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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 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 PA4311 Quantum Theory of Solids Last time… • Solve self-consistent Kohn-Sham single particle equations to find () for real interacting system 2 − 2 + = = , where, 2 , 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 Crystal = Bravais lattice + basis graphene unit cell 1 2 2 atom basis atoms at: 0,0 and 0 Primitive cell vectors: 1 = 2 = 3 1 , 2 2 0 3 −1 , 0 2 2 0 = 0.246 nm PA4311 Quantum Theory of Solids 1 ,0 3 2D crystal – many choices for unit cell Hexagonal lattice, 2 atom basis Primitive Primitive centred Non-primitive Wigner-Seitz (primitive) PA4311 Quantum Theory of Solids 3D crystal: zinc blende structure (diamond, Si, GaAs etc) FCC 2 atom basis (0,0,0) and 1 1 1 , , 4 4 4 0 Primitive cell vectors 1 = 0.5,0.5,0 0 2 = 0.5,0,0.5 0 3 = 0.5,0,0.5 0 wikipedia.org www.seas.upenn.edu PA4311 Quantum Theory of Solids Volume of cell, Ω = |1 ∙ (2 × 3 )| 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, = 1 1 + 2 2 + ⋯ 1 e.g. environment is the same at as it is at + 21 + 2 PA4311 Quantum Theory of Solids 1 2 Reciprocal lattice = 1 1 + 2 2 + ⋯ where reciprocal lattice vectors, 1 , 2 , …, satisfy = 2 Then, ⋅ = 2(1 1 + 2 2 + … ) Wigner-Seitz cell in reciprocal space = Brillouin zone 1 = 2 1 ,1 3 , 2 = 2 1 , −1 3 PA4311 Quantum Theory of Solids 1 2 FCC Reciprocal lattice = BCC = 1 1 + 2 2 + 3 3 2 1 = × 3 Ω 2 2 2 = × 1 Ω 3 2 3 = × 2 Ω 1 recip Volume of Brillouin zone = Ω Ω = 1 ⋅ 2 × 3 = 2 3 Ω Léon Brillouin (1889-1969): most convenient primitive cell in reciprocal space is the Wigner-Seitz cell - edges of BZ are Bragg planes. PA4311 Quantum Theory of Solids Brillouin Zone 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, Ω = 1 ⋅ 2 × 3 = hint: × × = . − . PA4311 Quantum Theory of Solids 2 3 Ω Example band structure for a Zinc Blende structure crystal Dispersion relation, , plotted along high symmetry lines in Brillouin zone L-G-X 5 4 conduction band = 2,3 valence band (heavy holes) band, = 1 doubly degenerate band (no spin orbit coupling) filled states, () = PA4311 Quantum Theory of Solids 2 Fourier representation of a periodic function If ( + ) = () then, ⋅ , = where, are reciprocal lattice vectors and 1 = −⋅ . Ω PA4311 Quantum Theory of Solids Bloch theorem If is an eigenstate of the single-electron Hamiltonian, 2 − 2 + , then + = ei⋅ . The Bloch states, (), are often written in the form, = ⋅ () plane wave part periodic 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 2 3 states is in i. a cuboid crystal ii. 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 = /2 PA4311 Quantum Theory of Solids