PHY 752 Solid State Physics

MWF 11 AM-11:50 PM

OPL 103

http://www.wfu.edu/~natalie/f15phy752/

Instructor: Natalie Holzwarth Phone:758-5510 Office:300 OPL e-mail:natalie@wfu.edu

Course schedule

(Preliminary schedule -- subject to frequent adjustment.)

	Date	F&W Reading	Topic	Assignment
1	Wed, 8/26/2015	Chap. 1.1-1.2	Electrons in a periodic one-dimensional potential	#1
2	Fri, 8/28/2015	Chap. 1.3	Electrons in a periodic one-dimensional potential	#2
3	Mon, 8/31/2015	Chap. 1.4	Tight binding models	#3
4	Wed, 9/02/2015	Chap. 1.6, 2.1	Crystal structures	#4
5	Fri, 9/04/2015	Chap. 2	Group theory	#5
6	Mon, 9/07/2015	Chap. 2	Group theory	#6
7	Wed, 9/09/2015	Chap. 2	Group theory	#7
8	Fri, 9/11/2015	Chap. 2	Group theory	#7
9	Mon, 9/14/2015	Chap. 2.4-2.7	Densities of states	#8`
10	Wed, 9/16/2015	Chap. 3	Free electron model	#9
11	Fri, 9/18/2015	Chap. 4	One electron approximations to the many electron problem	#10
12	Mon, 9/21/2015	Chap. 4	One electron approximations to the many electron problem	#11
13	Wed, 9/23/2015	Chap. 4	Density functional theory	#12
14	Fri, 9/25/2015	Chap. 5	Implementation of density functional theory	#13
15	Mon, 9/28/2015	Chap. 5	Implementation of density functional theory	#14
16	Wed, 9/30/2015	Chap. 5	First principles pseudopotential methods	#15
17	Fri, 10/02/2015	Chap. 6	Example electronic structures	#16
18	Mon, 10/05/2015	Chap. 6	Ionic and covalent crystals	#17
19	Wed, 10/07/2015	Chap. 6	More examples of electronic structures	#18
20	Fri, 10/09/2015	Chap. 1-6	Review	Start exam
	Mon, 10/12/2015		No class	Take-home exam
	Wed, 10/14/2015		No class	Exam due before 10/19/2015
	Fri, 10/16/2015		Fall break -- no class
21	Mon, 10/19/2015	Chap. 10	X-ray and neutron diffraction	#19
22	Wed, 10/21/2015	Chap. 10	Scattering of particles by crystals	#20
23	Fri, 10/23/2015	Chap. 11	Optical and transport properties of metals	#21
24	Mon, 10/26/2015	Chap. 11	Optical and transport properties of metals	#22
25	Wed, 10/28/2015	Chap. 11	Transport in metals	#23
26	Fri, 10/30/2015	Chap. 12	Optical properties of semiconductors and insulators
27	Mon, 11/02/2015	Chap. 7 & 12	Excitons	#24
28	Wed, 11/04/2015	Chap. 9	Lattice vibrations	#25
29	Fri, 11/06/2015	Chap. 9	Lattice vibrations	#26
30	Mon, 11/09/2015	Chap. 13	Defects in semiconductors	#27
31	Wed, 11/11/2015	Chap. 14	Transport in semiconductors	#28
32	Fri, 11/13/2015	Chap. 15	Electron gas in Magnetic fields	#29
33	Mon, 11/16/2015	Chap. 15	Electron gas in Magnetic fields	Prepare presentation.
34	Wed, 11/18/2015	Chap. 17	Magnetic ordering in crystals	Prepare presentation.
35	Fri, 11/20/2015	Chap. 18	Superconductivity	Prepare presentation.
36	Mon, 11/23/2015	Chap. 18	Superconductivity	Prepare presentation.
	Wed, 11/25/2015		Thanksgiving Holiday
	Fri, 11/27/2015		Thanksgiving Holiday
37	Mon, 11/30/2015	Chap. 18	Superconductivity	Prepare presentation.

	Wed, 12/02/2015		Student presentations I
	Fri, 12/04/2015		Student presentations II
	Mon, 12/07/2015		Begin Take-home final

PHY 752 -- Assignment #1

Aug. 26, 2015

Read Chapter 1 in GGGPP.

Work out some of the missing details of the Kronig-Penney model potential discussed in Lecture 1. In particular, verify the results on slides 13-15. You may wish to use the Maple file for this purpose.

PHY 752 -- Assignment #2

Aug. 28, 2015

Continue reading Chapter 1 in GGGPP.

GGGPP shows that Eq. 1.45 and 1.18 are equivalent, demonstrating that the eigenstates of an electron in a one-dimensional periodic potential can be equally well treated as a boundary value problem or as an electron tunneling phenomenon. Work out some of the intermediate steps of this derivation.

No Title

August 31, 2015

PHY 752 - Problem Set #3 PDF VERSION

Read Chapter 1.4 in GGGPP

Consider a one-dimensional tight-binding model system described by a tridiagonal Hamiltonian which has non-trivial matrix elements H_{n n′} of the form:

H_{n n} = α and H_{n (n±1)} = β

for all site indices n, where α and β are real energy parameters.
1. Consider the case where the site indices n,n′ take the values 1, 2, 3 exclusively and find the numerical values of the 3 eigenvalues.
2. Consider the case where the site indices n,n′ take the values 1 ... 8 exclusively and find the numerical values of the eigenvalues.
3. Consider the case where the site indices n,n′ have an infinite range (−∞ ≤ n,n′ ≤ ∞). Compare the energy range for this system with that of the previous two samples.

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PHY 752 -- Assignment #4

Sept. 2, 2015

Start to read Chapter 2 in GGGPP.

For each of the following Bravais lattices, using the primitive translation vectors given in your text, enumerate the nearest neighbor positions, find the primitive cell volume, and determine the reciprocal lattice vectors for the following structures:
1. simple cubic
2. face-centered cubic
3. body-centered cubic

PHY 752 -- Assignment #5

Sept. 4, 2015

Continue reading Chapters 2 in GGGPP.

Consider the 6 by 6 group multiplication table given above.
1. For each element of the group, find its inverse.
2. Find the subgroups of the 6-dimensional group.
3. Find the classes of the 6-dimensional group.

PHY 752 -- Assignment #6

Sept. 7, 2015

Continue reading Chapters 2 in GGGPP.

Consider a group of 4 elements described by the following multiplication table:

| E A B C

- | - - - -

E | E A B C

A | A E C B

B | B C E A

C | C B A E

Find the irreducible representations of this group.

PHY 752 -- Assignment #7

Sept. 9 & 11, 2015

Continue reading Chapters 2 in GGGPP.

Consider the following two groups. These are each based on a set of 8 transformations of a general point in space (x,y,z). For each, determine
1. The multiplication table of the group
2. The classes of the group
3. The number of irreducible representations and, if possible, the character table of the group.

Group I
1. (x,y,z)
2. (-x,y,z)
3. (x,-y,z)
4. (x,y,-z)
5. (-x,-y,z)
6. (-x,y,-z)
7. (x,-y,-z)
8. (-x,-y,-z)
Group II
1. (x,y,z)
2. (x,-y,z)
3. (x,y,-z)
4. (x,-y,-z)
5. (x,z,y)
6. (x,-z,y)
7. (x,z,-y)
8. (x,-z,-y)

PHY 752 -- Assignment #8

Sept. 14, 2015

Read Chapter 2.4-2.7 in GGGPP.

Consider the following two-dimensional electron systems, finding their densities of states and their Fermi levels (at temperature 0 K). Assume that in both cases, there is 1 electron within the unit cell area A. Your answer should depend on A and on the parameter X.
1. First consider the system where the electron energy E as a function of the radial wavevector k has the dispersion E(k)=X k².
2. Second consider the system where the electron energy E as a function of the radial wavevector k has the dispersion E(k)=X k.

PHY 752 -- Assignment #9

Sept. 16, 2015

Read Chapter 3 in GGGPP.

Verify Eq. 3.7 of your textbook.
Using the data given in your textbook, find both the Fermi energy and the ground-state energy at 0 K for Na and for Al.

No Title

September 18, 2015

PHY 752 - Problem Set #10 PDF VERSION

Read Chapter 4 in GGGPP

This problem is concerned with variationally estimating the ground state electronic energy of a two-electron atom with nuclear charge Ze in the Hartree-Fock approximation. The Hamiltonian for the two-electron system is

H(r₁,r₂) = −

ħ²

( ∇²₁ + ∇²₂)−Ze²

⎛
⎝

r₁

r₂

⎞
⎠

e²

|r₁ − r₂|

Assume that the spatial part of the two-electron wavefunction can be written in the form

Ψ(r₁,r₂)=ϕ(r₁) ϕ(r₂),

where

ϕ(r) = N e^−αr/a,

where N is the normalization factor, a = ħ²/(me²) is the Bohr radius, and α is a variational parameter.

Show that

E(α) ≡

〈Ψ| H | Ψ〉

〈Ψ| Ψ〉

ħ²

2m a²

⎛
⎝

α² − 2 α

⎛
⎝

Z−

⎞
⎠

Hint: Show that the two electron term involves the integral

⌠
⌡

∞

0

dr r² e^{−2 αr/a}

⎛
⎝

⌠
⌡

r

0

dr′r′² e^{−2 αr′/a} +

⌠
⌡

∞

r

dr′r′e^{−2 αr′/a}

⎞
⎠

= 2

⌠
⌡

∞

0

dr r e^{−2 αr/a}

⌠
⌡

r

0

dr′r′² e^{−2 αr′/a}.

Find the value of α that minimizes the Hartree Fock energy E(α) and the corresponding estimate of the ground state energy of the two-electron system.

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PHY 752 -- Assignment #11

Sept. 21, 2015

Read Chapter 4 including Appendix C in GGGPP.

With the help of Appendix C, supply the intermediate steps to obtain Eq. 4.43 and 4.44a.

No Title

September 23, 2015

PHY 752 - Problem Set #12 PDF VERSION

Finish reading Chapter 4 in GGGPP

This problem involves finding the functional form of an exchange potential by evaluating the functional derivative of the exchange energy expression with respect to the density. In class we noted that for

E_xc = ⌠
⌡ d³r f(n(r, |∇n(r)|),

the corresponding potential is given by

V_xc(r) = ∂f(n(r, |∇n(r)|)
∂n
− ∇· ⎛
⎝ ∂f(n(r), |∇n(r)|)
∂|∇n|
∇n
|∇n|
⎞
⎠ .

Suppose

f(n(r, |∇n(r)|) = − 3 e²
4 π
(3 π²)^1/3 ( n(r) )^4/3 ( 1 + β|∇n(r)|² ).

Here β represents a given constant. Also suppose that the system is spherically symmetric so that n(r) = n(r). Find the expression for V_xc(r) in terms of n(r) and its radial derivatives.

Note that the PBE-GGA form of the exchange contribution ( Phys. Rev. Lett. 77 3865-3868 (1996)) is somewhat more complicated than in this homework.

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PHY 752 -- Assignment #13

Sep. 25, 2015

Start reading Chapter 5 in GGGPP .

This problem concerns the second-order and Numerov methods of solving a two-point differential equation as shown in Lecture 14. Verify the results given on this slide and try one other choice of N.

PHY 752 -- Assignment #14

Sep. 28, 2015

Continue reading Chapter 5 in GGGPP .

Provide some of the steps to validate Eq. B.2 in Appendix B of Chapter 5. Demonstrate that the equation is true for l=0 and V(r)=0.

PHY 752 -- Assignment #15

Sep. 30, 2015

Continue reading Chapter 5 in GGGPP .

"Prove" Eq. (23) in Troullier and Martins paper. (Note that Troullier and Martins express the equations in Hartree units, while the lecture notes are in Rydberg units (a factor of 2 smaller unit of energy).)

PHY 752 -- Assignment #16

Oct. 2, 2015

Begin reading Chapter 6 in GGGPP .

From the data in Table 6.2 of your text, determine the lattice constants for the rare gas solids and the binding energy per volume.

PHY 752 -- Assignment #17

October 5, 2015

Continue reading Chap. 6 in GGGPP.

Using the Ewald summation methods developed in class, find the electrostatic interaction energy of a NaCl lattice having a cubic lattice constant a. Check that your result does not depend of the Ewald parameter η. Also check that your result is consistent with the Madelung constant given in you textbook for this structure. You are welcome to copy (and modify) the maple file used in class. A FORTRAN code is also available.

PHY 752 -- Assignment #18

October 7, 2015

Continue reading Chap. 6 in GGGPP.

Consider the simple band structure model of graphene given in Sec. 6.5 of your textbook.
1. Validate Eq. 6.38.
2. Assuming t and E_p are constants, find an expression for the energy bands from 6.37.
3. Show that your result is consistent with 6.39.

No Title

October 19, 2015

PHY 752 - Problem Set #19 PDF VERSION

Read Chapter 10 in GGGPP

The graph and table below shows simulated neutron diffraction data for ZnS in the zincblende structure (Fig. 2.9, pg 76-77 in GGGPP) showing the diffraction intensity versus d_hkl plane spacing, where the hkl Miller indices are based on the conventional cubic cell.


  h   k   l   d(hkl)    I/Imax

  1   1   1   3.07035    100.0%
  0   0   2   2.65900     16.8%
  0   2   2   1.88020     76.0%
  1   1   3   1.60344     22.3%
  2   2   2   1.53517      1.2%
  0   0   4   1.32950      9.5%
  1   3   3   1.22003      7.5%
  0   2   4   1.18914      2.7%
  2   2   4   1.08553      8.4%
  3   3   3   1.02345      1.2%
  1   1   5   1.02345      3.7%
  3  -3   3   1.02345      1.2%

Using the fractional atomic positions for the ZnS structure, explain the reason for at least two of the "missing" diffraction peaks.
From the table of neutron diffraction peaks given below, determine the lattice constant of ZnS and compare your result with that given in GGGPP.

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PHY 752 -- Assignment #20

October 21, 2015

Read Appendix A of Chapter 9 in GGGPP.

Write out some of the main ideas justifying the Bloch identity Eq. A.19 on page 434.

PHY 752 -- Assignment #21

October 23, 2015

Start reading Chapter 11 in GGGPP.

Using reference data available on the Web, estimate the plasma frequencies of fcc Al and bcc Na.

PHY 752 -- Assignment #22

October 26, 2015

Continue reading Chapter 11 in GGGPP.

Provide some of the steps necessary to derive Eq. 11.35 in your textbook.

PHY 752 -- Assignment #23

October 28, 2015

Continue reading Chapter 11 in GGGPP.

Consider a material having a 3-dimensional free-electron band of the form E(k)=ℏ² k²/m*. Evaluate the kinetic coefficients K_n for this case to leading order in T, following Eq. 11.67 and 11.68 of your textbook.

PHY 752 -- Assignment #24

November 2, 2015

Start reading Chapter 7 and continue reading Chapter 12 in GGGPP.

Estimate the exiton energy levels and effective Bohr radius of a material with a dielectric constant of 10 and reduced mass of 0.1 times the electron mass.

PHY 752 -- Assignment #25

November 4, 2015

Start reading Chapter 9 in GGGPP.

Consider the diatomic one-dimensional lattice discussed in Sec. 9.2 of your textbook. Suppose the M₁=M₂ ≡ M but there are two spring constants C₁ and C₂. Find the frequency ω(q) corresponding to Eq. 9.15a for normal modes of this system.

PHY 752 -- Assignment #26

November 6, 2015

Continue reading Chapter 9 in GGGPP.

For a system in the pure vibrational state |n>, find the expectation values of the following operators which are powers p of the displacement operator u: u^p. Explicitly consider p=3 and p=4.

PHY 752 -- Assignment #27

November 9, 2015

Continue reading Chapter 13 in GGGPP.

Consider the list of properties given in Table 13.1 for GaAs at 300K. For each property that depends on temperature, estimate its value at 0 K and at 500 K.

PHY 752 -- Assignment #28

November 11, 2015

Start reading Chapter 14 in GGGPP.

Derive Eqs. 14.9, 14.10a, and 14.10b.

PHY 752 -- Assignment #29

November 13, 2015

Start reading Chapter 15 in GGGPP.

Consider Sec. 15.2.1 of your textbook which derives the energy eigenstates for a two dimensional gas in the x-y plane in the presence of a magnetic field in the z with the vector potential given by 15.6. Determine the energy eigenstates for the same system with the vector potential A(r)=(-By/2, Bx/2, 0).

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Last modfied: Sunday, 15-Nov-2015 21:53:38 EST