Acoustic Intravalley Scattering

The expression for the scattering probability for electron intravalley scattering from acoustic phonons can be simplified by using the elastic and equipartition approximation. Within this approximation, the energy transfer in a scattering process is neglected, and the phonon population given by the Bose-Einstein statistics

$$N_q = \frac{1}{\exp{\left (\frac{\hbar \omega_q}{\ensuremath {{\mathrm{k_B}}}\ensuremath {T_\mathrm{L}}}\right )} - 1}$$ (5.16)

is represented by the equipartition expression $$N_q \simeq \frac{\ensuremath {{\mathrm{k_B}}}\ensuremath {T_\mathrm{L}}}{\hbar q u_\mathrm{s}} - \frac{1}{2}$$. Thus, (5.14) becomes

$$\{ S_\mathrm{ac}^{\tiny\shortstack{abs \\ [-2pt] emi }} \} ^{v}({... ...ta[E^{v}({\ensuremath{\mathitbf{k}}}') - E^{v}({\ensuremath{\mathitbf{k}}})]\ ,$$ (5.17)

where $v$ denotes the valley index, is the lattice temperature, $\Xi_\mathrm{adp}^{v}$ is the acoustic deformation potential of the $v$-th valley, $\ensuremath {{\mathrm{k_B}}}$ is Boltzmann's constant, $u_{\mathrm{s}}$ denotes the average sound velocity, and $\rho$ is the mass density of the crystal.

Since in the elastic approximation no distinction is made between absorption or emission processes, both transition probabilities can be added. In this approximation acoustic scattering is isotropic: any state ${\ensuremath{\mathitbf{k}}}'$ belonging to the equi-energy surface has the same probability of occurrence, independent of the angle of the initial state $\mathitbf{k}$. Thus, the rate for acoustic scattering is a function of energy only

$$\{S_\mathrm{ac} \}^v\left(E\right)=\frac{2\pi \ensuremath {{\math... ...\{\Xi_\mathrm{adp}^{v}\}^2}{\hbar u_{\mathrm{s}}^{2}\rho}g^{v}\left(E\right)\ ,$$ (5.18)

where $g^{v}\left(E\right)$ is the density of states per spin.

Table 5.1: Parameters for phonon scattering. The intervalley scattering parameters for the $\Delta$-valleys are taken from [Jacoboni83].

		Silicon	Units
Intra	$\rho$	2.33$^a$	g/cm$^3$
	$v_\mathrm{s}$	9.05$^b$	cm/sec
	$\Xi_{\mathrm{adp}}^\Delta$	9.0$^b$	eV
	$\hbar\omega_{\mathrm{op}}$	61.2$^c$	meV
	$D_tK_{L}$	1.75$^c$	10$^8$eV/cm
Inter $_{\Delta\Delta}$	$\hbar\omega_{\mathrm{g}1}$	12.06	meV
	$D_tK_{\mathrm{g}1}$	0.5	10$^8$eV/cm
	$\hbar\omega_{\mathrm{g}2}$	18.53	meV
	$D_tK_{\mathrm{g}2}$	0.8	10$^8$eV/cm
	$\hbar\omega_{\mathrm{g}3}$	62.04	meV
	$D_tK_{\mathrm{g}3}$	11.0	10$^8$eV/cm
	$\hbar\omega_{\mathrm{f}1}$	18.86	meV
	$D_tK_{\mathrm{f}1}$	0.3	10$^8$eV/cm
	$\hbar\omega_{\mathrm{f}2}$	47.39	meV
	$D_tK_{\mathrm{f}2}$	2.0	10$^8$eV/cm
	$\hbar\omega_{\mathrm{f}3}$	59.03	meV
	$D_tK_{\mathrm{f}3}$	2.	10$^8$eV/cm

$^\mathrm{a}$ [Jacoboni83], $^\mathrm{b}$ [Jungemann03], $^\mathrm{c}$ [Fischetti96b]

E. Ungersboeck: Advanced Modelling Aspects of Modern Strained CMOS Technology