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ex09 - Solid State Theory Exercise 9 FS 11 Prof M Sigrist...

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Solid State Theory Exercise 9 FS 11 Prof. M. Sigrist Corbino disk and the integer quantum Hall effect Exercise 9.1 The lowest Landau level in the Corbino geometry The Hamilton operator for an electron ( e < 0) restricted to the plane z = 0 and exposed to a magnetic field is given by H = 1 2 m * p - e c A 2 + U ( r ) (1) where r 2 = x 2 + y 2 . The annular potential U ( r ) = C 1 r 2 + C 2 r 2 + C 3 (2) with C 1 , C 2 > 0 yields a Corbino 1 geometry confining the electron on a two-dimensional ring. The constant term C 3 only leads to a shift of the energy which is why we neglect it in the following. Let the magnetic field B be homogeneous and directed along the z -axis for r > 0. In addition, a magnetic flux Φ = ν Φ 0 through the origin ( r = 0) that does not physically touch the electron is assumed: B = [ B + ν Φ 0 δ ( r )] e z , (3) with B, ν > 0. Φ 0 = hc/ | e | = 2 π ~ c/ | e | is the magnetic flux quantum. a) Show that the vector potential can be chosen in the symmetric gauge A = 1 2 B + ν Φ 0 πr 2 ( x e y - y e x ) . (4) b) Let us now solve this single-particle problem in the symmetric gauge. To begin with
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