... subsystem^2.1

In particular, this subsystem may be represented by one particle.

... domain^2.2

Here $\vec{r}$ and $\vec{p}=\hbar\vec{k}$ denote all coordinates and quasi-momenta of the subsystem. For an electron they are just $(x,y,z)$ and $(p_{x},p_{y},p_{z})$ (in Cartesian coordinates).

... function^2.3

Note that due to the fact that $\frac{\partial(\vec {r},\vec {k})}{\partial(\vec {r},\vec {k}_{c})}=1$ we can consider the phase space distribution as a function of the variables $(\vec {r},\vec {k}_{c})$ in the phase space of the variables which are canonically conjugated through the Hamiltonian (2.17). This will be assumed in the following.

... other^2.4

For particles this means the absence of scattering between them.

... space^2.5

Here $d\vec {r}_{1}d\vec {k}_{1}$ and $d\vec {r}_{2}d\vec {k}_{2}$ are elements of phase spaces of the two parts.

... system^2.6

In solids $\mu$ is the Fermi energy denoted as $\epsilon_{f}$.

... function^2.7

The Boltzmann distribution is valid when all $\langle n_{k} \rangle\ll 1$. Physically it corresponds to a dilute system.

... as^2.8

The factor $1/(2\pi)^{3}$ comes from the fact that in the quasi-classical case the phase space volume $(2\pi\hbar)^{3}$ corresponds to two quantum states of a particle. These states differ by spin orientation.

... as^2.9

The spin of a phonon is taken to be zero. Thus the pre-factor $g=2s+1=1$. Since the phonon number is not fixed and determined by the equilibrium condition, the chemical potential of the phonon gas is equal to zero: $\mu=0$.

... processes^2.10

This of course overestimates the efficiency of scattering needed to restore the equilibrium state.

... quasi-momentum^2.11

In the semiclassical approach collisions are considered as instantaneous events taking place at a given point in real space. Thus the semiclassical transport model only considers changes of the non-equilibrium distribution function which happen during time intervals longer in comparison with the collision duration and at distances longer than the collision domain.

... space^2.12

The phase space is considered in the sense of canonical conjugate variables.

... function^2.13

The fact that the equilibrium distribution is considered is shown by the absence of the time among the arguments.

... equation^2.14

In steady state there is no any explicit time dependence. Thus $\frac{\partial f}{\partial t}=0$.

... motion^2.15

This can be easily seen using the Hamiltonian form of the semiclassical equations of motion.

... subsystem^2.16

The other six independent integrals of motion are the components of momentum of the subsystem and components of its angular momentum.

... operator^2.17

The alternative symbol often used for the collision integral is $\mathbf{St}f$.

... follows^2.18

For the sake of simplicity the scattering within one band and without spin flipping is considered here.

... zero^2.19

This can easily be checked substituting (2.30) where $\epsilon(\vec{k})$ is replaced with $\epsilon(\vec {k})+U(\vec {r})$, $U(\vec {r})$ is the potential energy.

... periodicity^2.20

It should be noted that the periodicity is only a simplification. Real crystals never have an ideal periodicity due to impurities, thermal vibrations and so forth.

... branches^2.21

It is assumed that the discrete levels are continuous functions of the parameter $\vec{k}$.

... follows^2.22

Summation is understood under repeated indeces

... rank^2.23

It should be noted that $m_{\alpha\beta}^{-1}>0$ for a minimum, and $m_{\alpha\beta}^{-1}<0$ for a maximum.

... energy^2.24

The surface in the first Brillouin zone.

... respectively^2.25

For example for Si $X$ valleys along $z$ axes $m_{x}=m_{y}=m_{t}$, $m_{z}=m_{t}$.

... quasi-particles^2.26

For example electron-plasmon scattering.

... general^2.27

In the sense that the form of its eigenstate cannot be determined in contrary to the ideal crystal case where Bloch's theorem exists.

... equation^2.28

$\hat{H}$ denotes the full Hamiltonian: $\hat{H}=\hat{H}_{0}+\hat{H}_\mathrm{int}$.

... side^2.29

Matrix elements are considered as quantities of first order.

...WF_decomp)^2.30

The case when $s^{'}\ne s$ is considered. The probability $\vert a_{s}(t)\vert^{2}$ is obtained from (2.89).

... spectrum^2.31

Quasi-momentum components and phonon quasi-wave vectors are considered as discrete quantities.

... limit^2.32

Since it is assumed that an interaction decays much faster in comparison with the free-flight time, the error caused by the long interaction time limit is negligible.

... amplitude^2.33

It is also referred to as polarization.

... gas^2.34

Similarly to phonons plasmons represent bosons.

... field^2.35

The same is valid for two positively charged particles.

... charge^2.36

In our case the charge density of two ions.

... other^2.37

This is valid for a rather weak potential $V_{t}$.

... theory^2.38

Sometimes it is called the Random Phase Approximation (RPA).

... degeneracy^3.1

The original degeneracy is related to the symmetry of the unstrained Si and Ge crystals.

... principle^3.2

That is the statement about the radius of the atomic forces. This radius must be considered to be zero in the macroscopic theory.

... condition^3.3

The integration variables must be $x^{'}$. However, due to weak strain the derivative in (3.22) differs from the derivative with respect to $x^{'}$ by higher order terms.

... two^3.4

Moment of force is a vector. Its components can be expressed as $M_{i}=\frac{1}{2}e_{ijk}M_{jk}$, where $e$ is the unit antisymmetric tensor of rank three and $M_{jk}$ is the antisymmetric tensor of rank two which is integrand in (3.24).

... field^4.1

The quasi-momentum and electric field are assumed to be transformed to the Herring-Vogt space to spherical constant energy surfaces instead of ellipsoidal ones which are the case for Si and Ge.

... system^4.2

This is just an integral form of the Boltzmann kinetic equation discussed later in this chapter.

... events^4.3

In general a Monte Carlo trajectory is not identical with a real trajectory and depends on a specific Monte Carlo algorithm.

... rewritten^4.4

This is the standard form from the theory of integral equations.

...\space ^4.5

For inhomogeneous case it stands for $(\vec {k},\vec {r},t)$ that is a point in the seven dimensional space.

... series^4.6

Also known as the Neumann series [72].

... space^4.7

It should be noted that in our case this is the relative number of carriers determined by the perturbation $f_{1}(\vec{k},t)$.

... conservation^4.8

Or from the incompressibility of the phase space liquid.

... particles^A.1

The symbol $\vert...)$ stands for the tensor product of the single-particle states.