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K. v. Klitzing
discovery: 1980
Nobel prize: 1985
Oscillations of longitudina resistivity =Shubnikov-deHaas, minima close to 0.
Plateaux in Hall resistivity r=h/(ne2) with integer n correspond to the minima
(From Datta page 25)
1
Origin of the Oscillations of longitudina resistivity =Shubnikov-deHaas
N ( E , 0) 
m

( E  E0 )  N ( E , B) 
2
2eB
1
eB


(
E

E

n


),



0

 c c
h
2
m

r() without H
r (E) 
resistivity minima close to 0
Lx Ly 2 m
h
r() with H
( E )
E
E1
r ( E )  N L ( E )  ( E  (n  ) c )
2
LLL full for H  H t ; decreasing H one starts filling LL with  2. Number of filled LL :  
 The LL number ν is partially filled between H 
Ht
H
Ht
H
and H  t .
 1

2
Rectangular conductor very thin in z direction uniform in x direction confined
in y direction with B in z direction.
Assuming for the sake of argument that H is separable, the transverse
dimensions yield infinite solutions that are called subbands: let us assume
that only one z subband is occupied and the confinement along y is
described by U(y).
2
( px  eBy ) 2 p y
[

 U ( y )] ( x, y )  E  ( x, y )
2m
2m
( x, y)  eikx  ( y)
2
(k x  eBy ) 2 p y
[

 U ( y )] ( x, y )  E  ( x, y )
2m
2m
Expanding,
[
py 2
1 eB
 m( ) 2 ( y  yk ) 2  U ( y )] ( x, y )  E  ( x, y ),
2m 2
m
yk 
kx
eB
For =integer,..filled or empty LL  gap, no scattering  xx resistivity =0
For =0.5,1.5,2.5,.. Half filled LL  maximum xx resistivity
This is a paradox: one would expect minimum resistance when LL is at
Fermi energy, but it is maximum; how does the sample carry the current
when EF is between LL and there are no states at EF?
Reply: there are 1d states at the eges of the sample (hedge states ) that
carry the current!
The minimum resistivity is very low because of the suppression of
relaxation. Carriers that go to opposite directions are far away and never
meet.
r() with H
r() with H
but more like
this
Due to
impurities, the
DOS is not like
this
E
E
Due to disorder and impurities, it is possible to find the Fermi level away from the LL (otherwise it is unlikely
to find it where the DOS is small and a small charge can move EF). The conduction takes place through the
M hedge states that have very small resistance.
4
Origin of zero resistance (see also
Datta page 175)
We try to include confining potential U(y) along y as a perturbation, which is nearly
constant over the extent of the LL wavefunction.
Including confining potential in first order, states with k in x direction in the LL n
have energy:
1
E (n, k )  Es  (n  ) c  n, k U ( y ) n, k ,
2
Es  subband bottom, U ( y )  confining potential
n, k U ( y ) n, k  U ( yk ),
yk 
y
kx
eB
kx
In the middle of the sample, bulk-like eigenvalues and eigenvectors prevail, but
near the edges the levels are shifted by U yielding a quasi-continuum of levels,
also at fermi energy. Current in edge state can be evaluated by the group velocity:
1 E (n, k ) 1 U ( yk ) 1 U ( yk ) yk
1 U ( y )



,
k
k
y k eB y
edge states on opposite edges carry currents to opposite directions
v(n, k ) 
U ( y )
is opposite and the x component are assumed opposite.
y
No head-on collisions are possible.
since
5
No backscattering takes place.This situation when EF is between two LL,
otherwise the LL at Fermi level carries current within the sample with
scattering and maximum resistance (not an explanation but a description)
6
The Quantum of
conductance
h
2e2
Could be measured i n balistic conductors within a few % in QHE with better than ppb accuracy !
in QHE mm-sized electron mean free paths because curent carrying states in opposite directions
are localized on opposite sides  no backscattering
7
Quantized resistance
Degeneracy of each LL: D L 

,
0
number per spin N   N  
N el
2
Condition for exactly filled LLL (for each spin direction): N   N   D L 
 threshold field for having all N el electrons in LLL: H t S 
In terms of the surface density ns 
N el
,
Lx Ly
N el 
 .
2 0
N el
0 (LLL energy depends on spin)
2
Surface density per spin,
threshold field for having all N el electrons in LLL H t 
N
1
ns  el , .
2
Lx Ly
1 hc
ns .
2
e
Step hight
 r xx r xy 
B  0 1
Recall Hall resistance in Drude theory for   , r ( H )  



 r yx r yy  | e | ns  1 0 
B  0 1
H
since resistivity goes to 0
(r(H ) 
in
cgs
units).
Hall
resistance
r

xy


| e | nc  1 0 
ens c
For
H
2

H t ,   number of occupied LL, all others empty,
2 1 hc
ns

2
e  h
r xy 
ens c
e 2
and these are the quantized values if   integer.
8
Convincing explanation ? NO!
Drude theory ia rough, the result is extremely precise, and why the
plateaux?
Elaborated from a seminar by Michael Adler
9
Due to the applied bias, a current flows in tha sample; this produces the Hall field in the y
direction. So the upper edge states, where electrons go to the left , have the chemicel potential R
of the right electrode, the lower ones have the chemical potential L of the left electrode. The
potential drop VLongitudina along x for both is zero. Since the potential drop along y is VH,
R-L=VH.
Landauer formula along x direction : I 
2e
M (  L   R ),
h
eVHall   L   R
RLongitudinal 
VLongitudinal
I
 0 RHall 
VHall  L   R
h
25.8128K 

 2 
I
I
2e M
2M
Good, but a serious doubt remains.
We did several crude approximations. Why is the result so precise)?
10
Why the plateaux? Why so exact?
Hall resistivity quantized with r xy 

h
,
e 2
 =sharp integer
e 2
Hall conductivity quantized with  xy 
,  =sharp integer
h
Consider a Metal ribbon long side along x, magnetic field along z.
y
x
11
11
Assume noninteracting electrons. The electronic  obeys:
2
( px  eBy ) 2 p y
[

 U ( y )] ( x, y )  E  ( x, y )
2m
2m
Setting:
 ( x, y )  eikx  ( y )
one obtains
 [
2
(k x  eBy ) 2 p y
[

 U ( y )] ( x, y )  E  ( x, y ),
2m
2m
py 2
1 eB
 m( ) 2 ( y  yk ) 2  U ( y )] ( x, y )  E  ( x, y ),
2m 2
m
yk 
kx
eB
Laughlin does not even insert confining potential U(y) which plays no role in his argument.
Now add an electric field along y.
2
( px  eBy ) 2 p y
H

 eE0 y,
 kn  eikxn ( y  y0 )
2m
2m
py 2 1
kx2
eB
2 2
H
 mc y 
 ( k xc  eE0 ) y, c 
. Next find y 0 such that
2m 2
2m
mc
1
1
1
2 2
2
2
mc y  ( k xc  eE0 ) y  mc ( y  y0 )  mc2 y0 2
2
2
2
1
1
1
2 2
2
2
mc y  ( k xc  eE0 ) y  mc ( y  y0 )  mc2 y0 2
2
2
2
1
1
1
1
2 2
2 2
2
2
2
mc y  ( k xc  eE0 ) y  mc y  mc y0  mc yy0  mc2 y0 2
2
2
2
2
kx
eE0
E0
1 k
y0 

 [ c ]
2
mc mc c m
B
1
m 2 2
E0 just shifts oscillator :   n  ( n  ) c  c y0
2
2
13
Consider closing it as a ring pierced by a flux, with opposite sides
connected to charge reservoirs of infinite capacity , each as the same
potential as the side to which it is connected. A current I flows
around, the Hall potential VH exists between reservoirs.
Next replace E by a time-dependent flux  inside; it
produces a e.m.f that excites a current I around (along x)
I c
U
, U  total energy

but the magnetic field along z then produces a Hall electric field VH, along the ribbon, that
will transfer charge q from one reservoir to the other, contributing qVH to the energy.
If the flux is one fluxon, the system is completely restored in previous state .
Current due to flux piercing the ring: I  c
U
U
c


for a cycle, using a macroscopic ring,  0
U  (total energy)  qVH , since charge is transfered and system is restored
I c
qVH
hc
( )
e

Hall conductivity
I
eq
 .
VH
h
15
In order to have the ribbon in same state as before the fluxon is
applied, q=ne with n integer. Hence,
I
e2
 Hall conductivity
n
VH
h
is exactly quantized !!
Laughlin argues that the same holds true even in the presence of
interactions.
Laughlin’s argument has been criticized on the grounds that different cycles of the
pump may transport different amounts of charge, since q is not a conserved quantity. It
is the mean transferred charge that must be quantized.
It appears to me that the criticism is rather sophistic, because if the average is an exact
integer that does imply that every measurement gives an integer. One could envisage a
situation where two exactly equally likely outcomes are 0.80000 and 1.20000 (sharp!)
and so the average is 1,0000 without having integer outcomes each time. One should
discover fractionally charged real electrons before accepting this explanation.
However some authors of the above criticisms have produced a remarkable
alternative explanation.
16
A
According to the Authors, Laughlin’s argument is short in one important
step, namely, the inclusion of topological quantum numbers.
H  H ( ,  )   flux in the ring,   flux in the ammeter A that measures current along y.
Parameter space: ( ,  ).

Current flowing through A:
I c
H


 
H  i
i
t
 t
17
The reading of the ammeter does not change energy




 H  0 
H 
 H  H






Inserting I  c
H , setting c=1,

 

0  I  2 Re 
 H 
 

 


 I  2 Re 
 H    2 Im i  H  .

 


 
inserting H  i
i
and restoring c
t
 t
I
2

  2
Im i  i

c

 t c

 


 t
18



I K
, where K  2 Im


t
 
 curvature of
the bundle of
Hall conductance  c 2 K
1
2

S

ground states
Berry !
KdA  integer


where K  2 Im


 
S= parameter space
is a Chern number! What is it?
19
Gauss and Charles Bonnet formula
1
KdA  2(1  g )

S
2
K  curvature, g  number of
handles
Chern formula
1
2

S
KdA  integer (Chern number)
K  2 Im




 
20
The QHE is the prototype topological insulator
Topological Insulators
• (band) insulator with a nonzero gap to
excitated states
• topological number stable against any
(weak) perturbation
• gapless edge mode
• Low-energy effective theory of
topological insulators
= topological field theory (Chern-Simons)
Bi2Se3
is a 3d topological insulator
http://www.riken.go.jp/lab-www/theory/colloquium/furusaki.pdf
http://wwwphy.princeton.edu/~yazdaniweb/Research_TopoInsul.php
Fractional Quantum Hall
effect
D.C. Tsui, H.L. Störmer and A.C. Gossard, prl (1982): quantization of Hall
conductance at = 1/3 and 2/3 below 1 Kelvin
The fractional quantum Hall effect (FQHE) is a physical phenomenon in
which a certain strongly correlated system at T under a very strong magnetic field
behaves as if it were composed of particles with fractional charge (1998 Nobel
Prize).
22
experiments performed on gallium
arsenide heterostructures
23
Strong fields :
 N el

 all electrons in LLL
0
2
h
25.8

k , M  integer number
2e 2 M 2M
interpretation: M  number of edge states
Integer QHE : r xy quantized in units of
Somewhat Stronger fields :

 N el  all electrons with spin  in LLL
0
(typically8Tesla) , but integer QHE
Still Stronger fields :FQHE :
h
1 2 4
r xy quantized in units of 2 , p  , , ,
pe M
3 5 7
Denominators almost ever odd. The FQHE is a different phenomenon, requiring a different
explanation.
It is believed that the effect is due to the Coulomb interaction.
All electrons in LLL treating interactions as a perturbation that tends to lower the symmetry.
Wave functions for the LLL in polar coordinates
Using the Curl in cylindrical coordinates


1 

1 




rotA  rˆ 
Az  A   ˆ  Ar  Az   zˆ  (rA ) 
Ar 
z 
r 
r  r
 
 r 
 z
one finds a Gauge :
Ar  0,
A 
1
rB,
2
Az  0 such that rotA  (0, 0, B)
2
2
2 1  1 2
To write the SE we also need the Laplacian
 2 2
 2
.
2
2
x y
r
r r r 
1 
1  e


ie 2
p  p  eA
i
 i
 rB, that is,


r B
r 
r  2

 2
2
2

1 
1  
ie 2  



 2

r B   ( r , )  E ( r , ) .
 2
2m 
r r r   2
 
 r

2
Specialize for the LLL n  0, set
 ( r , )  Nz l e

| z|2
2
4 lM
z  x  iy  rei and define magnetic length: lM 
mc

eB
  ( r ,  )  eil , with integer l
25
setting

=il, angular momentum l,

 ( r ,  )  Nz l e
where
rl 

the wavefunction of an harmonic oscillator at large r,
| z|2
2
4 lM
z  x  iy, lM2 
2l
 2l lM . Minimum at r=rl
eB
eB
A circle of this radius encloses l fluxons.
Laughlin sought thee Many-Body wave function for LLL in the form
N
 ( z1 , z2 ,..., z N )  f ( z1 , z2 ,..., z N )  e

| zi |2
2
4 lM
i 1
where
, zi  xi  iyi , lM2 
f ( zi , z j )   f ( z j , zi ) ensures Pauli principle.

i
We want it to be an eigenfunction of total angular momentum
i
eB
,

 i
Laughlin ansatz :
N
 ( z1 , z2 ,..., z N )  
j k
Pauli  q odd
(z
 zk ) e
q
j


i
| zi |2
2
4 lM
,
26
For 3 electrons Laughlin Wave Function :
z12  z22  z32
 ( z1 , z2 , z3 )  ( z1  z2 ) ( z1  z3 ) ( z2  z3 ) exp[
].
2
4lM
q
q
q
Physical Picture:
Outer electron encloses lmax fluxons
lmax


0
R. Laughlin
This is the maximum power that any z j can have.
27
N
 ( z1 , z2 ,..., z N )  
j k
(z
 zk ) e
q
j


i
| zi |2
2
4 lM
, q odd
Let us evaluate q. Expanding the product in Laughlin ansatz the
maximum power of z of any particle evidently turns out to be:
lmax= Nelq
 lmax




 N el q  q  0  number of quanta per electron in sample.
0
N el
Equivalently, the number of electrons per fluxon in sample=
1
q
In terms of such wavefunctions one can estimate correlations and compute
successfully the relevant quantities. Quasiparticles of fractional charge e/q obey
anyon statistics (exchange of two quasiparticles brings a phase , which can be
calculated as a Berry phase).
Very accurate wave function when compared with numerical estimates. q=1 yields
integer QHE wavefunction
28
From a seminar by Michael Adler
29
Some CONCLUSIONS
The dimensionality conditions the many-body behaviour already at the
classical level- e.g. phase transitions
At the quantum level, the transport properties are very different when
nanoscopic objects are considered, and this requires new intriguing
concepts and funny mathematical methods, many of which involve the Berry
phase too. Here I just recall some. The subject is in rapid evolution, and new
applications are also under way.
Ballistic conduction, nonlinear magnetic behaviour, possibility of various
kinds of pumping.
The role of correlation effects is much more critical in 2d (e.g. QHE, charge
fractionalization, anyons , Kosterlitz-Thouless ) and above all in 1d (Peierls
transition and charge fractionalization, ) and there was no time to introduce
others, that would require the Luttinger Liquid formalism….
30
Charge-spin separation
                       
1  d antiferromagnet
            
1  d antiferromagnet with hole
         
spins are assumed to jump freely to empty sites
       
hole has moved
              
       
hole has moved
holon
              
spinon
31
       
hole has moved
              
holon
spinon
nearby sites are assumed to exchange spins
       
hole has moved
              
holon
spinon
propagation
holon also travels
Fermi particle becomes a pair of boson excitations that can propagate with different
speeds (in 1d only)
Bosonization
Fermion operators can be expressed in terms of bosons:
spinons and holons and problem is solved
32
Typical power-law dependence of the DOS and many quantities in 1d
r (q ,  )
(  v
)

spinon
1
2
 1

(  vholon ) 2 2 (  vholon ) 2

  interaction-dependent exponent
Separation Realized
experimentally!
33