Homework Set 1

Problem 1

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 $U_{50}=2$ eV in your Hamiltonian. Recompute the charge density at two different temperatures used in (b).

Problem 2

The dimensionality of a quantum system can be effectively reduced by confining its particles 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?

Problem 3

Electrons in an one-dimensional nanowire patterned within 2DEG are found to be in the mixed state, which is 25% plane wave with wave vector $k_{1}$ and 75% in the plane wave with the wave vector $k_{2}$ along the $x$-axis. This type of state is described by by the density matrix:

${\hat {\rho }}={\frac {1}{4}}|k_{1}\rangle \langle k_{1}|+{\frac {3}{4}}|k_{2}\rangle \langle k_{2}|$

where $\langle x|k\rangle =e^{ikx}/{\sqrt {L}}$ assuming wire of length $L$ with periodic boundary conditions.

Using the current density operator derived in the class:

${\hat {j}}(x)={\frac {e}{2m}}(|x\rangle \langle x|{\hat {p}}+{\hat {p}}|x\rangle \langle x|)$

find its expectation value $j(x)=\mathrm {Tr} [{\hat {j}}(x)\cdot {\hat {\rho }}]=\sum _{k}\langle k|{\hat {j}}(x)\cdot {\hat {\rho }}|k\rangle$ that can be measured. What is the spatial dependence of $j(x)$?

Problem 4

The 2DEG in semiconductor heterostructures with structural inversion asymmetry in the growth direction (perpendicular to the 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 effects in solids (enhanced, when compared to corrections in vacuum, by the band structure effects).

One of the important relativistic effects for 2DEGs is the linear-in-momentum Rashba SOC 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 current density operator, ${\hat {\mathbf {j} }}(\mathbf {r} )$.

(c) Using your result in (a), construct the expressions for the spin current density operator, ${\hat {j}}_{\alpha }^{S_{\beta }}$.