quantum hall effectlampx.tugraz.at/~hadley/ss2/lectures20/qhe.pdfthe hall effect (diffusive regime)...

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