Boltzmann Equation Handout

University of California, BerkeleyEE230 - Solid State Electronics Prof. J. Bokor

Transport Theory(read Lundstrom 3.1 - 3.4)

Aim Develop a general approach for relating microscopic description of carrier motion to mac-roscopic description.

Drift-Diffusion equation:

Classical approach to device design

material properties: μ , D +Boundary conditions

microphysics is buried devicein here structure

Device behavior described by independent specification of material and structure.

Modern devices

μ, D: can’t bury microphysics (i.e. ballistic transport)also depend explicitly on device structure (i.e. nonlocal transport)

Carrier energy & momentum distributions vary strongly in devices

Carrier scattering: key to transport.

Paradox: no scattering --> net J=0!E

k

consider simple band E = Eo - Eo cos(ka)

1 ∂E 22 = --------

h∂K

=Eoa-------- sin( ka)

hBoltzmann equation Handout.fm


periodic motion - Bloch oscillation. This has been experimentally observed in superlattices [“Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor super-lattice,” C. Waschke, et al., Phys. Rev. Lett. 70, 3319 - 3322 (1993). PDF copy is posted on the class web site.]

Electrons oscillate through BZ [Bragg diffraction] → localized in real space → no net cur-rent!

Scattering damps oscillation resulting in a net motion in an applied field. Yet the motion is damped by scattering. The message is that transport depends on the balance between the applied driving force and dissipation by scattering forces.

Boltzmann transport equation

Semi-classical approach. In semiconductors, all the QM is buried in m*. Electron motion will be described by classical mechanics, except that scattering probabilities will also be derived using quantum mechanics.

Complete description: could solve Newton equations for each particle (electron):

dpi ( , , )

------- = qE + F r p t dt

random forces due to phonons, impurities

for all i = 1, …, N . This is clearly not feasible.

Instead, we take a statistical approach:define distribution function (single particle):

f(r, p, t): probability of finding a particle at r , with momentum p at time t.

“phase space”: 6 dimensional space - x, y, z, px, py, pz

The particle density is found as:

Since momentum is discrete, we show a summation over allowed values of p. However, it is

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more convenient to convert to an appropriate integral.1

∑ ( , ) →

-- f r p V

p

12 π 3

-- -----h 2 L

≅

1 ⁄ V

vol----------

------of

---

one state in

p-space

13

∞

_3 ∫ ∫ ∫f( r, p4π h

–∞

(since f is well localized inside the 1st BZ)

[This general method of converting from a sum in p-space to an integral is important, and will come up again and again.]

Example: MB distribution:

f

p

energy

f(

=

exp

E

E

⁄

kT

–

2

m

to get the carrier density, integrate over momentum space

--------------

F4π

3 _3

h1st BZ

=1

– E

) ⁄ kT]

∫e

-------------- exp[ ( E

F4π

_3

hfor reasonable T, only populated near bottom of band, so we can extend the integral over all p-space

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∫∫∫e–p2

⁄ 2m∗kT

p2dpdΩ

p-space

Thus, the integral is:

Now use:

∞

–α

x2 1 π

∫

x e=

dx

------

--

4α α

0

Finally:

n =

1

2m∗

kT

3 ⁄ 2

--

---------------- exp[ ( E F – Ec) ⁄ kT]

4 π2

= Nc exp [ ( E F – Ec) ⁄ kT]

which is our previous result.

Average kinetic energy density:

W( r ) =1

∫

p

2

– E

– p

⁄ 2m∗

3

p---------

--------------- exp{ [ E F

c] ⁄ kT}d

4π

_3 2m

∗h

1∞

–p2 ⁄

2m∗kT

=4πexp[ ( E

– E

) ⁄ kT]

∫

dpp e

--------------

F8π

_3

h 0

31 ⁄ 2 ∗

5 ⁄ 2

--π ( 2 m kT)

8

3 1

2 m∗kT

3 ⁄ 2

=--

---------------- exp[ ( E

– E ) ⁄ kT]-- kT

F2 4 _2

πh

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W 3So we can say that the energy / electron = ---- = --kT . Again, a familiar result.

n 2

How about current density?

J( r )q

∫

pE

– E

p2

⁄ kT

3

p= -------------------- exp

F

– ---------- d

_3 ∗ ∗

4π h

p

m 2m

= 0

thermal equilibrium, f is even→ J

= 0

Very common and useful guess for f in nonequilibrium conditions is the “drifted” Max-wellian:

p –

pd 2

f( r, p, t) = exp F ( r , t) – E

c–

-------------------

⁄ kT

2m∗

quasi-Fermi level

“drift” velocity

f

pdzpz

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Consider heterojunction bipolar transistor (HBT) band diagram

Ec

ballistic peak

scattered electrons

emitter base

collector

electron distribution function at

collector

f

2 componentst h er m al i z e dcomponent

“ballistic” peak

pz' = 2m∗ Ec

How co

uld we

calculate such a distribution?

“Conservation of probability” --> continuity

eqn for f.

Balance forces and scattering

p

z

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7 -

University of California, Berkeley

EE230 - Solid State ElectronicsProf. J. Bokor

df( r , p , t) ∂f---------------------- = ---

dt ∂t G – Rtotal derivative: driving terms:

all changes in f generation, recombination, scattering

To see how this is used, expand total derivative using the chain rule:

df

=

dr ∇ dp ∂f

----

----

f + ----- ∇pf + ----

dtdt r dt ∂t

dp

Use F = ----- . dt

This is known as the Boltzmann transport equation (BTE). Compare this to the carrier con-tinuity eqn:

∂n 1∇ ⋅ J =

∂n

----- + --

-----

∂t q ∂t

G – R

We can rewrite the BTE to highlight the analogy to carrier continuity:

∂f∂f

--- + ∇r ⋅ ( v f ) + ∇p ⋅ ( Ff ) =

---

∂t ∂t

G – R

This shows how the BTE is a continuity equation for f. The vf term is analogous to an intu-

itively appealing probability current in real space. The less intuitive element here is the Ff

term, which represents the probability current in momentum space.

∂fNow let’s examine the term: ---

∂t G – R

2 processes contribute.1. Actual generation-recombination processes such as photogeneration, defect recombination, stimulated emission, ...

explicitly write s(r, p, t) for these

2. Carrier scattering . This takes carriers from one momentum state to another - creates sources-sinks in momentum space.

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EE230 - Solid State ElectronicsProf. J. Bokor

∂ffor these processes.Write ---

∂tcoll

For transport problems, we will be mainly interested in collisions.

Collision integral

Consider the effect of collisions on the distribution at a particular point in phase space: f(p, r, t). Two types of collision processes affect f at this point:

(1) carriers at p′ scatter into p (2) carriers at p scatter out to p′

The collision rate for process (1) is given by:

f( p ′)[ 1 – f( p )]S( p ′, p)

prob. of findingprob. there

a carrier at p′is no carrierat p

Similarly, the rate for process (2) is given by:

transition rate for thecollision p′ → p(Fermi’s golden rule)

The total rate of change at p due to collisions is then the sum over all possible processes scattering scattering particles into p minus the sum of all processes scattering particles

out of p:

∂f( r , p , t)----------------------∂t

S( p, p′) could be a sum of several processes.

Given Sppthe BTE is an

“integro-differential” eqn for f (r, p, t). (i.e. a mess!) The name of the game is finding suitable approximations and computational techniques.

Relaxation time approximation (RTA)

Simplest approximation for the collision term:

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where fo represents the equilibrium distribution.

This simply says that equilibrium is restored in time τ. We will use this approximation heavily.

The solution then is simply:

f( t ) = f0 + [ f ( t=0) – f0]e–t

⁄ τ

Conductivity in the RTA

Assume n-type semiconductor, applied field E .

Further assume system is spatially uniform so that ∇rf = 0 .∂f

Steady state implies --- = 0 .∂t

Then

Rewrite:

f = f0 – τqE ∇⋅ pf

For low field, assume small change in the momentum gradient. That is: ∇pf ≅ ∇pf0 . Then:

f = fo – τqE ∇⋅ pfo (1)

To improve on the first order approximation, we could iterate. That is, put this f in ∇pf .

This would give a term O( E 2 ) in the solution for f. We will neglect this for now.

Take E along zˆ .

This looks like the first 2 terms of a Taylor series. Then, to 1st order: (assuming τ constant)

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which is the drifted Maxwellian!We can derive another useful form. Let:

θ ≡

p2

⁄ ( kT)

EC – EF + ----------

2m∗

fo =

e–θ

then

∇

p

f

∂f

o∇pθ

o

= ------

∂θ

–fop v

------------- = -----

m∗kT kT

Equation (1) becomes:

f = f

qτE ⋅ v

o

+ -----f

kT

≡ fo + f′ (2)

with f′ « fo .Now calculate drift velocity

∑vz( f o + f′)

⟨ v ⟩ = ------------------------------p (3)

z∑( f o + f′)

p

If τ is constant (independent of p or E), then by inspection of drifted Maxwellian:

Current density is given by:

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University of California, BerkeleyEE230 - Solid State Electronics

=

nq2

τ-----------E

z∗

m

= nqμEz

This is the familiar result with

Prof. J. Bokor

w

I

v

2⟨v

1∑fo

insert (2)

By spherical symmetry,

v

ing these relations, we find:

.

University of California, BerkeleyEE230 - Solid State Electronics

If we write: ⟨ v z⟩ = μEz , then this defines μ more generally:

μq ⟨ ⟨ τ ⟩⟩

= ---------------- .

m∗

This special ensemble average is defined by:Often τ( E ) is given by a power law:

τ( E ) = τ0[ E ⁄ ( kT)]s

Then:

1∑ ( p 2 ⁄ 2m∗)( p

2 ⁄

2m∗kT)se

–p2 ⁄ 2m∗kT

2⟨ ⟨ τ⟩⟩ =

τ0-------------------------------------------------------------------------------------------p

1∑ ( p 2 ⁄

2m∗)e–p2 ⁄ 2m∗kT

∞

∫( p 2 ⁄ 2m∗kT)se

–p2 ⁄

2m∗kTp

4dp

= τ------------------------------------------

--------------------------------0

∞0

∫e–p2 ⁄

2m∗kTp4dp0

let y ≡ p2 ⁄ 2m∗kT dy =

----------------2pdp =

--------------------

2y1 ⁄

2

2m∗kT

( m ∗kT)1

⁄ 2

Then

∞ 1

s –--

⟨ ⟨ τ⟩⟩ = τ

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EE230 - Solid State Electronics

∞s +

3

--

∫y

2 e–ydy= τ

0

----------------------------

∞

3

∫

--

0y2 e

–ydy

Using the Γ function:

Γ( n )

Γ( p + 1)

Finally:

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Boltzmann Equation Handout

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