Self study material in Solid State Electronics using Multimedia

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Hans-Erik Nilsson

2. Electron transport in semiconductor materials


2.1 Crystal vibrations in lattices with monatomic basis

In this section we should look upon simple crystal structures with monatomic basis. We want to understand how atom plans can vibrate in relation to each other. There are three modes of vibrations, one longitudinal and two of transverse polarization.
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?

2.2 Crystal vibrations in lattices with two atoms per primitive basis

In this case we will concentrate to a simple picture with a lattice of primitive basis containing two different atoms. In this case we have the same modes as in the previus case. In addition the planes with different atoms can oscillate in phase or in opposite phase (180 degrees phase difference). An illustration of this situation can be found in the animation under figure 2.2.1. If the two atoms carry opposite charges, we may excite a motion of this type with the lectric field of a light wave, so that the branch is called the optical branch. The branch discussed in section 2.1 is called the accustic branch.

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)?

2.3 Interaction between the electron and crystal vibrations

The band structure of a solid material depends on the location of each atom in the lattice. Crystal vibration will be transformed into the reciprocal space and as a consequence the band structure will vibrate. The band structure vibration will interact with the electron in a strong way.

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 phonons.

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.

Figure 2.3.2. Plot of the approximated phonon energy vs. phonon wave vector

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.

2.4. Simulation of electron transport in semiconductor materials

The electron transport in semiconductor materials can be described using a statistical distribution function. This function f(r,k,t) describes the probability that an electron is located at a point k in the reciprocal space and a point r in the real space at a certain time t. The distribution function gives a full description of the electron state and we can use this function in order to extract macroscopic parameters as mean velocity, current density and the total current in a semiconductor component. The distribution function has to satisfy certain constrains. These constrains can be formulated as the Boltzmann Transport Equation (BTE):

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.

Figure 2.4.1. Illustration of the balance condition used in order to derive the BTE 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.

Figure 2.4.2. Illustration of an arbitrary function f surounded by a 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.

Figure 2.4.3. Monte Carlo simulation of electron transport in a pure silicon crystall at 300 Kelvin (real-space).

Figure 2.4.4. Monte Carlo simulation of electron transport in a pure silicon crystall at 300 Kelvin (reciprocal-space).

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:

  • The average life time of a momentum state is the average time it takes for all carriers in the ensemble to scatter once.
  • The momentum relaxation time is the average time it takes to completely randomize the initial state in momentum.
  • The energy relaxation time is the average time it takes to completely randomize the initial momentum and energy state.