2.5.2.4 Intervalley Phonon Scattering

An electron can be scattered from one valley to another one both by acoustical and optical phonons. Intervalley scattering can be treated as a deformation-potential interaction [30] in the same way as intravalley scattering by optical phonons.

2.5.2.4.1 Equivalent X-X Intervalley Scattering

This scattering process is subdivided into f-type and g-type processes. A process is referred to as f-type, if the initial and final orientations are different, otherwise as g-type process. The transition probability of this mechanism is given by:

$$\lambda(\epsilon_{i})=Z_{f}\frac{\pi D^{2}_{XX}}{\rho\omega_{XX}}\left(N_{XX}+\frac{1}{2}\pm\frac{1}{2}\right)g_{X}(\epsilon_{f}),$$ (2.115)

where $\epsilon_{f}$ is:

$$\epsilon_{f}=\epsilon_{i}\mp\hbar\omega_{XX}.$$ (2.116)

$N_{XX}$ is the equilibrium phonon number of the involved phonon type:

$$N_{XX}=\frac{1}{\exp\bigl(\frac{\hbar\omega_{XX}}{k_{B}T_{L}}\bigr)-1}.$$ (2.117)

$Z_{f}$ is the number of possible equivalent final valleys of the same type. For f-type scattering $Z_{f}=4$ and for g-type scattering $Z_{f}=1$. $D_{XX}$ is the coupling constant, $\hbar\omega_{XX}$ is the corresponding phonon energy.

The numerical values of the coupling constants and phonon energies [18,20] are shown in Table 2.4.

Table 2.4: Numerical values for the intervalley X-X scattering rate.

	Silicon	Germanium
$D_{XX}^{g_{1}}$	$0.5\times 10^{8}$ eV/cm	$0.488\times 10^{8}$ eV/cm
$\hbar\omega_{XX}^{g_{1}}$	$0.01206$ eV	$0.005606$ eV
$D_{XX}^{g_{2}}$	$0.8\times 10^{8}$ eV/cm	$0.79\times 10^{8}$ eV/cm
$\hbar\omega_{XX}^{g_{2}}$	$0.01853$ eV	$0.00861$ eV
$D_{XX}^{g_{3}}$	$1.1\times 10^{9}$ eV/cm	$9.5\times 10^{8}$ eV/cm
$\hbar\omega_{XX}^{g_{3}}$	$0.06204$ eV	$0.03704$ eV
$D_{XX}^{f_{1}}$	$0.3\times 10^{8}$ eV/cm	$0.283\times 10^{8}$ eV/cm
$\hbar\omega_{XX}^{f_{1}}$	$0.01896$ eV	$0.00992$ eV
$D_{XX}^{f_{2}}$	$2.0\times 10^{8}$ eV/cm	$1.94\times 10^{8}$ eV/cm
$\hbar\omega_{XX}^{f_{2}}$	$0.04739$ eV	$0.02803$ eV
$D_{XX}^{f_{3}}$	$2.0\times 10^{8}$ eV/cm	$1.69\times 10^{8}$ eV/cm
$\hbar\omega_{XX}^{f_{3}}$	$0.05903$ eV	$0.03278$ eV

2.5.2.4.2 Equivalent L-L Intervalley Scattering

For this type of scattering there is no separation into f- and g-type processes. The scattering rate is given as:

$$\lambda(\epsilon_{i})=Z_{L}\frac{\pi D^{2}_{LL}}{\rho\omega_{LL}}\left(N_{LL}+\frac{1}{2}\pm\frac{1}{2}\right)g_{L}(\epsilon_{f})$$ (2.118)

where $\epsilon_{f}$ is

$$\epsilon_{f}=\epsilon_{i}\mp\hbar\omega_{LL}.$$ (2.119)

$N_{LL}$ is the equilibrium phonon number of the involved phonon type:

$$N_{LL}=\frac{1}{\exp\bigl(\frac{\hbar\omega_{LL}}{k_{B}T_{L}}\bigl)-1}.$$ (2.120)

$Z_{L}=\frac{7}{2}$ for the transition between two different orientations and $Z_{L}=\frac{1}{2}$ for scattering within the same orientation, $D_{LL}$ denotes the corresponding coupling constant and $\hbar\omega_{LL}$ is the energy of the phonon involved in the scattering process.

The numerical values of the coupling constants and phonon energies [18,20] for this type of scattering are shown in Table 2.5.

Table 2.5: Numerical values for the intervalley L-L scattering rate.

	Silicon	Germanium
$D_{LL}$	$5.26\times 10^{8}$ eV/cm	$3.0\times 10^{8}$ eV/cm
$\hbar\omega_{LL}$	$0.02395$ eV	$0.02756$ eV

2.5.2.4.3 Non-Equivalent Intervalley Scattering

This process involves transitions between all possible valleys in the conduction band. The scattering rate is given by:

$$\lambda(\epsilon_{i})=Z_{j}\frac{\pi D^{2}_{ij}}{\rho\omega_{ij}}\left(N_{ij}+\frac{1}{2}\pm\frac{1}{2}\right)g_{j}(\epsilon_{f}),$$ (2.121)

where $\epsilon_{f}$ is:

$$\epsilon_{f}=\epsilon_{i}\mp\hbar\omega_{ij}-\Delta \epsilon_{ij}.$$ (2.122)

$N_{ij}$ is the equilibrium phonon number of the involved phonon type:

$$N_{ij}=\frac{1}{\exp\left(\frac{\hbar\omega_{ij}}{k_{B}T_{L}}\right)-1},$$ (2.123)

and $\delta\epsilon_{ij}$ is given as

$$\Delta \epsilon_{ij}=\epsilon_{j,{\min}}-\epsilon_{i,{\min}}.$$ (2.124)

Indices $i$ and $j$ stand for the initial and final valley, respectively, $Z_{j}$ is the number of possible equivalent final valleys, $D_{ij}$ is the corresponding coupling constant, $\hbar\omega_{ij}$ is the respective phonon energy, $\epsilon_{i,{\min}}$ and $\epsilon_{j,{\min}}$ are the energy minima of the initial and the final valley, respectively.

The numerical values of the coupling constants and phonon energies [18,20] for this type of scattering are shown in Table 2.6.

Table 2.6: Numerical values for the non-equivalent intervalley scattering rate.

	Silicon	Germanium
$D_{GX}$	$0.0$ eV/cm	$10^{9}$ eV/cm
$\hbar\omega_{GX}$	$0.0$ eV	$0.02756$ eV
$D_{GL}$	$0.0$ eV/cm	$2.0\times 10^{8}$ eV/cm
$\hbar\omega_{GL}$	$0.0$ eV	$0.02756$ eV
$D_{XL}$	$4.65\times 10^{8}$ eV/cm	$4.1\times 10^{8}$ eV/cm
$\hbar\omega_{XL}$	$0.02283$ eV	$0.02756$ eV

S. Smirnov: