, SOLUTIONS MANUAL FOR
QUANTUM THEORY OF MATERIALS
Efthimios Kaxiras
Department of Physics and
School of Engineering and Applied Sciences
Harvard University
John D. Joannopoulos
Department of Physics
Massachusetts Institute of Technology
Published by Cambridge University Press, June 2019
, PREFACE
The book authors are enormously grateful to all the students and other members of their research groups (post-
docs, visitors) who contributed to the compilation of the solutions of the problems.
Some of the solutions date back to when one of the authors (EK) first learned the subject in a course taught by
the other author (JDJ), almost four decades ago! Those solutions were of course hand-written (often illegible) and
the ownership trail has unfortunately been lost. For the more recent ones, where the person who provided the last
version is known, we provide this information with gratitude. The rest, we thankfully attribute to the “Unknown
Student”, who toiled away somtimes for days, hopefully gaining something very valuable in the process.
WARNING: While we have made every effort to check the accuracy and completeness of the
solutions, we cannot guarantee that they are free of errors. We provide them here as a usufeul guide
only. The user should check every detail carefully to make sure that they meet their own standards,
hopefully gaining an equally valuable experience as the students who toiled to put them together!
1
,Contents
1 From Atoms to Solids 5
1.1 Packing Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Madelung Energy of 2D Bipartite Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Simultaneous Energy and Momentum Eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Bond Formation in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6 Bonding in Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.7 Excitations of the Free-Electron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.8 Attractive Component of Lennard-Jones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.9 Approximate Morse Potential Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.10 Kronig-Penney Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 Electrons in Crystals: Translational Periodicity 35
2.1 Reciprocal Lattice Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Periodic Part of the Single-Particle Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Effective Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4 Fermi Sphere of a 2D Square Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 Band Structures of FCC Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.6 Orthogonality of Bloch States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.7 Band Width and Number of Nearest Neighbors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.8 Bloch Theorem for Many-Body Wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.10 1D and 2D free-electron DOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.11 Critical Points in 1D and 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.12 DOS at Dirac point in 1D, 2D, 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.13 Graphene Band-Structure in Tight-Binding Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.14 Tight-Binding Model of Copper(II) Peroxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3 Symmetries Beyond Translational Periodicity 71
3.1 Proper Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 Symmetries of the 2D Square Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3 Symmetries of the 2D Honeycomb Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4 Special k-Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5 Point-Group Symmetries of the Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.6 Optical Transitions of NV Center in Diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4 From Many Particles to the Single-Particle Picture 89
4.1 Hydrogen Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2 Many-Body States of the NV Center in Diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3 Hartree-Fock Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.4 Single Particle Spectrum of Hartree-Fock Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.5 Bulk Modulus of Free-Electron Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.6 Reduced Density Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.7 Meaning of Single-Particle Eigenvalues in DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.8 Exchange-Correlation Effects in the Wigner Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.9 Lindhard Dielectric Response Function for Free-Electron Gas . . . . . . . . . . . . . . . . . . . . . . . 117
4.10 Thomas-Fermi Screening Length at Zero-Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.11 Pseudopotential Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.12 Ewald Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.13 Madelung Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.14 Ewald Summation of Madelung Energy for NaCl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
2
, 4.15 Car-Parrinello Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5 Electronic Properties of Crystals 129
5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2 Graphene Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.5 Electron and Hole Concentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.6 Position of Fermi Level in Band Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.7 Semiconductor Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.8 More Realistic p-n Junction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.9 Band Bending at Metal-Semiconductor Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6 Electronic Excitations 154
6.1 Plasma Frequency of Uniform Electron Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.2 Dielectric Function From Inter-Band Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.3 Dielectric Function From Intra-Band Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.4 Lorentz Model of Dielectric Function, Application to Si . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.5 Drude Model of Dielectric Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.6 Static Dielectric Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.7 Sum Rule for Imaginary Part of Dielectric Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.8 Frenkel Exciton Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.9 Wannier Exciton Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.10 Frenkel Excitons: Wavefunction and Bloch Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7 Lattice Vibrations and Deformations 181
7.1 Phonons in Bulk Si Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.2 Harmonic Oscillator Potential and Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
7.3 Phonon Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
7.4 Thermal Expansion Coefficient, Grüneisen Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
7.5 Phonon Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
7.6 UBER bulk modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7.9 Elastic Constants of Isotropic Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7.10 Energy Density of Isotropic Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
7.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7.13 Phonons of Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8 Phonon Interactions 219
8.1 Debye-Waller Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8.2 Raman Scattering and Albrecht terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8.3 Cooper Pair Commutation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.4 BCS Number of Particles Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.5 BCS Gap Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
9 Dynamics and Topological Constraints 226
9.1 Resistivity Tensor in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
9.2 Time-Reversal Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
9.3 Exponential Form of Time-Reversal Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
9.4 Berry Curvature of a Magnetic Monopole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
3
, 9.5 Expectation Value For Electron Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
9.6 Polarization of Finite System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
9.7 Electron Velocity and Berry’s Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
9.8 Berry Curvature of 2D Honeycomb Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
9.9 Rice-Mele Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
9.10 T -Symmetry Breaking in Honeycomb Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
9.11 Dirac Points in Square Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
10 Magnetic Behavior of Solids 243
10.1 Hund’s Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
10.2 Magnetization and Band Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
10.3 Stoner Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
10.4 Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
4
, Thanks to Laura Kulowski for undertaking a significant amount of compiling, equation number checking, and
formatting of submitted solutions.
1 From Atoms to Solids
1.1 Packing Ratio
Solution by Robert Hoyt (2016) and Cedric Flamant (2018)
An important consideration in the formation of crystals is the so-called packing ratio or filling fraction. For each
of the elemental crystals (simple cubic, face-centered cubic, body-centered cubic), calculate the packing ratio,
that is the percentage of volume occupied by the atoms modeled as touching hard spheres.
Simple Cubic
Note that the simple cubic structure, shown in Figure 1 (which can be tiled
to cover all space) represents a unit volume taken up by the crystal. The
cut spheres are the atoms. If the atom has radius r, then we see from the
diagram that the side length of the unit block is 2r. Thus, the volume of
the block, Vs , is
3
Vs = (2r) = 8r3 .
The volume taken up by the atom can be determined by noting that there
are 8 octants of the atom in the unit cell, thus giving the total volume of
one atom. Thus,
4 3
Va = πr .
3
This leaves a packing ratio of the simple cubic structure ρsc of
Va π Figure 1: Simple Cubic
ρsc = = ≈ 0.5236. (1)
Vs 6
5
,Face-Centered Cubic
In the FCC structure we see that the radius of the unit cell is determined
by the diagonal line of the face coinciding with four copies of the sphere’s
radius. Thus,
√ 3
Vs = 2 2r .
There are also 8 octants and 6 hemispheres for a total of 4 spheres in this
unit cell. Thus, there are a total of 4 spheres in the unit volume and
16 3
Va = πr .
3
This gives a packing ratio of
π
ρfcc = √ ≈ 0.7405. (2)
3 2 Figure 2: Face-Centered Cubic
Body-Centered Cubic
In the BCC structure we see that the diagonal of the cube is equal to four
of the sphere’s radius. Hence,
√
3a2 = 4r
r
16 2 4
a= r = √ r,
3 3
where a gives the cube side length, implying that
43
Vs = a3 = √ r 3 .
3 3
There are 8 octants and 1 full sphere in the unit cell for a total of 2 spheres.
Thus,
8 3 Figure 3: Body-Centered Cubic
Va = πr .
3
This gives a packing ratio of
√
π 3
ρbcc = ≈ 0.6802. (3)
8
For the NaCl and CsCl structures, assuming that each type of ion is represented by a hard sphere and the
nearest-neighbor spheres touch, calculate the packing ratio as a function of the ratio of the two ionic radii and
find its extrema and the asymptotic values (for the ratio approaching infinity or zero, that is, with one of the two
ions being negligible in size relative to the other one). Using values of the ionic radii for the different elements,
estimate the filling fractions in the actual solids.
6
,NaCl Structure
The rock salt structure is technically two FCC lattices of Na and Cl shifted
and superimposed on one another, as seen in Figure 4, but notice that
simply an octant of this unit cell will suffice for analysis by symmetry.
The first thing to notice about the rock salt structure is that there will
be three regimes. Letting sodium be atom A and chloride be atom B and
rA and rB their corresponding radii, we intuit that the contact points of the
spheres will be different in the regions of rA
rB , rA ∼ rB (in a manner
yet to be made precise), and rA
rB .
Regime where rA
rB
In this regime, we see that the A atoms (blue) are much smaller than the
B atoms (green). The size of this characteristic cell is determined by the
B atoms coming into contact with each other at the faces of the cube while Figure 4: Rock salt structure
the A atoms simply take up the free space in between the larger B atoms.
We notice that the diagonal on√the face of the cube is coincident
√ with 2 of √ 3
atom B’s radii, so this means 2a = 2rB , implying a = 2rB and Vs = a3 = 2 2rB .
In total there are 4 octants of each atom in the cell, so
14 3 14 3 2 3 3
Va = 4 πrA + 4 πrB = π rA + rB .
83 83 3
This gives a packing ratio of
Va π r3 + r3 π
= √ A 3 B = √ 1 + x3 ,
ρNaCl,
=
Vs 3 2 rB 3 2
where x ≡ rrB
A
. Now we have to determine the limit of this regime. This
occurs when atom A is large enough that it comes into contact with atom
B along the faces as well—if it grows any bigger it will force atom B to
shrink to make room. At this point, the side length of the cell will also be
equal to rA + rB : Figure 5: Regime where rA
rB
√ rA √
rA + rB = 2rB ⇒= x = 2 − 1.
rB
√
Hence, this regime is described by x < 2 − 1.
Regime where rA ∼ rB
In this regime, the size of the cell is determined by the contact of atom A and atom B along the edges of the cell.
We see that the cell edge is equal to the sum of the two radii giving
3
Vs = (rA + rB ) .
Hence we are left with a packing efficiency of
2 3 3
Va 3 π rA + rB 2 x3 + 1
ρNaCl,∼ = = 3 = 3π 3.
Vs (rA + rB ) (x + 1)
Note that if we continue to increase the size of rA (blue) eventually we will reach the end of this regime—atom A
will come into contact with its other selves. With similar reasoning to the last section, this happens when
√ rA 1
rA + rB = 2rA ⇒ =x= √ ,
rB 2−1
7
, √
which we could have predicted from the symmetry between atoms A and B. So, this regime exists when 2−1 ≤
1
x ≤ √2−1 .
Regime where rA
rB
Now we consider the final regime where atom A has grown to dominate and
hence determines the size of the cell. By symmetry, we recoginize that the
packing efficiency must be
π r3 + r3
Va π 1
ρNaCl,
= = √ A 3 B = √ 1+ 3 .
Vs 3 2 rA 3 2 x
Now, putting the complete packing ratio as a function of ionic radii
ratio together, we obtain
π 3
√
3√2 1 + x
x < 2 − 1,
x3 +1
√
ρNaCl (x) = 2π 3 (x+1) 3
1
2 − 1 ≤ x ≤ √2−1 ,
Figure 6: Regime where rA ∼ rB
π
1 + x13 √1
√
3 2 2−1
< x.
Plotting the above function, we get Figure 8, where the black dot shows the
actual location of NaCl, using the ionic radii data found on the Wikipedia
page of CsCl (it has a discussion of the CsCl compared to NaCl where the
ionic radii at the relevant coordination number are mentioned) of
Na+ radius 102 pm
xNaCl = = .
Cl− radius 181 pm
Explicitly, the estimated packing ratio of NaCl is
ρNaCl = 0.646. (4) Figure 7: Regime where rA
rB
It is worth mentioning that the packing ratio curve is understandable intuitively—consider Figure 9: In the first
configuration, we see that the packing is not as efficient as it could be since the blue atoms (sodium) can be made
bigger so that they completely fill the space between the chloride ions. Next, when the sodium atoms get too big to
fill the holes, they start fighting for space with the chlorides and as such we see the formation of more gaps and a
decrease in packing efficiency. Finally, when the sodium atoms are quite larger than the chlorides, we get a symmetric
repeat of the first diagram.
NaCl Packing Efficiency vs. Atomic Radii Ratio
0.80
0.75
0.70
0.65
0.60
0.55
1 2 3 4
Figure 8: NaCl Packing Efficiency Curve. The black dot indicates the actual packing efficiency of NaCl. The top
green line shows the highest packing efficiency, and the lower orange line shows the asymptotic packing achieved
when one atom is negligible in size relative to the other species.
8
QUANTUM THEORY OF MATERIALS
Efthimios Kaxiras
Department of Physics and
School of Engineering and Applied Sciences
Harvard University
John D. Joannopoulos
Department of Physics
Massachusetts Institute of Technology
Published by Cambridge University Press, June 2019
, PREFACE
The book authors are enormously grateful to all the students and other members of their research groups (post-
docs, visitors) who contributed to the compilation of the solutions of the problems.
Some of the solutions date back to when one of the authors (EK) first learned the subject in a course taught by
the other author (JDJ), almost four decades ago! Those solutions were of course hand-written (often illegible) and
the ownership trail has unfortunately been lost. For the more recent ones, where the person who provided the last
version is known, we provide this information with gratitude. The rest, we thankfully attribute to the “Unknown
Student”, who toiled away somtimes for days, hopefully gaining something very valuable in the process.
WARNING: While we have made every effort to check the accuracy and completeness of the
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1
,Contents
1 From Atoms to Solids 5
1.1 Packing Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Madelung Energy of 2D Bipartite Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Simultaneous Energy and Momentum Eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Bond Formation in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6 Bonding in Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.7 Excitations of the Free-Electron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.8 Attractive Component of Lennard-Jones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.9 Approximate Morse Potential Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.10 Kronig-Penney Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 Electrons in Crystals: Translational Periodicity 35
2.1 Reciprocal Lattice Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Periodic Part of the Single-Particle Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Effective Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4 Fermi Sphere of a 2D Square Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 Band Structures of FCC Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.6 Orthogonality of Bloch States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.7 Band Width and Number of Nearest Neighbors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.8 Bloch Theorem for Many-Body Wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.10 1D and 2D free-electron DOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.11 Critical Points in 1D and 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.12 DOS at Dirac point in 1D, 2D, 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.13 Graphene Band-Structure in Tight-Binding Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.14 Tight-Binding Model of Copper(II) Peroxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3 Symmetries Beyond Translational Periodicity 71
3.1 Proper Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 Symmetries of the 2D Square Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3 Symmetries of the 2D Honeycomb Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4 Special k-Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5 Point-Group Symmetries of the Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.6 Optical Transitions of NV Center in Diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4 From Many Particles to the Single-Particle Picture 89
4.1 Hydrogen Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2 Many-Body States of the NV Center in Diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3 Hartree-Fock Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.4 Single Particle Spectrum of Hartree-Fock Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.5 Bulk Modulus of Free-Electron Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.6 Reduced Density Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.7 Meaning of Single-Particle Eigenvalues in DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.8 Exchange-Correlation Effects in the Wigner Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.9 Lindhard Dielectric Response Function for Free-Electron Gas . . . . . . . . . . . . . . . . . . . . . . . 117
4.10 Thomas-Fermi Screening Length at Zero-Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.11 Pseudopotential Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.12 Ewald Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.13 Madelung Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.14 Ewald Summation of Madelung Energy for NaCl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
2
, 4.15 Car-Parrinello Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5 Electronic Properties of Crystals 129
5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2 Graphene Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.5 Electron and Hole Concentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.6 Position of Fermi Level in Band Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.7 Semiconductor Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.8 More Realistic p-n Junction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.9 Band Bending at Metal-Semiconductor Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6 Electronic Excitations 154
6.1 Plasma Frequency of Uniform Electron Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.2 Dielectric Function From Inter-Band Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.3 Dielectric Function From Intra-Band Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.4 Lorentz Model of Dielectric Function, Application to Si . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.5 Drude Model of Dielectric Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.6 Static Dielectric Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.7 Sum Rule for Imaginary Part of Dielectric Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.8 Frenkel Exciton Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.9 Wannier Exciton Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.10 Frenkel Excitons: Wavefunction and Bloch Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7 Lattice Vibrations and Deformations 181
7.1 Phonons in Bulk Si Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.2 Harmonic Oscillator Potential and Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
7.3 Phonon Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
7.4 Thermal Expansion Coefficient, Grüneisen Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
7.5 Phonon Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
7.6 UBER bulk modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7.9 Elastic Constants of Isotropic Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7.10 Energy Density of Isotropic Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
7.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7.13 Phonons of Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8 Phonon Interactions 219
8.1 Debye-Waller Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8.2 Raman Scattering and Albrecht terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8.3 Cooper Pair Commutation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.4 BCS Number of Particles Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.5 BCS Gap Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
9 Dynamics and Topological Constraints 226
9.1 Resistivity Tensor in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
9.2 Time-Reversal Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
9.3 Exponential Form of Time-Reversal Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
9.4 Berry Curvature of a Magnetic Monopole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
3
, 9.5 Expectation Value For Electron Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
9.6 Polarization of Finite System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
9.7 Electron Velocity and Berry’s Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
9.8 Berry Curvature of 2D Honeycomb Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
9.9 Rice-Mele Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
9.10 T -Symmetry Breaking in Honeycomb Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
9.11 Dirac Points in Square Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
10 Magnetic Behavior of Solids 243
10.1 Hund’s Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
10.2 Magnetization and Band Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
10.3 Stoner Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
10.4 Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
4
, Thanks to Laura Kulowski for undertaking a significant amount of compiling, equation number checking, and
formatting of submitted solutions.
1 From Atoms to Solids
1.1 Packing Ratio
Solution by Robert Hoyt (2016) and Cedric Flamant (2018)
An important consideration in the formation of crystals is the so-called packing ratio or filling fraction. For each
of the elemental crystals (simple cubic, face-centered cubic, body-centered cubic), calculate the packing ratio,
that is the percentage of volume occupied by the atoms modeled as touching hard spheres.
Simple Cubic
Note that the simple cubic structure, shown in Figure 1 (which can be tiled
to cover all space) represents a unit volume taken up by the crystal. The
cut spheres are the atoms. If the atom has radius r, then we see from the
diagram that the side length of the unit block is 2r. Thus, the volume of
the block, Vs , is
3
Vs = (2r) = 8r3 .
The volume taken up by the atom can be determined by noting that there
are 8 octants of the atom in the unit cell, thus giving the total volume of
one atom. Thus,
4 3
Va = πr .
3
This leaves a packing ratio of the simple cubic structure ρsc of
Va π Figure 1: Simple Cubic
ρsc = = ≈ 0.5236. (1)
Vs 6
5
,Face-Centered Cubic
In the FCC structure we see that the radius of the unit cell is determined
by the diagonal line of the face coinciding with four copies of the sphere’s
radius. Thus,
√ 3
Vs = 2 2r .
There are also 8 octants and 6 hemispheres for a total of 4 spheres in this
unit cell. Thus, there are a total of 4 spheres in the unit volume and
16 3
Va = πr .
3
This gives a packing ratio of
π
ρfcc = √ ≈ 0.7405. (2)
3 2 Figure 2: Face-Centered Cubic
Body-Centered Cubic
In the BCC structure we see that the diagonal of the cube is equal to four
of the sphere’s radius. Hence,
√
3a2 = 4r
r
16 2 4
a= r = √ r,
3 3
where a gives the cube side length, implying that
43
Vs = a3 = √ r 3 .
3 3
There are 8 octants and 1 full sphere in the unit cell for a total of 2 spheres.
Thus,
8 3 Figure 3: Body-Centered Cubic
Va = πr .
3
This gives a packing ratio of
√
π 3
ρbcc = ≈ 0.6802. (3)
8
For the NaCl and CsCl structures, assuming that each type of ion is represented by a hard sphere and the
nearest-neighbor spheres touch, calculate the packing ratio as a function of the ratio of the two ionic radii and
find its extrema and the asymptotic values (for the ratio approaching infinity or zero, that is, with one of the two
ions being negligible in size relative to the other one). Using values of the ionic radii for the different elements,
estimate the filling fractions in the actual solids.
6
,NaCl Structure
The rock salt structure is technically two FCC lattices of Na and Cl shifted
and superimposed on one another, as seen in Figure 4, but notice that
simply an octant of this unit cell will suffice for analysis by symmetry.
The first thing to notice about the rock salt structure is that there will
be three regimes. Letting sodium be atom A and chloride be atom B and
rA and rB their corresponding radii, we intuit that the contact points of the
spheres will be different in the regions of rA
rB , rA ∼ rB (in a manner
yet to be made precise), and rA
rB .
Regime where rA
rB
In this regime, we see that the A atoms (blue) are much smaller than the
B atoms (green). The size of this characteristic cell is determined by the
B atoms coming into contact with each other at the faces of the cube while Figure 4: Rock salt structure
the A atoms simply take up the free space in between the larger B atoms.
We notice that the diagonal on√the face of the cube is coincident
√ with 2 of √ 3
atom B’s radii, so this means 2a = 2rB , implying a = 2rB and Vs = a3 = 2 2rB .
In total there are 4 octants of each atom in the cell, so
14 3 14 3 2 3 3
Va = 4 πrA + 4 πrB = π rA + rB .
83 83 3
This gives a packing ratio of
Va π r3 + r3 π
= √ A 3 B = √ 1 + x3 ,
ρNaCl,
=
Vs 3 2 rB 3 2
where x ≡ rrB
A
. Now we have to determine the limit of this regime. This
occurs when atom A is large enough that it comes into contact with atom
B along the faces as well—if it grows any bigger it will force atom B to
shrink to make room. At this point, the side length of the cell will also be
equal to rA + rB : Figure 5: Regime where rA
rB
√ rA √
rA + rB = 2rB ⇒= x = 2 − 1.
rB
√
Hence, this regime is described by x < 2 − 1.
Regime where rA ∼ rB
In this regime, the size of the cell is determined by the contact of atom A and atom B along the edges of the cell.
We see that the cell edge is equal to the sum of the two radii giving
3
Vs = (rA + rB ) .
Hence we are left with a packing efficiency of
2 3 3
Va 3 π rA + rB 2 x3 + 1
ρNaCl,∼ = = 3 = 3π 3.
Vs (rA + rB ) (x + 1)
Note that if we continue to increase the size of rA (blue) eventually we will reach the end of this regime—atom A
will come into contact with its other selves. With similar reasoning to the last section, this happens when
√ rA 1
rA + rB = 2rA ⇒ =x= √ ,
rB 2−1
7
, √
which we could have predicted from the symmetry between atoms A and B. So, this regime exists when 2−1 ≤
1
x ≤ √2−1 .
Regime where rA
rB
Now we consider the final regime where atom A has grown to dominate and
hence determines the size of the cell. By symmetry, we recoginize that the
packing efficiency must be
π r3 + r3
Va π 1
ρNaCl,
= = √ A 3 B = √ 1+ 3 .
Vs 3 2 rA 3 2 x
Now, putting the complete packing ratio as a function of ionic radii
ratio together, we obtain
π 3
√
3√2 1 + x
x < 2 − 1,
x3 +1
√
ρNaCl (x) = 2π 3 (x+1) 3
1
2 − 1 ≤ x ≤ √2−1 ,
Figure 6: Regime where rA ∼ rB
π
1 + x13 √1
√
3 2 2−1
< x.
Plotting the above function, we get Figure 8, where the black dot shows the
actual location of NaCl, using the ionic radii data found on the Wikipedia
page of CsCl (it has a discussion of the CsCl compared to NaCl where the
ionic radii at the relevant coordination number are mentioned) of
Na+ radius 102 pm
xNaCl = = .
Cl− radius 181 pm
Explicitly, the estimated packing ratio of NaCl is
ρNaCl = 0.646. (4) Figure 7: Regime where rA
rB
It is worth mentioning that the packing ratio curve is understandable intuitively—consider Figure 9: In the first
configuration, we see that the packing is not as efficient as it could be since the blue atoms (sodium) can be made
bigger so that they completely fill the space between the chloride ions. Next, when the sodium atoms get too big to
fill the holes, they start fighting for space with the chlorides and as such we see the formation of more gaps and a
decrease in packing efficiency. Finally, when the sodium atoms are quite larger than the chlorides, we get a symmetric
repeat of the first diagram.
NaCl Packing Efficiency vs. Atomic Radii Ratio
0.80
0.75
0.70
0.65
0.60
0.55
1 2 3 4
Figure 8: NaCl Packing Efficiency Curve. The black dot indicates the actual packing efficiency of NaCl. The top
green line shows the highest packing efficiency, and the lower orange line shows the asymptotic packing achieved
when one atom is negligible in size relative to the other species.
8
