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Electrons in the band structure

After deriving the basic ideas of the origin of the bandstructure and the bandgaps we want to see how we can actually accomodate electrons into these bands. Like at the beginning for the free electrons in the rectangular slab, we fill all the available states with one electron each. In the semiconductor this will result in a situation where the states on all bands below the band gap are filled and all states above the gap are empty. This is a rather boring situation because every electron sits in its state and cannot move because all other states are occupied.

What we have considered so far is essentially the zero temperature state of the electrons. All the energy that the electrons have in their respective states is something like the ground state of an oscillator. It is there, but no use can be made of it. For any significant change we have to move the electrons into available states above the gap.

We must introduce some energy (some temperature) in order to make states above the gap available for electrons. At finite temperature some electrons will have sufficient energy to be excited into free states in the conduction band. Our problem is now to determine how many.

We need to know the number (or density) of available states at given energy, then multiply with the probalility of occupation and sum everything up. Generally, the band structure is quite complicated and any attempts to perform calculations in detail immediately become tedious. We try to get the picture by making some helpful approximations.

Parabolic band approximation

Let us first consider the probability of occupation. Electrons are Fermions, for a given energy E measured with respect to the Fermi energy E_F they obey the Ferm-Dirac distribution function.

At zero temperature the Fermi-Dirac distribution is a step function. It reflects a situation where the occupation probability for all states below the Fermi level is 1 and all states above the Fermi level are free. At finite temperature there is some probability for occupied states above the Fermi level. In the figure below the green curve illustrates the Fermi-Dirac function for finite temperature (plotted upwards). Note that correctly speaking the Fermi level should be called chemical potential because the term Fermi level refers to the energy of the highest occupied electron level; as we do not have electron levels within the gap of ideal semiconductors, there are cannot be any electrons with this energy and the Fermi level would always be at the valence band edge. Unfortunately the term is used a little sloppy in semiconductor physics.

We observe that in the presence of the bandgap the possible states in the valence band and the conduction band overlap only with the tails of the distribution. For energies far away from the band edges the distribution function drops off exponentially and those states are hardly ever populated. Thus, it is allowed to ignore the exact band structure and expand the energy dependence in Taylor series of second order in k around the valence band maximum (VBM) and the conduction band minimum (CBM), respectively. This is known as the parabolic band approximation.

The curvature of the energy with respect to k in the proximity of the band edges take over the part of an inverse mass, the so called effective masse m*. This description is equivalent to the free electron approximation, all the difficulties of crystal periodicity and electron interaction are now packed into one single parameter, the effective mass. A more complicated treatment allows for an ellipsoidal rather than a spherical dependence and consequently assumes up to three different effective masses in the respective directions of k-space.

Next, we deal with the Fermi-Dirac statistics because it complicates the integrals we have to calculate. We can make an approximation because the band gap forbids any states between the bands, and all allowed states are located "far" away from the Fermi energy where we feel only the tails of the Fermi-Dirac distribution. Thus, we can neglect the number 1 in the denominator and proceed with Maxwell Boltzmann statistics.

Density of states

How many states are avaliable on a band? We know that the density of states in k-space is constant because k-states correspond to the points of a regular grid. For the calculation, however, the density in dependence on energy is more convenient.

g(E)dE = 1/L³ g(k_x, k_y, k_z)d³k

The factor 1/L³ is the crystal volume and is needed to yield a density. We stick to the effective mass approximation with the spherical energy surface. The density of states in k-space is determined by the volume allotted to each k-point and is given by by (2π/L)³. The volume element d³k is conveniently converted into spherical coordinates where it equals 4πk²dk.

g(E)dE = 1/L^{3 .} (2π/L)³ πk²dk

Finally, we can use the equation of the quasi-free electron energy to convert k² and dk to the respective terms in dependence on energy E.

Only we must use the effective mass m* instead of the free electron mass. The density of states is then given by:

Density of electrons

We are ready to calculate the number densisty of electrons in the conduction band at a given temperature T.

With the Maxwell-Boltzmann distribution function and the parabolic density of states the integration is easy and yields:

Here we have introduced the so called effective density of states N_C.

Later we will learn about controlling the density of electrons by doping. In that case it is often more important to calculate the position of the Fermi level from a given electron density n.

Holes

When we excite an electron from the valence band into the conduction band we leave behind an unoccupied state. One of the other electrons in the valence is free move into this state. By doing so it leaves behind an unoccupied state somewhere else. Basically, now we have a possibility ot transport charge by moving one electron after another, all of them exclusively in the valence band. If we want to know how many free states we have below the gap at given temperature we must go through exactly the same steps as above. There are two differences:

We must use a probability of the form 1-f_FD, describing free rather than occupied states. Mathematically this distribution function takes a form very similar to the original Fermi-Dirac distribution but the energy scale counts downwards from the Fermi level.
The band curvature will generally not be the same, thus we must distinguish between the effective masses in the conduction band and the valence band, respectively. The effective masses are often writen as m_e, the electron mass, and m_h, the hole mass.

The whole description is similar to the case of electrons. In fact it is so similar that we no longer think of unoccupied states but of a new charge carrier called hole. The hole carries a positive charge and is treated as charge carrier equivalent to the electron.

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