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Quantum Theory of Solids
Mervyn Roy (S6)
www2.le.ac.uk/departments/physics/people/mervynroy
PA4311 Quantum Theory of Solids
Course Outline
1. Introduction and background
2. The many-electron wavefunction
- Introduction to quantum chemistry (Hartree, HF, and CI methods)
3. Introduction to density functional theory (DFT)
- Framework (Hohenberg-Kohn, Kohn-Sham)
- Periodic solids, plane waves and pseudopotentials
4. Linear combination of atomic orbitals
5. Effective mass theory
6. ABINIT computer workshop (LDA DFT for periodic solids)
Assessment:
70% final exam
30% coursework – mini ‘project’ report for ABINIT calculation
www.abinit.org
PA4311 Quantum Theory of Solids
Last time…
• Solve self-consistent Kohn-Sham single particle equations to find
𝑛(𝒓) for real interacting system
𝛻2
−
2
+ 𝑣𝑠 𝒓
𝑛 𝒓 =
𝑖
𝜓𝑖 = 𝐸𝑖 𝜓𝑖 , where,
𝜓𝑖 2 , and 𝐸 𝑛 = 𝑇𝑠 𝑛 + ∫ 𝑣 𝒓 𝑛 𝒓 𝑑𝒓 + 𝐸𝐻 [𝑛] + 𝐸𝑋𝐶 [𝑛]
𝛿𝐸𝑋𝐶
𝛿𝑛
• 𝑣𝑠 𝒓 = 𝑣 𝒓 + 𝑣𝐻 𝒓 + 𝑣𝑋𝐶 (𝒓), where 𝑣𝑋𝐶 =
• Know 𝑣𝑋𝐶 exactly for uniform electron gas – use LDA for real
materials
• Many different 𝑣𝑋𝐶 functionals available
• In principle, Kohn-Sham 𝐸𝑖 and 𝜓𝑖 are meaningless (except the HOMO). In
practice, often give decent band structures, effective masses etc
• DFT band gap problem – extend DFT (GW or TDDFT) to get excitations right
PA4311 Quantum Theory of Solids
Periodic structures and plane waves
223 course notes
Solid state text books – e.g.
• Tanner, Introduction to the Physics of Electrons in Solids,
Cambridge University press
• Hook and Hall, Solid State Physics 2nd Ed., John Wiley and Sons
• Ashcroft and Mermin, Solid State Physics, Holt-Saunders
PA4311 Quantum Theory of Solids
Crystal = Bravais lattice + basis
graphene unit cell
2 atom basis
atoms at:
𝒂1
𝒂2
0,0 and
𝑎0
Primitive cell vectors:
𝒂1 =
𝒂2 =
3 1
, 𝑎
2 2 0
3 −1
,
𝑎0
2 2
𝒂0 = 0.246 nm
PA4311 Quantum Theory of Solids
1
,0
3
2D crystal – many choices for unit cell
Hexagonal lattice, 2 atom basis
Primitive
Primitive centred
Non-primitive
Wigner-Seitz
(primitive)
PA4311 Quantum Theory of Solids
3D crystal: zinc blende structure (diamond, Si, GaAs etc)
FCC
2 atom basis
(0,0,0) and
1 1 1
, ,
4 4 4
𝑎0
Primitive cell vectors
𝒂1 = 0.5,0.5,0 𝑎0
𝒂2 = 0.5,0,0.5 𝑎0
𝒂3 = 0.5,0,0.5 𝑎0
wikipedia.org www.seas.upenn.edu
PA4311 Quantum Theory of Solids
Volume of cell,
Ω𝑐𝑒𝑙𝑙 = |𝒂1 ∙ (𝒂2 × 𝒂3 )|
Any function f(r), defined in the crystal which is the same in each
unit cell (e.g. electron density, potential etc.) must obey,
𝑓 𝒓+𝑻 =𝑓 𝒓 ,
where,
𝑻 = 𝑛1 𝒂1 + 𝑛2 𝒂2 + ⋯
𝒂1
e.g. environment is
the same at 𝒓 as it
is at 𝒓 + 2𝒂1 + 𝒂2
PA4311 Quantum Theory of Solids
𝒂1
𝒓
𝑻
𝒂2
Reciprocal lattice
𝑮 = 𝑚1 𝒃1 + 𝑚2 𝒃2 + ⋯
where reciprocal lattice vectors,
𝒃1 , 𝒃2 , …, satisfy
𝒂𝑖 𝒃𝑗 = 2𝜋𝛿𝑖𝑗
Then,
𝑮 ⋅ 𝑻 = 2𝜋(𝑛1 𝑚1 + 𝑛2 𝑚2 + … )
Wigner-Seitz cell in reciprocal
space = Brillouin zone
𝒃1 = 2𝜋
𝑎
1
,1
3
, 𝒃2 = 2𝜋
𝑎
1
, −1
3
PA4311 Quantum Theory of Solids
𝒃1
𝒃2
FCC Reciprocal lattice = BCC
𝑮 = 𝑚1 𝒃1 + 𝑚2 𝒃2 + 𝑚3 𝒃3
2𝜋
𝒃1 =
𝒂 × 𝒂3
Ω𝑐𝑒𝑙𝑙 2
2𝜋
𝒃2 =
𝒂 × 𝒂1
Ω𝑐𝑒𝑙𝑙 3
2𝜋
𝒃3 =
𝒂 × 𝒂2
Ω𝑐𝑒𝑙𝑙 1
recip
Volume of Brillouin zone = Ω𝐵𝑍
Ω𝐵𝑍 = 𝒃1 ⋅ 𝒃2 × 𝒃3 =
2𝜋 3
Ω𝑐𝑒𝑙𝑙
Léon Brillouin (1889-1969): most convenient primitive cell in reciprocal
space is the Wigner-Seitz cell - edges of BZ are Bragg planes.
PA4311 Quantum Theory of Solids
Brillouin
Zone
Question 3.1
a. Calculate the reciprocal lattice vectors for an FCC structure
Show that the FCC reciprocal lattice is body centred cubic
b. Calculate the reciprocal lattice vectors for graphene
c. Construct the graphene BZ, labelling the high symmetry points
d. Show that, in 3 dimensions, Ω𝐵𝑍 = 𝒃1 ⋅ 𝒃2 × 𝒃3 =
hint: 𝑨 × 𝑩 × 𝑪 = 𝑨. 𝑪 𝑩 − 𝑨. 𝑩 𝑪
PA4311 Quantum Theory of Solids
2𝜋 3
Ω𝑐𝑒𝑙𝑙
Example band structure for a
Zinc Blende structure crystal
Dispersion relation, 𝐸 𝒌 ,
plotted along high symmetry
lines in Brillouin zone L-G-X
5
4
conduction band
𝑛 = 2,3
valence band (heavy holes)
band, 𝑛 = 1
doubly degenerate band (no spin orbit
coupling)
𝒌
filled states, 𝑛(𝑟) =
PA4311 Quantum Theory of Solids
𝑛𝑘
𝜓𝑛𝑘
2
Fourier representation of a periodic function
If 𝑓(𝒓 + 𝑻) = 𝑓(𝒓) then,
𝑓𝑮 𝑒 𝑖𝑮⋅𝒓 ,
𝑓 𝒓 =
𝑮
where, 𝑮 are reciprocal lattice vectors and
1
𝑓𝑮 =
𝑓 𝒓 𝑒 −𝑖𝑮⋅𝒓 𝑑𝒓.
Ω𝑐𝑒𝑙𝑙 𝑐𝑒𝑙𝑙
PA4311 Quantum Theory of Solids
Bloch theorem
If 𝜓𝑛𝒌 is an eigenstate of the single-electron Hamiltonian,
𝛻2
−
2
+ 𝑣 𝒓 , then
𝜓𝑛𝒌 𝒓 + 𝑻 = ei𝒌⋅𝑻 𝜓𝑛𝒌 𝒓 .
The Bloch states, 𝜓𝑛𝒌 (𝒓), are often written in the form,
𝜓𝑛𝒌 𝒓 = 𝑒 𝑖𝒌⋅𝒓 𝑢𝑛𝒌 (𝒓)
plane wave part
periodic part - 𝑢𝑛𝒌 has the periodicity of
the lattice so 𝑢𝑛𝒌 𝒓 + 𝑻 = 𝑢𝑛𝒌 (𝒓)
Orthogonality - the 𝑢𝑛𝒌 are orthonormal within one unit cell, the 𝜓𝑛𝒌 are only
orthogonal over the whole crystal
PA4311 Quantum Theory of Solids
Question 3.2
a. If 𝑉 is the crystal volume, show that the spacing between k
2𝜋 3
𝑉
states is
in
i. a cuboid crystal
ii. a non-cuboid crystal
b. Show that the number of states in the first BZ for a single band
is 𝑁, where 𝑁 is the number of unit cells in the crystal
c. If there are 𝑁𝑎 atoms in the basis and 𝑁𝑒 electrons per atom,
show that the band index of the highest valence band is 𝑛 =
𝑁𝑎 𝑁𝑒 /2
PA4311 Quantum Theory of Solids