Download Final Exam from Spring 2005

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.