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Second Exam Physics 555 Introduction to Solid State Physics You may use your textbook, a calculator and a ruler during the exam but no other material/notes are allowed. Name: SSN: , March-31-2004, 11:10 AM till 12:25 PM (If students need more time, please be flexible and give them more time). Note: Show all steps and identify all symbols that you use. Write in ink on both sides of this paper and extra papers if necessary. Choose four problems out of the total of six problems. 1. (25 points) The Free and Independent Electron Gas in Two dimensions i) What is the relation between n and kF in two dimensions? (5 points) 3 1/ 3 j) What is the relation kF and rs in two dimensions, where rs [ = ( ) )]is the radius of 4π n the free electron sphere? (5 points) k) Prove that in two dimensions the free electron density of state D (ε ) is a constant independent of ε for ε >0, and 0 for ε <0. What is the constant? (7 points) l) Show that because D(ε ) is constant, every term in the Sommerfeld expansion for n vanishes except the T=0 term. Deduce that µ = ε F at any temperature (8 points). 2. (25 points) Try the jigsaw puzzle task of fitting the second through fifth zones of a hexagonal lattice into the hexagonal size and shape of the reduced zone. 3. (25 points) Consider a set of energy bands given by Ek = ± = 2 k 2 ∆ / m * + ∆ 2 with all the positive-energy states empty, and all negative-energy states full. a. At time zero add an electron with k x = k0 , k y = k z = 0, and a field ε x = ε y = 0; ε z = ε . Obtain the current as a function of time. Note particularly the limit as t goes to infinity. (Do exactly rather than by making the band parabolic). b. At time zero empty instead a state in the lower band at the same k as in part a, and obtain the current as a function of time with the same field applied. 4. (25 points) Consider a one-dimensional crystal, for which energy varies with wavevector in accordance with ε = ε1 + (ε 2 − ε1 ) sin 2 ( ak x / 2) Populating this band with a single electron, which is not scattered. Discuss the behavior of the effective mass, the electron velocity, and the electron position in real space under the influence of a time-independent electric field, assuming that Bragg reflection occurs at the zone boundaries as expected by Houston. If a = 1 angstrom, then for how long must a field of 100 V/m be applied to make this electron execute one complete oscillation in space? If the band is 1eV wide, what is the range of distance covered in this oscillation? 5. (25 points) Consider the problem of an extremal cyclotron orbit in the k x k y plane of the Fermi surface for a non-isotropic solid. The equation for the Fermi energy is 2 2 ε = (α k x + β k y ) . Apply a magnetic induction Bz, and describe the rate at which k changes during an orbit when α and β are unequal. Describe the orbit performed in real space for this situation. 6. (25 points) Consider a one-dimensional crystal with primitive lattice translation a, and let the energy of an electron be given by 1 =2 7 − cos ka + cos 2ka 2 ma 8 8 Determine the effective mass at the bottom of the band and at the top from quadratic expansion of E in the departure of k from these points. Determine an effective inertial mass for electrons by applying a force F and writing dv 1 = F dt mi 1 Determine an effective velocity mass such that v = =k (The velocity mass enters mv expressions for the conductivity). Note that the number of states per unit length of crystal in the wavenumber interval dk is dk/2π, Write the density of states per unit energy in terms of k and a densityof-states mass, md. (The formula should give the correct value for free electrons in one dimension if md is taken equal to m). E (k ) = a. b. c. d.