2.5.2.1 Phonon Concept

From the quantum theory point of view instead of waves (2.98) phonons are introduced through the second quantization formalism (see Appendix A). Phonons represent quasi-particles moving in the crystal. The energy of a phonon is

$$\epsilon=\hbar\omega,$$ (2.99)

where $\omega$ is the frequency of the classical wave (2.98). The wave vector $\vec{k}$ in (2.98) determines a phonon quasi-momentum as follows:

$$\vec{p}=\hbar\vec{k}.$$ (2.100)

This quantity is not unique as any quasi-momentum $\hbar\vec{k}+\hbar\vec{b}$, where $\vec{b}$ is a reciprocal lattice vector, is physically equivalent to $\hbar\vec{k}$. The velocity of a phonon is determined as the group velocity of the corresponding classical waves $\vec{v}_\mathrm{gr}=\partial\omega/\partial\vec{k}$ and has the form:

$$\vec{v}=\frac{\partial\epsilon(\vec{p})}{\partial\vec{p}}.$$ (2.101)

All the properties of the classical wave spectrum are valid for the energy spectrum of phonons. In particular it has $3n$ branches, three of which are acoustic ones. The wave density is now interpreted as the phonon density of states.

The free wave motion is considered as the free motion of non-interacting phonons. Inclusion of the anharmonicity leads to scattering processes in the phonon gas. These scattering processes restore the thermal equilibrium of the phonon gas. The processes conserve the quasi-momentum. However, this is only valid within an addition of a reciprocal lattice vector $\hbar\vec{b}$.

It should be noted that the phonon concept appears only as the quantum mechanical description of the collective atomic motion in a crystal and that phonons cannot be identified with individual atoms.

As applied to the electron transport in semiconductors the interaction between the electron gas and the phonon gas plays an important role. In this work only covalent semiconductors are considered. In this case the electron-phonon interaction can be successfully described using the deformation-potential approach [25]. In this case the interaction Hamiltonian in (2.95) is given as:

$$\hat{H}_\mathrm{el-ph}=\Xi_{ij}\frac{\partial u_{i}}{\partial x_{j}},$$ (2.102)

where $\Xi_{ij}$ is the deformation-potential tensor describing the shift of a band per unit deformation. Using the continuous medium approximation the ion displacement takes the form:

$$\vec{u}=\sum_{\vec{q}}\biggl(\frac{\hbar}{2\rho V\omega_{\vec{q}}... ...ggr)^{\frac{1}{2}}(a_{\vec{q}}+a_{-\vec{q}}^{+})\exp(i\vec{q}\vec{r})\,\vec{e},$$ (2.103)

where $\rho$ is the density of the crystal and $a$ and $a^{+}$ stand for the phonon annihilation and creation operators, respectively. Therefore the interaction Hamiltonian is:

$$\hat{H}_\mathrm{el-ph}=\sum_{\vec{q}}\biggl(\frac{\hbar}{2\rho V\... ...{1}{2}}(a_{\vec{q}}+a_{-\vec{q}}^{+}) \exp(i\vec{q}\vec{r})\Xi_{ij}iq_{j}e_{i}.$$ (2.104)

It is convenient to interpret this equation in terms of phonon emission and absorption processes. If in (2.95) $\vert s\rangle$ stands for $\vert N_{\vec{q}_{1}},N_{\vec{q}_{2}},...,N_{\vec{q}},N_{\vec{q}^{'}},...\rangle\vert\vec{k}\rangle$ then in the sum over $\vec{q}$ only two terms will contribute: one from $\langle...,N_{\vec{q}}-1,...\vert a_{\vec{q}}\vert...,N_{\vec{q}},...\rangle$ with pre-factor $N_{\vec{q}}$ and the second from $\langle...,N_{\vec{q}}+1,...\vert a_{\vec{q}}^{+}\vert...,N_{\vec{q}},...\rangle$ with pre-factor $N_{\vec{q}}+1$. Thus using (2.95) the phonon differential scattering rates in the kinetic equation (2.60) take the form:

$$S^\mathrm{ab}(\vec{k},\vec{k}^{'},\vec{r},t)=\frac{N_{\vec{q}}I_\... ...vec{q}}} \delta[\epsilon(\vec{k}^{'})-\epsilon(\vec{k})-\hbar\omega_{\vec{q}}],$$

$$S^\mathrm{em}(\vec{k},\vec{k}^{'},\vec{r},t)=\frac{(N_{\vec{q}}+1... ...vec{q}}} \delta[\epsilon(\vec{k}^{'})-\epsilon(\vec{k})+\hbar\omega_{\vec{q}}],$$ (2.105)

where $N_{\vec{q}}$ is given by (2.34), that is, within this work the phonon gas is assumed to be in the equilibrium which is not valid in general [26,27,28]. $I_\mathrm{ov}$ is the overlap integral:

$$I_\mathrm{ov}=\biggl\vert\int_\mathrm{el.cell}d\vec{r}\,u^{*}_{\vec{k}^{'}}(\vec{r})u_{\vec{k}}(\vec{r})\exp(i\vec{G}\cdot\vec{r})\biggr\vert^{2},$$ (2.106)

where $\vec{G}$ is a reciprocal lattice vector. For intervalley transitions the angle between initial and final states depends mainly on the valleys involved in the transition, $I_\mathrm{ov}$ is nearly constant [29] and can be taken into account by renormalizing the corresponding coupling constant.

The most important phonon scattering processes for covalent semiconductors can be described by (2.105). Those of them which are important for Si, Ge and SiGe are briefly given below.

S. Smirnov: