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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 www.abinit.org 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 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.

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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 14.00 Tuesday November 25 th – room G Work through tutorial tasks (based on online abinit tutorial at www.abinit.org) www.abinit.org 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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