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

Consider a metallic quantum dot containing $N$ electrons. Find the energy of the ground state ("ground state" means at zero temperature $T=0$) of the dot as $N$ varies from 1 through 15 (that is, find ground state energy for dot charged with 1 electron, 2 electrons, ...). Assume that electrons within the dot are free particles whose eigenfunctions $\Psi (\mathbf {r} )={\frac {1}{\sqrt {V}}}e^{i\mathbf {k} \cdot \mathbf {r} }$ are subjected to periodic boundary conditions, so that their wave vector is

$\mathbf {k} ={\frac {2\pi }{L}}(n_{x},n_{y},n_{z});\ n_{x},n_{y},n_{z}=0,\pm 1,\pm 2,...$

and the corresponding single particle energy levels are given by:

$E(\mathbf {k} )={\frac {\hbar ^{2}\mathbf {k} ^{2}}{2m^{*}}}$.

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: Datta's textbook, pages 138-140.

Problem 4

A researcher in spintronics is investigated two devices in order to generate spin-polarized currents. One of those devices has spins comprising the current described by the density matrix:

${\hat {\rho }}_{1}={\frac {|\uparrow \rangle \langle \uparrow |+|\downarrow \rangle \langle \downarrow |}{2}}$,

while the spins comprising the current in the other device are described by the density matrix

${\hat {\rho }}_{2}=|u\rangle \langle u|$ , where $\ |u\rangle ={\frac {e^{i\alpha }|\uparrow \rangle +e^{i\beta }|\downarrow \rangle }{\sqrt {2}}}$.

Here $|\uparrow \rangle$ and $|\downarrow \rangle$ are the eigenstates of the Pauli spin matrix ${\hat {\sigma }}_{z}$:

${\hat {\sigma }}_{z}|\uparrow \rangle =+1|\uparrow \rangle ,\ {\hat {\sigma }}_{z}|\downarrow \rangle =-1|\downarrow \rangle$.

What is the spin polarization of these two currents? Comment on the physical meaning of the difference between the spin state transported by two currents. (HINT: Compute the x, y, and z components of spin using both of these density matrices to evaluate the quantum-mechanical definition of an average value $\langle \sigma _{x,y,z}\rangle =\mathrm {Tr} \,[{\hat {\rho }}{\hat {\sigma }}_{x,y,z}]$.)

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.