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Hall effect
1
Hy
z
x
J
VH
x
Low Field Magnetoresistance, Hall effect in Drude Theory
steadystate : (rateof loss=rateof gain from field)
v v B , v drift velocity, scattering time
scattering field
dp dpdt dt
m e E
(0,0, ), ( , ,0), no force along , so forget about third component.
v B (v , v ,0) v (n electrons per unit area)x y
y x s s
B B E E E z
B B J e n
2
Hy
z
x
J
VH
x
vFrom the steady state equation of motion v B
we obtain electric field versus v,but since one measures the current we rewrite in terms of J:
v 1v
x x
y y
m e E
m BE eE m eB
e
11 ,
1| |
| | Drude mobility (velocity/electric field)
x x
y ys s
m BB J JeJ Jmn e nB B
eem
We may rewrite , where:0
1 1 ,| | | |
x xx xy x
y yx yy y
xx yx xys s
E JE J
B Be n e n
3
longitudinal resistance= due todissipation as usual
linearHall resistance in B= due to Lorentz force
xxx
x
yyx
x
EJ
EJ
Hall resistance
, , Hall field0
1 1 , | | | |
0 1for , ( )
1 0| |
x xx xy xx xx x y yx x
y yx yy
xx yx xys s
xx xy
yx yy s
E JE J E J
E
B Be n e n
BHe n
�
4
4
One measures the voltage drop along y and the current along x, not a resistance
Hall resistance is proportional to magnetic field since the phenomenon isdue to the Lorentz force and the conductor is linear and inversely proportional to nsince the Lorentz force goes with the velocity, not with J, and J=neV.
1
3 4
2 (red arro (blue arrows)
ws)x x
H yVV V V E L
V V E W
Hy
z
x
J
VH
Vx
321
5
Quantized Hall effect of the two-dimensional electron gas in GaAs- AlxGa1-xAs heterojunctions at low temperatures to 50 mK. In the small-current and low-temperature limit sharp steps connecting the quantized Hall resistance plateaus. The diagonal resistivity ρxx decreases with decreasing T at the Shubnikov—de Haas peaks, as well as at the dips, and is vanishingly small at magnetic fields above 40 kG
Hall resistance is proportional to magnetic field in Drude theory, however…..
2d electron gas in (x,y) plane in magnetic field H parallel to z axis
second Landau Ga
( ,0,0) (does not treat x,y on equal footing)
(0, ,0) (also does not treat x,y on equal footing)1 ( , ,0) ( treats x,y on equal fo
u
oting
Landau G
2
aug
ge
)
eA H y
A H x
A H y x
6
H
y
z
x
Main alternative Choices of the gauge
22
We use thee ( ,0,0) in a 2d box of sides L,pbcalong x
1ˆ 2
2
Landau Ga
ˆ[ , ] 0 plane w
ug
avealong , , *
e
x y
x x x x
A H y
eHH p y pm c
p H x p k k nL
We start with the Theory for spinless electrons
22
2 22 2 222 2 2
0
1ˆThen oscillator Hamiltonian along y,2
w cyclotronhere f
x y
x xx c
c
eHH p y pm c
c k c keH eH eHy p y m y m y yc c eH mc eH
eHmc
7 20 0 0
00
, where flux quantum 4.13
req
610 *2 2
Dimensions : [ ] [ ] momentum [ ]
uency
1 [ ] [ ]
.
./
x x x
xx
c k k khc hcy Gauss cmeH e H H e
keH y k y Lc eHy c
0
40
Width of gaussian
(If 1 Tesla 10 Gauss 257Angstrom)
c
cl eHm eHmmc
H l
01 1LL (Landau levels) Amplitudes ( , ) ( )2
x
x
ik x harmonicn c nk n
x
E n x y e y yL
7
y
kx
0 is thecenter of oscillator, proportional to k .xy
2 2
( ) ( )2k
kE Em
2 2
2 2
2
2
2 2( ) ( )1dimension: ( ) ( )
2 | |2
2( ) 22 ( ) ( )2
mE mEk kLE dk Ed kdk m
mEkL L mdk E Ek h E
m
Free electron density of states at B=0 (continuous model)
2 2
2
2 2
2 2
2( )2 : ( ) (2 ) ( )
2 | |
2 2( ) ( ) ( ), the case of interest.2
mEkLd E kdk Ek
mL mE L mdk k E E
h
33 32 22 32 3 2 2 2
2( ) 4 2 23 : ( ) (4 ) ( ) ( ) ( ) 2 ( )2 (2 )| |
mEkL L m mE md E k dk E dkk k E L E Ek h
m
( 1)
is the degeneracy of each Landau level : integrating ( ) over one LL,
1( ) ( ) ( ( ) ) .2
c
c
L
n
L c Ln
D H E
E D E E n dE E D H
without H
E
with H
E
2( ) ( )x yL L mE E
h
1( ) ( ) ( ( ) )2L cE D E E n
The energy levels do not depend on kx, so their degeneracy is just the number of kx points: it is found by imposing that y0 cannot exceed sample size.
0
0
The same for all
2 2, ,
where . LL.
x yx y x y L
x x
x yL
eHL Lc n c ny k L k L n DeH L eH L hc
eHL L eHSDhc hc
9
y
kx
el
0
Let us put all the N electrons in the LLL.All electrons fill exactly the LLL if each gets a fluxonel L elN D N
0
t
0 t 0
11 2 4
.
Given , what is the H such that all the N electrons fill exactly the LLL ?
=N Thresh
Thus, thedegeneracyof Land
old field H
au Levels is
thre
Typical: 10 10 Gau
shold field
L
el el
elel x y
elt
D
NN S L LS
N cm HS
ss
10
Лев Васи́льевич Шу́бников
0
t
per spin.
Given , what is the H such that all the N electrons fill exactly the LLL ?
that electrons have spin up and electrons
Thed
e
have
generacy of Landau Levels is
threshold field
spi2 2
Assuming
L
el el
el el
D
NN N
0 t 0
n down, the system is not in equilibrium
1= N Threshold field H . 2 2
elel
x y
NL L
11
0 el el
00
N N umber of filled is the n2 2
2
Lt t
Lelt
DH S H
D HNHL
S
L
is full for ; H onestarts filling LL
partially fi
d
l
ecreasing
led between
wit
d1
2
a
.
.
h
nt t
tLLL H H
H HH H
Including spin
12
partially filled between and .1t tH H
H H
Ground state energy – 2DEG 1For All electrons in LLL with n=0 .
2 2t c el el B eleH H E N HN HNmc
1 31 occupied other LL empty ( ) 2 ( ), (factor 2 for spin).2 2 2t
c LH
For H n E D H
For partially occupied.1t tH HH n
0
The filled LL contribution to E amounts to 2 ( ) electrons for each filled LL
1filled LL contribution to E= 2 ( ) ( )2
L
L cn
N H
N H n
The energy contribution from the partially filled LL :1 there are 2 ( ) electrons each with energy ( + ) .2el L c
n
N D H
13
Ht H
1
0
1 1total energy 2 (n+ ) [N -2 ]( + ) ,2 2L c el L c
n
E N D
1
0
1 12 (n+ ) [N -2 ]( + ) ,2 2L c el L c
n
E N D
21
0
1 ( 1)since (n+ ) ,2 2 2 2n
2
2
Inserting 2 and ,2
( ) (2 1) ( ) .
elc B L
t
B el tt t
N HH DH
H HE H N HH H
2
2
12 [N -2 ]( + )2 2
1=N ( + ) 2 ( ).2 2
L el Lc
el Lc
E N D
E D
0 el el
00
N NRecall: 2 2
2
Lt t
Lelt
DH S H
D HNH
S
15
2
2
2
2
2
(2 1) ( )
(2 1) ( )right of interval:
The minima haveequal values: indeed,
1
(2 1) (left
1
(
of interval:
)
)
1
(
1
B el tt t
B e
t
t
tt
l
tH HH
E H
H
H HN HH H
H NE H
H
H
H
2
) 1( 1)
( ) B el tH NE H
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2
2( )
minima
(2 1) )
( )
(B el tt t
B el t
E H
E
H HN HH H
N HH
2 2
2
:
2 1 0
2 1 1 1 12 12
t
Position of MaximaH dx x xH dx
x midway
22
22 22 2
2 1 2 1(2 1) ( ) (2 1) ( ) 2 14 ( 1)2
Values of max
2
ima
( ) B el tB el t B el t
t t
N HH HN NE HH HH H
Note!
All electrons contribute , not only the open LL
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01 EMagnetic moment per unit surface S M
S H
2
2(2 1) (( ) )B el tt t
E H H HN HH H
18
MMagnetic susceptivity per unit surface S MH
2
2(2 1) (( ) )B el tt t
E H H HN HH H
K. v. Klitzing
Used the depletion layer of a GaAs MOSFET as a 2d electron gas
Oscillations of longitudina resistivity =Shubnikov-deHaas, minima close to 0.Plateaux in Hall resistivity =h/(ne2) with integer n correspond to the minima
20(From Datta page 25)
discovery: 1980Nobel prize: 1985
K. v. Klitzing