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 calculationwww.abinit.org

Course Outline

PA4311 Quantum Theory of Solids

Last time…• Solve self-consistent Kohn-Sham single particle equations to find

for real interacting system, where,, 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

1

2

graphene unit cellCrystal = Bravais lattice + basis

2 atom basisatoms at: and

Primitive cell vectors:

nm

PA4311 Quantum Theory of Solids

2D crystal – many choices for unit cellHexagonal lattice, 2 atom basis

Wigner-Seitz(primitive)

Primitive

Primitive centred

Non-primitive

PA4311 Quantum Theory of Solids

wikipedia.org

3D crystal: zinc blende structure (diamond, Si, GaAs etc)

FCC 2 atom basis and

Primitive cell vectors

Volume of cell,

www.seas.upenn.edu

PA4311 Quantum Theory of Solids

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,

𝒓e.g. environment is the same at as it is at

𝒂1

𝒂1 𝒂2

𝑻

PA4311 Quantum Theory of Solids

Reciprocal lattice

where reciprocal lattice vectors, , satisfy

Then,

Wigner-Seitz cell in reciprocal space = Brillouin zone

𝒃1

𝒃2

,

PA4311 Quantum Theory of Solids

FCC Reciprocal lattice = BCC

Volume of Brillouin zone =

recip

Léon Brillouin (1889-1969): most convenient primitive cell in reciprocal space is the Wigner-Seitz cell - edges of BZ are Bragg planes.

Brillouin Zone

PA4311 Quantum Theory of Solids

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, hint:

PA4311 Quantum Theory of Solids

Example band structure for a Zinc Blende structure crystal

Dispersion relation, plotted along high symmetry lines in Brillouin zone L-G-X

conduction band

valence band (heavy holes)

doubly degenerate band (no spin orbit coupling)

band,

𝑛=2,34

5

filled states,

𝒌

PA4311 Quantum Theory of Solids

Fourier representation of a periodic function

If then,

where, are reciprocal lattice vectors and

PA4311 Quantum Theory of Solids

Bloch theoremIf is an eigenstate of the single-electron Hamiltonian, , then.

The Bloch states, , are often written in the form,

plane wave partperiodic 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 states is ini. a cuboid crystalii. 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