Homework Set 1

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 an essential role in the pursuit of "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 coupling (SOC) which represent manifestations of relativistic quantum mechanics in solids (enhanced, when compared to corrections in vacuum, by the band structure effects).

One of the important SOCs for 2DEGs is the linear Rashba one encoded by the following effective mass Hamiltonian:

${\hat {H}}={\frac {{\hat {p}}_{x}^{2}+{\hat {p}}_{y}^{2}}{2m^{*}}}+{\frac {\alpha }{\hbar }}\left({\hat {p}}_{y}{\hat {\sigma }}_{x}-{\hat {p}}_{x}{\hat {\sigma }}_{y}\right),\ (1)$

where $\alpha$ measures the strength of the Rashba coupling. Here $({\hat {p}}_{x},{\hat {p}}_{y})$ is the two-dimensional momentum operator and ${\hat {\mathbf {\sigma } }}=({\hat {\sigma }}_{x},{\hat {\sigma }}_{y},{\hat {\sigma }}_{z})$ is the vector of Pauli spin matrices.

(a) Find the expression for the velocity operator $\mathbf {v}$ in Rashba 2DEG.

(b) Using your result in (a), construct expressions for the charge ${\hat {\mathbf {j} }}(\mathbf {r} )$ and spin ${\hat {\mathbf {j} }}$ current density operators.

Problem 2

Torque ...

Problem 3

The dimensionality of a system can be reduced by confining the electrons in certain directions. A two-dimensional electron gas (2DEG) is produced in semiconductor heterostructures and is used for the investigation of the quantum Hall effect, creation of semiconductor quantum dots, quantum point contacts, nanowires, etc.

Consider a simplified model of a 2DEG where electron gas (infinite in the x and y directions; you can assume periodic boundary conditions in these directions) is subjected to an external potential $V=0$ for $|z|<d/2$ and $V=V_{0}$ for $|z|>d/2$.

(a) What is the density of states (DOS) as a function of energy for $V_{0}\rightarrow \infty$? Discuss what happens at low energies and how DOS behaves in the limit of high energies.

(b) Assume $V_{0}\rightarrow \infty$ and $d=100\mathrm {\AA}$. Up to what temperature $T$ can we consider the electrons to be two-dimensional? (HINT: The electrons will behave two-dimensionally if $k_{B}T$ is less then the difference between the ground and first excited energy levels in the confining potential along the $z$-axis.)

(c) In real systems we can only produce a finite potential well. This puts a lower limit on the 2DEG thickness $d$ since the ground state must be a bound state in the z direction with a clear energy gap up to the first excited state. If we can produce a potential of $V_{0}=100$ meV and reach a temperature of 20 mK, what is the range of thicknesses $d$ feasible for the study of such two-dimensional electron gas?

REFERENCE: Ihn textbook Chapter 9.

Problem 4

An experimentalist has fabricated a thin film of silver, which is $10^{6}\ \mathrm {\AA} {}$ wide and $10^{6}$ \AA{ } long along the $x$ and $y$ axis, respectively, and has thickness of $4.1\$\mathrm {\AA} {}$. In order to contact film with electrodes of external circuit, one has to know its Fermi energy and the Fermi energy of electrodes.

(a) By using the fact that the density of electrons in bulk silver is $n=5.86\cdot 10^{22}\$electrons/cm\$^{3}$, find the Fermi energy of bulk silver in eV.

(b) Consider the thin film of sliver as a free Fermi gas and demand that the wave function vanishes at the boundaries along the $z$-axis. Find the difference between the energies of the lowest and highest occupied single-particle states, and compare the difference to the Fermi energy in bulk silver

HINT: In the case of thin metal film $L_x=L_y=L_r \gg L_z$, the electron motion is confined in the $z$-direction, which can be simply modeled by requiring that wave function vanishes in the boundaries along the $z$-axis. At the same time we assume that electrons are free in the $xy$-plane so that final model to which we apply the Schrodinger equation is that of a thin layer periodically repeated only in the $x$ and $y$ directions. Therefore, to solve (b) start by showing that the energy spectrum of a single electron in such thin film is:

$\varepsilon (n_{x},n_{y},n_{z})={\frac {\hbar ^{2}}{2m}}\left[\left({\frac {2\pi }{L_{r}}}\right)^{2}(n_{x}^{2}+n_{y}^{2})+\left({\frac {\pi }{L_{z}}}\right)^{2}n_{z}^{2}\right]$

where $n_{x},n_{y}=\ldots ,-3,-2,-1,0,1,2,\ldots$, while $n_{z}=1,2,\ldots$.

Problem 5

Problem E.4.2. in the textbook. In addition to reproducing panels (b)-(f), repeat calculations in panels (e) and (f) with two additional impurities at sites $x=25$ and $x=75$ of the same potential as the one placed at site $x=50$ in the textbook.