Temp

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 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} .

Problem 2

Consider a metallic quantum dot containing 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 } 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 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 \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 is in 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 temperatures 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 level in the confining potential).

(c) In real systems we can only produce a finite potential well. This puts a lower limit on 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 feasible for the study of such two-dimensional electron gas?