Quantized Hall effect
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Transcript of Quantized Hall effect
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.