23. Kinetic theorylampx.tugraz.at/~hadley/ss1/lectures18/jun19.pdfUmklapp scattering Ballistic...
Transcript of 23. Kinetic theorylampx.tugraz.at/~hadley/ss1/lectures18/jun19.pdfUmklapp scattering Ballistic...
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).