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Institute of Solid State PhysicsTechnische Universität Graz
Quantum Hall Effect
Quantum Hall Effect
Shubnikov-De Haas oscillations
xxx
x
yxy
x
EjEj
Resistance standard25812.807557(18)
Klaus von Klitzing
Heterostructure
pn junction formed from two semiconductors with different band gaps
MODFET (HEMT)
Modulation doped field effect transistorHigh electron mobility transistor
Modulation doped field effect transistor (MODFET)High electron mobility transistor (HEMT)
The magnetic field can be at an angle to the 2DEG. The Landau splitting experiences the component perpendicular to the plane. The Zeeman splitting experiences the full field.
Inversion
Institute of Solid State Physics
MOSFETs Technische Universität Graz
Classical equipartition:
At 300 K, vth ~ 107 cm/s.
Drift
2 312 2th Bmv k T
mean free path: = vthsc ~ 10 nm ~ 200 atoms
The electrons scatter and change direction after a time sc.
Drift (diffusive transport)
<v0> = 0 <t - t0> = sc
* *sc
deE eEvm m v
, , d n n d p pv E v E drift velocity:
* * dvF eE m a mdt
0 0* ( )eEv v t tm
t0
t+t0v0
v
time between two collisions
Review of the Hall effect
sc
dvF e E v B m
sc
dvF ma eE m
If B is in the z-direction, and E is in the x- direction, the three components of the force are
sc
dxx dy z
ve E v B m
sc
dydx z
vev B m
sc
0 dzvm
diffusive regime
tan zH sc
eBm
Hall angle
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 drift velocity decreases as the B field increases.
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
vd,x=- jx/ne
The Hall Effect (diffusive regime)
xxx
x
yxy
x
Ej
Ej
RH=Ey/jxBz = - 1/ne
y zxy
x
E Bj ne
multiply both sides by Bz
The Hall resistivity is proportional to the magnetic field.
In 2D, j has units of A/m and n has units of 1/m2.
In 3D, j has units of A/m3 and n has units of 1/m3.
Quantum Hall Effect
Shubnikov-De Haas oscillations
xxx
x
y zxy
x
EjE Bj ne
Resistance standard25812.807557(18)
Quantum hall effect
y zxy
x
E Bj ne
Each Landau level can hold the same number of electrons. 0
2 20
zxy
hDB hne ve D ve
0 2c zm eBD
h
FE
If the Fermi energy is between Landau levels, the electron density n is an integer v times the degeneracy of the Landau level n = D0v
zc
eBm
0z
hDBe
Quantum hall effect
2xyh
ve
S. Koch, R. J. Haug, and K. v. Klitzing,Phys. Rev. B 47, 4048–4051 (1993)
Quantum Hall effect
Ibach & Lueth (modified)
On the plateaus, resistance goes to zero because there are no states to scatter into.
Edge states are responsible for the zero resistance in xx