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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

The two-dimensional electron gas (2DEG) in semiconductor heterostructures with structural inversion asymmetry in the growth direction (perpendicular to the 2DEG plane) plays and essential role in "spintronics without magnetism" since the spin of an electron in nanostructures made of such 2DEGs can be controlled by electrical fields (which can be controlled on much smaller spatial and temporal scales than traditional cumbersome magnetic fields). Such control is made possible by the spin-orbit (SO) couplings which represent manifestations of relativistic quantum mechanics in solids (enhanced, when compared to corrections in vacuum, by the band structure effects).

The important SO coupling for 2DEGs are the linear Rashba and Dresselhaus ones, encoded by the following effective mass Hamiltonian:

${\hat {H}}={\frac {{\hat {p}}_{x}^{2}+{\hat {p}}_{y}^{2}}{2m^{*}}}+{\frac {\alpha }{\hbar }}\left({\hat {p}}_{y}{\hat {\sigma }}_{x}-{\hat {p}}_{x}{\hat {\sigma }}_{y}\right)+{\frac {\beta }{\hbar }}\left({\hat {p}}_{x}{\hat {\sigma }}_{x}-{\hat {p}}_{y}{\hat {\sigma }}_{y}\right)$,

where $\alpha$ measures the strength of the Rashba coupling and $\beta$ measures the strength of the Dresselhaus coupling.

a)

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 Schrodinger equation with the Dirichlet [i.e., $\Psi (\mathbf {r} )=0$] boundary conditions along the lateral edges of the strip.