2. The Semiclassical Transport Model

The semiclassical transport model represents a generalization of the theory of free electrons in the case of a spatially-periodic potential. In the free electron theory electrons move between two collisions according to the classical equations of motion. From the quantum mechanical point of view these equations of motion actually describe the behavior of wave packets constructed using energy levels of a free electron. This can be generalized for the case of electrons in an arbitrary periodic potential where plane waves are replaced by Bloch's waves. The proof of this generalization is a difficult mathematical task. However it removes various contradictions of the free electron theory. In particular, in the semiclassical model electron collisions with motionless periodic ions do not influence the resistivity of a solid because now the electron interaction with fixed periodic lattice has been taken into account in the original Sroedinger's equation which Bloch's wave is obtained from. In other words, within Bloch's theory the classical point of view about scattering on fixed periodic ions is not valid any longer. From the quantum mechanical point of view this means that in a periodic structure of scattering centers a wave can move without any damping [14].

The semiclassical model based on the concept of Bloch's wave packets correctly works only when the electron position is measured with an accuracy of the wave packet width. The fact that the wave packet width must be less than the size of the Brillouin zone can be used to estimate the size of Bloch's wave packets and the limitations of the semiclassical model. Similar to the theory of free electrons, the wave packet is constructed using energy levels of an electron. But now an electron is represented by Bloch's wave:

$$\Psi_{n}(\vec{r},t) = \sum_{\vec{k}^{'}}A_{\vec{k}^{'}}\psi_{n\vec{k}^{'}}(\vec{r})\exp\biggl(-\frac{i}{\hbar}E_{n}(\vec{k}^{'})t\biggr),$$ (2.1)

where $n$ numerates bands and $A_{\vec{k}^{'}}\approx 0$ for $\vert\vec{k}^{'}-\vec{k}\vert > \Delta k$. Changing the position of an electron from $\vec{r}$ to $\vec{r}+\vec{R}$, where $\vec{R}$ is a Bravais lattice vector, gives:

$$\Psi_{n}(\vec{r}+\vec{R},t) = \sum_{\vec{k}^{'}}A_{\vec{k}^{'}}\p... ... \exp\biggl(i\vec{k}^{'}\cdot\vec{R}-\frac{iE_{n}(\vec{k}^{'})t}{\hbar}\biggr).$$ (2.2)

As a function of $\vec{R}$ this expression is just a superposition of plane waves with another weight. Thus the wave packet must be localized in a domain with a characteristic size $\Delta R\approx 1/\Delta k$. The characteristic size $\Delta k$ is less than the size of the Brillouin zone which is of the order of $1/a$, where $a$ is the lattice constant. This gives the condition $\Delta R > a$, that is, the wave packet built from Bloch energy levels and which has a wave vector correctly determined in a domain smaller than the size of the Brillouin zone is smeared over a large number of elementary crystal cells in the coordinate space. Therefore the conclusion is that the semiclassical model describes the electron response to external electric and magnetic fields which slowly change within the wave packet's width and thus more slowly within an elementary cell. In this model such fields are responsible for the creation of classical forces which determine the evolution of the wave packet's coordinate and its wave vector.

It should be noted that the semiclassical model is more complex than the classical limit of free electrons. This difference comes from the fact that the characteristic length at which the lattice periodic potential considerably changes is smaller than the wave packet's width and thus this potential cannot be treated classically. Consequently, the semiclassical model represents only in part the classical limit as the external fields are only considered classically while the ion periodic field is treated quantum mechanically.

An additional element of the quantum mechanical nature of the semiclassical model is that the electron wave vector is only accurate to the vector $\vec{k}_{b}$ of the reciprocal lattice. Thus two sets of variables $(n,\vec{r},\vec{k})$ and $(n,\vec{r},\vec{k}+\vec{k}_{b})$ specify the same electron state. This means that all physically distinguishable wave vectors within the same band are located within the elementary cell of the reciprocal lattice.

The fact that in the semiclassical model the band number represents an integral of motion allows to formulate some restrictions on the external fields. For the amplitude of the electric field $E$ the condition reads:

$$qEa\ll\frac{\Delta \epsilon(\vec{k})^{2}}{\epsilon_{f}},$$ (2.3)

and for the amplitude of the magnetic field $H$ the condition has a similar form:

$$\frac{\hbar qH}{mc}\ll\frac{\Delta \epsilon(\vec{k})^{2}}{\epsilon_{f}},$$ (2.4)

where $\Delta\epsilon(\vec{k})=\epsilon_{n}(\vec{k})-\epsilon_{n^{'}}(\vec{k})$ and $\epsilon_{f}$ is the Fermi energy. The frequency of the external electromagnetic field must satisfy the following condition:

$$\hbar\omega\ll\Delta\epsilon(\vec{k}),$$ (2.5)

which prevents that a photon with a high enough energy causes an interband transition. For the wave length of the external electromagnetic field the condition

$$\lambda\gg a$$ (2.6)

is necessary to make the concept of wave packets meaningful.

Subsections

• 2.1 Equations of Motion
  • 2.1.1 Real Space Equation of Motion
    • 2.1.1.1 Equation of Motion
    • 2.1.1.2 Velocity-Band Structure Relationship
  • 2.1.2 Time Evolution of the Quasi-Momentum
    • 2.1.2.1 Equation of Motion
    • 2.1.2.2 Force Expression
  • 2.1.3 General Properties of Semiclassical Dynamics
    • 2.1.3.1 Phase Space Domain Evolution.
    • 2.1.3.2 Constant External Electric Field
  
  
• 2.2 Distribution Function
  • 2.2.1 Equilibrium Distribution
    • 2.2.1.1 Statistical Distribution
    • 2.2.1.2 Distribution of Fermions
    • 2.2.1.3 Distribution of Bosons
  • 2.2.2 Non-Equilibrium Distribution
    • 2.2.2.1 Non-Equilibrium State
    • 2.2.2.2 Interpretation Within the Relaxation Time Approximation
  
  
• 2.3 Boltzmann's Transport Equation
  • 2.3.1 Liouville's Theorem
  • 2.3.2 Collision Integral
    • 2.3.2.1 Scattering Probability
    • 2.3.2.2 Pauli Exclusion Principle
    • 2.3.2.3 Out-Scattering and In-Scattering Terms
  • 2.3.3 Principle of Detailed Balance
  
  
• 2.4 Band Structure
  • 2.4.1 Electron in a Periodic Potential
    • 2.4.1.1 Bloch's Theorem
    • 2.4.1.2 Energy Bands
    • 2.4.1.3 Band Structure Periodicity
  • 2.4.2 Analytical Band Structure
    • 2.4.2.1 Effective Mass Tensor
    • 2.4.2.2 Nonparabolic Band Structures
    • 2.4.2.3 Analytical Band Structures of Si and Ge
    • 2.4.2.4 Herring-Vogt Transformation
  
  
• 2.5 Scattering Mechanisms
  • 2.5.1 Perturbation Theory
    • 2.5.1.1 First Order Perturbation
    • 2.5.1.2 Fermi's Golden Rule
  • 2.5.2 Scattering on Phonons
    • 2.5.2.1 Phonon Concept
    • 2.5.2.2 Intravalley Scattering by Acoustic Phonons
    • 2.5.2.3 Intravalley Scattering by Optical Phonons
    • 2.5.2.4 Intervalley Phonon Scattering
      • 2.5.2.4.1 Equivalent X-X Intervalley Scattering
      • 2.5.2.4.2 Equivalent L-L Intervalley Scattering
      • 2.5.2.4.3 Non-Equivalent Intervalley Scattering
  • 2.5.3 Plasmon Scattering
    • 2.5.3.1 Plasmon Concept
    • 2.5.3.2 Scattering Rate
  • 2.5.4 Alloy Scattering
  • 2.5.5 Ionized Impurity Scattering
    • 2.5.5.1 Brooks-Herring Model
    • 2.5.5.2 Two-Ion Scattering
    • 2.5.5.3 Screening Theory
    • 2.5.5.4 Second Born Correction
    • 2.5.5.5 Ridley's Model

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