Homework Set 1

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Problem 1

Consider electrons in a toy model of 1D nanowire modeled on a discrete lattice of 100 points which are spaced by Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a=0.2 } nm. Hard wall boundary conditions are modeling edges of the wire. Write Python script that constructs the Hamiltonian matrix Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{H} } of the dot and the corresponding equilibrium density matrix Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\rho}_\mathrm{eq} = f(\mathbf{H} - \mu\mathbf{I}) } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu=0.25 } eV.
(b) Using the diagonal elements of the equilibrium density matrix Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\rho}_\mathrm{eq} } , compute electron density within the dot and make a plot Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n(x) } vs. Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x } at room temperature Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_B T =0.025 } eV, as well as at ten times lower temperature Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_B T =0.0025 } eV. Explain the difference in Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n(x) } as the temperature is increased (at Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x=50} ), which can be modeled by a large on-site repulsive potential Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_{50}=2 } eV in your Hamiltonian, as well as two additional impurities at positions positions Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x=25} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x=75} . Recompute the charge density at two different temperatures used in (b), while exploring different potentials of two "side" impurities.

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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V=0} for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |z| < d/2} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V=V_0} for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |z| > d/2} .

(a) What is the density of states (DOS) as a function of energy for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_0 \rightarrow \infty} ? Discuss what happens at low energies and how DOS behaves in the limit of high energies.
(b) Assume Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_0 \rightarrow \infty} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d = 100 \AA} . Up to what temperature Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T } can we consider the electrons to be two-dimensional? (HINT: The electrons will behave two-dimensionally if Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_BT} is less then the difference between the ground and first excited energy levels in the confining potential along the Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z} -axis.)
(c) In real systems we can only produce a finite potential well. This puts a lower limit on the 2DEG thickness Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_0=100} meV and reach a temperature of 20 mK, what is the range of thicknesses Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_1 } and 75% in the plane wave with the wave vector Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_2 } along the Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x } -axis. This type of state is described by by the density matrix:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\rho} = \frac{1}{4} |k_1\rangle \langle k_1| + \frac{3}{4} |k_2\rangle \langle k_2| }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \langle x|k \rangle=e^{ikx}/\sqrt{L} } assuming wire of length Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L } with periodic boundary conditions.

Using the current density operator derived in the class

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{j}(x) = \frac{e}{2m}(|x\rangle \langle x|\hat{p} +\hat{p}|x\rangle \langle x|) }

find its expectation value Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle j(x) } ? Note that Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{p}|k_{1,2} \rangle = \hbar k_{1,2} |k_{1,2}\rangle } .

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:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{H} = \frac{\hat{p}_x^2 + \hat{p}_y^2}{2 m^*} + \frac{\alpha}{\hbar} \left( \hat{p}_y \hat{\sigma}_x - \hat{p}_x \hat{\sigma}_y \right), \ (1) }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \alpha } measures the strength of the Rashba coupling. Here Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\hat{p}_x,\hat{p}_y) } is the two-dimensional momentum operator and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\boldsymbol{\sigma}} = (\hat{\sigma}_x,\hat{\sigma}_y,\hat{\sigma}_z) } is the vector of the Pauli matrices.

(a) Find the expression for the velocity operator Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{v} } in Rashba 2DEG.
(b) Using your result in (a), construct the expressions for the charge current density operator, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbf{j}}(\mathbf{r}) } .
(c) Using your result in (a), construct the expressions for the spin current density operator, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{j}^{S_\beta}_\alpha } .
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