Quantized Hall effect

Experimental systems

• MOSFET’s (metal-oxide-semiconductor-field-effect-transistor.)

• Two-dimensional electron gas on the “capacitor plates” which can move laterally.

Experimental systems

• GaAs heterostructures: higher mobility. 2D electron gas confined to the interface of the heterostructures because of the band offset.

Experimental results

• RH: xy

• R: xx

• Integer vs fractional QHE.

Experiment was done under a high magnetic field

• The energy of the 2D electrons are quantized under a large magnetic field. The density of states is illustrated on the right. There are gaps between the Landau levels.

Topics to be covered:

• Physics of MOSFET’s• Landau levels• Transport. (We address this first.)

Relationship between conductivity and resistivity

• Ji=jijEj; Ei=jijJj.

• xx=yy/ [xxyy-xy2]. When i

=0 in

between the Landau levels, ii=0 also!

• xy=-xy/ [xxyy-xy2] remains finite even

when ii=0.

Conductivity

,= 0% dv eiu

[j(u),j(0)]>/in0e2,/m

Hall conductivity

x,y= 0% dt ei t [ a|jx(t)|b><b|jy(0)|a>-<a| jy(0)|

b><b|jx(t )|a>] [fa-fb] /

a|jx(t)|b>=<a|eitHjxe-itH|b> =<a|eitEajxe-itEb|b> = eit(Ea-Eb) <a|jx|b>

x,y= 0% dt ei t [eit(Ea-Eb) a|jx|b><b|jy|a>- eit(Eb-Ea)

<a| jy|b><b|jx|a>] [fa-fb] /

x,y=i [ a|jx|b><b|jy|a>/(+ Ea-Eb) - <a| jy|b><b|jx|a>/(+ Eb-Ea)] [fa-fb] /

Hall conductivity

• Zero frequency limit, L’Hopital’s rule, differentiate numerator and denominator with respect to , get

x,y=i [ a|jx|b><b|jy|a> - <a| jy|b><b|jx|a>] [fa-fb] /( Ea-Eb)2

Topological consideration

• J= i ki/m (=1, e=1), H=i ki2/2m+V(r );

• Jx= H/ kx

x,y=i dk [ a| H/ kx |b><b| H/ ky |a> - <a| H/ ky|b><b| H/ kx|a>] [fa-fb]/ /( Ea-Eb)2

Perturbation theory: |a> =j |j><j| H|a> /(Ej-Ea) ; for a change in wave vector k, H= k( H/ k). Hence |a>/ kx =j |j><j| H/ kx|a>/(Ej-Ea);

Hall conductivity

x,y=i dkdr [ ( a( r)/ kx)( a(r)/

ky ) - ( a *(r)| / ky)( a (r)/ kx ) ] f(a). The above contain contributions with both |a> and |b> occupied but those contributions cancel out.

• From Stokes’s theorem, the volume integral in k can be converted to a surface integral:

Hall conductivity• Stokes: d

2 k k x g = s d k .g . Consider g = * k .

x,y=i dr sdk . k( r) k(r)/ k . The

surface integral is over the perimeter of the Brillouin zone.

• This expression is also called the Berry phase in previous textbook.

• Let =u exp(i). Then =[ u+u i ] ei. Now dr * = dr u2 =1. Hence dr uk u=0. dr *k = dr u2 ik.

Topological Invariant

• In general (r+a)=exp(ika)(r). At the zone boundary, Ga=. Exp(iGa)=-1 is real. At the zone boundary, the phase is not a function of r. x,y=i

, dk . dr u2 ik/

k =- dk . k/ k =2 n. • Crucial issues are that n need not be

zero; the electrons are not localized.

Berry phase: For H as functions of parameters R

• Substitute (3) into (1). LHS =E. RHS=(E-t +i<n|R[n(R)]> t R).

• We thus get -t +i<n|R[n(R)]> t R=0.

x,y=i dk .[ drk( r) k(r)/ k.]

The quantity in the square bracket corresponds to a Berry phase. k is the parameter is this case.