#### Transcript Lecture 6

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 𝑛(𝒓) for real interacting system 𝛻2 − 2 + 𝑣𝑠 𝒓 𝑛 𝒓 = 𝑖 𝜓𝑖 = 𝐸𝑖 𝜓𝑖 , where, 𝜓𝑖 2 , and 𝐸 𝑛 = 𝑇𝑠 𝑛 + ∫ 𝑣 𝒓 𝑛 𝒓 𝑑𝒓 + 𝐸𝐻 [𝑛] + 𝐸𝑋𝐶 [𝑛] 𝛿𝐸𝑋𝐶 𝛿𝑛 • 𝑣𝑠 𝒓 = 𝑣 𝒓 + 𝑣𝐻 𝒓 + 𝑣𝑋𝐶 (𝒓), where 𝑣𝑋𝐶 = • Know 𝑣𝑋𝐶 exactly for uniform electron gas – use LDA for real materials • Many different 𝑣𝑋𝐶 functionals available • In principle, Kohn-Sham 𝐸𝑖 and 𝜓𝑖 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 𝒂1 𝒂2 2 atom basis atoms at: 0,0 and 𝑎0 Primitive cell vectors: 𝒂1 = 𝒂2 = 3 1 , 𝑎 2 2 0 3 −1 , 𝑎0 2 2 𝒂0 = 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 𝑎0 Primitive cell vectors 𝒂1 = 0.5,0.5,0 𝑎0 𝒂2 = 0.5,0,0.5 𝑎0 𝒂3 = 0.5,0,0.5 𝑎0 wikipedia.org www.seas.upenn.edu PA4311 Quantum Theory of Solids Volume of cell, Ω𝑐𝑒𝑙𝑙 = |𝒂1 ∙ (𝒂2 × 𝒂3 )| Any function f(r), defined in the crystal which is the same in each unit cell (e.g. electron density, potential etc.) must obey, 𝑓 𝒓+𝑻 =𝑓 𝒓 , where, 𝑻 = 𝑛1 𝒂1 + 𝑛2 𝒂2 + ⋯ 𝒂1 e.g. environment is the same at 𝒓 as it is at 𝒓 + 2𝒂1 + 𝒂2 PA4311 Quantum Theory of Solids 𝒂1 𝒓 𝑻 𝒂2 Reciprocal lattice 𝑮 = 𝑚1 𝒃1 + 𝑚2 𝒃2 + ⋯ where reciprocal lattice vectors, 𝒃1 , 𝒃2 , …, satisfy 𝒂𝑖 𝒃𝑗 = 2𝜋𝛿𝑖𝑗 Then, 𝑮 ⋅ 𝑻 = 2𝜋(𝑛1 𝑚1 + 𝑛2 𝑚2 + … ) Wigner-Seitz cell in reciprocal space = Brillouin zone 𝒃1 = 2𝜋 𝑎 1 ,1 3 , 𝒃2 = 2𝜋 𝑎 1 , −1 3 PA4311 Quantum Theory of Solids 𝒃1 𝒃2 FCC Reciprocal lattice = BCC 𝑮 = 𝑚1 𝒃1 + 𝑚2 𝒃2 + 𝑚3 𝒃3 2𝜋 𝒃1 = 𝒂 × 𝒂3 Ω𝑐𝑒𝑙𝑙 2 2𝜋 𝒃2 = 𝒂 × 𝒂1 Ω𝑐𝑒𝑙𝑙 3 2𝜋 𝒃3 = 𝒂 × 𝒂2 Ω𝑐𝑒𝑙𝑙 1 recip Volume of Brillouin zone = Ω𝐵𝑍 Ω𝐵𝑍 = 𝒃1 ⋅ 𝒃2 × 𝒃3 = 2𝜋 3 Ω𝑐𝑒𝑙𝑙 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, Ω𝐵𝑍 = 𝒃1 ⋅ 𝒃2 × 𝒃3 = hint: 𝑨 × 𝑩 × 𝑪 = 𝑨. 𝑪 𝑩 − 𝑨. 𝑩 𝑪 PA4311 Quantum Theory of Solids 2𝜋 3 Ω𝑐𝑒𝑙𝑙 Example band structure for a Zinc Blende structure crystal Dispersion relation, 𝐸 𝒌 , plotted along high symmetry lines in Brillouin zone L-G-X 5 4 conduction band 𝑛 = 2,3 valence band (heavy holes) band, 𝑛 = 1 doubly degenerate band (no spin orbit coupling) 𝒌 filled states, 𝑛(𝑟) = PA4311 Quantum Theory of Solids 𝑛𝑘 𝜓𝑛𝑘 2 Fourier representation of a periodic function If 𝑓(𝒓 + 𝑻) = 𝑓(𝒓) then, 𝑓𝑮 𝑒 𝑖𝑮⋅𝒓 , 𝑓 𝒓 = 𝑮 where, 𝑮 are reciprocal lattice vectors and 1 𝑓𝑮 = 𝑓 𝒓 𝑒 −𝑖𝑮⋅𝒓 𝑑𝒓. Ω𝑐𝑒𝑙𝑙 𝑐𝑒𝑙𝑙 PA4311 Quantum Theory of Solids Bloch theorem If 𝜓𝑛𝒌 is an eigenstate of the single-electron Hamiltonian, 𝛻2 − 2 + 𝑣 𝒓 , then 𝜓𝑛𝒌 𝒓 + 𝑻 = ei𝒌⋅𝑻 𝜓𝑛𝒌 𝒓 . The Bloch states, 𝜓𝑛𝒌 (𝒓), are often written in the form, 𝜓𝑛𝒌 𝒓 = 𝑒 𝑖𝒌⋅𝒓 𝑢𝑛𝒌 (𝒓) plane wave part periodic part - 𝑢𝑛𝒌 has the periodicity of the lattice so 𝑢𝑛𝒌 𝒓 + 𝑻 = 𝑢𝑛𝒌 (𝒓) Orthogonality - the 𝑢𝑛𝒌 are orthonormal within one unit cell, the 𝜓𝑛𝒌 are only orthogonal over the whole crystal PA4311 Quantum Theory of Solids Question 3.2 a. If 𝑉 is the crystal volume, show that the spacing between k 2𝜋 3 𝑉 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 𝑁, where 𝑁 is the number of unit cells in the crystal c. If there are 𝑁𝑎 atoms in the basis and 𝑁𝑒 electrons per atom, show that the band index of the highest valence band is 𝑛 = 𝑁𝑎 𝑁𝑒 /2 PA4311 Quantum Theory of Solids