1. Quantized Hall effect

2. Experimental systems MOSFET’s (metal- oxide-semiconductor- field-effect-transistor.) Two-dimensional electron gas on the “capacitor plates” which can move laterally.

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

4. Experimental results R H :  xy R:  xx Integer vs fractional QHE.

5. 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.

6. Topics to be covered: Physics of MOSFET’s Landau levels Transport. (We address this first.)

7. Relationship between conductivity  and resistivity  J i =  j  ij E j ; E i =  j  ij J j.  xx =  yy / [  xx  yy -  xy 2 ]. When   i =0 in between the Landau levels,  ii =0 also!  xy =-  xy / [  xx  yy -  xy 2 ] remains finite even when  ii =0.

8. Conductivity  ,  =  0 % dv e i  u  [  j  (u),j  (0)]>/  in 0 e 2  ,  /m 

9. Hall conductivity  x,y =  0 % dt e i  t [  a|j x (t)|b> - ] [f a -f b ] /   a|j x (t)|b>= = = e it(E a -E b )  x,y =  0 % dt e i  t [e it(E a -E b )  a|j x |b> - e it(E b -E a ) ] [f a -f b ] /   x,y =i [  a|j x |b> /(  + E a -E b ) - /(  + E b -E a )] [f a -f b ] / 

10. Hall conductivity Zero frequency limit, L’Hopital’s rule, differentiate numerator and denominator with respect to , get  x,y =i [  a|j x |b> - ] [f a - f b ] /( E a -E b ) 2

11. Topological consideration J=  i k i /m (  =1, e=1), H=  i k i 2 /2m+V(r ); J x =  H/  k x  x,y =i  dk [  a|  H/  k x |b> - ] [f a -f b ]/ /( E a - E b ) 2 Perturbation theory:  |a> =  j |j> /(E j -E a ) ; for a change in wave vector  k,  H=  k(  H/  k). Hence  |a>/  k x =  j |j> /(E j -E a );

12. Hall conductivity  x,y =i  dk  dr [ (   a  ( r)  /  k x )(   a (r)/  k y ) - (   a * (r)| /  k y )(   a (r)/  k x ) ] 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:

13. Hall conductivity Stokes:  d 2 k  k x g =  s d k.g. Consider g =  *  k .  x,y =i  dr  s  dk.  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   ] e i . Now  dr  *  =  dr u 2 =1. Hence  dr u  k u=0.  dr  *  k  =  dr u 2 i  k .

14. 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  u 2 i  k /  k =-  dk.  k /  k =2  n. Crucial issues are that n need not be zero; the electrons are not localized.

15. Berry phase: For H as functions of parameters R

16. Substitute (3) into (1). LHS =E . RHS=(E-   t  +i    t R) . We thus get -   t  +i    t R=0.  x,y =i  dk.[  dr  k  ( r)   k (r)/  k.] The quantity in the square bracket corresponds to a Berry phase. k is the parameter is this case.