|Figure 2.1.1.||Figure 2.1.2.||Figure 2.1.3.||Figure 2.1.4.|
Q2.1.1 Which modes of vibration are described in figure 2.1.1
to figure 2.1.4?
Figure 2.2.1. Crystal vibration with two atoms per basis
Q2.2.1 How many modes of vibration is possible in a diamond lattice
(include both accustic and optical branches)?
Crystal vibrations are usually called phonons. A phonon can be regarded as a particle that can interact with holes and electrons. In room temperature a crystal is full of lattice vibrations and in the phonon representation it exits a phonon gas within the crystal.
Different types of vibrations will have different effect on the band structure and therefore the phonon mode will interact differently with the electron. The interaction between an electron and a phonon can be described in terms of transistion rates between one state to an other state in the reciprocal space. The transistion rate the number of transistions from a inital k0 to final k0' that will take place per unit time. A typical scattering rate is in the range of 10-100 scatterings per picosecond. The transistion rate can be calculated quantum mechanically. The input data for this type of calculation is a so called deformation potential and the phonon dispersion relation (relation between phonon energy and wave vector).
The deformation potential describes how much the band structure will change due to a change in lattice constant. There are methods that can be used in order to extracted the deformation potentials from experiments.
The dispersion relation can be looked upon as the band structure for
the phonon since it describes the relation between energy and wave vector
for the phonon. Phonon energy is often given in terms of phonon frequency.
Figure 2.3.1 shows a typical 1D dispersion relation for optical and accustical
Figure 2.3.1. Plot of phonon energy vs. phonon wave vector for a typical semiconductor
The dispersion relation for optical phonons are often considered to
be a constant value and is often given as a phonon temperature (se figure
2.3.2). Phonon frequency, phonon temperature and phonon energy is different
representation of the same physical quantity.
The accustic phonon dispersion relation is often approximated by the following relation:
In both cases the correctness of the approximation depends on weather
the phonon wave vector is small or large. In semiconductor modelling the
phonon wave vector is considered to be small and therefore the approximations
can be used without significant errors.
On the right hand side we have a source or sink function which can be used in order to describe generation or recombination of carriers and a collision term that describes the change in the distribution per unit time due to collisions with other electrons or particles. The BTE is a continuity equation for f(r,k,t) . The following picture describes the book-keeping of f(r,k,t) in a one dimensional system.
Conservation of carriers in the one dimensional description requires that:
Dividing both sides with dtdrdp and letting dt, dr and dp approach zero gives (s(r,p,t) has been neglected):
This is the BTE in a one dimensional system. The distribution function can be found if we solve the BTE using adequate boundary conditions. Unfortunately is this a very difficult equation to solve numerically or analytically. Monte Carlo simulation is very usefull in order to solve complicated equations and has become a standard method in order to solve the BTE.
The Monte Carlo simulation technique was first used by Fermi, Von Neumann and Ulam who developed it for the solution of problems related to neutron transport during the development of the atomic bomb. The name Monte Carlo is used since the method is based on the selection of random numbers. In this sense it is related to the gambling casinos at the city Monte Carlo in Monaco. The Monte Carlo method can be considered as a very general mathematical method to solve a great variety of problems. A simple example is to evaluate a difficult definite integral. In such a case the function that should be integrated can be limited by a rectangular box. In the figure below the integral of f on the interval [a,b] is less than the area of the bounding box.
Pairs of random numbers can be transformed into coordinates uniformly distributed within the box. The fraction of coordinates that falls below the function multiplied with the area of the limiting box, gives the solution of the integral. The accuracy of the solution depends on the number of random numbers used. The exact solution will be found within some interval around the result obtained by the Monte Carlo method. For an infinite number of coordinates the solution will be exact.
Solving the BTE using the Monte Carlo method can be used in order to study the behaviour of the electrons in a semiconductor material under the influence of an electric field.
In the following animations the Monte Carlo method has been used in order to simulate the electron tradjectories in silicon. The electric field applieded to the crystall is 100kV/cm. At the start of the animation 2000 electrons are injected at a specific point in the crystall and then accelerated in the electric field. Periodic boundaries has been forced on to the simulation so that electrons that leaves the cube at one side will be injected at the opposite side.
Q2.4.1 The electron population is divided into six clouds in the reciprocal space. Why?
Q2.4.2 It is much more difficult to follow the tradjectory of
an electron in the reciprocal animation than in the animation in real space.
Why? (The answer should be related to some physical phenomina, not related
to the resolution of the animation).
The transition rate can be used in order to calculate characteristic time constants of the carrier transport in a specific semiconductor. These constants describe the average time it takes for an ensemble of carriers with a given initial momentum state to scatter into a new final state. There are three major time constants that is often used: