage of the changing band structure near the bandedge, the charge density is lowered (by thinning the well layer to8 nm) and kept constant at 1! 1013/cm2 by variable applied bias evenas the strain is changed. These adjustments to thickness and chargematch closely the actual 2DHG environment in p-channel FETs.8 Inparts (b) and (c) of these figures, uniaxial in-plane strain is seen tosplit the topmost two-bands. Depending on the sign and orientationof the strain, the topmost band may become light or heavy in thecurrent 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 theavailable scattering DOS is reduced by the band-splitting, the netmobility 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 xand interband scattering is diminished. As such, these strain

conditions result in an enhanced mobility. The mobility improvementdue to x-compressive strain is more pronounced than that due toy-tensile strain because of the aforementioned effect which tensilestrain has on the VBO. However, these two enhancement mechanismsare not mutually exclusive: a fin relaxation and a compressive regrowthwould complement each other via the Poisson effect to reduce the totalamount of stress which each mechanism would have to apply.

Of course, any enhancements will still be limited by whateverextrinsic mechanisms are at play, and so Figs. 5 and 6 include twomobility curves: (1) a (solid blue) prediction which assumes the extrin-sic scattering element is similar to the present data and (2) a (dashedblue) intrinsic calculation using only phonon mechanisms, corre-sponding to space for further improvements in growth.

In conclusion, this work has combined solutions for both theoptical and acoustic phonon spectra in a GaN/AlN heterostructurewith a multiband description of the hole gas to model the hole mobil-ity. The model qualitatively matches experimental observations andcan be used to estimate what sort of intrinsic mobility enhancementsare possible by strain engineering, suggesting that a unixial compres-sion is the best approach for optimizing the GaN/AlN hole gas. Ahigh-mobility 2DHG could join the celebrated GaN 2DEG to realizethe future of energy-efficient complementary GaN-based circuits.

See the supplementary material for mathematical and numericalsimulation details.

This work was supported by Intel Corp, by AFOSR Grant No.FA9550-17-1-0048, and by NSF Grant Nos. 1710298/1534303. Theauthors appreciate discussions with Samuel Ponc"e, FelicianoGiustino, Tom"as Palacios, Nadim Chowdhury, Paul Fischer, andSansaptak Dasgupta.

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FIG. 5. (a) x-directed mobility due to all mechanisms (solid blue) and phonons only(dashed blue) vs strain along the current-flow direction (x). The y-direction is leftpseudomorphic (#2.4% y-strain in GaN). Bottom axis: x-strain applied to structure.Top axis: corresponding total x-strain in GaN. For compressive x-strain, thex-directed mobility is enhanced. The dispersion along is shown under (b) compres-sion and (c) tension as solid curves (dashed as-grown bands provided for compari-son). The x-directed mobility enhancement corresponds to a reduced effectivemass along kx of the topmost band in (b). (d) Such compression could be applied,for example, by regrown stressors on the source and drain.

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Appl. Phys. Lett. 114, 253501 (2019); doi: 10.1063/1.5099957 114, 253501-4

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Supplement to “Wurtzite Phonons and the

Mobility of a GaN/AlN 2D Hole Gas”

S1) Multiband kp model

The Multiband k.p matrix di↵erential equation1 can be split into four terms bythe order of derivatives

⇥C

0(z)� iCL(z)@z � i@zC

R(z)� @zC2(z)@z

⇤f(z) = �f(z) (S1)

where, for a wurtzite valence band f(z) is a six-component spinor. In the basis|X "i , |Y "i , |Z "i , |X #i , |Y #i , |Z #i, these position-dependent matrices canbe written

C0 = C

0L + C0D + C

0S (S2)

C0L = I2 ⌦

0

@kxL

u1kx + kyM

u1 ky kxN

+1 ky + kyN

�1 kx ·

kyN+1 kx + kxN

�1 ky kxM

u1 kx + kyL

u1ky ·

· · kxMu3 kx + kyM

u3 ky

1

A

(S3)

C0D =

0

BBBBBB@

�1 �i�2 · · · �3

i�2 �1 · · · �i�3

· · · ��3 i�3 ·· · ��3 �1 i�2 ·· · �i�3 �i�2 �1 ·

�3 i�3 · · · ·

1

CCCCCCA(S4)

C0S =

0

@l1exx +m1eyy +m2ezz n1exy n2exz

n1exy m1exx + l1eyy +m2ezz n2eyz

n2exz n2eyz m3exx +m3eyy + l2ezz

1

A

(S5)

CL = I2 ⌦

0

@· · kxN

+2

· · kyN+2

kxN�2 kyN

�2 ·

1

A (S6)

CR = I2 ⌦

0

@· · N

�2 kx

· · N�2 ky

N+2 kx N

+2 ky ·

1

A (S7)

1

C2 = I2 ⌦

0

@M

u2 · ·· M

u2 ·

· · Lu2

1

A (S8)

where I2 is the 2x2 identity matrix,

Lu

1 = A2 +A4 +A5, Lu

2 = A1

Mu

1 = A2 +A4 �A5, Mu

2 = A1 +A3, Mu

3 = A2

N+1 = 3A5 �A2 �A4 + U ; N

�1 = �A5 +A2 +A4 � U

N+2 =

p2A6 �A1 �A3 + U, N

�2 = A1 +A3 � U

with U = h2

2meand

l1 = D2 +D4 +D5, l2 = D1

m1 = D2 +D4 �D5, m2 = D1 +D3, m3 = D2

n1 = 2D5, n2 =p2D6

where Ai are the Rashba-Sheka-Pikus parameters and Di are the deformationpotentials. Notes:

• The A7 parameter is neglected as is common in heterostructural multi-band kp analysis1. The e↵ects of this negligence are known8

• Since none of the important scattering mechanisms interact with spinand the spin-splitting is a small energy scale in the band structure, onecan “cheat” that degree of freedom to speed up calculations. For fasterevaluation of mobility, the o↵-diagonal 3x3 blocks of C0D are zeroed anda sub-meV up-down term is added along the diagonal to ensure the spinshave a finite non-degeneracy. Then the mobility calculation can accountfor only one of the spins and assume the same for the other.

• The above calculation is a 6x6 kp method in which the valence bandsare accounted for directly and the conduction band appears only as aperturbation. Since GaN is a wide-bandgap material, this convenient sim-plification is reasonably accurate as compared to a slower 8x8 kp imple-mentation.

S2) Acoustic phonon dispersion

As given in the main text, the elastic continuum model joins the continuumNewton’s law with the material stress-strain relation7:

⇢@2ui

@t2=

@Tij

@rj, Tij = cijkl✏kl (S9)

2

where ui is the local displacement, ⇢ is the density, Tij is the stress tensor, cijklis the sti↵ness tensor, and ✏ijkl is the strain tensor ✏ij =

12

�@rjui + @riuj

�. The

latter equation can be re-expressed in Voigt notation

T↵ = c↵�✏� (S10)

where ↵,� run 1-6 and the Voigt tuples are related to the actual tensors by

T1 = Txx, ✏1 = ✏xx (S11)

T2 = Tyy, ✏2 = ✏yy (S12)

T3 = Tzz, ✏3 = ✏zz (S13)

T4 = Tyz, ✏4 = 2✏yz (S14)

T5 = Txz, ✏5 = 2✏xz (S15)

T6 = Txy, ✏6 = 2✏xy (S16)

For a wurtzite crystal, the c↵� can be written

c =

0

BBBBBB@

C11 C12 C13 0 0 0C12 C22 C13 0 0 0C13 C13 C33 0 0 00 0 0 C44 0 00 0 0 0 C44 00 0 0 0 0 1

2 (C11 � C12)

1

CCCCCCA(S17)

For a wurtzite structure uniform in the basal plane but possibly inhomoge-nous along z, we note that the elastic continuum model is rotationally symmetricin plane, so we can choose the in-plane wavevector along x and assume phononshave a form

u =

0

@ux(z)uy(z)uz(z)

1

A ei(qx�!t) (S18)

We will hide the explicit z dependence for now. The the strain can be evaluated

✏1 = iqux, ✏4 = @zuy (S19)

✏2 = 0, ✏5 = iquz + @zux (S20)

✏3 = @zuz, ✏6 = iquy (S21)

Then the two constitutive laws can be combined as

�⇢!2

0

@ux

uy

uz

1

A =

0

@�C11q

2ux + iqC13@zuz + @zC44iquz + @zC44@zux

� 12 (C11 � C12) q2uy + @zC44@zuy

�C44q2uz + C44iq@zux + iq@zC13ux + @zC33@zuz

1

A (S22)

This matrix can be split by order

⇢!2u = Cu, C = q

2C

0 � iqCL@z � iq@zC

R � @zC2@z (S23)

3

where

C0 =

0

@C11

12 (C11 � C12)

C44

1

A , C2 =

0

@C44

C44

C33

1

A (S24)

CL =

0

@C13

0C44

1

A , CR =

0

@C44

0C13

1

A (S25)

This generalized eigenvalue problem can be solved by the Finite Element Methodonce boundary conditions are prescribed. Specifically, the upper boundary willbe treated as free to vibrate (i.e. Neumann), so Tiz = 0. The bottom boundaryis unimportant in the limit of a thick bu↵er, but, to preclude the appearance ofan irrelevant bottom-boundary mode decaying upwards into AlN, one may setthe bottom boundary by Dirichelet ui = 0 condition.

We note here that in the above matrices, there is no coupling between the Y(second) component and the XZ (first and third) components, so the problemcould be further broken apart into a pair of problems if desired.

S2.1) Piezoelectric potential

Once the acoustic phonon modes are solved for, each mode can be considereda source of piezoelectric charge, which induces a further scattering potential.This approximation of treating the acoustic and electric problem separately iscommon practice6 given the low-frequency of zone-center acoustic phonons. Aphonon in a piezoelectric material induces a charge

⇢ = �r · ~P = �r ·

2

6666664

0

@0 0 0 0 e15 00 0 0 e15 0 0e31 e31 e33 0 0 0

1

A

0

BBBBBB@

✏xx

✏yy

✏zz

2✏yz2✏xz2✏xy

1

CCCCCCA

3

7777775(S26)

where e↵� are the piezoelastic moduli

= �r ·

0

@e15✏xz

e15✏yz

e31✏xx + e31✏yy + e33✏zz

1

A (S27)

= �e15(iqx✏xz + iqy✏yz)� @z[e31(✏xx + ✏yy) + e33✏zz] (S28)

= �e15(iqx@zux � q2xuz + iqy@zuy � q

2yuz) (S29)

� @z[ie31(qxux + qyuy) + e33@zuz] (S30)

= �e15(iq@zuL � q2uz)� @z[iqe31uL + e33@zuz] (S31)

= q2e15uz � iqe15@zuL � iq@ze31uL � @ze33@zuz (S32)

4

where uL is the in-plane longitudinal component of the displacement. Pluggingthis into the Poisson equation, we find a potential

�r ["r�] = ⇢ (S33)

q2"?�� @z"k@z� = ⇢ (S34)

So

q2"?�� @z"k@z� = q

2e15uz � iqe15@zuL � iq@ze31uL � @ze33@zuz (S35)

which can be written

C0�� @zC

2@z� = C

00uz � iC

L0@zux � i@zC

R0ux � @zC

20@zuz (S36)

whereC0 = q

2"?, C2 = "k (S37)

C00 = q

2e15, C

L0= qe15, C

R0= qe31, C

20 = e33 (S38)

and solved by the Finite Element Method.

S3) Polar optical phonon dispersion

As discussed in the main text, the interaction of the uniaxial polar atomiclattice with electromagnetic waves can be incorporated as a pair of frequency-dependent dielectric constants "k and "?, upon which solving the Poisson equa-tion gives all the modes which produce an electric potential [that is, all themodes which are important for POP scattering]. For in-plane wavevector q,

@z✏k@z� = q2✏?� (S39)

The normalization condition given in the main text, applied to both classes ofphonon, is expressed in terms of u, but, by solving for u as a function of �, thiscondition can be re-expressed directly in terms of �:

h

2!=

Zdz"1(!2

LO� !

2TO

)

0

@

@z�

!2TOk � !2

!2

+

✓q�

!2TO? � !2

◆21

A (S40)

Given the frequency-dependence of the parameters in the equation, a numericalsolution of the eigenvalue problem is actually somewhat involved. Fortunately,the discretely layered binary heterostructure structure lends itself to analyticsolutions. We will solve a single heterojunction structure 1/2 where materials 1and 2 are GaN or AlN in either order, and the bottom material is semi-infinite.The top surface at z = 0 is assumed Dirichelet. The thickness of the top layeris t1 and a normalization thickness of t2 is set for the bottom layer which willdiscretize the bottom-region-confined states.

5

In a given region, solutions are oscillating if "?"k < 0 and exponential if"?"k > 0. At an interface, the derivative switches signs i↵ "1k"2k < 0. We willuse the following convenient definitions, similar to the notation of Komirenko5

but for a factor of two in ↵

⇠i =q��"i?"ik

��, ↵i =q��"i?/"ik

�� (S41)

Then the vertical wavevector of a mode in a given region is ki = q↵i.If the solution is written in Region 1 with some normalization constant A

and Region 2 with some normalization constant B, then the first matching con-dition �(t�1 ) = �2(t

+1 ) gives us some expression for B/A, and the normalization

condition will be written for A:

A2 =

h

2!

�"�2k1�

2k1 + �

2?1�

2?1 +

✓B

A

◆2 ⇣�2k2�

2k2 + �

2?2�

2?2

⌘#(S42)

with

�2ki = "

1i

�!2LOi

� !2TOi

�

ki

!2TOki � !2

!2

(S43)

�2?i

= "1i

�!2LOi

� !2TOi

�✓ q

!2TO?i

� !2

◆2

(S44)

and

�2ki =

Z

i

dz

✓@z�

Aki

◆2

, �2?i

=

Z

i

dz

✓�

B

◆2

(S45)

From there on out, the solution is a simple mechanical procedure, dependingon the signs of the dielectric constants.

S3.1) Confined to Region 1

If the solution is oscillating in Region 1 and decaying in Region 2, we can write

�1 = A sin(k1z), �2 = Be�k2z (S46)

Matching interface conditions gives

q =1

↵1t1

⇥tan�1 (⇠1/⇠2) + ⇡n

⇤(S47)

with B/A = sin(k1t1)ek2t1 and

�k1 =1

2(t1 +

1

2k1sin(2k1t1)), �?1 =

1

2(t1 �

1

2k1sin(2k1t1)) (S48)

�k2 =1

2k2e�2k2t1 , �?2 =

1

2k2e�2k2t1 (S49)

6

S3.2) Confined to Interface

If the solution is decaying in both regions, we can write

�1 = A sinh(k1z), �2 = Be�k2z (S50)

Matching interface conditions gives

q =1

2↵t1log

⇠2 + ⇠1

⇠2 � ⇠1

�(S51)

with B/A = sinh(k1t1)ek2t1 and

�k1 =1

2

✓1

2k1sinh(2k1t1) + t1

◆, �?1 =

1

2

✓1

2k1sinh(2k1t1)� t1

◆(S52)

�k2 =1

2k2e�2k2t1 , �?2 =

1

2k2e�2k2t1 (S53)

S3.3) Confined to Region 2

If the solution is decaying in Region 1 and oscillating in Region 2, we can write

�1 = A sinh(k1z), �2 = B sin(k2z + ✓) (S54)

Matching interface conditions gives

✓ = tan�1

✓⇠2

⇠1tanh(k1t1)

◆� k2t1 (S55)

with B/A = sinh(k1t1)/ sin(k2t1 + ✓). The t2 thickness normalization givesk2 = ⇡(n+ 1)/t2, so

q =⇡(n+ 1)

↵2t2(S56)

Normalization is accounted for via

�k1 =1

2

✓1

2k1sinh(2k1t1) + t1

◆, �?1 =

1

2

✓1

2k1sinh(2k1t1)� t1

◆(S57)

�k2 =t2

2, �?2 =

t2

2(S58)

S4) Numerical details

Most material parameters are drawn from Vurgaftman and Meyer’s compila-tion11, but anisotropic dielectric constants are from the measurements of Kane3

and Kazan4, optical phonon parameters are from the compilation in Komirenko5

and the polarization-related quantities are computed from the calculations ofDreyer2. To ensure that the results do not depend on the specific choice of k · p

7

parameters, we performed the entire calculation with not only the k · p param-eters from Vurgaftman and Meyer but also ran a comparison with more recentparameters fitted by Rinke9 from atomistic GW calculations (but keeping the�SO since Rinke neglects this). Results and trends were quite similar despitethe di↵erences in individual parameters.

The first step in each mobility point is calculation of the band diagram by astandard Newton iteration of a charge model (including analytical derivatives)and the Poisson equation (as laid out in Tan’s work10). The Poisson equationincludes a Dirichelet boundary at the surface (SBH of 1.4 eV) and a Neumannboundary at the artificial bottom termination (500nm of AlN). The charge den-sity consists of polarization contributions (from the interface discontinuities),quantum carrier densities (from occupation of the MBKP-solved states in aregion extending from the surface to several nanometers below the interface),classical carrier densities (from occupation of the band-edge DOS in a regionextending from several nanometers below the interface down the rest of thedomain), and a small background (1 ⇥ 1017/cm3) of deep donor-like defectsthroughout. To achieve convergence over a wide range of conditions and tem-peratures despite extreme charges, the algorithm begins by artificially scalingup all dielectric constants by orders of magnitude to decouple the charge andfields, and then ramps the dielectric constants to their true values. For the sim-ulations with varied strain, the Fermi level at the surface is adjusted (mimickingapplication of a gate) to keep the hole sheet density at a particular set value.

Separately, the phonon energies are evaluated by the means described inthe above sections, with the caveat that the artificial bottom termination forthe phonon solution domain is 40nm into the AlN (rather than 500nm as usedfor electrical simulation). This abridgment drastically reduces the time andstorage requirements of the mobility solution, mainly by reducing the numberof pseudo-continuum modes which must be considered, but is deep enough thatthe artificial confinement e↵ects on the phonon spectra are at an energy scalewell below thermal energy over the entire temperature range where phononscontribute significantly to scattering. Specifically, the phonon states used arethe first 400 acoustic modes (including discrete and pseudo-continuum statesindiscriminately), the first 20 AlN-confined optical modes of each polarization,the first 30 GaN-confined modes of each polarization, and both polarizations ofthe interface optical mode.

With both phonon and electron states solved, we then solve the LinearizedBoltzmann Transport Equation. We first discretize k-space. In regions of k-space where the occupation changes rapidly with k (that is, essentially, regionswithin a kbT scale of the Fermi energy and with significant group velocity), it isvital to define a dense grid. However, since this equation is to be solved over awide range of temperatures and under many variations of the bandstructure, theprecise locations of these important regions are not known in advance. Thus,an adaptive mesh is formed by the following procedure.

The mesh will be uniform in the ✓ direction, but variably spaced along k. Thederivative D of circularly averaged occupation with respect to k is calculated(D = 1

2⇡@k

Rd✓f(k, ✓) ), and an upper bound kmax for the k-mesh is determined

8

by a point beyond which D has fallen o↵ to a negligible value compared to itspeak. The simplest strategy would be then to space mesh points evenly withrespect to k from zero to this upper bound. A more aggressive strategy is tospace mesh points evenly with respect to circularly averaged occupation, thusregions with a rapidly changing occupation have the higher density of points. Forthis work, a compromise is struck in which the k-points are placed evenly alonga weighted average (.7+.3) of 1

2⇡

Rd✓f(k, ✓) and k/kmax, ensuring a spread of k-

points which cluster around the vital regions. Employing this adaptive scheme,a relatively small grid of 25 k-values x 18 ✓ values (or, for simulations in whichhexagonal symmetry broken, 25x24) was found su�cient. This is the mesh onwhich transition matrix elements will be calculated.

This mesh is further refined by even subdivision (here by a factor of 4)for the computation of the energy-conserving � function factor, a more rapidlyvarying factor of position. For each subcell, all the other subcells into which thecenterpoint of that subcell may legally scatter are found, and the delta-functiondeterminant is computed using k-derivatives evaluated at the subcell centers.These contributions over subcells are then summed to produce a sparse matrixof the conservation factors between each cell of the original mesh described inthe previous paragraph. Then, on the original mesh, the necessary transitionmatrix elements can be calculated only where needed by the relevant interactionHamiltonian and the two factors are multiplied.

Explicitly, those transition matrix elements, the Hl(q) in Eq 5 of the main

text, are as follows. For acoustic phonons, H l = Hl

adp+ H

lpz. H

l

adp= C

0S(z)from Eq (S5) using the strain phasors eii(z) from the acoustic mode of levell solved for in Sec S2) and the material dependent Di(z) parameters. H

lpz

=�e�(z) where � is the piezoelectric potential generated by the acoustic mode oflevel l as solved in Sec S2.1). For polar optical phonons, H l

pop= �e�(z) where

�(z) is the potential coupled with the POP mode of level l. Finally, for the“generic extrinsic scatterer”, H l is just a constant (17.3 meV) tuned to fit thelow-temperature mobility.

The transition rates are arranged to form a transition matrix Rk0m

0

km, and

thus Eq (4) of the main text is solved by a least squares procedure to find theperturbation-per-electric-field of the carrier distribution function, from whichthe mobility is extracted by an occupation-weighted sum of the group velocities.

References

[1] Stefan Birner. “Modeling of semiconductor nanostructures and semicon-ductor – electrolyte interfaces”. PhD thesis. Technical University of Mu-nich, 2011. isbn: 9783941650350.

[2] Cyrus E. Dreyer et al. “Correct implementation of polarization constantsin wurtzite materials and impact on III-nitrides”. In: Phys. Rev. X 6.2(2016), p. 1. issn: 21603308. doi: 10.1103/PhysRevX.6.021038.

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[3] M. J. Kane et al. “Determination of the dielectric constant of GaN in thekHz frequency range”. In: Semiconductor Science and Technology 26.8(2011), p. 1. issn: 02681242. doi: 10.1088/0268/26/8/085006.

[4] M. Kazan et al. “Directional dependence of AlN intrinsic complex di-electric function, optical phonon lifetimes, and decay channels measuredby polarized infrared reflectivity”. In: J. App. Phys. 106.2 (2009). issn:00218979. doi: 10.1063/1.3177323.

[5] S. Komirenko et al. “Dispersion of polar optical phonons in wurtzite quan-tum wells”. In: Phys. Rev. B 59.7 (1999), p. 5013. issn: 0163. doi: 10.1103/PhysRevB.59.5013.

[6] Evgenii P. Pokatilov, Denis L. Nika, and Alexander A. Balandin. “Con-fined electron-confined phonon scattering rates in wurtzite AlN/GaN/AlNheterostructures”. In: J. App. Phys. 95.10 (2004), p. 5626. issn: 00218979.doi: 10.1063/1.1710705.

[7] Evghenii P. Pokatilov, Denis L. Nika, and Alexander A. Balandin. “Phononspectrum and group velocities in AlN/GaN/AlN and related heterostruc-tures”. In: Superlattices and Microstructures 33.3 (2003), p. 155. issn:07496036. doi: 10.1016/S0749(03)00069.

[8] G. B. Ren, Y. M. Liu, and P. Blood. “Valence-band structure of wurtziteGaN including the spin-orbit interaction”. In: App. Phys. Letters 74.8(1999), p. 1117. issn: 00036951. doi: 10.1063/1.123461.

[9] Patrick Rinke et al. “Consistent set of band parameters for the group-IIInitrides AlN, GaN, and InN”. In: Phys. Rev. B - Condensed Matter andMaterials Physics (2008). issn: 10980121. doi: 10.1103/PhysRevB.77.075202.

[10] I-H. Tan et al. “A self-consistent solution of Schrodinger–Poisson equa-tions using a nonuniform mesh”. In: J. App. Phys. 68.8 (Oct. 1990),p. 4071. issn: 00218979. doi: 10.1063/1.346245.

[11] I. Vurgaftman and J. R. Meyer. “Band parameters for nitrogen-containingsemiconductors”. In: J. App. Phys. 94.6 (2003), p. 3675. issn: 00218979.doi: 10.1063/1.1600519.

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