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Pick four out of six problems below. Students who try to solve all six problems will be given extra credit.

Problem 1

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Problem 2

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Problem 3

{\bf Classical point contact conductance:} The Sharvin formula for the electrical conductance of an extremely short contact area $A$ between two pieces of metal is given by

$G=\frac{2e^2}{h}\frac{k_F^{2}A}{4\pi }$

where $k_F$ is the Fermi wavelength. Derive the Sharvin formula by considering the total current flowing through a hole of area $A$ in a thin insulating barrier separating two free electron gases with different Fermi energies---the gas on the left has Fermi energy $\varepsilon_F^0 + eV$, while the gas on the right has Fermi energy $\varepsilon_F^0$, where $V$ is the applied voltage bias. Use purely macroscopic arguments.

In a free electron gas, the number of electrons with energies between $E$ and $E+dE$ traveling at an angle between $\theta$ and $\theta +d\theta$ with respect to a given axis is

$\frac{{\partial }^{2}n}{\partial E\partial \theta }dEd\theta =\frac{g(E)}{2}\sin \theta d\theta dE$,

where $g(E)$ is the density of states in three dimensions.

Problem 4

{\bf Quantum point contact conductance:} When the size of the contact from Problem 3 becomes comparable to Fermi wavelength ${\lambda }_F=2\pi /k_F$, the contact enters the quantum regime where its conductance follows from the Landauer formula:

$G=\frac{2e^2}{h}N$

where $N$ is the number of "conducting channels" assumed to have perfect transmission $T=1$ in ballistic transport. Find resistance in Ohm of such contact modeled by a two-dimensional wire (joining the macroscopic reservoirs) in the form of a strip of width $W = 1.75 \lambda_F$. Assume that conduction electrons in the wire can be described by the free-particle Schr\" odinger equation with Dirichlet [i.e., $\Psi({\bf r})=0$] boundary conditions along the lateral edges of the strip.