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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… Solved the single-electron Schrödinger equation 2 − + 2 = for and by expanding in a basis of plane waves Derived the central equation – an infinite set of coupled simultaneous equations Examined solutions when the potential was zero, or weak and periodic Reduced zone scheme, ’ = − PA4311 Quantum Theory of Solids Band gaps at the BZ boundaries Central equation … … … − 2 ⋮ ⋮ ⋮ − −2 … 2 − 2 − … + 2 − 2 ⋮ … 2 − 2 ⋮ ⋮ ⋮ − ⋮ =0 In a calculation – must cut off the infinite sum at some ’’ = Supply Fourier components of potential, , up to then calculate expansion coefficients (single particle wavefunctions) and energies The more terms we include, the better the results will be PA4311 Quantum Theory of Solids Pseudopotentials Libraries of ‘standard’ pseudopotentials available for most atoms in the periodic table () = () + []() + []() ( − , − ) = =1 =1 = =1 Ω () Ω () is independent of crystal structure - tabulated for each atom type en.wikipedia.org/wiki/Pseudopotential PA4311 Quantum Theory of Solids # Skeleton abinit input file (example for an FCC crystal) ecut 15 # cut-off energy determines number of Fourier components in # wavefunction from ecut = 0.5|k+G_max|^2 in Hartrees # “… an enormous effect on the quality of a calculation; …the larger ecut is, the better converged the calculation is. For fixed geometry, the total energy MUST always decrease as ecut is raised…” # Definition of unit cell acell 3*5.53 angstrom # lattice constant =5.53 is the same in all 3 directions rprim # primitive cell definition 0.00000E+00 0.50000E+00 0.50000E+00 # first primitive cell vector, a_1 0.50000E+00 0.00000E+00 0.50000E+00 # a_2 0.50000E+00 0.50000E+00 0.00000E+00 # a_3 # Definition of k points within the BZ at which to calculate E_nk, \psi_nk # Definition of the atoms and the basis # Definition of the SCF procedure # etc. PA4311 Quantum Theory of Solids Supercells • using plane waves in aperiodic structures Calculate for a periodic structure with repeat length, 0 = lim 2 →∞ If system is large in real space, reciprocal lattice vectors are closely spaced. So, for a given , get many more plane waves in the basis PA4311 Quantum Theory of Solids ABINIT tutorial • 14.00 Tuesday November 25th – room G • Work through tutorial tasks (based on online abinit tutorial at www.abinit.org) Assessed task • Calculate GaAs ground state density, band structure, and effective mass • Write up results as an ‘internal report’ 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 Semi-empirical methods 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 PA4311 Quantum Theory of Solids Semi-empirical methods Devise non-self consistent, independent particle equations that describe the real properties of the system (band structure etc.) Use semi-empirical parameters in the theory to account for all of the difficult many-body physics PA4311 Quantum Theory of Solids = 2 /2 = primary photoelectron KE Photoemission , Primary photoelectron ℏ (no scattering – ∴ must originate close to surface) = ℏ − − Vacuum level ℏω ⋮ Valence band Core levels Photoemission spectrum from Au, ℏ = 1487 eV Fermi edge, where = 0 Kinetic energy PA4311 Quantum Theory of Solids Angle-resolved photoemission spectroscopy Surface normal ℏ spectrometer electrons ⊥ ∥ ∥ = sin = 2 sin is conserved across the boundary Malterre et al, New J. Phys. 9 (2007) 391 PA4311 Quantum Theory of Solids Tight binding or LCAO method • Plane wave basis good when the potential is weak and electrons are nearly free (e.g simple metals) • But many situations where electrons are highly localised (e.g. insulators, transition metal d-bands etc.) • Describe the single electron wavefunctions in the crystal in terms of atomic orbitals (linear combination of atomic orbitals) • Calculate () for highest valence bands and lowest conduction bands • Solid State Physics, NW Ashcroft, ND Mermin • Physical properties of carbon nanotubes, R Saito, G Dresselhaus, MS Dresselhaus • Simplified LCAO Method for the Periodic Potential Problem, JC Slater and GF Koster, Phys. Rev. 94, 1498, (1954). PA4311 Quantum Theory of Solids Linear combination of atomic orbitals In a crystal, = + Δ is the single particle hamiltonian for an atom, = Construct Bloch states of the crystal, 1 , = ⋅ − , where , = , Expand crystal wavefunctions (eigenstates of = + Δ ) as Ψ (, ) = , labels different atomic orbitals and different inequivalent atom positions in the unit cell PA4311 Quantum Theory of Solids Expansion coefficients Use the variational method to find the best values of the Minimise subject to the constraint that Ψ is normalised = Ψ Ψ − Ψ Ψ − 1 ∗ ′ ′ − = ⋮ ′ ( − ) = 0 (H − I) = 0 PA4311 Quantum Theory of Solids ∗ ′ ′ − 1 ′ s-band from a single s-orbital Real space lattice – 1 atom basis 1 = (1,0,0) 1 = Reciprocal space lattice 2 (1,0,0) 1 atom basis, 1 type of orbital so = = , H is a 1 × 1 matrix and 1 = = ⋮ ′ ⋅(− ) − ′ ( − ) ′ = + 21 cos() PA4311 Quantum Theory of Solids