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Pick four out of six problems below. Students who try to solve all six problems will be given extra credit.

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 and essential role in "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 (SO) couplings which represent manifestations of relativistic quantum mechanics in solids (enhanced, when compared to corrections in vacuum, by the band structure effects).

The important SO coupling for 2DEGs are the linear Rashba and Dresselhaus ones, 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) + \frac{\beta}{\hbar} \left(\hat{p}_x \hat{\sigma}_x - \hat{p}_y \hat{\sigma}_y \right) } ,

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 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 \beta } measures the strength of the Dresselhaus 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 (\sigma_x,\sigma_y,\sigma_z) } is the vector of Pauli spin matrices. In GaAs quantum wells the two terms are of the same order of magnitude, while the Rashba SO coupling dominates in narrow band-gap InAs-based structures [the relative strength 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/\beta } can be extracted from, e.g., photocurrent measurements, Phys. Rev. Lett. 92, 256601 (2004)].

a) Assume a toy model of 1DEG with the Rashba coupling described by the 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}^{\rm 1D}_{\rm R}(k_x)=\frac{\hbar^2k_x^2}{2m^*}-\alpha k_x\hat{\sigma}_y }

Find its eigenstates and eigenvalues as a function 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 k_x } . Using Mathematics or Matlab, plot both branches 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 E(k_x) } .

b) For the Rashba-dominated 2DEG, 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 > 0 } , 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 \beta = 0 } , find eigenstates and eigenvalues of the Hamiltonian above and use Mathematica or Matlab to plot the corresponding 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 E(k_x,k_y) } dispersion surface.

c) What is the expectation value of spin 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 \hbar \mathbf{\sigma}/2 } in the eigenstates of spin-split 1DEG in a) and spin-split 2DEG in b)?

d)

REFERENCE: B. K. Nikolic, L. P. Zarbo, and S. Souma, Spin currents in semiconductor nanostructures: A nonequilibrium Green-function approach, Chapter 24 in Volume I of "The Oxford Handbook on Nanoscience and Technology: Frontiers and Advances" (Oxford University Press, Oxford, 2010).

Problem 2

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Problem 3

Classical point contact conductance: The Sharvin formula for the electrical conductance of an extremely short contact area 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 } between two pieces of metal is given 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 G=\frac{2e^2}{h}\frac{k_F^2 A}{4\pi} }

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 k_F } is the Fermi wavelength. Derive the Sharvin formula by considering the total current flowing through a hole of area $A$ in a thin insulating barrier separating two free electron gases with different Fermi energies---the gas on the left has Fermi energy $\varepsilon_F^0 + eV$, while the gas on the right has Fermi energy $\varepsilon_F^0$, where $V$ is the applied voltage bias. Use purely macroscopic arguments.

In a free electron gas, the number of electrons with energies between 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 E } 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 E+dE } traveling at an angle between 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 \theta} 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 \theta + d\theta} with respect to a given axis is

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 \frac{\partial^2 n}{\partial E \partial \theta} dE d\theta = \frac{g(E)}{2} \sin \theta d\theta dE } ,

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 g(E) } is the density of states in three dimensions.

Problem 4

Quantum point contact conductance: When the size of the contact from Problem 3 becomes comparable to Fermi wavelength 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 \lambda_F = 2\pi/k_F } , the contact enters the quantum regime where its conductance follows from the Landauer formula:

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 G = \frac{2e^2}{h} \sum_{n=1}^N T_n = N }

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 N } is the number of "conducting channels" assumed to have perfect transmission 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_n =1 } in ballistic transport at zero temperture. Find resistance in Ohm of such contact modeled by a two-dimensional wire (joining the macroscopic reservoirs) in the form of a strip of width 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 W = 1.75 \lambda_F } . Assume that conduction electrons in the wire can be described by the free-particle Schrodinger equation with hard-wall [i.e., 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 \Psi(\mathbf{r})=0 } ] boundary conditions along the lateral edges of the strip.

Problem 5

Suppose that a quantum wire can be modeled by a 2D strip with hard-wall boundary conditions in the direction transverse to the current flow. The width of the wire is 10 nm and the effective mass of electrons is 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 m^* = 0.067m_0 } of the bare electron mass 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 m_0 } (which is the case for GaAs heterostructures).

a) Plot the conductance of such a wire as function of the Fermi energy for a number of temperatures 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 < E_F } . The Landauer formula for conductance at finite temperature is:

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 G = \frac{2e^2}{h} \sum_{n=1}^N \left(\frac{-\partial f }{\partial E} \right) T_n(E) }

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(E) } is the Fermi-Dirac distribution function 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 T_n(E) = \theta (E-E_n) } [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 \theta(x) <\math> is the step function] since in ballistic wires a channel is either open (when the Fermi energy is above the bottom of the corresponding subband) or closed (otherwise). b) Determine characteristic temperature(s) at which features of the quantization plateaux begin to disappear. == Problem 6 == {\bf Two quantum point contacts in series:} Two ballistic quantum point contacts are connected in series, as shown in the Figure below. When measured independently, the conductance of the first contact is <math> \frac{2e^2}{h} } and that of the second one is 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 4e^2 /h } . The whole structure is still ballistic. What is the conductance of the whole structure?