Institute of Solid State PhysicsTechnische Universität Graz

23. Kinetic theory

June 19, 2018

describe electrons as a gas of particles

vF = 108 cm/s.

kinetic theory

The average time between scattering events sccan be calculated by Fermi's golden rule

mean free path: l = vFsc ~ 1 nm - 1 cm

Electrons as waves or particles

Scattering of electrons can be thought of as transitions between k states or as collisions between particles.

Umklapp scattering of electrons by phonons makes large changes in the momentum of the electrons because of the reciprocal lattice vector G.

el el phk k k

el el phk k k G

phonon emitted

Umklapp scattering

Ballistic transport

dvF ma eE mdt

electrons in an electric field follow a parabola.

electrons in a magnetic field move in a spiral

electrons crossed electric and magnetic fields spiral along the direction perpendicular to the electric and magnetic fields

0eEtv vm

2

0 02eEtx v t x

m

If no forces are applied, the electrons diffuse.The average velocity moves against an electric field. In just a magnetic field, the average velocity is zero.In an electric and magnetic field, the electrons move in a straight line at the Hall angle.

Vacuum diodes

diode

Diffusive transport

<v0> = 0 <t - t0> = sc < average time between scattering events

* *sc

dF

eE eEvm m v

dv E

drift velocity:

* * dvF eE m a mdt

0 0* ( )eEv v t tm

t0

t+t0v0

v

time between two collisions

t

v - v0

t0sc

=dj nev ne E E Ohm's law:

Matthiessen's rule

, ,

1 1 1

sc sc lattice sc impurity

phonons, temperature dependent

mostly temperature independent

1 1 1

lattice impurity

Ballistic transport in transistors

The mean free path ~100 nm > gate length ~ 20 nm

v not proportional to E

j not proportional to E j E

v E

nonlocal response

Electrons bend in a magnetic field like they do in vacuum.

Magnetic field (diffusive regime)

sc

dvF ma e E v B m

sc

dvF ma eE m

If B is in the z-direction, the three components of the force are

sc

dxx dy z

ve E v B m

sc

dyy dx z

ve E v B m

sc

dzz

ve E m

scd

e E vm

Magnetic field (diffusive regime)

If Ey = 0, Ez = 0

, ,x sc z

d x sc d yeE eBv v

m m

, ,y sc z

d y sc d x

eE eBv vm m

,z sc

d zeEv

m

, ,z

d y sc d xeBv vm

tan zH sc

eBm

,d yv

,xdvH

Crossed E and B fields

Ballistic transportE

<v>

Diffusive transportE

<v>

H

1tan z scH

eBm

Hall angle:

B

F ma e E v B

If no forces are applied, the electrons diffuse.The average velocity moves against an electric field. In just a magnetic field, the average velocity is zero.In an electric and magnetic field, the electrons move in a straight line at the Hall angle.

The Hall Effect (diffusive regime)

Ey = vd,xBz = VH/W = RH jxBz VH = Hall voltage, RH = Hall Constant

If vd,y = 0,

, ,x sc z

d x sc d yeE eBv v

m m

, ,y sc z

d y sc d x

eE eBv vm m

,z sc

d zeEv

m

RH=Ey/jxBz = - 1/ne

jx=- nevd,x

Kittel

Diffusion equation/ heat equation

2dn D ndt

j D n

dn jdt

21 exp44

rnDtDt

Continuityequation

Fick's law

Diffusion constant

Einstein relation

In equilibrium, drift = diffusion

0en E eD n

Boltzmann factor( )

( ) exp pot

B

U xn x A

k T

1 exp potpot pot

B B B B

U n enEn A U Uk T k T k T k T

2 0B

nEen E e Dk T

Bk TDe

potF U eE

Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, A. Einstein (1905).