• Quantum Hall effect

• Classic (3D)• Bz, Ex in a conductor!

!E = "!jresistivity tensor

!yx = !!xy =Bz

ne= !Ey

jxHall resistivity, transverse

RH =Ey

jxBz= ! 1

ne

Hall coefficient

51

EH = Ey = ! jx

neBz

• Quantum Hall effect

52

• 2D electron gas

!xy = ! Bz

nae; !xx = ! me

nae2"

!xx = !yy ; !xy = !!yx

areal electron densitytransverse

magnetoresistivity average time

between collisions

d ! !F

jx = !xxEx + !xyEy

jy = !yxEx + !yyEy

•But at B ! (low T) ! steps in Hall resistivity, different behaviour !!

INTEGER QUANTUM HALL EFFECT(!c" >> 1 , !!c >> kBT )

!c =eB

me

cyclotron

resonance

frequency

(e.g AlGaAS/GaAS modulation doped

heterostructure)

field independent

magnetoresistivity

• Inverting eqs.conductivity tensor

!xx ="xx

"2xx + "2

xy

; !xy =!"xy

"2xx + "2

xy

• Quantum Hall effect

53

• Integer QH effect

transverse

longitudinal

10

2

3

4

56

8

INTEGER QUANTUM HALL EFFECT

(von Klitzing et al, 1980 for Si MOSFET)Nobel Prize 1985

• Quantum Hall effect

54

•Values of Hall resistivity quantized in units of

•and sharp peaks in in the jumps ; (" Shubnikov-de Haas effect)

• in the B ranges of the plateaux

• , zero longitudinal resistance

!xy =!h

pe2 p = 1, 2, 3...

(h/e2 = 25812, 807! ! 26 k!)

!xx(B)

!xx = 0

•Steps more evident at B !

!xy != 0 , !xx " 0 , "xx " 0

•Explanation " 2D behaviour of Landau levels (explained later)

• Quantum Hall effect

55

Skipping cyclotron orbits

Four-terminal sample configuration to measure

the Hall and longitudinal resistivities

• Quantum Hall effect

56

•For a given plateau not a perfect conductor,

!xx = 0 , !xy != 0"electrons move with zero longitudinal resistance.

•Electron cyclotron orbits confined to edges of the sample: skipping orbits do not permit back-scattering !edge-channel transport resistanceless

!Relationship between edge states and contacts analogous to

quantized point-contact conductance (Landauer formalism)

• Quantum Hall effect

57

•B in 3D electron gas !collapse of allowed k states onto Landau tubes.

!allowed energy levels:

!modified DOS (as in 1D)

free movement in z direction (B)

!n =!2k2

z

2me+ (n +

12)!"c

!c =eB

mecyclotron resonance frequency

n = 0, 1, 2....

•Discrete energy levels ! oscillations in magnetization and other properties: Shubnikov de Haas effect

• Landau levels (3D behavior)

• Quantum Hall effect

58

• 2D behaviour of Landau levels

ground-state level for the well

! = !1 + (n +12)!"c ± µBB

spin

µB =e!

2me

Bohr magneton

quantized n Landau level

•DOS: series of delta functions, spin

doublet for every Landau level

•When m!c = me , !!c = 2µBB

cyclotron effective mass

!c =eB

m!c

, Zeeman (spin) and cyclotron

splitting the same

•B !!degeneracy per unit area of Landau levels, gn

!c ! , gn =eB

h!" QH effect from dependenceµB(B)

chemical potential

Zeeman energy, +- for spin

Levels move upwards in energy

• Quantum Hall effect

59

• , dependenceof chemical potential with magnetic fieldµB(B)B2 > B1

•Consider 2D (density na) e- gas in B, with n=2 ! Landau level half-

filled ! µ pinned to n=2 ! position.

•Increase B : B !" !2! ! , " µ ! but gn(<2) !! the part-filled level must be depleted of electrons

! µ# discontinuously to 1# !

Discontinuous jumps in whenever an integral number of Landau levels are

completly occupiedµ(B)

• Quantum Hall effect

60

! na

gn= p , integer

! Bp =hgn

e=

hna

pe

! !xy = ! Bp

nae= ! h

pe2

!xx = 0 , since g("F ) = 0 when all Landau levels completely empty

or completely filled

• Condition:

• Quantum Hall effect

61

• But this picture doesn’t account for the ranges of B corresponding to the plateaux.

! Disorder in the 2D system (structural defects at heterojunction)

! - Bands in g(!) broadened.

- electron states spatially localized (high disorder)

! µ(B) oscillatory but smoothly varying

When µ(B) in band of localized states ! “Fermi glass”,

in a B range (T=0)!xx = 0no hopping conduction

• Very high B (only one Landau level)!

integral QH disappears,

but p=n/m !FRACTIONAL QH EFFECTn, m integers,

n<m

• Two effects:

• Quantum Hall effect

• Fractional or non-integer QH effect

62

• Quantum Hall effect

63

•Due to e--e - interactions in 2D ! “Incompressible quantum fluid”

For fractional Landau filling factor p=1/m, quasiparticle excitations have charge

Q=e/m fraction of electronic charge

•Theory ! Laughlin, Phys. Rev. Lett. 50, 1395 (1982)

• Nobel Prize with Stormer & Tsui, 1998

•Experimental confirmation ! - de Picciotto et al., Nature 38, 168 (1997)

- Saminadeyar, Phys. Rev. Lett. 79, 2526 (1997)

Measurement of noise in the current through a constrictionof a 2D gas at

high B

(fig. next page)

•Very different mechanism to integer QH.

• Quantum Hall effect

•Split-gate electrode ! 1D confinemrnt of 2D electron gas (QP contact)

e/3

Strong pinch-off

weak pinch-off

Shot noise weak pinch off,, p=1/3

fitted to eq. of

only is Q=e/3 assumed

!(I2)

64

•No uniform flow of charge carriers ! fluctuations in number of carriers (shot noise)

!(I2) = 2QI0!f ! determine Q

averagecurrent

frequency interval

(Approximate for T=0 and weak transmission)

•More generally,

!(I2) = 2Gt(1! t)!f [QV coth(QV/2kBT )! 2kBT ] + 4kBTG0t!f

G0 = Qe/hVt

thermal noise (Nyquist theorem)

quantized conductance

applied voltage

transmission

•Two regimes, depending on Vg:

1.- Vg# (weak pinch-off)" 2D gas between two electrodes

2.- Vg! (strong pinch-off) "tunnel of electrons in multiples of e (2D gas

separated in two)

• Quantum Hall effect

65

• Applications

•Semiconductor transistors (bipolar, field-effect,

modulation-doped devices)

•Opto-electronic devices (solar cells, photodetectors, light-

emitting diodes, semiconductor lasers)

Possible assignments for final presentations...

66

• Summary

67

•We have studied the main features taking place when two different materials (metal or

semiconductors) are put into contact, paying attention to the current conduction through the

junction.They have important applications for electronic devices.

•The effects of confinement in thin slabs produce discrete states which change the optical

absorption pattern with respect to the bulk, and make them interesting for optoelectronic

devices.

•Artificial structures can be prepared by MBE, producing periodic arrays of two alternate

materials. The period and width can be tuned to produce multiple QWs or superlattices,

which have very different properties from bulk materials. Using gradual doping, nipi

structures are produced with interesting photoluminiscence properties.

•Finally we have studied the effect of magnetic fields on 2D electron gas structures, giving

rise to Landau levels, and we have described the integer and fractional quantum Hall effect.