3.2.1 Material Properties

The choice on the bandgap energies for GaN is based on a publication [139]. The particular setup for the masses has negligible impact within the available range, thus an average value [140] is chosen.

An interesting result of the literature search is the fact that in almost all MC simulations the piezoelectric scattering mechanisms were modeled assuming a cubic crystal structure. This is a correct approach to most of the technologically significant semiconductors, whereas for wurtzites the hexagonal structure has to be accounted for in the relevant piezoelectric scattering model.

The role of piezoelectric interaction in bulk wurtzite GaN has been analyzed by Kokolakis et al. [141]. In particular, the effect of acoustic piezoelectric scattering is taken in consideration, and the scattering rates have been calculated including the effect of screening. In accordance with their simulations, present results show that the piezoacoustic rates are higher in the wurtzite phase than in the cubic phase, and they are very sensitive to the background doping of the sample. Since nitrides exhibit the largest piezoelectric constants among all of the III-V semiconductors, an accurate modeling of piezoelectric scattering is especially important. In this work a piezoelectric scattering model similar to [142,141] is used, assuming equipartition, valid at temperatures over one Kelvin and considering non-parabolicity and screening in terms of the Thomas-Fermi inverse length $q_0$.

Table 3.1: Summary of material parameters of wurtzite GaN for Monte Carlo simulation.

Bandgap energy			Electron mass			Non-parabolicity			Scattering models						Ref.
$\Gamma_{1}$	U	$\Gamma_{3}$	$\ensuremath{\mathrm{m}}_{\Gamma 1}$	$\ensuremath{\mathrm{m}}_\ensuremath{\mathrm{U}}$	$\ensuremath{\mathrm{m}}_{\Gamma 3}$	$\alpha_{\Gamma 1}$	$\alpha_\ensuremath{\mathrm{U}}$	$\alpha_{\Gamma 3}$	ADP	$\hbar \omega _\ensuremath{\mathrm{ij}}$	$\hbar \omega _\ensuremath{\mathrm{LO}}$	$\rho$	$\ensuremath{\ensuremath{\epsilon}_\ensuremath{\mathrm{r}}}$	$\ensuremath{\ensuremath{\epsilon}_{\infty}}$
eV	eV	eV	m$_0$	m$_0$	m$_0$	1/eV	1/eV	1/eV	eV	meV	meV	g/cm$^3$	-	-
3.5	-	-	0.19	-	-	0.187	-	-	12.0	-	99.5	6.1	9.5	5.35	[143]
3.5	5.00	-	0.19	1.00	-	0.187	-	-	12.0	-	92.0	6.1	9.5	5.35	[144]
3.5	5.00	-	0.19	0.7	-	0.187	-	-	12.0	-	92.0	6.1	9.5	5.35	[145]
3.4	5.50	5.60	0.19	-	-	-	-	-	10.1	-	92.0	6.095	9.5	5.35	[146]
3.5	5.50	5.60	0.20	0.40	0.60	0.183	0.065	0.029	8.3	92.9	92.9	6.1	8.9	5.35	[147]
3.5	5.5	5.6	0.19	0.4	0.6	0.187	0.065	0.029	12.0	-	92.0	6.1	9.5	5.35	[148]
3.39	5.39	5.59	0.20	0.40	0.60	0.189	0.067	0.029	8.3	91.2	91.2	6.15	8.9	5.35	[149]
3.5	4.99	5.25	0.20	0.24	0.40	0.19	0.17	0	7.8	65.0	92.0	6.095	9.5	5.35	[110]
3.39	5.49	5.29	0.20	1.00	1.00	0.189	0	0	8.3	91.2	91.2	6.15	8.9	5.35	[150]
3.5	5.50	5.60	0.19	0.40	0.60	0.183	0.065	0.029	10.1	92.0	92.0	6.1	8.9	5.35	[151]
3.5	5.45	5.60	0.21	0.25	0.40	0.19	0.1	0	8.0	65.0	92.0	6.095	9.5	5.35	[140]
3.36	-	-	0.20	-	-	-	-	-	10.1	-	92.0	6.095	9.5	5.35	[152]
3.52	5.77	5.87	0.212	-	-	0.37	-	-	8.3	65.8	90.88	6.087	9.7	5.28	[153]
3.5	4.5	4.6	0.186	0.40	0.60	0.189	0.065	0.029	8.3	-	99.5	6.15	9.5	5.35	[116]
3.52	5.77	5.87	0.212	0.493	0.412	-	-	-	8.3	-	90.88	6.087	9.7	5.28	[154]
3.5	5.60	3.90	0.20	0.60	0.22	0.183	0.029	0.065	8.3	80.0	92.2	6.15	9.95	5.35	[155]
3.39	5.49	5.29	0.21	1.00	1.0	0.189	0	0	8.3	92.0	92.0	6.15	8.9	5.35	[139]
3.39	5.49	5.29	0.2	1.0	1.0	0.189	0	0	8.3	91.2	91.2	6.15	8.9	5.35	[156]
3.39	5.49	5.29	0.2	1.0	1.0	0.189	0	0	8.3	92.0	92.0	6.15	8.9	5.35	[157]
3.39	5.29	5.49	0.20	0.30	0.40	0.189	0	0	8.3	91.0	92.0	6.1	8.9	5.35

Material parameters as used by different group are given in Table 3.1, while Table 3.2 summarizes the experimental and theoretical values of the elastic constants $c_{11}$, $c_{12}$, and $c_{44}$, available for wurtzite GaN in the literature. From these the corresponding values for $c_\ensuremath{\mathrm{L}}$, $c_\ensuremath{\mathrm{T}}$, $v_\ensuremath{\mathrm{sl}}$, and $v_\ensuremath{\mathrm{st}}$ are calculated. The latest experimental values for GaN [158] are adopted in the MC simulation [159].

Table 3.2: Summary of elastic constants of GaN and the resulting longitudinal and transverse elastic constants and sound velocities.

$c_{11}$	$c_{12}$	$c_{44}$	Data	Refs.	$c_\ensuremath{\mathrm{L}}$	$c_\ensuremath{\mathrm{T}}$	$v_\ensuremath{\mathrm{sl}}$	$v_\ensuremath{\mathrm{st}}$
GPa	GPa	GPa			GPa	GPa	m/s	m/s
296	120	24	exp.	[160]	245	50	6342	2855
374	106	101	exp.	[161]	348	114	7557	4331
390	145	105	exp.	[162]	376	112	7859	4290
377	160	81	exp.	[163]	355	92	7637	3888
365	135	109	exp.	[164]	360	111	7693	4278
370	145	90	exp.	[165]	364	108	7733	4212
373	141	94	exp.	[158]	355	103	7641	4110
369	94	118	calc.	[166]	353	126	7620	4546
396	144	91	calc.	[167]	368	105	7775	4153
367	135	95	calc.	[168]	350	103	7585	4122
350	140	101	calc.	[169]	347	103	7548	4106

Table 3.3 summarizes the experimental and theoretical values of the piezoelectric coefficients $e_{15}$, $e_{31}$, and $e_{33}$, available for GaN in the literature. In cases, where $e_{15}$ is not available, $e_{15}=e_{31}$ is assumed. From these, the corresponding $\langle e_\ensuremath{\mathrm{L}}^2 \rangle$ and $\langle e_\ensuremath{\mathrm{T}}^2 \rangle$ are calculated, which are necessary to obtain the coupling coefficient $K_{av,WZ}$ taking into account the wurtzite structure [159].

Table 3.3: Summary of piezoelectric coefficients of GaN for Monte Carlo simulation
of piezoelectric scattering.

$e_{15}$	$e_{31}$	$e_{33}$	Data	Refs.	$\langle e_\ensuremath{\mathrm{L}}^2 \rangle$	$\langle e_\ensuremath{\mathrm{T}}^2 \rangle$
C/m$^2$	C/m$^2$	C/m$^2$	Data	Refs.	C$^2$/m$^4$	C$^2$/m$^4$
-0.30	-0.36	1.00	exp.	[170]	0.103	0.123
-	-0.55	1.12	exp.	[171]	0.175	0.234
-	-0.33	0.65	calc.	[143]	0.061	0.082
-	-0.49	0.73	calc.	[172]	0.118	0.149
-0.22	-0.22	0.43	calc.	[173]	0.027	0.036
-	-0.32	0.63	calc.	[169]	0.058	0.077
-	-0.44	0.86	calc.	[174]	0.109	0.145