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 ${\hat {\boldsymbol {\sigma }}}=({\hat {\sigma }}_{x},{\hat {\sigma }}_{y},{\hat {\sigma }}_{z})$ 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 the expressions for the charge ${\hat {\mathbf {j} }}(\mathbf {r} )$ and spin ${\hat {j}}_{\alpha }^{S_{\beta }}$ current operators.

Problem 2

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 3

An experimentalist has fabricated a thin film of silver, which is $10^{5}$ nm wide and $10^{5}$ nm long along the $x$ and $y$ axis, respectively, and has thickness of $0.41$ nm in the $z$-direction. In order to contact the film with electrodes of external circuit, one has to know its Fermi energy and the Fermi energy of electrodes in order to avoid too much charge transfer and the ensuing contact resistance.

(a) By using the fact that the density of electrons in bulk silver is $n=5.86\cdot 10^{22}\ \mathrm {electrons/cm^{3}}$, 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 4

Consider electrons in a toy model of 1D quantum dot modeled on a discrete lattice of 100 points which are spaced by $a=0.2$ nm. Hard wall boundary conditions are modeling edges of the dot. Write a MATLAB script that constructs the Hamiltonian matrix $\mathbf {H}$ of the dot and the corresponding equilibrium density matrix ${\boldsymbol {\rho }}=f(\mathbf {H} -\mu \mathbf {I} )$ where $f(x)$ is the Fermi-Dirac distribution function.

(a) Plot the energy eigenvalues of the dot Hamiltonian (in eV) as the function of eigenvalue number. Add horizontal line on this plot for the chemical potential $\mu =0.25$ eV.

(b) Using the diagonal elements of the equilibrium density matrix ${\boldsymbol {\rho }}$, compute electron density within the dot and make a plot $n(x)$ vs. $x$ at room temperature $k_{B}T=0.025$ eV, as well as at ten times lower temperature $k_{B}T=0.0025$ eV. Explain the difference in $n(x)$ as the temperature is increased (at $T=0$ K one would get a result - probability density for eigenfunctions in an infinite potential well - familiar from textbook quantum mechanics).

(c) Add an impurity in the center of the quantum dot (at position $x=50$), which can be modeled by a large on-site repulsive potential $U_{5}0=2$ eV in your Hamiltonian. Recompute the charge density at two different temperatures used in (b).