Table 2 The formulas of major scattering mechanisms used in 2DEG mobility calculations

Acoustic phonon: piezoelectric[15–17]

${\mu }_{\text{PE}}=\frac{\pi {\epsilon }_{\mathrm{s}}{\hslash }^{3}k}{e{K}^2{k}_{\mathrm{B}}T{m}^{*2}}\frac{1}{J_{\text{PE}}\left(k\right)}$

K, electromagnetic coupling coefficient; J_PE(k), electron wave vector dependent integral.

$J_{\text{PE}}\left(k\right)={\displaystyle {\int }_0^{2k}\frac{F_{11}\left(q\right)}{4k^2{\left(q+q_{\mathrm{s}}\right)}^2\sqrt{1-{\left(q/2k\right)}^2}}{q}^{3}dq}$

$K^2=\frac{{\epsilon }_{\text{LA}}^2}{{\epsilon }_{\mathrm{s}}{c}_{\text{LA}}}+\frac{{\epsilon }_{\text{TA}}^2}{{\epsilon }_{\mathrm{s}}{c}_{\text{TA}}}$

Acoustic phonon: deformation[11, 18] potential

${\mu }_{\mathit{\text{DP}}}=\frac{16\rho e{v}_s^2{\hslash }^3}{3{\Xi }^2{k}_{B}T{m}^{*2}b}\frac{1}{J_{\mathit{\text{DP}}}\left(k\right)}$

ρ, crystal density; v _s , longitudinal acoustic phonon velocity; Ξ_, deformation potential constant; m*, electron effective mass; J_DP(k), electron wave vector dependent integral. b, Fang-Howard expression; q_s, reciprocal screening length; f(0), occupation probability; F₁₁(q), ground-state Fang-Howard wave function.

$J_{\text{DP}}\left(k\right)={\displaystyle \underset{0}{\overset{2k}{\int }}\frac{1}{2k{\pi }^3{\left(q+q_{\mathrm{s}}\right)}^2\sqrt{1-{\left(q/2k\right)}^2}}}{q}^{4}dq$

$q_{\mathrm{s}}=\frac{e^2{m}^{*}}{2\pi {\hslash }^2{\epsilon }_{\mathrm{s}}}{F}_{11}\left(q\right)f\left(0\right)$

$b={\left(\frac{33e^2{m}^{*}{n}_{2\mathrm{D}}}{8{\epsilon }_{\mathrm{s}}{\hslash }^2}\right)}^{1/3}$

$F\left(q\right)=b\left(8b^2+9qb+3q^2\right)/8{\left(b+q\right)}^3$

Polar optical phonon[17–19]

${\mu }_{\text{PO}}=\frac{4\pi {\epsilon }_{\text{s}}{\hslash }^2}{e\omega m^{*2}{Z}_0}\left[e^{\hslash {\omega }_{\text{LO}}/k_{\mathrm{B}}T}-1\right]$

$\hslash {\omega }_{\mathit{\text{LO}}}$, polar optical phonon energy;${\epsilon }_{\infty }$ and${\epsilon }_{\mathrm{s}}$, high- and low-frequency dielectric constant; Z₀, effective width of triangular well formed at the Ga_xIn_1-xN/GaN interface and is given in terms of Fermi wave vector.

$\frac{1}{{\epsilon }_{\mathrm{P}}}=\frac{1}{{\epsilon }_{\infty }}-\frac{1}{{\epsilon }_{\mathrm{s}}}$

$Z_0=\frac{2\pi }{k_{\mathrm{F}}}=\sqrt{\frac{2\pi }{n_{2\mathrm{D}}}}$

Interface roughness[11, 15, 20]

${\mu }_{\text{IFR}}={\left(\frac{2{\epsilon }_{\mathrm{s}}}{n_{2\mathrm{D}}\Delta \Lambda }\right)}^2\frac{{\hslash }^3}{e^3{m}^{*2}}\frac{1}{J_{\text{IFR}}\left(k\right)}$

Δ, lateral size of the roughness; _Λ, correlation length between fluctuations; J_IFR(k), correlation length and the lateral size-dependent integral; n_2D, 2D electron density.

$J_{\text{IFR}}\left(k\right)={\displaystyle \underset{0}{\overset{2k}{\int }}\frac{exp\left(-q^2{\Lambda }^2/4\right)}{2k^3{\left(q+q_{\mathrm{s}}\right)}^2\sqrt{1-{\left(q/2k\right)}^2}}}{q}^{4}dq$

$q=2ksin\left(\theta /2\right)$

$q_{\mathrm{s}}=\frac{e^2{m}^{*}}{2\pi {\epsilon }_{\mathrm{s}}{\hslash }^2}F\left(q\right)$

Alloy disorder[20]

${\mu }_{\text{Alloy}}=\frac{16e{\hslash }^3}{3bx\left(1-x\right)m^{*2}{\Omega }_0{U}_{\mathrm{A}}^2}$

x, Ga fraction; Ω₀, the volume occupied by one atom; U_A, alloy potential.

Dislocation[21–23]

${\mu }_{\mathit{\text{Dis}}}=\frac{30\sqrt{2\pi }{\epsilon }^2{c}^2{\left(k_{B}T\right)}^{3/2}}{e^3{N}_{\mathit{\text{Dis}}}{f}^2{\lambda }_D\sqrt{m^{*}}}$

N_Dis, dislocation density per unit area which is taken as a fitting parameter; λ_D, Debye screening length; c, lattice constant of Ga_xIn_1-xN. f, the fraction of filled traps that are assumed fully occupied.

${\lambda }_{\mathrm{D}}={\left({\epsilon }_{\mathrm{s}}{k}_{\text{B}}T/e^2{n}_{2\mathrm{D}}\right)}^{1/2}$