PHY 752 Solid State Physics

MWF11-11:50 PM OPL 107 http://www.wfu.edu/~natalie/s11phy752/

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

Homework Assignments

Problem Set #1 (1/12/2011)
Problem Set #2 (1/14/2011)
Problem Set #3 (1/21/2011)
Problem Set #4 (1/24/2011)
Problem Set #5 (1/26/2011)
Problem Set #6 (1/28/2011)
Problem Set #7 (1/31/2011)
Problem Set #8 (2/02/2011)
Problem Set #9 (2/04/2011)
Problem Set #10 (2/07/2011)
Problem Set #11 (2/18/2011)
Problem Set #12 (2/21/2011)
Problem Set #13 (2/28/2011)
Problem Set #14 (3/14/2011)
Problem Set #15 (3/16/2011)
Problem Set #16 (3/28/2011)
Problem Set #17 (3/30/2011)
Problem Set #18 (4/06/2011)
No Title
Jan 12, 2011
PHY 752 - Problem Set #1
PDF version
Read Chapters 1-3 in Martin: homework is due Friday Jan 14, 2011.
  1. Consider an N ×N Hamiltonian matrix of the form:
    H = ⎛
    ⎜
    ⎜
    ⎜
    ⎜
    ⎜
    ⎝
    E0
    v
    0
    0
    v
    E0
    v
    0
    0
    v
    E0
    v
    0
    0
    v
    E0
    		⎞
    ⎟
    ⎟
    ⎟
    ⎟
    ⎟
    ⎠     		,
    		(1)
    where E0 and v are fixed constants.
    1. Using your favorite software, find the eigenvalues of H for at least 3 different choices of N.
    2. Show that the eigenvalues are consistent with the analytic form we derived in class and find the limiting eigenvalues for N → ∞.


File translated from TEX by TTH, version 3.88.
On 12 Jan 2011, 10:06.

PHY 752 -- Assignment #2

January 14, 2011

Continue reading Chap. 3 in Martin. The following problem will be due Fri, Jan. 21, 2011.

  1. Work problem 3.23 in Martin.

PHY 752 -- Assignment #3

January 21, 2011

Continue reading Chap. 4 in Martin. The following problem will be due Mon, Jan. 24, 2011.

  1. Consider Fig. 4.5 Martin. On the right is shown a diagram of a two-dimension graphite sheet, where the filled and empty circles represent equivalent C sites. Using the definitions of a1 and a2 and of τ1 and τ2 given in the text find:
    1. The nearest neighbors to the atom at the origin.
    2. The next-nearest neighbors to the atom at the origin.
    3. For each neighbor, show that it can be expressed in the form
      R = τi + n1 a1 + n2a2.
Note: This problem should be carried out by head (not computer).

PHY 752 -- Assignment #4

January 24, 2011

Continue reading Chap. 4 in Martin. The following problem will be due Wed, Jan. 26, 2011.

For this problem, you can will need to use the crystallography software program mercury which you can install on your laptop from the website http://www.ccdc.cam.ac.uk/products/mercury/ or you can use the installed version on the cluster which is available from the directory /home/natalie/ForPHY752/pgms/mercury_2.3/bin/mercury. You will also need to use the crystallagraphy data file in "cif" format 1010467.cif and you may also wish to refer to the published paper (which gives slightly different X-ray results) Li2SO4.1975.a13255.pdf . For sharing your visual results, you may email, print (try not to print with the black background), or save your results on your cluster account and give me read access.

  1. Visualize the crystal structure of β-Li2SO4. Save an image of the structure in jpg or png format.
  2. Use the mercury program to find the X-ray powder pattern for this material and save the graph.
  3. Find the primitive lattice and reciprocal lattice vectors for this material.
  4. In terms of your reciprocal lattice vectors, find the G which corresponds to the largest X-ray peak.

PHY 752 -- Assignment #5

January 26, 2011

Continue reading Chap. 4 in Martin. The following problem will be due Fri, Jan. 28, 2011.

  1. Using the Ewald summation method, determine the electrostatic interaction energy of MgO and of ZnS as a function of the cubic lattice constant a. The basis vectors and Bravais lattice parameters for both of these are given the the Ewald lecture notes. An example Ewald evaluation is also given in the lecture note directory.

PHY 752 -- Assignment #6

January 28, 2011

Read Chapter 14 in Martin and also read through the literature papers on diamond and graphite given in the handouts and also available on the webpage http://www.wfu.edu/~natalie/s11phy752/lecturenote/.

PHY 752 -- Assignment #7

January 31, 2011

Continue reading Chapter 14 in Martin. This homework is due February 2, 2011.

  1. Consider an s-electron tight-binding model of material with a single atom per unit cell in a FCC lattice with cubic lattice constant a. Write an expression for the band dispersion function ε(k) in terms of the nearest-neighbor hopping parameter w. Plot the band disperson along various lines in the Brillouin zone including the X point at (0,0,2 π/a), the Γ point at (0,0,0), and the L point at ( π/a, π/a, π/a).
  2. Consider an s-electron tight-binding model of material with a single atom per unit cell in a BCC lattice with cubic lattice constant a. Write an expression for the band dispersion function ε(k) in terms of the nearest-neighbor hopping parameter w. Plot the band disperson along various lines in the Brillouin zone including the H point at (0,0,2 π/a), the Γ point at (0,0,0), and the P point at ( π/a, π/a, π/a).

    PHY 752 -- Assignment #8

    February 2, 2011

    Continue reading Chapter 14 in Martin. This homework is due February 4, 2011.

    1. Consider the tight-binding model of the π bands of 2-dimensional graphite that we discussed in class, including the nearest-neighbor terms with interaction parameter wnn and also next-nearest-neighbor terms with interaction parameter wnnn. Write a general expression for the two π band energies as a function (kx, ky ).
      1. In terms of wnn and wnnn find the band energies at the special k points (in cartesian coordinates in units of 2π/a): K (2/3,0), Γ (0,0), and M (0,1/√3), or their equivalents.
      2. Evaluate the expressions for wnnn/wnn = 1/4. In particular, determine how does the next-nearest interaction contribution affect the special behavior of the bands near the K-point?

    PHY 752 -- Assignment #9

    February 4, 2011

    Read beginning of Chapter 12 in Martin. This homework is due February 7, 2011.

    1. Using the free-electron model, find the Fermi level of Al using the lattice constant 4.05 Å. Compare your result to that of Segal's value in in the paper http://link.aps.org/doi/10.1103/PhysRev.124.1797
    No Title
    Feb 4, 2011
    PHY 752 - Problem Set #10
    PDF VERSION
    Read Chapters 6-7 in Martin: homework is due Wednesday, Feb. 9, 2011. This assignment is designed to explore the power of the variational method for a simple Hamiltonian problem.
    1. Consider the Hamiltonian for the electron in a H atom:
      H(r) = − ħ2
      2m
      		 ∇2 − e2
      r
      		 .
      		(1)
      For the following two functional forms, use the variational principle to find the best estimate of the ground state energy. That is, in each case, find the optimal value of α which minimizes the energy
      E(α) ≡ 〈Ψ(r,α) | H(r) | Ψ(r,α) 〉
      〈Ψ(r,α) | Ψ(r,α) 〉
      		 .
      		(2)

      1. | Ψ(r,α) 〉 ≡ e−αr.
        		(3)

      2. | Ψ(r,α) 〉 ≡ e−αr2.
        		(4)


    File translated from TEX by TTH, version 3.88.
    On 4 Feb 2011, 13:54.

    PHY 752 -- Assignment #11

    February 18, 2011

    Read Chapter 16 in Martin. This homework is due February 21, 2011.

    1. Derive equation 16.6.

    PHY 752 -- Assignment #12

    February 21, 2011

    Read Chapters 16 and 17 in Martin. This homework is due February 28, 2011.

    1. Use the WIEN2k code to duplicate the results for diamond that we discussed in class plus one other material of your choice. Some helpful information is available on the lecture note webpage. Note:
      1. Use the working directories /wfurc6/classes/phy752-spr11/$User/material1 and /wfurc6/classes/phy752-spr11/$User/material2, where $User is your login and material1 and material2 are your two material names. Make sure that the permissions of these directories are:
        drwxrwsr-x
        so that I will be able to access your results.
      2. For both materials, successfully complete the self consistent field (SCF) calculation by running the program run_lapw. Make sure that your calculation has converged by running
        >>> grep :ENE *scf
        to see that the total energies have converged.
      3. (Extra credit) Determine the DOS and bandstructure diagram of your materials. Note that after you have created each plot, you can save the plot in postscript format. You will see a file something like 127970-8771.ps. In order to easily print this result, use the command
        >>> ps2pdf 127970-8771.ps
        Now you will have a file in PDF format with the name 127970-8771.pdf. You can print that file directly on the cluster or transfer it to your laptop and print from there.

    PHY 752 -- Assignment #13

    February 28, 2011

    Read Chapter 10 in Martin. This homework is due March 14, 2011 (Monday after spring break.)

    1. Use the program graphatom to calculate the electronic structure of two different atoms of your choice. This program is available from the directory /home/natalie/ForPHY752/pgms/graphatom/. Example inputs and outputs are available from the directory /home/natalie/ForPHY752/pgms/graphatom/examples. The exchange-correlation options are
      1. LDA-PW corresponding to the LDA parameterization of Perdew and Wang
      2. GGA-PBE corresponding to the GGA parameterization of Perdew, Burke, and Ernzerhof
      To complete the assignment for each atom:
      1. Report the list of one-electron energies and the total energy.
      2. Use gnuplot or other plotting software to make plots of the electron density (stored in the density.GA file) and the electron wave functions (stored in the GAwfn0, GAwfn1, GAwfn2, .. files).
      3. (Extra credit) Within the "Delta-SCF" approximation find the electron removal energy for your atom. In some cases, you can compare your calculated results with experiment using the www.NIST.gov databases.

    PHY 752 -- Assignment #14

    March 14, 2011

    Read Chapter 11 in Martin and the paper by Gonze, Stumpf, and Scheffler. This homework is due March 16, 2011

    1. Verify that Eq. (22) is a solution to the radial Kohn-Sham equation (17) for angular momentum l.

    PHY 752 -- Assignment #15

    March 16, 2011

    Read Chapter 11 in Martin and the paper by Peter Blöchl describing the PAW method. This homework, due March 18, 2011, has a choice of two problems (extra credit for doing both.

    1. Show that the partitioning of the electron density in Eq. (14) into a sum of smooth and one-center terms, follows as a very good approximation from the wavefunction transformation given in Eq. (9) of that paper.
    2. Using Maple or some other mathematical software, demonstrate that the radial electrostatic potential generated by a spherically symmetric charge distribution ρ1(r) is the same as that generated by a different spherically symmetric charge distribution ρ2(r) under the following conditions
      1. While the shapes of the distributions are different, the total charges are the same.
      2. Assuming the two distributions are confined within a radius r < a, the two electrostatic potentials become equal outside the radius r > a.

PHY 752 -- Assignment #16

March 28, 2011

  1. In separate appropriately named directories on your /wfurc6/classes/phy752-spr11/[login]/ partition, use the atompaw code in the directory /home/natalie/ForPHY752/pgms/graphatom/src to generate atomicdata files for C and one other element of our choice. Plot out the wave functions, pseudo wavefunctions, and projectors as well as the logarithmic derivatives. For C, you can compare your results with examples in the directories /home/natalie/ForPHY752/examples/atompaw/Clda2p and /home/natalie/ForPHY752/examples/atompaw/Clda1p.

PHY 752 -- Assignment #17

March 30, 2011

  1. In separate appropriately named directories on your /wfurc6/classes/phy752-spr11/[login]/ partition, use the PWscf code to generate the electronic structure of diamond and one other material of your choosing. Your results should include the binding energy as a function of lattice constant and/or a determination of the optimal geometry using "vc-relax". (Extra credit for comparison of your results obtained using wien2k code with results obtained using the PWscf code.) Example files and scripts available in the directories /home/natalie/ForPHY752/examples/PWscf/diamond-scan and /home/natalie/ForPHY752/examples/PWscf/diamond-relax.

PHY 752 -- Assignment #18

April 6, 2011
This assignment is due Friday April 8th.

  1. This assignment is correlated with the "Third exam" for this course which will be accomplished during the week of April 11-15. Each student should choose one of the published papers provided on the webpage http://www.wfu.edu/~natalie/s11phy752/lecturenote/Casestudies/. (Each choice should be unique. Note: it is possible to use alternate papers of your own choice. In that case, please make two copies of that paper and make an appointment to discuss it with NAWH before Friday, April 8.) Once you have made your choice, prepare your atomic data files and PWscf input files. Set up a trial run for PWscf to make sure that it will run correctly. Send an email to natalie@wfu.edu with the directory names of your trial runs. Please consult with natalie@wfu.edu if you run into any difficulties.
[an error occurred while processing this directive]
Last modfied: Wednesday, 06-Apr-2011 08:55:08 EDT