Department of physics, NTNU TFY4340 Mesoscopic Physics Spring 2010

Exercise 9

Assistance: Tuesday April 20.

Consider the 4–terminal device shown in the figure, with ideal contacts, and with a geometrical constriction of width w.

We assume that the perpendicular magnetic field B is strong, so that l_{B} = \sqrt{h̄/eB} ≪ W. Here, W is the typical channel width, except at the constriction. However, B is not sufficiently strong to create edge states in the constriction. In other words, we have w < l_{B}.

a) The Fermi energy is E_{F}. We assume that the number of edge states N is so large that we may ignore the zero point energy h̄ω_{c}/2. Write down an expression for N. (Outside the constriction, of course.)

b) Suppose n states (n < N) may transmit through the constriction. Ignore tunneling and back–scattering from impurities. Use the Büttiker–Landauer equations to find the longitudinal resistance R_{L} = R_{13,24} (in terms of N and n), i.e., with current from 1 to 3 and the voltage measured between 2 and 4. (Set e.g. V_{3} = 0.)

c) How will R_{L} depend on B. Compare with figure 50 in the ”book” by Beenakker and van Houten, and/or the paper in Phys Rev B 37, 8534 (1988).

d) Use the results of b) to find R_{2t} = R_{13,13}. Is this result as expected?