Transcript Document

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
Semi-empirical methods
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
PA4311 Quantum Theory of Solids
Last time…
Pseudopotentials and supercells
Semi-empirical methods used for describing large systems >> few hundred atoms
Use non-self consistent, independent particle, equations to calculate 𝐸(𝒌), 𝜓𝑛𝒌
- modify parameters in the theory semi-empirically to match experiment
LCAO method – physically motivated expansion of Ψ𝑗 (𝒌, 𝒓) in Bloch
functions made from atomic orbitals, 𝜙𝑛 𝒌, 𝒓 =
Ψ𝑗 𝒌, 𝒓 =
𝑐𝑗𝑛 𝒌 𝜙𝑛 𝒌, 𝒓
1
N
𝑹𝑒
𝑖𝒌⋅𝑹 𝜓 (𝒓 −
𝑛
𝑹)
𝑛 labels different orbitals
and different basis sites in
the expansion
𝑛
(𝐻𝑚𝑛 −𝜖𝑗𝒌 𝛿𝑚𝑛 )𝑐𝑗𝑛 = 0
𝑛
𝐻𝑚𝑛 = 𝜙𝑚 𝐻 𝜙𝑛
1
=
𝑁
′
𝑒 𝑖𝒌⋅(𝑹−𝑹 ) 𝜓𝑚 (𝒓 − 𝑹′ ) 𝐻 𝜓𝑛 (𝒓 − 𝑹)
𝑹
𝑹′
PA4311 Quantum Theory of Solids
s-band from a single s-orbital
Real space lattice – 1 atom basis
𝒂1 = 𝑎(1,0,0)
𝒃1 =
Reciprocal space lattice
2𝜋
(1,0,0)
𝑎
1 atom basis, 1 type of orbital so 𝑛 = 𝑚 = 𝑠, H is a 1 × 1 matrix and
1
𝜖𝒌 = 𝐻𝑠𝑠 =
𝑁
⋮
′
𝑒 𝑖𝒌⋅(𝑹−𝑹 ) 𝜓𝑠 𝒓 − 𝑹′ 𝐻 𝜓𝑠 (𝒓 − 𝑹)
𝑅
𝑅′
𝜖𝑘 = 𝜖𝑠 + 2𝛾1 cos(𝑘𝑎)
PA4311 Quantum Theory of Solids
Band width
Real space lattice – 1 atom basis
𝒂1 = 𝑎(1,0,0)
𝜖𝑘 = 𝜖𝑠 + 2𝛾1 cos(𝑘𝑎)
Overlap integral, 𝛾1 = 𝜓𝑠 (𝒓 ± 𝒂1 ) 𝐻 𝜓𝑠 (𝒓) ,
treated as an empirical parameter used to fit
experiment
Band width proportional to 𝛾1
𝛾1 falls off rapidly with atomic separation, 𝑎
2
e.g. diamond: 𝛾1 = 𝐴𝑎−2 𝑒 −𝛼𝑎
- Hu et al, J. Phys. CM 4, 6047 (1992)
PA4311 Quantum Theory of Solids
4𝛾1
Question 4.1
A 2D rectangular lattice has primitive cell vectors 𝑎1 (1,0,0) and 𝑎2 0,1,0 .
i. show that the reciprocal lattice vectors are 𝒃1 = 2𝜋/𝑎1 (1,0,0) and 𝒃2 =
2𝜋/𝑎2 (0,1,0)
ii. sketch the real space and reciprocal lattices
iii. calculate 𝐸(𝑘𝑥 , 𝑘𝑦 ) for a single s-band and sketch 𝐸(𝑘𝑥 , 0) and 𝐸(0, 𝑘𝑦 )
within the first Brillouin zone
iv. state the band width
PA4311 Quantum Theory of Solids
𝝅 bands of trans-polyacetylene
𝒂1 = 𝑎(1,0,0)
Real space lattice – 2 atom basis
Plan view
H
C
‘A’
‘B’
𝒃1 =
𝑎/2
2𝜋
𝑎
1,0,0 , reciprocal space
Unit cell
pz orbitals
Real space lattice – 2 atom basis
side view
𝑎/2
‘B’ lattice site
𝒂1 = 𝑎(1,0,0)
‘A’ lattice site
PA4311 Quantum Theory of Solids
𝝅 bands of trans-polyacetylene
|𝐇 − 𝐸𝐈| = 0
2 atom basis, so 2 types of Bloch state,
𝜙𝐴 , 𝜙𝐵 , and 𝐇 is a 2 × 2 matrix
𝐻𝐴𝐴 − 𝐸
𝐻𝐴𝐵
𝐻𝐴𝐵
=0
𝐻𝐵𝐵 − 𝐸
𝑘𝑎
𝜖𝑘 = 𝜖𝑝 + 2𝛾1 cos 𝑘𝑎 ± 2𝛾 cos
2
𝛾 ≫ 𝛾1 because overlap integral falls off
rapidly with separation
Select 𝛾, 𝛾1 to fit experiment: usefulness of empirical tight binding determined by
transferability of overlap integral parameters
PA4311 Quantum Theory of Solids
𝝅 bands of Graphene
only 2pz contributes to conduction/valence 𝜋 bands
1 orbital, 2 atoms - 2 × 2 hamiltonian matrix
𝐸 𝑘𝑥 , 𝑘𝑦 = ±𝛾 1 + 4 cos
𝑘𝑦 𝑎
𝑘𝑦 𝑎
3𝑘𝑥 𝑎
2
cos
+ 4 cos
2
2
2
𝐸+ solution
linear 𝐸 𝒌
near 𝑘 = 𝐾
𝐾
PA4311 Quantum Theory of Solids
Question 4.2
A 3 dimensional face centred cubic crystal has lattice constant, 𝑎.
i. show that the dispersion relation of the band arising from a single s-orbital is,
𝐸 𝑘
𝑘𝑦 𝑎
𝑘𝑦 𝑎
𝑘𝑥 𝑎
𝑘𝑧 𝑎
cos
+ cos
cos
2
2
2
2
𝑘𝑧 𝑎
𝑘𝑥 𝑎
+ cos
cos
.
2
2
= 𝜖𝑠 + 4𝛾1 cos
ii. what is the band width?
Question 4.3
A 1D crystal with lattice constant, 𝑎, has a 2 atom basis with atoms at 0 and at 𝑎/4.
i. Calculate 𝐸 𝑘 for the LCAO bands arising from a single s-orbital on each site.
ii.Show that the band gap at the zone boundary is 2 2𝛾 where 𝛾 is the overlap
integral.
PA4311 Quantum Theory of Solids