Temp HW 1

Problem 1

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.

Problem 2

Consider a metallic quantum dot containing $N$ electrons. Find the energy of the ground state ("ground state" means at zero 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=0} ) of the dot as 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} 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 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}) = \frac{1}{\sqrt{V}} e^{i \mathbf{k} \cdot \mathbf{r}}} are subjected to periodic boundary conditions, so that their wave vector 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 \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:

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(\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 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?

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:

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}_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

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}_2 = |u \rangle \langle u|} , 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 \ |u\rangle = \frac{e^{i\alpha} |\uparrow\rangle + e^{i\beta}|\downarrow\rangle}{\sqrt{2}}} .

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 |\uparrow\rangle } 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 |\downarrow\rangle } are the eigenstates of the Pauli spin 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{\sigma}_z } :

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{\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 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 \sigma_{x,y,z}\rangle =\mathrm{Tr}\, [\hat{\rho} \hat{\sigma}_{x,y,z}] } .)