5.3.2 Scattering Mechanisms in the 2DEG

In analogy with the 3D case, the scattering rates for electrons residing in subband $n$ of valley $v$ are obtained from an integration over all possible two-dimensional $\mathitbf{k'}$ states after scattering

$$S_{n}^v({\ensuremath{\mathitbf{k}}}) = \frac{A}{4\pi^2} \int_{\ma... ...'} S_{n',n}^{v',v}({\ensuremath{\mathitbf{k'}}},{\ensuremath{\mathitbf{k}}})\ .$$ (5.22)

Here, $A$ denotes the area, and a summation over all subbands $n'$ and valleys $v'$ after scattering is performed. The density of states per spin for the 2DEG is obtained as

$$g_{n}\left(E\right)=\frac{1}{(2\pi)^{2}}\int\delta[E-E_{n}({\ensuremath{\mathitbf{k}}})]\,\mathrm{d}^2k\ .$$ (5.23)

As in the three dimensional case the energy subband dispersion $E_{n}^v({\ensuremath{\mathitbf{k}}})$ of subband $n$ in a valley $v$ can be approximated close to its minimum using a nonparabolic relation [Ando82,Laux88]

$$E(1 + \alpha_n^v E) = \frac{\hbar^2}{2}\left ( \frac{k_1^2}{m_{\shortparallel,1}^v} + \frac{k_2^2}{m_{\shortparallel,2}^v}\right )\ ,$$ (5.24)

where $E=E_{n}^v({\ensuremath{\mathitbf{k}}}) - E_{n,0}^v$ denotes the energy with respect to the subband minimum $E_{n,0}^v$, and $\alpha_n^v$ is the nonparabolicity coefficient. The integration over the energy in (5.23) can be performed analytically yielding

$$g_{n}^v\left(E\right) = \frac{\{m_\mathrm{dos}\}_n^v}{2\pi \hbar^2} (1+2\alpha_n^v E) \Theta[E]\ ,$$ (5.25)

for the density of states of the subband $n$ of valley $v$. Here, $\Theta$ denotes the Heaviside step function, and $\{m_\mathrm{dos}\}^v = \sqrt{m_{\shortparallel,1}^v m_{\shortparallel,2}^v}$ is the density of states mass. In Si the subband dispersion is frequently assumed to be independent of the subband index $n$ [Jungemann93,Fischetti02,Roldan96]. The masses $m_{\shortparallel,1}^v$ and $m_{\shortparallel,2}^v$ for three substrate orientations of Si are listed in Table 4.1.

The bulk phonon scattering model for the six valleys along $\Delta$ has been adapted for the 2DEG following the treatment of Price [Price81]. The well accepted model for the bulk phonon spectrum of Jacoboni [Jacoboni83] can be used for the Si inversion layer.

Subsections

• Acoustic Intravalley Scattering
• Intervalley Scattering
• Surface Roughness Scattering

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