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Homework07 - each well is U and its width a/5 a Calculate...

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Homework # 7 (due Tuesday, 29 March) 1. Consider the free electron energy bands of an fcc crystal lattice in the reduced zone scheme in which all k 's are transformed to lie in the first Brillouin zone. Plot roughly in the [111] direction the energies of all bands up to six times the lowest band energy at the zone boundary at k = (2 π /a)(½,½,½). Explain what happens with these bands in the presence of a weak crystal potential. 2. Suppose that the crystal potential in a one-dimensional lattice of lattice constant a is composed of a series of rectangular wells which surround the atom. Suppose that the depth of
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Unformatted text preview: each well is U and its width a /5. a. Calculate the values of the first three energy gaps. Compare the magnitudes of these gaps. b. Evaluate these gaps for the case of U = 5 eV and a = 4Å. 3. Using the solution for the energy bands near the zone boundary in the presence of a weak crystal potential show that the electron velocity is parallel to the Bragg plane. Since the gradient is perpendicular to the surfaces on which a function is constant, this fact allows us to conclude that the constant energy surfaces (such as the Fermi surface) at the Bragg plane are perpendicular to that plane....
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