Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Final 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: , May-4-2005, 10:15 AM till 12:15 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) Show that the geometric structure factor of the F. C. C. lattice is zero for (100) and (110) reflections, but non-zero for the (111) reflection. Show that the reciprocal lattice of simple cubic is also simple cubic. What is the angle between G100 and G111? To what plane of the direct lattice is G100 × G111 perpendicular? 2. (25 points) Consider an electron moving in a one-dimensional solid with the periodic 2π 4π potential V ( x ) = V0 cos x + V1 cos x. The lattice constant is “a”. Assume that V0 and a a V1 are small so that the nearly-free-electron model is valid. Plot schematically the band structure showing all gaps. What are the magnitudes of the gaps? 3. (25 points) The figure below give the phonon spectrum for a certain solid. The zoneboundary wavevector are 0.88 Å-1 along [1,0,0] etc. a) What is the approximate velocity of ordinary longitudinal sound in this crystal? b) How many atoms are there per unit cell? c) Suppose that this hypothetical compound shows superconductivity at Tc = 100 K. Argue why or why not this superconductivity could be plausibly caused by electronphonon coupling within the BCS theory. 0.77 Å-1 0.88 Å-1 4. (25 points) Consider the two-dimensional triangular lattice with the nearest-neighbor tightbinding Hamiltonian, H = ∑ α n n + ∑ β { n n ' + n ' n } n,n’ are nearest neighbors n {nn '} and “a” is the lattice constant. Obtain the energy dispersion relation for the electron states. 5. (25 points) Consider an interface between an insulator and a semiconductor, as in a metaloxide-semiconductor transistor or MOSFET. With a strong electric field applied across the interface, the potential energy of a conduction electron may be approximated by V ( x ) = eE x for x > 0 and +∞ for x ≤ 0 . The wave function Ψ ( x ) = 0 for x < 0 and may be written as Ψ ( x ) = u ( x ) exp(ik y y + ik z z ). u( x ) satisfies the differential equation = 2 d 2u + V ( x )u( x ) = ε u( x ). With the model potential for V ( x ) , the exact eigenfunctions 2m dx 2 are Airy functions, but we can find a fairly good ground state energy from the variational trial function, u ( x ) = x exp( − ax ) for x > 0 where a is the variational parameter. With this trial wave function obtain the ground state energy as a function of E. − 6. (25 points) Imagine a band for a simple cubic structure (with cubic edge a) given by Ek = − E0 (cos k x a + cos k y a + cos k z a ) . Let an electron at rest (k=0) at t = 0 feel a uniform electric field E, constant in time. a) Find the trajectory in real space. This can be specified by giving x (t ), y (t ), and z (t ). b) Sketch the trajectory for E in a [120] direction.