Acoustic Intravalley Scattering

In the 2DEG scattering with long-wavelength acoustic phonons causes intra- and intersubband transitions within the same valley. The transition rate of electrons in the inversion layer due to interaction with acoustic phonons is given by [Price81]

$$\{ S_\mathrm{ac} \}^{v}_{n'n}({\ensuremath{\mathitbf{k}}}',{\ensu... ...{n'}^{v}({\ensuremath{\mathitbf{k}}}') - E_n^v({\ensuremath{\mathitbf{k}}})]\ .$$ (5.26)

Here, $n$ and $n'$ denote the subband indices, $v$ the valley index, and $u_\mathrm{l}$ is the longitudinal sound velocity. In (5.26) an overlap integral occurs,

$$\frac{1}{b_{n'n}^{v'v}} = \int_0^\infty \vert\zeta_{n'}^{v'}(z)\vert^2 \vert\zeta_{n}^{v}(z)\vert^2 \mathrm{d}z\ ,$$ (5.27)

where $\zeta_{n'}^{v'}(z)$ denote the one dimensional envelope wave functions being solutions of the Schrödinger equation (4.13). The parameters $b_{n'n}^{v'v}$ have the unit of length and are usually referred to as effective widths. It can be understood as the effective extent of the interaction region between electrons in different subbands and valleys in the $z$ direction.

Using the nonparabolic density of states (5.25) the scattering rate can be written as

$$\{S_\mathrm{ac} \}_{n}^v(E) = \frac{2\pi}{\hbar} \frac{\ensuremat... ...}^v\}^2}{\rho u_l^2} \sum_{n'} \frac{1}{b_{n'n}^{vv}} g_{n'}^{v}(E-E_{n'}^v)\ .$$ (5.28)

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