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PA4311 Quantum Theory of Solids Quantum Theory of Solids Mervyn Roy (S6) www2.le.ac.uk/departments/physics/people/mervynroy

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PA4311 Quantum Theory of Solids 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 Course Outline

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PA4311 Quantum Theory of Solids Last time… Band gaps at the BZ boundaries

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PA4311 Quantum Theory of Solids Central equation

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PA4311 Quantum Theory of Solids Pseudopotentials en.wikipedia.org/wiki/Pseudopotential

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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 E E E+00# first primitive cell vector, a_ E E E+00# a_ E E E+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.

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PA4311 Quantum Theory of Solids Supercells using plane waves in aperiodic structures

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PA4311 Quantum Theory of Solids ABINIT tutorial Tuesday November 25 th – room G Work through tutorial tasks (based on online abinit tutorial at Assessed task Calculate GaAs ground state density, band structure, and effective mass Write up results as an ‘internal report’

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PA4311 Quantum Theory of Solids 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 Course Outline Semi-empirical methods

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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

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PA4311 Quantum Theory of Solids Photoemission Kinetic energy Core levels Valence band Vacuum level

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PA4311 Quantum Theory of Solids Surface normal electrons spectrometer Angle-resolved photoemission spectroscopy Malterre et al, New J. Phys. 9 (2007) 391

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PA4311 Quantum Theory of Solids Tight binding or LCAO method

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PA4311 Quantum Theory of Solids Linear combination of atomic orbitals

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PA4311 Quantum Theory of Solids Expansion coefficients

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PA4311 Quantum Theory of Solids s-band from a single s-orbital Real space lattice – 1 atom basis Reciprocal space lattice

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