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The Boltzmann equation

The description of a moving particle like an electron in an electric field requires the knowledge of its position in space as well as its velocity. Thus, we need a total of six coordinates and we can represent the single particle by a point in a six dimensional phase space. If we want to describe many particles as it is the case for conduction electrons, we also need a distribution function for the calculation of averages. For electrons this will generally be the Fermi-Dirac distribution, for semiconductors with low doping levels we may approxmate with the Maxwell-Boltzmann distribution.

If an external field is applied, the distribution functions will change from their equilibirium values, and their correct form must be determined by solving of the Boltzmann transport equation (see, e.g. [1],pp. 17-29).

The term on the left hand side is the total time derivative of f and all its variables. It was first given by Liouville and is a formulation of the continuity equation. The term on the right hand side was added by Boltzmann and is called collision term.

A general solution of this equation is a hard task, but as a first approximation we may assume steady state conditions and spatial uniformity throughout the bulk of the investigated material. Thus, the first and the third term are zero. Note that that the remaining term on the left hand side contains a derivative for the velocity coordinates. If we describe the electrons in the solid in terms of their wavefunctions, we can recognize the semiclassical relationship v = ħk/m and apply the chain rule to convert this term into a derivative for k. We find:

In the following, we will go through a series of simplifications and approximations. Let's start on the right hand side:

The collision term

The distribution function f(k) describes the probability to find an electron in the infinitesimal space element d3k around k. The value of f(k) is increased by scattering of electrons from some state k' into k or decreased by scattering from k into some state k'. The function S(k',k) describes the probability of a scattering event taking place in unit time. For the calculation of the collision term we must sum over all possible initial states k', weighted by its occupancy, and we must take into account the availability of the final state.

Let us briefly assume that the system is in thermal equilibrium without external field. We denote this by a function f0 which is given by the Fermi-Dirac distribution. This allows us to derive the following relation:

This statement is called principle of detailed balance.

Elastic scattering and the relaxation time approximation

Often a perturbation approach is used to obtain an approximate solution. Without externally applied field, the system is described by the Fermi-Dirac distribution function f0. We assume that applied field yields only a small perturbation, resulting in a small correction f1

f = f0 + f1

Elastic scattering means that we do not change the energy of the particles, but only their direction. Then, E(k) is equal to E(k') and the detailed balance tells us that also the scattering functions S(k,k') and S(k',k) must be equal.

We obtain a massive simplification of the collision integral:

The quantity τ(k) is called transport relaxation time. We could also write τk,k', but an exchange of variables shows that this is equal to τk',k. Take care, τ(k) is NOT the same as the relaxation time as the one we used in the definition of the mobility. See [1] for more on the different definitions.

Next, we look at the left hand side of the Boltzmann equation. The remaining term contains a velocity derivative. We can relate this acceleration to the force that the external electric field exerts on the charge:

As further approximation, we apply the gradient operation only to the undisturbed distribution function f0. The result can be solved for the nonequilibrium correction f1:

Approximation for parabolic bands

Close to the band edge, it is very often sufficient to use a parabolic band approximation where the relation between wavevector and energy is given by E(k) = ħ2k2/(2m*). This resembles the one of free particles except that the mass is replaced by an effective mass which is related to the curvature of the band. We can apply the chain rule and obtain the following:

In the parabolic band approximation, the gradient of the energy yields a term that is proportional to the wavevector. Then, the vectorial product with the field will yield the cosine of the angle between these two directions. Nevertheless, this looks still like an integral equation because the term τ(k) depends f1. Let's spell it out:

Inserting the relation for f1 into the definition of τ(k) above, we have to evaluate the ratio that appears in the square brackets. We can use the assumption of elastic scattering once more; since the scattering process does not change the energy but only the direction of the particles, the derivative of f0 with respect to energy is the same for E(k) and E(k'); consequently the derivative cancels out of the ratio. Moreover, we asserted further up that the relaxation times are also the same for k and k'. They drop out of the ratio as well!

All things together, the unwieldy fraction reduces to the following:

Remember that all the simplifications so far work towards the calculation of the integral that appears in the calculation of the transport relaxation time. The problem is now reduced to resolving the angles between the the applied field and the wavevectors of the initial and of the final state. We choose a coordinate system where the initial velocity is along the z direction and the electric field is contained in the yz plane.

We can read off the spherical coordinates from the illustration above and resolve the vectorial products (see e.g. [2], pp. 47-48).

Carrying out the integration over k space in spherical coordinates, the dependence on the polar angle will be averaged out. We obtain the following:

Thus, the big term in the square brackets reduces to [1-cos θ], where θ is the angle between the directions before and after the scattering event. We can think of this cosine-term as a weighting function which tells us that scattering events with large angles contribute most to the scattering integral (not really a big surprise!).

The scattering probability

The function S(k,k') was defined as the probability that a scattering event which takes an electron from a state k' into a state k does take place in unit time. We want to relate this function to the scattering cross section which has the unit of an area. Assume the time τ between two collisions. In this time an electron with velocity v travels the distance v τ. If we multiply this distance with the cross section σ we get a volume, and because one scattering event took place within τ, we conclude that this volume must contain one scattering center. Thus, N=1/σvτ, if we denote the density of scatterers with N.

The probability for scattering after a time τ is virtually one, but with respect to unit time it is just 1/τ. However, this is exactly the definition of S(k',k):

S(k',k) = Nvσ(θ).δ(E',E)

If we know the differential cross section of our scattering process, we can determine the transport relaxation time. Here, we replaced the dependence on k' and k by the scattered angle θ. We explicitely write the δ-function to express that we are dealing with elastic scattering processes that conserve the energy. As such, it comes out of Fermi's Golden Rule when the differential cross secion is determined from the scattering potential. Sometimes it is dropped in the literature, but it is then a bit surprising why the integral for the transport relaxation time is extended only over the angular part of d3k without further explanation.

We are through! After solving the integral for the transport relaxation time, we can obtain the first-order correction to the distribution function in the presence of an electric field with the expression that was given above:

Carrying out the derivative for the Fermi-Dirac distribution is a bit messy. Below is an approximation using the Maxwell-Boltzmann distribution which applies generally for non-degenerate situations. We revert to the expression for particles with kinetic energy and A is a suitable normalization constant. Note that f1 changes sign with the direction of the electric field.

The current integral

We have now all things in place to calculate the current that flows when we apply an electric field to our system. Macroscopically, we can relate this current to the conductivity or the mobility. Coming from a microscopic description, each charge carrier makes a contribution to the current density that is equal to the product of its speed and its charge. We have to sum up all contributions and as usual, we replace the sum by an integral over all momentum states which we weight with the distribution function. Additonally, we multiply with a factor of 2 because each momentum state can be occupied by two electrons of oppsosite spin.

Note that the integral weighted with the equilibrium function f0 will give zero because there is no current without field; the only contribution will come from the f1 term.

Sometimes the explicit route via the integral of the the current density is not spelled out, but the authors calculate the mobility relaxation time by averaging the transport relaxation time with a weighting factor of E3/2 (e.g. eq. (5.4) in [3] or eq. (35) in [4]). The form of this particular weighting factor comes from the fact that we pick up a wavevector from the gradient term and another one from the current density integral, resulting in k2 which is proportional to E. An additional factor of E1/2 comes from the usual conversion of d3k into dE.

[1] P. L. Rossiter, The electrical resisitvity of metals and alloys, Cambridge University Press, 1991
[2] T. M. Tritt,
Thermal Conductivity, Springer, 2005
[3] H. Brooks, Adv. Electr. Elecron Phys. 7, 85 (1955)
[4] R. B. Dingle, Phil. Mag. 46, p831 (1955)
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last changed: 6/7/2015