### Lecture 6

```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
() for real interacting system
2
−
2
+
=

=   , where,
2 , and   =   + ∫      +  [] +  []

•   =   +   +  (), where  =
• Know  exactly for uniform electron gas – use LDA for real
materials
• Many different  functionals available
• In principle, Kohn-Sham  and  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
1
2
2 atom basis
atoms at:
0,0 and
0
Primitive cell vectors:
1 =
2 =
3 1
,
2 2 0
3 −1
,
0
2 2
0 = 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
0
Primitive cell vectors
1 = 0.5,0.5,0 0
2 = 0.5,0,0.5 0
3 = 0.5,0,0.5 0
wikipedia.org www.seas.upenn.edu
PA4311 Quantum Theory of Solids
Volume of cell,
Ω = |1 ∙ (2 × 3 )|
Any function f(r), defined in the crystal which is the same in each
unit cell (e.g. electron density, potential etc.) must obey,
+ =  ,
where,
= 1 1 + 2 2 + ⋯
1
e.g. environment is
the same at  as it
is at  + 21 + 2
PA4311 Quantum Theory of Solids
1

2
Reciprocal lattice
= 1 1 + 2 2 + ⋯
where reciprocal lattice vectors,
1 , 2 , …, satisfy
= 2
Then,
⋅  = 2(1 1 + 2 2 + … )
Wigner-Seitz cell in reciprocal
space = Brillouin zone
1 = 2

1
,1
3
, 2 = 2

1
, −1
3
PA4311 Quantum Theory of Solids
1
2
FCC Reciprocal lattice = BCC
= 1 1 + 2 2 + 3 3
2
1 =
× 3
Ω 2
2
2 =
× 1
Ω 3
2
3 =
× 2
Ω 1
recip
Volume of Brillouin zone = Ω
Ω = 1 ⋅ 2 × 3 =
2 3
Ω
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, Ω = 1 ⋅ 2 × 3 =
hint:  ×  ×  = .   − .
PA4311 Quantum Theory of Solids
2 3
Ω
Example band structure for a
Zinc Blende structure crystal
Dispersion relation,   ,
plotted along high symmetry
lines in Brillouin zone L-G-X
5
4
conduction band
= 2,3
valence band (heavy holes)
band,  = 1
doubly degenerate band (no spin orbit
coupling)

filled states, () =
PA4311 Quantum Theory of Solids

2
Fourier representation of a periodic function
If ( + ) = () then,
⋅ ,
=

where,  are reciprocal lattice vectors and
1
=
−⋅ .
Ω
PA4311 Quantum Theory of Solids
Bloch theorem
If  is an eigenstate of the single-electron Hamiltonian,
2
−
2
+   , then
+  = ei⋅   .
The Bloch states,  (), are often written in the form,
=  ⋅  ()
plane wave part
periodic part -  has the periodicity of
the lattice so   +  =  ()
Orthogonality - the  are orthonormal within one unit cell, the  are only
orthogonal over the whole crystal
PA4311 Quantum Theory of Solids
Question 3.2
a. If  is the crystal volume, show that the spacing between k
2 3

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 , where  is the number of unit cells in the crystal
c. If there are  atoms in the basis and  electrons per atom,
show that the band index of the highest valence band is  =
/2
PA4311 Quantum Theory of Solids
```