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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>.
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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 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 \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 is in 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 temperatures 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 level in the confining potential).
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| :(c) In real systems we can only produce a finite potential well. This puts a lower limit on <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 feasible for the study of such two-dimensional electron gas?
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