1. PA4311 Quantum Theory of Solids Quantum Theory of Solids Mervyn Roy (S6) www2.le.ac.uk/departments/physics/people/mervynroy

2. PA4311 Quantum Theory of Solids 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 Course Outline

3. PA4311 Quantum Theory of Solids Last time…

4. 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 2 nd Ed., John Wiley and Sons Ashcroft and Mermin, Solid State Physics, Holt-Saunders

5. PA4311 Quantum Theory of Solids graphene unit cell Crystal = Bravais lattice + basis

6. PA4311 Quantum Theory of Solids 2D crystal – many choices for unit cell Hexagonal lattice, 2 atom basis Wigner-Seitz (primitive) Primitive Primitive centred Non-primitive

7. PA4311 Quantum Theory of Solids wikipedia.org 3D crystal: zinc blende structure (diamond, Si, GaAs etc) www.seas.upenn.edu

8. PA4311 Quantum Theory of Solids

9. Wigner-Seitz cell in reciprocal space = Brillouin zone

10. PA4311 Quantum Theory of Solids FCC Reciprocal lattice = BCC 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

11. PA4311 Quantum Theory of Solids Question 3.1

12. PA4311 Quantum Theory of Solids conduction band valence band (heavy holes) doubly degenerate band (no spin orbit coupling)

13. PA4311 Quantum Theory of Solids Fourier representation of a periodic function

14. PA4311 Quantum Theory of Solids Bloch theorem plane wave part

15. PA4311 Quantum Theory of Solids Question 3.2