HOMEWORK SET 5 (70 points) Mesoscopic Physics Lecture J. Fabian starred (*) problems are voluntary

1. (30 points) QPC in a transverse magnetic field. Consider a quantum point contact in the presence of an external magnetic field, B, applied transverse to the plane of the 2d electron gas:

We have seen in the lecture that, for B = 0, the conductance decreases in steps as a function of the increasing magnitude of the (negative) gate voltage which squeezes the constriction region. What would you expect to happen for B 6= 0. Would the steps survive? If yes, would they be wider or narrower compared to B = 0? Would they have the same height? Are the modes still doubly degenerate? After you persuade yourself about the answer, consult B. J. van Wees,“Quantized conductance of magnetoelectric subbands in ballistic point contacts,” Phys. Rev. B 38, 3625 (1988), for an experimental result.

2. (20 points) Repeat the calculation of the current in a mesoscopic system containing a scatterer, given in the lecture, now looking at the current in the left lead close to the left contact.

3. (10 points) Chemical potential profile in a mesoscopic system. Sketch the profile of the average chemical potential for a mesoscopic two-contact and two-lead system containing a scatterer in the middle with the average scattering probability T per mode. Considering that the electrostatic potential can have no discontinuities (it has to satisfy the Poisson equation which has a second derivative) discuss the carrier distribution around the scatterer. Is there charge accumulation or depletion?

4. (10 points) Two QPCs in series. Suppose we have two quantum point contacts in series:

Assume that the quantum point contacts, QPC1 and QPC2, are identical and that there is no scattering in the region between the contacts. What will be the conductance of the system? Is Ohm’s law valid? What do you think would happen if the contacts were not identical?

5.∗ Read about nonlinear current versus voltage effects at large applied biases, in L. P. Kouwenhoven, et al., “Nonlinear conductance of quantum point contacts,” Phys. Rev. B 39, 8040 (1989). 6.∗ Phonon waveguides. Conductance quanta are not limited to electrical current. Consider a mesoscopic system (made of undoped silicon, for example) in which the thermal transport is carried by phonons. The two contacts are held at different temperatures, TL and TR:

As a result of the temperature gradient, the thermal current will flow, due to transmission of phonons in the lead. Assume an ideal lead.

(a) Convince yourself that the thermal current, which is the current of energy, is 1 X j = ¯hω v n , (1) th L k k k k where ωk is the frequency of the phonon of the wave number k, vk is the phonon’s group velocity, and nk is the phonon distribution function. We assume that only one phonon branch is present, for simplicity.

(b) Show that if the phonon transport is coherent, the thermal current becomes Z ∞ 1 0 0 jth = dk¯hωkvk[nL(ωk) − nR(ωk)], (2) 2π 0 where

0 1 nL,R(ωk) = , (3) exp(¯hωk/kBTL,R) − 1 is the Bose-Einstein distribution function for phonons in contacts L and R.

(c) For a small difference in temperature, ∆T = TL − TR ¿ TL,R, show that the thermal conductance κ, defined as j κ = th , (4) ∆T becomes 2 Z ∞ x kBT 2 e κ = dxx x 2 . (5) h x0 (e − 1)

Here the variable x = ¯hω/kBT , and x0 =hω ¯ 0/kBT , with ω0 the minimum phonon frequency in the lead (compare with the confinement energy for electron transverse modes).

(d) As T → 0, the lower limit x0 → ∞, for any minimum frequency ω0 > 0. There are, however, phonon modes of zero frequency. These come from translational and rotational invariance. Translating or rotating a wire along its axis, for example, costs no energy. These modes are also called Goldstone modes, since they appear as a result of the symmetry breaking (by placing the wire somewhere in space, one breaks the translational invariance by choosing a specific position). Which zero modes actually propagate through the lead depend on the geometry and coupling of the lead to the contacts. Show, by putting x0 = 0, that each zero frequency mode contributes one quantum of thermal conductance, π2k2 T κ = B (6) 3h For an experiment detecting this quantum see K. Schwab et al, “Measurement of the quantum of thermal conductance,” Nature, 404, 974 (2000).