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Solid
State Physics |
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LAST UPDATE: 2/24/07
Topics
Poster/Power Point Poster Format Power Point Format
Grading
Sheet Definitions Formulas
Comp.
Prob. #1 Comp. Prob.#2 Comp. Prob. #3
Solid State Physics Lecture Notes
Thermal Equilibrium
Properties
Non-Thermal
Equilibrium Properties
University of Massachusetts Lowell
Department
of Physics and Applied Physics
Suggested Topics for Solid State Physics
Poster
Papers / Power Point Presentations
95.472
/ 572
Dr. A.S. Karakashian
1. Josephson
Effect
2. Adiabatic Demagnetization Method of Cooling
3. Alfven Waves
4. BCS Theory of Superconductivity
5. Helicons
6. Solar Cells
7. Color Centers
8. Cyclotron Resonance
9. Curie-Weiss Theory
10. Electromagnetic Properties of Superconductors
11. De Haas-van Alphen Effect
12. Exciton Waves
13. Solid State Lasers
14. Gunn Effect
15. Non-linear Optical Materials
16. Schottky-Barrier Diodes
17. Buckminister Fullerenes
18. Kondo Effect
19. Magnons
20. Plasmons
21. Mossbauer Effect
22. Raman Scattering
23. Antiferromagnetic Resonance
24. Thermodynamic Properties of Superconductors
25. Quantum Well Devices
A topic must be chosen by each student by the end of the 4th week of classes, and the title, outline and at least three references which you have used must be submitted to me for approval. Topics other than those listed above may be chosen with approval.
University of Massachusetts Lowell
Department
of Physics and Applied Physics
Format for Solid State Physics
Poster
Papers Presentations
Poster Presentation:
Please use large type, double-spaced with one inch margin on the left side on 8 1/2 x 11 inch white paper.
1. Title page (1 page, large type)
____________________________________________________________
Solid State Physics Paper
"Title of Paper"
by
"Author"
"Author affiliation"
____________________________________________________________
2. Abstract (Summary) page (1 page)
3. Narrative pages (3 to 5 pages, reference all sources)
4. Conclusions (1 page)
5. References (1 page)
6. Figures (Draw in ink with ruler and compass or computer graphics and/or clear Xerox of original figures). Include captions explaining the figure. (1 figure plus caption per page)
7. Total number of pages should be about 9 or 10.
8. See papers posted on the walls in the Physics Department area on the first floor of the Olney
Science Center for examples.
9. A copy of the pages on the poster must
be given to the instructor and class at the presentation.
University of Massachusetts Lowell
Department
of Physics and Applied Physics
Format for Solid State Physics
Power Point Presentations
Power Point Presentation:
Please use a plain template for your presentation with your name and affiliation at the bottom of each slide.
1. Title Slide
____________________________________________________________
Solid State Physics Presentation
"Title of Presentation"
by
"Author"
"Author affiliation"
____________________________________________________________
2. Outline of Presentation Slide (1 slide)
3. Narrative slides (3 to 5 slides, reference all sources)
4. Conclusions Slide (1 slide)
5. References (1 slide)
6. Figures (Use draw program or scan in original figures). Include captions explaining the figure. (1 figure plus caption per slide)
7. Total number of slides should be about 9 or 10.
8. See papers posted on the walls in the Physics Department area on the first floor of the Olney
Science Center for examples.
9. A copy of the slides in the powerpoint
must be given to the instructor and class at the presentation.
University
of Massachusetts Lowell
Department
of Physics and Applied Physics
Poster Paper /
Power Point Grading Sheet
NAME:______________________________________________________
TITLE:_____________________________________________________
Total Percent Score __________
A. Poster Paper / Power Point (30) A. Poster Paper / Power Point
1. Appearance (5) 1. ___________
2. Mechanics 2. Mechanics
i) Format (5) i) __________
ii) Spelling (2) ii) __________
iii) Grammar (2) iii)__________
iv) Punctuation (2) iv) __________
3. Content
i) Presentation (5) i) ___________
ii) Accuracy (5) ii) __________
iii) Originality (2) iii)__________
4. Difficulty of Topic (2) 4. ___________
Paper Total ____________
B. Oral Questions/Answers (20) B. Oral
1. Number of questions (5) 1. ___________
2. Difficulty of questions (5) 2. ___________
3. Accuracy of answers (5) 3. ___________
4. Clarity of answers (5) 4. ___________
Oral Total ___________
Lateness Penalty_________
(1 point/day)
Total _____________
University
of Massachusetts Lowell
Department of Physics and Applied Physics
Solid State Physics
DEFINITIONS
OF SOLID STATE
TERMINOLOGY
1. Solid state system is a set of atoms or molecules confined to a small enough volume that the boundaries remain fixed over macroscopic time intervals and are independent of the size and shape of the container.
2. Ion core is the nucleus of an atom plus its non-valence electrons.
3. Valence electron is an electron in the outer atomic shell which has the least binding energy to the atom.
4. Static lattice approximation assumes that the ion cores are fixed in their periodic equilibrium positions.
5. Mean field approximation assumes that all the electrons in the solid see the same average, periodic potential as they move through the solid.
6. Single particle Schroedinger equation for a solid is the Schroedinger equation which is obtained from the exact multi-particle Schroedinger equation by assuming the Mean Field Approximation and separation of variables.
7. Fermi-Dirac distribution is the distribution function that gives the probability that in a system of identical odd half-integer spin particles (or fermions) the single particle state with energy E is occupied at absolute temperature T.
8. Chemical potential is the parameter in the Fermi-Dirac distribution function which is determined by setting the total probability for finding a fermion in all possible single particle energy states equal to 1.
9. Fermi energy is the energy of the highest filled energy state in a free electron system at 0 0K.
10. Drude model assumptions 1.In a metal the valence electrons of the atoms are free and form a free electron gas within the metal. 2. The ion cores scatter the electrons thus providing a mechanism for thermal equilibrium to be established. 3. The electron gas is treated by using the kinetic theory of gases.
11. Failures of the Free Electron (Drude) Model 1.The electrical conductivity of metals is not independent of temperature. 2.The electrical conductivity of some non-cubic metals is not isotropic. 3.In the Weidemann-Franz Law the proportionality constant is not independent of temperature.
12. Specific heat at constant volume is the partial derivative of the internal energy with respect to temperature at constant volume.
13. Density of free electron states is the number of energy states per unit energy per unit volume (in 3-D).
14. Periodic wave function boundary conditions The value of the wave function at the origin repeats at a large number of lattice spacings from the origin.
15. Momentum transport equation d<p>/dt +<p>/t = Fext(t) where t is the time, <p> is the average momentum, t is the relaxation time due to collisions and Fext(t) is the externally applied force.
16. Sommerfeld Model assumptions 1.Free Electron Model 2.Electrons satisfy quantum mechanics. 3.Electrons satisfy Fermi-Dirac statistics.
17. Electrical conductivity is the ratio of the current density in a conductor to the applied electric field.
18. Electron mean free path is the average distance an electron travels before it is scattered.
19. Cyclotron frequency is the angular frequency of rotation of a moving charged particle around the magnetic field lines.
20. Hall effect experiment is the application of a uniform magnetic field perpendicular to the current in a conductor while measuring the transverse induced electric field.
21. Hall coefficient is the ratio of the induced electric field to the product of the current density and magnetic field.
22. Wiedemann-Franz law For metals at not too low temperatures the ratio of the thermal conductivity to the electrical conductivity is directly proportional to the temperature, and the proportionality constant is independent of the metal.
23. Direct lattice translation vector is a linear combination of integer multiples of the three primitive translation vectors
24. Lattice is a set of points generated by the direct lattice translation vector.
25. Atomic basis is set of atoms identical in number, composition, arrangement and orientation.
26. Crystal structure is a structure formed by associating an atomic basis with every lattice point.
27. Primitive translation vector is the smallest translation vector which can be used to generate the lattice.
28. Primitive lattice cell is the smallest unit cell which has primitive lattice vectors as its sides and contains only one lattice point.
29. Wigner-Seitz cell is a unit cell in direct space which is formed by bisecting nearest and next-nearest neighbor lattice points until the smallest volume closed cell is obtained.
30. Bravais lattice is a lattice that has both translational and point group symmetry.
31. Miller indices are a set of three integers which determine a set of parallel crystal planes obtained from the intercepts of the planes with the coordinate axes.
32. Braggs law is nl = 2d sin q where n is an integer, l is the x-ray wavelength, d is the smallest spacing between a set of parallel crystal planes, and q is the incident angle of the x-rays measured from the surface which is parallel to the crystal planes and which gives the position of the reflected x-ray maxima.
33. X-ray scattering amplitude is the integral over the volume of the crystal of the electron density times the phase factor due to the incident and scattered waves.
34. Geometrical structure factor is the scattering amplitude for the x-ray scattering due to the ion cores at the lattice sites.
35. Atomic form factor is the scattering amplitude for the x-ray scattering due to the electrons inside the ion core.
36. Reciprocal lattice vector is the translation vector in reciprocal (or wave vector space) which generates the reciprocal lattice associated with a direct lattice.
37. First Brillouin zone is the Wigner-Seitz Cell in reciprocal space.
38. Gamma point in
the first Brillouin zone is the point at the center of the 1 st Brillouin
Zone.
39. Blochs theorem is the statement that an eigenfunction of the Schroedinger equation for a periodic potential at a point r plus a direct lattice vector equals the eigenfunction at r times a phase factor which depends on the direct lattice vector.
40. Bloch Wave - is the product of a plane wave times a function with the periodicity of the direct lattice translation vector.
41. Pseudo-potential is the sum of the periodic ion core potential and an average electron-electron interaction potential which is weak enough to use in the Nearly free electron model.
42. Pseudo-potential coefficient is the Fourier coefficient in the Fourier series expansion of the pseudo-potential.
43. Nearly-free electron model is the Sommerfeld Model modified by the presence of a weak periodic pseudo-potential.
44. Electron energy bands are the ranges of energy in which propagating solutions of the electron wavefunctions are allowed in a crystal.
45. Empty lattice approximation is the assumption that the periodic pseudo-potential is zero, but the resulting free electron gas still has periodicity in reciprocal space.
46. Two band model is the assumption that there is only one non-zero pseudo-potential coefficient in the Fourier series expansion of the pseudo-potential.
47. Energy bands in reduced zone scheme is the energy vs. momentum plot where the momentum vector is restricted to the 1 st Brillouin Zone, and the energy is a multi-valued function of momentum.
48. Energy bands in extended zone scheme - is the energy vs. momentum plot where the momentum vector is not restricted to the 1 st Brillouin Zone, and the energy is a single-valued function of momentum.
49. Energy bands in periodic zone scheme - is the energy vs. momentum plot where the momentum vector is not restricted to the 1 st Brillouin Zone, and the energy is a multi-valued function of momentum.
50. Energy gap - are the ranges of energy in which only non-propagating solutions of the electron wavefunctions are allowed in a crystal.
51. Conduction and valence bands The conduction band is a partially filled energy band in a crystal, and the valence band is a completely filled energy band in a crystal.
52. Electron state is an occupied electron state in the conduction band of the crystal.
53. Hole state is a vacant electron state in the valence band of the crystal.
54. Effective mass is the modification of the electron or hole mass in the crystal due to the curvature in the band structure near a minimum or maximum point in the corresponding band.
55. Indirect bandgap semiconductor is a semiconductor in which the maximum energy point in the valence band is not at the same value of the momentum vector as the minimum energy point in the conduction band.
56. Direct bandgap semiconductor - is a semiconductor in which the maximum energy point in the valence band is at the same value of the momentum vector as the minimum energy point in the conduction band.
57. Doping is the process of deliberately introducing small concentrations of a foreign impurity atom into a crystal in order to change its properties.
58. Donor impurity is a foreign impurity atom such as phosphorous in silicon which will contribute extra electrons to the conduction band of the crystal.
59. Acceptor impurity - is a foreign impurity atom such as boron in silicon which will contribute extra holes to the valence band of the crystal.
60. Intrinsic semiconductor is an undoped semiconductor which is an insulator at 0 0K.
61. Extrinsic semiconductor is a semiconductor which is doped with either donor or acceptor impurity atoms.
62. Mobility is the magnitude of the drift velocity of the charge carrier (electron or hole) per unit applied electric field.
63. Drift current is the contribution to the total current of the charge carriers in the presence of an applied electric field.
64. Diffusion current - is the contribution to the total current of the charge carriers in the presence of a gradient in the charge carrier concentration.
University of Massachusetts Lowell
Department of Physics and Applied Physics
Solid State Physics
fFD(e) = 1/{exp[(e - m)/kBT] +1}
yk(r)= A exp(ik·r) + B exp(-ik·r)
eF = (
2/2m)(3p2N/V)2/3
D(e) = dN/de = V/2p2 (2m/
)3/2 e1/2
d<p>/dt + <p>/t = Fext = -e(E + 1/c (v x B))
s = ne2t/m
wC =
eB/m*c
RH = Ey/jxB
Tn = n1 a1 + n2 a2 + n3 a3
Gm = m1 b1 + m2
b2 + m3 b3
bi = 2p (aj
x ak)/ [ai · (aj x ak)]
ai · bj
= 2p dij
nl = 2dhkl sin
qn
k k = Ghkl
2 k · Ghkl= ½ Ghkl ½2
n(r) = εG n(Ghkl) exp(i Ghkl· r)
SG = ςcell dV
n(r)exp(-i Ghkl· r)
fj = ς dV nj(r) exp(-i Ghkl· r)
U(rij) = A/rijn ± q2/rij
a = εj(±1)/pij
vg = 1/
Ρke(k)
D(w) = V/(2p)3 ς dSw /½vg½ = (Vk2/2p2)dk/dw
fBE(w) = 1/{exp(
w/kBT) 1}
yk(r) = uk(r)exp(ik·r)
(
2k2/2m - e)C(k) + εG VG
C(k-G) = 0
dk/dt = Fext
(1/m*)ij = (1/
2) Ά2e/ΆkiΆkj
Ed = e4m*/2k2
2
ad = k 
2/m*e2
kh = -ke
eh(kh) = -ee(ke)
vh = ve
mh = -me
dkh/dt = e[E +
1/c (vh x B)]
dke/dt = -e[E +
1/c (ve x B)]
n » (n0Nd)1/2
exp(-Ed/2kBT)
University
of Massachusetts Lowell
Department of Physics and Applied Physics
Solid
State Physics
95.472/572
Computer Problem #1
1. Consider the generation of the direct and reciprocal two dimensional rectangular Bravais lattice with spacing, a, parallel to the x-axis and, 2a, parallel to the y-axis ( a primative unit cell of sides a and 2a) and the first Brillouin zone and high symmetry directions in reciprocal space.
2. Write a program to calculate the coordinates of each lattice point in both direct and reciprocal space out to a given (input) distance from the origin. Then calculate the coordinates of the corners of the Wigner-Seitz cell and the first Brillouin zone. Now determine the coordinates of the corner and the equations of the lines from the origin to the corner, from the origin to the center of the edge and from the center of the edge to the corner in the first quadrant (high symmetry directions) for both the Wigner-Seitz cell and first Brillouin zone.
3. Input:
Set the input distance equal to 3 times the nearest neighbor distance.
4. Run Program:
5. Output:
a) Listing of program
b) Tables for coordinates of each lattice point in the direct and reciprocal lattices out to the input distance.
c) Computer generated drawings of the Wigner-Seitz and first Brillouin zone including the high symmetry directions labeled.
6. Turn in the derivations, listing of program and outputs.
University
of Massachusetts Lowell
Department of Physics and Applied Physics
Solid
State Physics
95.472/572
1. Consider the vibrational normal mode frequencies of a two dimensional Bravais lattice with spacing, a, parallel to the x-axis and, 2a, parallel to the y-axis ( a primative unit cell of sides a and 2a) with alternating masses m and M at successive lattice sites.
2. Write a program to calculate the dispersion relation for both the acoustic and optical normal modes for longitudinal oscillations. Use the nearest neighbor approximation with spring constant, k, connecting nearest neighbor masses.
3. Input:
a) Let (m/M) = 0.2,
b) Let w0 = Φ(k/m),
c) Let K0 = (p/a),
d) Let the wavenumber K = integer x (0.1K0).
4. Run the program.
5. Output:
a) Listing of program,
b) Tables for frequency vs. wavenumber (w / w0 vs. K / K0) for each high symmetry line,
c) Plots of frequency vs. wavenumber (w / w0 vs. K / K0) for each high symmetry line.
d) Check to show that your results agree with the diatomic quasi-one dimensional case in the appropriate limit.
University
of Massachusetts Lowell
Department of Physics and Applied Physics
Solid
State Physics
95.472/572
1. Consider the electron energy bands of a two dimensional Bravais lattice with spacing, a, parallel to the x-axis and, 2a, parallel to the y-axis ( a primative unit cell of sides a and 2a).
2. Write a program to calculate the energy bands in the Empty Lattice approximation and reduced zone scheme for the lowest 3 bands in the first Brillouin zone along the high symmetry directions.
3. Input:
a) Let e0 = (h2 / 2ma2), where h = Plancks constant and m = free electron mass,
b) Let K0 = (p/a),
c) Let the wavenumber K = integer x (0.1K0).
4. Run the program.
5. Output:
a) Listing of program,
b) Tables for energy vs. wavenumber (e / e0 vs. K / K0) for each high symmetry line,
c) Plots of energy vs. wavenumber (e / e0 vs. K / K0) for each high symmetry line.
d) Check to show that your results agree with the free electron model in the extended zone scheme.
SOLID
STATE PHYSICS
LECTURE NOTES
LECTURE NOTES
BY: A. S. KARAKASHIAN
Professor, Physics and Applied Physics
NOTES TYPED BY:
MEG NOAH
Graduate Student,
Physics and Applied Physics
2007
Solid
State Physics Overview
Thermal Equilibrium Properties
Solid State Physics
System A set of N atoms or molecules which are confined to a small enough volume that the boundaries remain fixed over macroscopic time intervals and are independent of the size and shape of the container. → Bonding occurs through valence electrons (covalent-ionic).
Fundamental Interaction Electromagnetic force (gravity neglected when charged particles are present, nuclear forces are short range ~ nuclear distances)
If 
is an operator corresponding to a physical property, then:
Problem:
1. What are the properties of the system?
2. How do you predict these properties?
Experiment: (Answers 1.)
\
\
Vary the properties of probe, measure corresponding response.
Theory: (Answers 2.)
1.) Model of system → predict response to probe → compare to experiment
a. If successful, apply to another experiment
b. If not successful, modify the model
Independent particle model:
1. 
where 
contains coordinates
of only 1 particle.
2. 
where
3. 
Thermal Expectation Value:
Consider:
Separate electrons and ions:
Static Lattice Approximation 
equil ion core
positions.
and
Constant absorbed in
definition of potential
Mean Field Approximation (Independent Electrons):
Assume Slater Determinant Form for Electron Wavefunction:
For a single electron in the state qi:
Solid State Physics
Overview
Non-Equilibrium
Transport Properties
Transport Theory for Non-Equilibrium Phenomena
1.) Wave-Packet Assumption: 
and 
are well-defined for packet.
2.) Local Equilibrium Assumption:
at equilibrium where 
=non-equilibrium
distribution function
Expand 
to 1st order in 
by chain-rule:
Note: If 
independent of 
and 
Let 
Solution
Electric Current:
Thermal Current:
From Thermodynamics:
where: 
Thermal Conductivity: 
Solution of Boltzman
Equation:
If 
and 
where 
if 
and 
=electrical
conductivity
=thermal conductivity
Wiedeman-Franz Law
From above result:
Momentum Equation From Boltzman Equation
Boltzman equation in
relaxation approximation:
Assumptions:
No conc. gradients
Consider:
Note: Fore long-range
(Coulomb) interaction forces, the collision term in Boltzman equation is not
valid. Therefore, the Boltzman equation
should be modified by letting:
