3.3.1 Material Properties

In recent years Indium Nitride (InN) has attracted much attention due to the considerable advancement in the growth of high quality crystals. Furthermore, several new works on the material properties proposed a bandgap of $\approx$0.7 eV [215],[216],[217] instead of $\approx$1.9 eV [4]. Here a Monte Carlo approach is used to investigate the electron transport, considering two band structures [218],[219]. The calculations include the three lowest valleys of the conduction band (depending on the chosen band structure, see Table 3.4) and account for non-parabolicity effects. Several stochastic mechanisms such as acoustic phonon, polar optical phonon, inter-valley phonon, Coulomb, and piezoelectric scattering are considered and their impact is assessed [8]. The parameter values for the acoustic deformation potential (ADP $\Xi$=7.1 eV), polar-optical phonon scattering ( $\hbar \omega _\ensuremath{\mathrm{LO}}$=73 meV or 89 meV), inter-valley scattering ( $\hbar\omega_\ensuremath{\mathrm{iv}}$= $\hbar \omega _\ensuremath{\mathrm{LO}}$), mass density ($\rho$=6.81 g/cm$^{3}$), and static and high-frequency dielectric constants ( $\varepsilon_{\mathrm{s}}$=15.3 and $\varepsilon_{\mathrm{\infty}}$=8.4) are adopted from [219],[221]. In addition, the influence of another set of dielectric constants ( $\varepsilon_{\mathrm{s}}$=11.0 and $\varepsilon_{\mathrm{\infty}}$=6.7) recently proposed in [222] in conjunction with the narrow bandgap and lower effective mass is studied.

Table 3.4: Summary of material parameters of wurtzite InN for MC simulation.

Bandgap energy			Electron mass			Non-parabolicity			Scattering models			Ref.
$\Gamma_{1}$	A	$\Gamma_{2}$	$\ensuremath{\mathrm{m}}_{\Gamma 1}$	$\ensuremath{\mathrm{m}}_\ensuremath{\mathrm{A}}$	$\ensuremath{\mathrm{m}}_{\Gamma 2}$	$\alpha_{\Gamma 1}$	$\alpha_\ensuremath{\mathrm{A}}$	$\alpha_{\Gamma 2}$	$\hbar \omega _\ensuremath{\mathrm{LO}}$	$\varepsilon_{\mathrm{s}}$	$\varepsilon_{\mathrm{\infty}}$
eV	eV	eV	m$_0$	m$_0$	m$_0$	1/eV	1/eV	1/eV	meV	-	-
1.89	4.09	4.49	0.11	0.4	0.6	0.419	0.088	0.036	-	-	-	[218]
1.89	-	-	0.11	-	-	-	-	-	89	15.3	8.4	[176]
1.89	4.09	4.49	0.11	0.4	0.6	0.419	0.88	0.036	89	15.3	8.4	[223]
1.89	-	-	0.11	-	-	0.419	-	-	89	15.3	8.4	[224]
0.8	3.0	3.4	0.042	1.0	1.0	0.419	-	-	89	15.3	8.4	[225]
1.89	4.09	4.49	0.11	0.4	0.6	0.419	0.88	0.036	89	15.3	8.4
$\Gamma_{1}$	$\Gamma_{2}$	M-L	$\ensuremath{\mathrm{m}}_{\Gamma 1}$	$\ensuremath{\mathrm{m}}_{\Gamma 2}$	$\ensuremath{\mathrm{m}}_\ensuremath{\mathrm{ML}}$	$\alpha_{\Gamma 1}$	$\alpha_{\Gamma 2}$	$\alpha_\ensuremath{\mathrm{ML}}$	$\hbar \omega _\ensuremath{\mathrm{LO}}$	$\varepsilon_{\mathrm{s}}$	$\varepsilon_{\mathrm{\infty}}$
eV	eV	eV	m$_0$	m$_0$	m$_0$	1/eV	1/eV	1/eV	meV	-	-
0.69	2.47	3.399	0.04	0.25	1	1.413	0	0	73	15.3	8.4	[221]
0.7	-	-	0.07	-	-	-	-	-	-	9.3	6.7	[226]
0.69	2.47	3.399	0.04	0.25	1	1.413	0	0	75/89	11.0	6.7

An accurate piezoelectric scattering model, which accounts for non-parabolicity and wurtzite crystal structure, is also employed [227]. Table 3.5 summarizes experimental values for the elastic constants ($c_{11}$, $c_{12}$, and $c_{44}$) of wurtzite InN. From these the corresponding longitudinal and transversal elastic constants ( $c_\ensuremath{\mathrm{L}}$ and $c_\ensuremath{\mathrm{T}}$) and sound velocities ( $v_\ensuremath{\mathrm{sl}}$ and $v_\ensuremath{\mathrm{st}}$) are calculated.

Table 3.5: Summary of elastic constants of InN and the resulting longitudinal and transverse elastic constants and sound velocities.

$c_{11}$	$c_{12}$	$c_{44}$	$c_\ensuremath{\mathrm{L}}$	$c_\ensuremath{\mathrm{T}}$	$\nu_\ensuremath{\mathrm{sl}}$	$\nu_\ensuremath{\mathrm{st}}$	Ref.
GPa	GPa	GPa	GPa	GPa	m$/$s	m$/$s
-	-	-	265	44	6240	2550	[176]
223	115	48	218	50	5660	2720	[168]
190	104	10	163	23	4901	1845	[228]
271	124	46	248	57	6046	2893	[167]
258	113	53	242	61	5966	2987	[229]

Table 3.6 gives theoretical values of the piezoelectric coefficients $e_{31}$ and $e_{33}$ available in the literature and the calculated corresponding $<e^2_\ensuremath{\mathrm{L}}>$ and $<e^2_\ensuremath{\mathrm{T}}>$ ($e_{15}$=$e_{31}$ is assumed). Choosing the set of elastic constants from [168] and piezoelectric coefficients from [230] results in a coupling coefficient $K_\ensuremath{\mathrm{av}}$=0.24.

Table 3.6: Summary of piezoelectric coefficients of InN for MC simulation of piezoelectric scattering.

$e_{31}$	$e_{33}$	$<e^2_\ensuremath{\mathrm{L}}>$	$<e^2_\ensuremath{\mathrm{T}}>$	Ref.
C/m$^2$	C/m$^2$	C$^2$/m$^4$	C$^2$/m$^4$
-0.57	0.97	0.17	0.72	[230]
-0.11	0.81-1.09	0.13	0.16-0.58	[174]