Introduction to Solid State Physics — Spring 2014

Introduction to Solid State Physics — Spring 2014
Satisfactory homework solutions must include complete development of mathematical aspects, illustrative diagrams, and brief explanations of the reasoning that guides your method of solution.
Your solutions should be neatly organized and flow logically down the page.
Homework grading scale 1-6:
1 = something relevant is started?
2 = some evidence that an actual attempt at the problem was made.
3 = solution has major flaws or is so difficult to follow that a reader can’t tell.
4 = possibly or even probably correct, but little effort made to make the solution logical and coherent for another person to read and understand, or may have significant flaws or omissions.
5 = correct solution, clear effort made to explain reasoning but could be improved by more attention to organization and presentation; or may have a minor flaw, but clearly presented so error is
easily detected.
6 = correct and convincingly presented solution, with reasoning explained and well organized.
For Monday, February 3:
(1) The density of an element is ρ. It forms an FCC lattice, and has molar mass of M . (a) Find
an expression for the length of the side on the conventional (cubic) unit cell, a, in terms of ρ, M ,
and whatever other constants are needed. (b) For copper ρ = 8.96 g/cm3 and M = 63.4 g/mole.
Evaluate your expression to find a for copper. Give your result in nanometers. (c) What is the
nearest neigbor distance for copper?
(2) Sidebottom 1.4
(3) Sidebottom 1.5
(4) The volume of a parallelpiped defined by three vectors is V = |~a1 · (~a2 × ~a3 )|. Use the 3 (corrected) primitive vectors from Sidebottom fig 1.9 to find the volume of a primitive unit cell for the
body-centered lattice. (b) Explain how to get the same result without quite so much algebra by
considering the conventional (cubic) unit cell.
(5) In the diamond lattice, a C atom at (0,0,0) has 4 nearest neighbors. (a) What are the coordinates (in terms of the conventional cubic cell lattice constant, a) of these nearest neighbors? (b)
Compute via the dot product the angle between the lines connecting the (0,0,0) atom and any two
of its nearest neighbors - i.e. find the tetrahedral angle.
(6) Sidebottom 1.9. (For part (a) identify which of the 14 3-D Bravais lattice families this belongs
to.)
For Monday, February 10:
(7) Sidebottom 2.1
(8) Sidebottom 2.2
(9) Sidebottom 2.4
(10) Using the program “bravais” in Simulations for Solid State Physics (hereafter sss) work through
exercises 2.3, and 2.5. See http://www.d.umn.edu/∼jmaps/phys5531/ for info on sss.
For Monday, February 17:
(11) Sidebottom 3.1
(12) Sidebottom 3.4
(13) Sidebottom 3.6
(14) Compute the sums used in the model for van der Waals or molecular crystals
N
X
j6=i
1
pij
!12
and
N
X
j6=i
1
pij
!6
accurate to 4 significant figures for the BCC lattice.
(a) First develop an an expression for the distance or (distance)2 between two lattice points, the
first serving as an origin and the second being another arbitrary lattice point. Make clear what
you use for primitive vectors for the lattice and how you arrvive at your expression for distance.
(b) Use that expression to evaluate the sums. You can do this by hand (not recommended), or any
convenient programming environment - Mathematica, Matlab, C/C++, Python, etc. Include any
programming code used to do the sums. Explain how your code works.
(c) Approximately what minimum number of lattice points must be included to achieve this accuracy? Describe what you did to determine this. What’s the approximate radius of space you’ve
summed over, in terms of the conventional unit cell lattice constant?
Sidebottom problem 3.4 contains the expected values for the sums.
For Monday, February 24:
(15) Sidebottom 5.3
(16) Sidebottom 5.4
(17) Sidebottom 5.7
(18) Exercises 2.9, 2.10 using bravais in sss.
(19) Shade in examples of the specified planes as in the example on the left. Show two adjacent
(111) planes. Find the (perpendicular) distance (in terms of the cubic lattice constant a) between
your (111) planes.
For Monday, March 3:
(20) Sidebottom 6.3
(21) Sidebottom 6.5
(22) Sidebottom 6.6
(23) Sidebottom 6.8
(24) Do sss exercises 2.12, 13, 17 in bravais for practice; and do 2.18 and 2.24 using bravais to
hand in. Note that for 2.24 sss includes the atomic form factors as part of its “structure factor,” so
S(∆~k), in eqns. 2.4, 2.5, is what Sidebottom identifies as fcell (~q) (Sidebottom eqn. 6.1). Exercise
2.24 is interested in extinctions arising from fcell . (∆~k in sss corresponds to Sidebottom’s ~q.)
For Monday, March 24:
(25) Sidebottom 10.1
(26) Sidebottom 10.2
(27) Sidebottom 10.6
(28) Do sss 4.2, 4.5, 4.10 using born to hand in.
For Monday, March 31:
(29) Sidebottom 11.1
(30) Sidebottom 11.5
(31) Apply the Debye approximation to a 2D monatomic crystal lattice with a speed of sound
vo . The density of states from 11.1 will be useful. (a) What is the Debye frequency and Debye
temperature? (b) Show that the specific heat is proportional to T 2 at low temperatures and agrees
with a 2D version of the Dulong-Petit result at high temperatures, C = 2R.
(32) Consider a 3-D solid with a density of states consistent with the Debye approximation. Show
that the most probable phonon frequency, ωp , satisfies
eβ¯hωp [1 −
β¯hωp
]=1
2
(The number of phonons of a given frequency involves both the Planck distribution for the occupation number < n > and the Debye density of states.)
(33) Use the “born” program in sss and work through exercises 4.11, 4.13, 4.15, 4.16, and 4.17.
You need only write up 4.16 and 4.17 to hand in.
For Monday, April 14:
(34) Silver has a resistivity at room temperature (T = 300 K) of 1.62 µΩ cm. Assume it’s a monovalent metal (one conduction electron per atom).
(a) Calculate the electron scattering time, τ . (Look up and use the density, molar mass, and/or
lattice parameter to first calculate the conduction electron number density.)
(b) Assume the conduction electrons’ average speed is given by the classical equipartition result:
1
3
2
2 mv = 2 kB T . Find the electrons’ mean free path.
(c) Assume instead that Ag is well described by the Sommerfeld model of a free electron gas. Calculate the fermi energy, fermi temperature and the fermi velocity (vF = h
¯ kF /m).
(d) What is the mean free path of an electron near the fermi surface, based on the collision time
calculated in (a)?
(35) Sidebottom 12.2
(36) Sidebottom 12.5
(37) Using the Drude program in sss work through exercises 6.1, 6.7, 6.22, 6.23. (Remember, you
can slow the simulations down by reducing the speed to a minimum and reduce the “graph updates”
setting under “configure...”
(38) Using the Sommerfeld program in sss work through exercises 7.2, 7.5, 7.7, 7.8
For Monday, April 21:
(39) Consider the free electron model in a simple cubic lattice. (a) Show that if the metal is
monovalent (each atom contributes exactly one electron to the conduction electron pool) the fermi
sphere fits entirely in the first Brillouin zone in reciprocal space. (b) For a trivalent atom (three
electrons from each atom) find the radius of the fermi sphere. How does kF compare with the
distance from the origin to the zone boundary, π/a (a=lattice constant) along the x axis? What
about the distance to the corner of the first zone at (π/a, π/a, π/a)? Sketch the fermi sphere and
the first Brillouin zone to illustrate relative sizes.
(40) In some very thin films of materials at low temperatures, the conduction electrons may be
treated as a 2-dimensional fermi gas. The fermi sphere from 3-D now becomes a circle. For a 2-D
fermi gas in a sample of area A, total number of conduction electrons N , with a number density
n = N/A (number per unit area), find: (a) the density of states g(E), (b) the radius kF of the
fermi circle, and (c) the fermi energy.
(41) Equivalency of two statements of Bloch’s theorem: Prove that if the wavefunction of an electron
in a periodic potential can be written as
~
ψ(~r) = eik·~r u(~r) with u(~r) satisfying u(~r) = u(~r + T~ ),
then it is also true that
~ ~
ψ(~r + T~ ) = eik·T ψ(~r)
where T~ is a Bravais lattice vector in the real space (direct) lattice.
(42) Using the Bloch program in sss work through exercises 8.3 through 8.5.
For Monday, May 5:
(43) Sidebottom 13.3
(44) Sidebottom 13.8
(45) Sidebottom 13.9
(46) In sss exercise 8.10 using bloch, but make more comparisons. Examine and sketch |ψ|2 for
k values near 0, about halfway to the zone boundary, and near the zone boundary in each of the
three lowest energy bands. Summarize any patterns you detect.
(47) In sss exercises 9.2, 9.6 using ziman. Calculate the group velocity from the dispersion relation
E(kx , ky ) given as eq 9.1. Explore the behavior of the electron in presets 1 (no external fields) and
2 (with an electric field). Sketch out the real space motion of a Bloch oscillation described following
exercise 9.6 Modify preset 2 by turning off the E field and turning on a 1 T magnetic field and
describe the electron’s motion in real space and k space with a B field.

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