Download Quantum Hall effect

• Quantum Hall effect
• Classic (3D)
EH = E y = −
• Bz, Ex in a conductor!
jx
Bz
ne
! = ρ!j
E
resistivity tensor
ρyx = −ρxy =
Bz
Ey
=−
ne
jx
Hall resistivity, transverse
Hall coefficient
RH =
Ey
1
=−
jx Bz
ne
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• Quantum Hall effect
• 2D electron gas
field independent
magnetoresistivity
d ! λF
(e.g AlGaAS/GaAS modulation doped
heterostructure)
ρxy = −
transverse
magnetoresistivity
• Inverting eqs.
Bz
me
; ρxx = −
na e
na e2 τ
areal electron density
average time
between collisions
ρxx = ρyy ; ρxy = −ρyx
conductivity tensor
jx = σxx Ex + σxy Ey
jy = σyx Ex + σyy Ey
σxx =
ρxx
−ρxy
; σxy = 2
2
+ ρxy
ρxx + ρ2xy
ρ2xx
•But at B ! (low T) ! steps in Hall resistivity, different behaviour !!
(ωc τ >> 1 , !ωc >> kB T )
ωc =
eB
me
INTEGER QUANTUM HALL EFFECT
cyclotron
resonance
frequency
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• Quantum Hall effect
INTEGER QUANTUM HALL EFFECT
• Integer QH effect
(von Klitzing et al, 1980 for Si MOSFET)
transverse
2
longitudinal
3
4
10
8
6
5
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• Quantum Hall effect
•Values of Hall resistivity quantized in units of(h/e2 = 25812, 807Ω ! 26 kΩ)
ρxy =
−h
pe2
p = 1, 2, 3...
•and sharp peaks in ρxx (B) in the jumps ; (" Shubnikov-de Haas effect)
•ρxx = 0
•
in the B ranges of the plateaux
ρxy != 0 , ρxx → 0 , σxx → 0
, zero longitudinal resistance
•Steps more evident at B !
•Explanation " 2D behaviour of Landau levels
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(explained later)
Nobel Prize 1985
• Quantum Hall effect
Skipping cyclotron orbits
Four-terminal sample configuration to measure
the Hall and longitudinal resistivities
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• Quantum Hall effect
ρxx = 0 , ρxy != 0 ⇒ not a perfect
conductor, electrons move with zero longitudinal resistance.
•For a given plateau
•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)
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• Quantum Hall effect
• Landau levels (3D behavior)
free movement in z direction (B)
•B in 3D electron gas !collapse of allowed k states onto Landau tubes.
!allowed energy levels:
!modified DOS (as in 1D)
!n =
!2 kz2
1
+ (n + )!ωc
2me
2
n = 0, 1, 2....
eB
cyclotron resonance frequency
me
and other properties: Shubnikov de Haas effect
•Discrete energy levels ! oscillations in magnetization
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ωc =
• Quantum Hall effect
• 2D behaviour of Landau levels
1
! = !1 + (n + )!ωc ± µB B
Zeeman energy,
2
spin
ground-state
level for the well
quantized n
Landau level
+- for spin
Bohr magneton
µB =
e!
2me
•DOS: series of delta functions, spin
doublet for every Landau level
•When m!c = me , !ωc = 2µB B
ωc =
cyclotron effective
eB mass , Zeeman
m!c
(spin) and cyclotron
splitting the same
•B !!degeneracy per unit area of Landau levels, gn
ωc ↑ , gn =
eB
↑⇒ QH effect from µB (B) dependence
h
Levels move upwards in energy
chemical potential
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• Quantum Hall effect
• µB (B)
, dependenceof chemical potential with magnetic field
B2 > B1
•Consider 2D (density na) e- gas in B, with n=2 ! Landau level halffilled ! 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 µ(B) whenever an integral number of Landau levels are
completly occupied
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• Quantum Hall effect
• Condition:
!
na
= p , integer
gn
!
Bp =
!
hgn
hna
=
e
pe
ρxy = −
Bp
h
=− 2
na e
pe
σxx = 0 , since g("F ) = 0
when all Landau levels completely empty
or completely filled
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• Quantum Hall effect
• But
this picture doesn’t account for the ranges of B corresponding to the plateaux.
! Disorder in the 2D system (structural defects at heterojunction)
• Two effects:
! - 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”,
σxx = 0 in a B range (T=0)
no hopping
conduction
• Very high B (only one Landau level)!
integral QH disappears,
but p=n/m !FRACTIONAL QH EFFECT
n, m integers,
n<m
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• Quantum Hall effect
• Fractional or non-integer QH effect
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• Quantum Hall effect
•Very different mechanism to integer QH.
•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)
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• Quantum Hall effect
•Split-gate electrode ! 1D confinemrnt of 2D electron gas (QP contact)
Shot noise weak
pinch off,, p=1/3
fitted to eq. of
∆(I 2 )
only is Q=e/3
assumed
Strong pinch-off
e/3
weak pinch-off
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• Quantum Hall effect
•No uniform flow of charge carriers ! fluctuations in number of carriers (shot noise)
∆(I 2 ) = 2QI0 ∆f
! determine Q
frequency
average interval
current
(Approximate for T=0 and weak transmission)
•More generally,
∆(I 2 ) = 2Gt(1 − t)∆f [QV coth(QV /2kB T ) − 2kB T ] + 4kB T G0 t∆f
G0 = Qe/h
V
t
quantized conductance
thermal noise
(Nyquist theorem)
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)
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• Applications
Possible assignments for final presentations...
•Semiconductor transistors (bipolar, field-effect,
modulation-doped devices)
•Opto-electronic devices (solar cells, photodetectors, lightemitting diodes, semiconductor lasers)
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• Summary
•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.
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