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| == Problem 1 ==
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| 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 <math>x=25</math> and <math>x=75</math> of the same potential as the one placed at site <math>x=50</math> in the textbook.
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| == Problem 2 ==
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| Consider a metallic quantum dot containing <math> N </math> electrons. Find the energy of the ground state ("ground state" means at zero temperature <math>T=0</math>) of the dot as <math>N</math> 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 <math> \Psi(\mathbf{r}) = \frac{1}{\sqrt{V}} e^{i \mathbf{k} \cdot \mathbf{r}}</math> are subjected to periodic boundary conditions, so that their wave vector is
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| <math>\mathbf{k} = \frac{2\pi}{L}(n_x,n_y,n_z); \ n_x,n_y,n_z=0,\pm 1, \pm 2, ... </math>
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| and the corresponding single particle energy levels are given by:
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| <math> E(\mathbf{k}) = \frac{\hbar^2 \mathbf{k}^2}{2m^*} </math>.
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| == Problem 3 ==
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| 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.
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| 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 <math>V=0</math> for <math> |z| < d/2</math> and <math> V=V_0</math> for <math>|z| > d/2</math>.
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| :(a) What is the density of states (DOS) as a function of energy for <math>V_0 \rightarrow \infty</math>? Discuss what happens at low energies and how DOS behaves in the limit of high energies.
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| :(b) Assume <math>V_0 \rightarrow \infty</math> and <math> d = 100 \AA</math>. Up to what temperature <math> T </math> can we consider the electrons to be two-dimensional? (HINT: The electrons will behave two-dimensionally if <math>k_BT</math> is less then the difference between the ground and first excited energy levels in the confining potential along the <math>z</math>-axis.)
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| :(c) In real systems we can only produce a finite potential well. This puts a lower limit on the 2DEG thickness <math> d </math> 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 <math>V_0=100</math> meV and reach a temperature of 20 mK, what is the range of thicknesses <math> d </math> feasible for the study of such two-dimensional electron gas?
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| REFERENCE: Datta's textbook, pages 138-140.
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| == Problem 4 ==
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| 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:
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| <math> \hat{\rho}_1 = \frac{|\uparrow \rangle \langle \uparrow| + |\downarrow \rangle \langle \downarrow|}{2} </math>,
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| while the spins comprising the current in the other device are described by the density matrix
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| <math> \hat{\rho}_2 = |u \rangle \langle u|</math> , where <math> \ |u\rangle = \frac{e^{i\alpha} |\uparrow\rangle + e^{i\beta}|\downarrow\rangle}{\sqrt{2}}</math>.
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| Here <math> |\uparrow\rangle </math> and <math> |\downarrow\rangle </math> are the eigenstates of the Pauli spin matrix <math> \hat{\sigma}_z </math>:
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| <math> \hat{\sigma}_z |\uparrow \rangle = +1 |\uparrow \rangle, \ \hat{\sigma}_z |\downarrow \rangle = -1 |\downarrow \rangle </math>.
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| 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 <math> \langle \sigma_{x,y,z}\rangle =\mathrm{Tr}\, [\hat{\rho} \hat{\sigma}_{x,y,z}] </math>.)
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