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.