pa4311 quantum theory of solids quantum theory of solids mervyn roy (s6)
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PA4311 Quantum Theory of SolidsPA4311 Quantum Theory of Solids
Quantum Theory of SolidsMervyn Roy (S6)www2.le.ac.uk/departments/physics/people/mervynroy
PA4311 Quantum Theory of Solids
1. Introduction and background2. 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 orbitals5. Effective mass theory6. ABINIT computer workshop (LDA DFT for periodic solids)
Assessment: 70% final exam 30% coursework – mini ‘project’ report for ABINIT calculation
Course Outline
Semi-empirical methods
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 - modify parameters in the theory semi-empirically to match experiment
LCAO method – physically motivated expansion of in Bloch functions made from atomic orbitals,
labels different orbitals and different basis sites in the expansion
PA4311 Quantum Theory of Solids
s-band from a single s-orbital
𝒂1=𝑎(1,0,0)
Real space lattice – 1 atom basis
Reciprocal space lattice
𝒃1=2𝜋𝑎
(1,0,0)
1 atom basis, 1 type of orbital so , H is a matrix and
PA4311 Quantum Theory of Solids
Band width
𝒂1=𝑎(1,0,0)Real space lattice – 1 atom basis
Overlap integral, , treated as an empirical parameter used to fit experiment
Band width proportional to
falls off rapidly with atomic separation, e.g. diamond:
- Hu et al, J. Phys. CM 4, 6047 (1992) 4𝛾1
PA4311 Quantum Theory of Solids
Question 4.1
A 2D rectangular lattice has primitive cell vectors and
i. show that the reciprocal lattice vectors are and
ii. sketch the real space and reciprocal lattices
iii. calculate for a single s-band and sketch and within the first Brillouin zone
iv. state the band width
PA4311 Quantum Theory of Solids
𝑎 /2
bands of trans-polyacetylene𝒂1=𝑎(1,0,0)
Real space lattice – 2 atom basisPlan view
Unit cell
C
H
𝒂1=𝑎(1,0,0)
pz orbitals
Real space lattice – 2 atom basisside view
𝑎 /2‘B’ lattice site
‘A’ lattice site
reciprocal space‘A’
‘B’
PA4311 Quantum Theory of Solids
bands of trans-polyacetylene
|
2 atom basis, so 2 types of Bloch state, and is a matrix
because overlap integral falls off rapidly with separation
Select to fit experiment: usefulness of empirical tight binding determined by transferability of overlap integral parameters
PA4311 Quantum Theory of Solids
bands of Graphene
only 2pz contributes to conduction/valence bands
1 orbital, 2 atoms - hamiltonian matrix
𝐸 (𝑘𝑥 ,𝑘𝑦 )=±𝛾 [1+4 cos(√3𝑘𝑥𝑎2 )cos(𝑘𝑦𝑎
2 )+ 4cos2(𝑘𝑦𝑎2 ) ]
linear near
𝐾
solution
PA4311 Quantum Theory of Solids
Question 4.2A 3 dimensional face centred cubic crystal has lattice constant,
i. show that the dispersion relation of the band arising from a single s-orbital is,
ii. what is the band width?
Question 4.3A 1D crystal with lattice constant, has a 2 atom basis with atoms at and at .
i. Calculate for the LCAO bands arising from a single s-orbital on each site.
ii.Show that the band gap at the zone boundary is where is the overlap integral.