The Boltzmann Transport Equation

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 • show how particles are distributed as a function of momentum in equilibrium • we need to extend this concept to non-equilibrium distributions

4. define a phase-space density for N interacting and indistinguishable particles • the distribution function satisfies for all times • where

5. the phase-space density for all particles is too unwieldy to work with • consider single particle distribution function • where

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

7. equivalently written as • now use Hamilton's equations of motion • 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

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

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. 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'

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 • this is a tough quantity to work with so often the relaxation time approximation is employed

13. 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 • so

15. What can we calculate? • electron density • current density

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

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

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.