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

16

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

17

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