Department of physics, NTNU TFY4340 Mesoscopic Physics Spring 2010

Exercise 10

Assistance: Tuesday April 20.

Consider the single electron transistor shown in the figure, with a metallic island coupled to the outside world through two tunneling junctions (C_{1}, R_{1}) and (C_{2}, R_{2}), and an ideal capacitor C_{g} with tunable gate voltage U.

The tunneling junctions both have a large resistance compared to the quantum of resistance R_{Q} = h/e^{2}. The island has a net charge q = −ne, corresponding to n ”extra” electrons. Bias voltages ±V/2 are applied to the tunneling junctions, as illustrated in the figure.

a) Use Kirchhoff’s voltage law to show that

Q_{1} = \frac{C_{1}}{C_{Σ}} [(C_{2} + C_{g}/2) V + C_{g}U + q]
Q_{2} = \frac{C_{2}}{C_{Σ}} [(C_{1} + C_{g}/2) V − C_{g}U − q]
Q_{g} = \frac{C_{g}}{C_{Σ}} [\frac{1}{2} (C_{1} − C_{2}) V + (C_{1} + C_{2}) U − q]

with C_{Σ} ≡ C_{1} + C_{2} + C_{g}.

b) Show that the q–dependent part of the equilibrium electrostatic energy, including the work done by the charges on the surroundings (i.e., on the voltage sources) is

E_{eq} = \frac{1}{2C_{Σ}} [C_{g}U − ne + V (C_{2} + C_{g}/2)]^{2} .

c) Assume all three capacitances are equal, i.e., C_{1} = C_{2} = C_{g} = C. Sketch a ”stability diagram”, i.e., the U −V plane (or rather, the C_{g}U/e−C_{Σ}V/e plane) where you indicate regions stable and unstable with respect to tunneling of an electron. Use this diagram to explain how this ”transistor” can be used to create a DC current I = ef. Here, f is the frequency of an AC voltage U(t) = U_{0} + U_{1} cos(2πft).