Lecture 8
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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 ðŧ2 − + ðĢð ð 2 ððð = ðļðð ððð for ðļðð and ððð by expanding ððð 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, ð’ = ð − ðŪ PA4311 Quantum Theory of Solids Band gaps at the BZ boundaries Central equation … … … ð−ð 2 âŪ âŪ âŪ ðĢ−ð ðĢ−2ð … ð2 − ðļðð 2 ðĢ−ð … ð+ð 2 − ðļðð 2 âŪ … 2 − ðļðð ðĢð ðĢ2ð ðĢð âŪ âŪ âŪ ð−ð ðð ðð âŪ =0 In a calculation – must cut off the infinite sum at some ðŪ’’ = ðŪðððĨ Supply Fourier components of potential, ðĢðŪ , up to ðŪðððĨ then calculate expansion coefficients ððŪ (single particle wavefunctions) and energies ðļðð 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 ðĢð (ð) = ðĢ(ð) + ðĢðŧ [ð](ð) + ðĢððķ [ð](ð) ððĄðĶðð ðð ðĢ ð (ð − ðð ,ð − ðŧ) ðĢ ð = ð =1 ð=1 ðŧ ððĄðĶðð ðĢðŪ = ð =1 Ωð ð ð ðŪ ðĢ ð (ðŪ) Ωðððð ðĢ ð (ðŪ) 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, ð0 = lim 2ðŋ ðŋ→∞ If system is large in real space, reciprocal lattice vectors are closely spaced. So, for a given ðļððĒðĄ , 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 ðļ = ð 2 /2 = primary photoelectron KE Photoemission ðļ, Primary photoelectron âð (no scattering – ∴ must originate close to surface) ðļ = âð − ðĩ − ð Vacuum level âω ð ðĩ ðļðđ âŪ Valence band Core levels Photoemission spectrum from Au, âð = 1487 eV Fermi edge, where ðĩ = 0 Kinetic energy PA4311 Quantum Theory of Solids Angle-resolved photoemission spectroscopy Surface normal âð spectrometer ð electrons ð⊥ ðâĨ ðâĨ = ð sin ð = 2ðļ sin ð 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 ðļ(ð) 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, ðŧ = ðŧððĄ + Δð ð ðŧððĄ is the single particle hamiltonian for an atom, ðŧððĄ ðð ð = ðð ðð ð Construct Bloch states of the crystal, 1 ðð ð, ð = ð ðð⋅ðđ ðð ð − ðđ , ð ðđ where ðŧððĄ ðð ð, ð = ððð ðð ð, ð Expand crystal wavefunctions (eigenstates of ðŧ = ðŧððĄ + Δð ð ) as Ψð (ð, ð) = ððð ð ðð ð, ð ð ð 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 ððð ð Minimise ðļðð subject to the constraint that Ψð is normalised ðļðð = Ψð ðŧ Ψð − ððð Ψð Ψð − 1 ∗ ððð′ ððð ðð′ ðŧ ðð − ððð ðļðð = âŪ ð′ ð (ðŧðð −ððð ðŋðð )ððð = 0 ð (H − ðļI)ð = 0 PA4311 Quantum Theory of Solids ∗ ððð′ ððð ðð′ ðð − 1 ð′ ð s-band from a single s-orbital Real space lattice – 1 atom basis ð1 = ð(1,0,0) ð1 = Reciprocal space lattice 2ð (1,0,0) ð 1 atom basis, 1 type of orbital so ð = ð = ð , H is a 1 × 1 matrix and 1 ðð = ðŧð ð = ð âŪ ′ ð ðð⋅(ðđ−ðđ ) ðð ð − ðđ′ ðŧ ðð (ð − ðđ) ð ð ′ ðð = ðð + 2ðū1 cos(ðð) PA4311 Quantum Theory of Solids
