PHY 752 Solid State Physics 11-11:50 AM MWF Olin 107 Plan for Lecture 9: Reading: Chapter 8 in MPM; Electronic Structure 1. Linear combination of atomic orbital (LCAO) method 2. Slater and Koster analysis 3. Wannier representation 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 1 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 2 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 3 Linear combinations of atomic orbitals (LCAO) methods for analyzing electronic structure a R Bloch wave: nk r e ik r u nk r periodic function Let R a τ a T lattice translation basis vector Bloch wave identity: LCAO basis functions with Bloch symmetry: nk r T e ik T nk r a nlm k (r ) e ik τ a T a nlm r τ a T T 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 4 LCAO methods -- continued LCAO basis functions with Bloch symmetry: a ka nlm (r ) eik Tnlm r τ a T Example for nlm (r) 100 (r) Ce r T k=0 k=p/2a k=p/a 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 5 LCAO methods -- continued – angular variation http://winter.group.shef.ac.uk/orbitron/ l=0 l=1 l=2 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 6 LCAO methods -- continued – angular variation While, for atoms the “z” axis is an arbitrary direction, for diatomic molecules and for describing bonds, it is convenient to take the “z” axis as the bond direction. Atom l=0 l=1 l=2 2/04/2015 Bond symbol m=0 s m=0 p m= ±1 p m=0 d m= ± 1 d m= ± 2 d l=0 l=1 l=2 PHY 752 Spring 2015 -- Lecture 9 l=0 l=0 l=1 l=0 l=1 l= 2 symbol s s p s p d 7 LCAO methods -- continued – bond types pps sss ppp 2/04/2015 ddp PHY 752 Spring 2015 -- Lecture 9 8 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 9 LCAO methods -- continued – Slater-Koster analysis LCAO basis functions with Bloch symmetry: a nlm k (r ) e ik τ a T a nlm r τ a T T Approximate Bloch wavefunction: k (r ) X aknlm ka nlm (r ) a nlm In this basis, we can estimate the electron energy by variationally computing the expectation value of the Hamiltonian: k H k E k k k Terms in this expansion have the form: e ik τ a τ b T na' l ' m ' r τ b H nla m r τ a T T 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 10 LCAO methods – Slater-Koster analysis -- continued e ik τ a τ b T a a na' l ' m ' r τ b H nlm r τ T T Notation in Slater-Koster paper k τ a τ b T l m n na' l ' m ' r τ b H nalm r τ a T : 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 11 LCAO methods – Slater-Koster analysis -- continued Simple cubic lattice on site NN ( s / s ) Es , s (000) 2 Es , s (100) cos cos cos NNN +4Es , s (110) cos cos cos cos τ a l cos cos ... τ b T apxˆ aqyˆ razˆ p p2 q2 r 2 2/04/2015 m q p2 q2 r 2 n r p2 q2 r 2 PHY 752 Spring 2015 -- Lecture 9 12 LCAO methods – summary LCAO basis functions with Bloch symmetry: a nlm k (r ) e ik τ a T a nlm r τ a T T Approximate Bloch wavefunction: k (r ) X a nlm k ka nlm (r ) a nlm In this basis, we can estimate the electron energy by variationally computing the expectation value of the Hamiltonian: E k k H k k k Terms in this expansion have the form: e ik τ a τ b T a na' l ' m ' r τ b H nlm r τa T T and also e ik τ a τ b T a na' l ' m ' r τ b nlm r τa T T 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 13 LCAO methods – summary LCAO basis functions with Bloch symmetry: a nlm k (r ) e ik τ a T a nlm r τ a T T Is there a “best choice” for atom-centered functions? Introduction to the Wannier representation 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 14 Wannier representation of electronic states Note: This formulation is based on the relationship between the Bloch and Wannier representations and does not necessarily imply an independent computational method. Bloch wave: nk r e Bloch wave identity: ik r u nk r nk r T e ik T nk r Orthonormality: nk r n ' k ' r d nn d 3 k k ' Wannier function in lattice cell T, associated with band n is given by Wn (r T) 2/04/2015 V 2p 3 ik T d k e nk (r ) 3 PHY 752 Spring 2015 -- Lecture 9 15 Wannier representation of electronic states -- continued Wannier function in lattice cell T, associated with band n is given by: Wn (r T) V 2p 3 ik T d k e nk (r ) 3 Note that: Wn (r T) Wn ' (r T ') d nn d TT ' Comment: Wannier functions are not unique since the the Bloch function may be multiplied by a k-dependent phase, which may generate a different function Wn(r-T). 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 16 Example from RMP 84, 1419 (2012) by Mazari, Mostofi, Yates, Souza, and Vanderbilt 2/04/2015 PHY 752 Spring 2015 -- Lecture 9 17