Transcript Lecture 5
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… Can’t calculate full 𝑁-electron wavefunction, Ψ0 , for 𝑁 > ~10… Hohenberg-Kohn • Can, in principle, determine everything from 𝑛0 (𝒓) - Simple proof that 𝑛0 (𝒓) uniquely determines 𝑣(𝒓) • Can, in principle, calculate 𝑛0 (𝒓), by finding the density that minimises the functional 𝐸 = 𝐹𝐻𝐾 𝑛 + ∫ 𝑣 𝒓 𝑛 𝒓 𝑑𝒓 − 𝜇(∫ 𝑛(𝒓)𝑑𝒓 – 𝑁) - But, can’t find 𝐹𝐻𝐾 [𝑛] without finding Ψ0 … PA4311 Quantum Theory of Solids 𝜇 = chemical potential Last time… Kohn-Sham • Map the difficult interacting problem onto an auxiliary noninteracting problem that has the same 𝑛0 𝒓 (and is easy to solve) • Single-particle KS equations for each Kohn-Sham orbital with all the difficult many-body terms expressed in terms of the exchange and correlation energy 𝐸𝑋𝐶 PA4311 Quantum Theory of Solids Kohn-Sham Want to find 𝑛0 𝒓 by minimising 𝐸 𝑛0 = 𝑇 𝑛0 + 𝑊 𝑛0 + ∫ 𝑛0 𝒓 𝑣 𝒓 𝑑𝒓 Invent auxiliary non-interacting system with same density, 𝐻𝑆 Ψs = 𝐸Ψs where 𝐻𝑆 = 𝑖 −𝛻𝑖2 2 + 𝑣𝑠 𝒓𝑖 , Ψs is a single determinant and 𝑁 𝑖 𝜓𝑖 𝐸 𝑛0 = 𝑇 𝑛0 + 𝑊 𝑛0 + ∫ 𝑛0 𝒓 𝑣 𝒓 𝑑𝒓 1 𝑛 𝑟 𝑛 𝑟′ ′ + 𝐸 [𝑛 ] 𝐸 𝑛0 = 𝑇𝑠 𝑛0 + ∫ 𝑛0 𝑟 𝑣 𝑟 𝑑𝑟 + ∫ ∫ 𝑑𝑟𝑑𝑟 𝑋𝐶 0 2 𝑟 − 𝑟′ Non-interacting system KE External potential = 𝑛0 (𝒓) Exchange and correlation energy makes up the difference Hartree interaction 𝐸𝑋𝐶 𝑛 = 𝑇 𝑛 − 𝑇𝑠 𝑛 + 𝑊 𝑛 − 𝐸𝐻 𝑛 PA4311 Quantum Theory of Solids 2 Trick is that 𝐸𝑋𝐶 is small Atom 𝐸 𝑇𝑠 ∫ n𝟎 𝒓 𝒗 𝒓 𝒅𝒓 𝑬𝑯 𝑬𝑿𝑪 He -2.83 2.77 -6.63 2.00 -0.97 Ne -128.23 127.74 -309.99 65.73 -11.71 Ar -525.95 524.97 -1253.13 231.46 -29.24 Kr -2750.15 2747.81 -6577.87 1171.72 -91.82 Xe -7228.86 7225.10 -17159.16 2880.92 -175.71 Total ground state energy, 𝐸 (Hartrees.), calculated within the LDA – CA Ullrich, Table 2.1 PA4311 Quantum Theory of Solids Exchange correlation potential 1 𝑛 𝑟 𝑛 𝑟′ ′ 𝐸 𝑛 = 𝑇𝑠 𝑛 + ∫ 𝑛 𝑟 𝑣 𝑟 𝑑𝑟 + ∫ ∫ 𝑑𝑟𝑑𝑟 + 𝐸𝑋𝐶 [𝑛] 2 𝑟 − 𝑟′ where 𝐸𝑋𝐶 𝑛 = 𝑇 𝑛 − 𝑇𝑠 𝑛 𝐻𝑠 = 𝑖 + 𝑊 𝑛 − 𝐸𝐻 𝑛 −𝛻𝑖2 + 𝑣𝑠 𝒓𝑖 2 Single particle equations are where, 𝑣𝑠 𝒓 = 𝑣 𝒓 + −𝛻2 2 + 𝑣𝑠 𝒓 𝑛 𝒓 𝑑𝒓′ + 𝑣𝑋𝐶 [𝑛] 𝒓 ′ 𝒓−𝒓 PA4311 Quantum Theory of Solids 𝜓𝑖 = 𝜖𝑖 𝜓𝑖 , Exchange and correlation potential Local density approximation 𝑣𝑋𝐶 𝛿𝐸𝑋𝐶 𝑛 𝑟 = 𝛿𝑛 𝑣𝑋𝐶 can be accurately calculated for a uniform electron gas - exchange part known exactly – 𝑣𝑋 𝑛 𝑟 ∝ 𝑛(𝑟)1/3 𝒓1 , 𝑛1 Use uniform electron gas result at density 𝑛1 for 𝑣𝑋𝐶 (𝒓1 ) 𝒓2 , 𝑛2 Use uniform electron gas result at density 𝑛2 for 𝑣𝑋𝐶 (𝒓2 ) Expect LDA should work well for simple metals where density variations are slow ie. where 𝛻𝑛 𝒓 𝑛 𝒓 ≪ 𝑘𝐹 (𝒓) PA4311 Quantum Theory of Solids Local density approximation LDA works surprisingly well in a wide range of materials • Atomic and molecular ground state energies within ~ 1-5% • Molecular equilibrium distances and geometries within ~ 3% • Fermi surfaces of bulk metals within ~ few % • Lattice constants of solids within ~ 2% • Vibrational frequencies and phonon energies within ~ 2% PA4311 Quantum Theory of Solids Self consistent field (SCF) Use new 𝑛(𝒓) Iterative procedure - essentially 𝑖𝑛 𝑛𝑖+1 = 𝛼𝑛𝑜𝑢𝑡 + 1 − 𝛼 𝑛𝑖𝑖𝑛 Pre-condition or ‘seed’ SCF loop The 𝐾𝑆 single particle 𝑖𝑛 Guess 𝑛 (𝒓) equations must be solved Set maximum allowed number of steps self-consistently Calculate effective potential 𝑣𝑠 𝒓 = 𝑣(𝒓) + 𝑣𝐻 [𝑛] + 𝑣𝑋𝐶 [𝑛] 1 Solve KS Eq.s - − 2 𝛻𝑖2 − 𝑣𝑠 𝒓𝑖 Calculate new 𝑛𝑜𝑢𝑡 (𝒓) = No Self consistent? Yes PA4311 Quantum Theory of Solids 𝜓 𝒓𝑖 = 𝐸𝑖 𝜓 𝒓𝑖 𝑖 𝜓𝑖 (𝒓) 2 Set acceptable tolerance for self consistency e.g. difference in total energies < 10−6 H Finished, output 𝐸[𝑛] and 𝑛(𝑟) # Skeleton abinit input file (example for H2) #Definition of the atoms ntypat 1 # There is only one type of atom znucl 1 # atomic number natom 2 # There are two atoms typat 1 1 # They both are of type 1, that is, Hydrogen xcart # cartesian co-ordinates for each atom -0.7 0.0 0.0 # Triplet giving the cartesian coordinates of atom 1, in Bohr 0.7 0.0 0.0 # Triplet giving the cartesian coordinates of atom 2, in Bohr #Definition of the SCF procedure nstep 10 # Maximal number of SCF cycles toldfe 1.0d-6 # Will stop when, twice in a row, the difference between two # consecutive total energies differ by less than toldfe (in Hartree) diemac 2.0 # precondition the SCF cycle (see documentation) iscf 7 # default - use Pulay mixing of the density (update info from 7 steps) PA4311 Quantum Theory of Solids Beyond the LDA – acronym zoo (ii) exact RPA Unoccupied orbitals ℎ𝑦𝑏𝑟𝑖𝑑 Hyper-GGA hybrids Exact exchange 𝐸𝑋𝐶 = 𝑎𝐸𝑋𝑒𝑥𝑎𝑐𝑡 + 1 − 𝑎 𝐸𝑋𝐺𝐺𝐴 + 𝐸𝐶𝐺𝐺𝐴 e.g. B3LYP Meta-GGA 𝛻2𝑛 𝑟 , 𝜏 PKZB, TPSS, VSXC etc. GGA 𝛻𝑛(𝑟) Hundreds of GGA functionals: PBE, BLYP, PW91 etc. etc. LDA 𝑛(𝑟) Slater 𝑋𝛼, LSDA Hartree 0 CA Ullrich, Fig. 2.7 PA4311 Quantum Theory of Solids LDA + U Kohn-Sham eigenvalues −𝛻 2 + 𝑣𝑠 𝒓 2 𝜓 𝑖 = 𝜖𝑖 𝜓 𝑖 𝑛 𝒓 ′ 𝑣𝑠 𝒓 = 𝑣 𝒓 + 𝑑𝒓 + 𝑣𝑋𝐶 [𝑛] 𝒓 ′ 𝒓−𝒓 Eigenvalues, 𝜖𝑖 , have no physical meaning • except - energy of highest occupied state 𝜖𝑁 𝑁 = −𝐼 Band gap problem with DFT • KS band gap 𝜖𝑁+1 𝑁 − 𝜖𝑁 𝑁 is not the correct gap • 𝐸𝑔𝑎𝑝 = 𝜖𝑁+1 𝑁 + 1 − 𝜖𝑁 (𝑁) • Problem related to lack of derivative discontinuities in 𝑣𝑋𝐶 • Extend DFT (TDDFT, GW etc.) to get excitations right PA4311 Quantum Theory of Solids DFT in crystals A crystal is a periodic structure Specified by • types and positions of atoms in one repeat • rules that describe the repetitions crystal = Bravais lattice + basis PA4311 Quantum Theory of Solids 2D crystal graphene unit cell 2 atom basis atoms at: a1 a2 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 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 0.5,0.5,0 𝑎0 0.5,0,0.5 𝑎0 0.5,0,0.5 𝑎0 www.seas.upenn.edu wikipedia.org PA4311 Quantum Theory of Solids