Boltzmann Transport Equation

Unit-I

Boltzmann Transport Equation

• The BTE is a statement that in the steady state, there is no net change in the distribution function

• Which determines the probability of finding an electron at position , crystal momentum

and time t.r

k

),,( tkrf

• Therefore we get a zero sum for the changes in due to the 3 processes of diffusion, the effect of forces and fields and collisions:

•

• ………(1)

),,( tkrf

0|),,(

|),,(

|),,( =

∂∂+

∂∂+

∂∂

collisionsfieldsdifussion t

tkrf

t

tkrf

t

tkrf

• The differential form of the diffusion process can be substituted as follows:

• This equation expresses the continuity equation in real space in the absence of forces, fields and collisions.

)2........(),,(

).(|),,(

r

tkrfkv

t

tkrfdifussion

∂∂−=

∂∂

• The forces and fields equation can be written as:

• The Boltzmann equation can be obtained from these.

)3....(..........),,(

.|),,(

k

tkrf

t

k

t

tkrffeilds

∂∂

∂∂−=

∂∂

• The BTE is:

)4........(|),,(

),,(),,().(

),,(

collisionsr

tkrf

k

tkrf

t

k

r

tkrfkv

t

tkrf

∂∂=

∂∂

∂∂+

∂∂+

∂∂

• The BTE includes derivatives of all the variables of the distribution function on the left hand side and of the equation and the collision terms appear on the right hand side of this equation.

• The first term in the equation (4) gives the explicit time dependence of the distribution function.

• This is needed for the solution of the ac driving forces or for impulse perturbations.

• BTE is solved using the following approximations:– (1) The perturbations due to the external fields

and forces is assumed to be small so that the distribution function can be linearized as:

)5).......(,()(),(10krfEfkrf +=

• Where f0 (E)is– The equilibrium distribution function (Fermi

function) which depends only on the energy E

• f1(r,k) is the perturbation term giving the departure from equilibrium.

– (2) the collision term in the BTE is written in the relaxation time so that the system returns to the equilibrium uniformly:

– Where τ - relaxation time, is in general a function of crystal momentum i.e. τ = τ(k).

)6..(..........)(

| 10

ττfff

t

fcollisions

−=−−=∂∂

• The physical interpretation of the relaxation time is the time associated with the rate of return to the equilibrium distribution when the external fields or thermal gradients are switched off.

• The solution for (6) when the fields are switched off at t=0 leads to:

• The solutions are:

• f(t)=f0[f(0)-f0]e-t/τ

• ---------------(8)

)7..(..........)(

| 0 =−−=∂∂

τff

t

fcollisions

• Where f0 is the equilibrium distribution and f(0) is the distribution function at time t=0

• The relaxation in (8) follows a Poisson distribution – the collisions relax distribution function exponentially to f0 with a time constant τ

• These approximations help us in solving BTE

• The Boltzmann equation is solved to find the distribution function which in turn determines the number density and current density.

• The current density is given by:

• Every element of size h (Planck’s constant) in phase space can accommodate one spin ↑ and one spin ↓ electron.

)9......(),,()(4

),( 3

3kdtkrfkv

etrj ∫=

π

• The carrier density n( r, t) is thus simply given by integration of the distribution function over k - space.

• Where d3k is an element of 3D wave vector space.

)10....(),,(4

1),( 3

3 ∫= kdtkrftrn

π