1. The Boltzmann Transport Equation
Matt Krems
Physics 211A
Dec. 10, 2007

2. Introduction
Classical theory of transport processes based on the Boltzmann formalism
The rate of electron collisions depends critically on the distribution of other electrons
The Boltzmann equation can be derived by considering the time derivative of the distribution function

3. The Distribution Function
A distribution function describes how electrons or other types of particles are distributed in real and momentum space through the course of time
As an example, consider the Fermi-Dirac distribution -
-remind everyone what the terms mean
show how particles are distributed as a function of momentum in equilibrium
we need to extend this concept to non-equilibrium distributions

4. the distribution function satisfies for all times
define a phase-space density for N interacting and indistinguishable particles
the distribution function satisfies for all times
where -
-dOmega is an infinitesimal element of phase space spanned by the coordinates and momenta of all particles and DdOmega is the probability that at time t, the particles are found in a volume dOmega centered at (r_1,p_1;..r_N,p_N)
-the 1/N! factor is for the indistinguishability of the particles

5. the phase-space density for all particles is too unwieldy to work with
consider single particle distribution function
where
-carrying around all the information of all the particles would require incredible computational resources and in practice is ridiculous
-f(r,p,t) is the number of particles at time t found in drdp around (r,p) -

6. The Boltzmann Equation
consider a set of N non-interacting particles subject to an external potential
since we are dealing with non-interacting particles we can use the single particle distribution function with no approximations
Liouville's theorem -
-the total distribution function D factorizes into a product of N one particle functions
-Liouville’s theorem of incompressible phase volume flow

7. equivalently written as
now use Hamilton's equations of motion
-the gradient operator with respect to the space and momenta
-recall from classical mechanics
-this is the collisionless Boltzmann equation
to get

8. now consider an aperiodic lattice or the Coulomb interaction
impurities or crystal defects
intrinsic deviations from periodicity in a perfect crystal, due to thermal vibrations
the presence of these interaction changes particles momenta via scattering processes such that the particle can scatter in and out of the phase space volume
the distribution function is no longer a conserved quantity -
-change in phase space volume must be offset by a change in the distribution function due to collisions, this quantity I[f] is called the collision integral

9. The Boltzmann Equation
we considered the time derivative of a single particle non- interacting distribution function
we used Hamilton's equations of motion to insert physics
we turned on an interaction and accounted for this with the collision integral
-this is a general form of the Boltzmann equation

10. The Collision Integral
takes into account electronic collisions due to the aperiodicity of a real lattice
the collision integral can be calculated exactly in principle
but for a two body potential it depends on the two particle distribution function which depends on the three particle distribution function and so on
this is called the BBGKY hierarchy
we need to account for the scattering in and out of phase space volume

11. defined in terms of a quantity Wp,p'
need to find the probability per unit time that an electron with momentum p will suffer a collision
defined in terms of a quantity Wp,p'
assumes all levels p' are unoccupied
must be reduced by a factor (1-f(r,p',t))
the total probability per unit time then is obtained by summing over all p' -
-these quantities W, are called the scattering rates and are typically found from something like Fermi’s Golden rule
-W assumes that all states p’ are unoccupied and available for the states p to scatter in to, but this is not true due to the Pauli exclusion principle
-the actual rate of transition must be reduced by the fraction of unoccupied levels forbidden by the Paul exclusion principle which is simply 1 – f(r,p’,t)
-1/tau is called the relaxation time

12. contribution to the collision integral for a particle scattering out of dp in the neighborhood of p is then
total contribution for scattering in and out of dp in the neighborhood of p is then -
-I[f]_in is found with similar arguments that we used to find I[f]_out
-this is a potentially tough quantity since the relaxation time itself is dependent on the distribution function f
this is a tough quantity to work with so often the relaxation time approximation is employed

13. the Boltzmann equation in the relaxation time approximation
assume that the relaxation time no longer depends on the distribution function itself but is a specified function of p
the Boltzmann equation in the relaxation time approximation -

14. The Relaxation Time Approximation
this assumes that the rate at which f returns to the equilibrium distribution, feq is proportional to the deviation of f from feq
assume distribution function and external potential do not have large spatial variations -> collision integral dominates -
-can see this by looking at the boltzmann equation
-all we do is integrate, this gives one an idea and motivation for the term relaxtion time, in a time tau the system is essentially back to the equilibrium distribution for this simple examples so

15. What can we calculate? electron density current density
-in transport, we are often interest in the carrier density and current density, one we have the distribution function we can easily find them

16. Electrical Conductivity
suppose we have an electric field E in an infinite medium at a constant temperature
solve for f
use expression for current density to arrive at

17. there is no current associated with feq
at least for a metal behaves like a delta function so we can write

18. compare
with Ohm's law

19. Example: Spin-valve GMR
first principles model based on a semi-classical study of electronic transport in Co/Cu/Co spin valves
a spin valve consists of two magnetic layers separated by a spacer layer
the magnetic orientation of one layer is “pinned” in one direction by adding a strong antiferromagnet layer
when a weak magnetic field passes beneath, the magnetic orientation of the unpinned magnetic layer rotates relative to that of the pinned layer, generating a significant change in electrical resistance
In an antiferromagnet, unlike a ferromagnet, there is a tendency for the intrinsic magnetic moments of neighboring valence electrons to point in opposite directions. When all atoms are arranged in a substance so that each neighbor is 'anti-aligned', the substance is antiferromagnetic. Antiferromagnets have a zero net magnetic moment, meaning no field is emitted by them. Antiferromagnets are less common compared to the other types of behaviors, and are mostly observed at low temperatures. In varying temperatures, antiferromagnets can be seen to exhibit diamagnetic and ferrimagnetic properties.

20. Example: Spin-valve GMR
electronic transport within a layer is modelled with the Boltzmann equation
in a linear response regime, the change in the distribution function is given by electron drift, scattering in and scattering out terms, as well as the acceleration due to an electric field
the Wpp' terms are best fit to experimentally determined resitivites
DFT calculations are used to calculate electronic states
the Fermi energy electronic structure of Cu and Co are obtained as well as the energies and velocities of the Bloch states

21. Example: Spin-valve GMR
the solution to the Boltzmann transport equation is matched within each layer by knowing the Bloch wave scattering reflection and transmission matrices for interfaces formed between Co and Cu, obtained from a DFT method

22. Other Calculations Hall effect thermal conductivity thermopower
model for magneto-resistance
viscosity
transport coefficients
H-theorem
MOSFET gate leakage current
MOSFET - metal–oxide–semiconductor field-effect transistor

23. References
J.M. MacLaren, L. Malkinski, and J.Q.Wang. “First principles based solution to the Boltzmann Transport Equation for Co/Cu/Co Spin Valves.” Material Research Society, 2000.
J.M. MacLaren, X.-G. Zhang, W.H. Butler, and Xindong Wang. “Layer kkr approach to bloch-wave transmission and reflection: Application to spindependent tunneling.” Phys. Rev. B, 59(8):5470–5478, 1999.
Neil W. Ashcroft and N. David Mermin. Solid State Physics. Thomson Learning, 1st edition, 1976.
Charles Kittel. Introduction to Solid State Physics. Wiley, 7th edition, 1996.
J.M. Ziman. Principles of the Theory of Solids. Cambridge, 2nd edition,1972.