Presentation on theme: "Quantum Theory of Solids"— Presentation transcript:
1 Quantum Theory of Solids Mervyn Roy (S6)www2.le.ac.uk/departments/physics/people/mervynroy
2 Course Outline Introduction and background The many-electron wavefunction- Introduction to quantum chemistry (Hartree, HF, and CI methods)Introduction to density functional theory (DFT)- Framework (Hohenberg-Kohn, Kohn-Sham)- Periodic solids, plane waves and pseudopotentialsLinear combination of atomic orbitalsEffective mass theoryABINIT computer workshop (LDA DFT for periodic solids)Assessment: 70% final exam30% coursework – mini ‘project’ report for ABINIT calculation
3 Last time… Solved the single-electron Schrödinger equation − 𝛻 𝑣 𝑠 𝒓 𝜓 𝑛𝑘 = 𝐸 𝑛𝑘 𝜓 𝑛𝑘for 𝐸 𝑛𝒌 and 𝜓 𝑛𝒌 by expanding 𝜓 𝑛𝒌 in a basis of plane wavesDerived the central equation – an infinite set of coupled simultaneous equationsExamined solutions when the potential was zero, or weak and periodicReduced zone scheme,𝒌’=𝒌−𝑮Band gaps at the BZ boundaries
4 Central equation⋮ ⋮ ⋮ … 𝒌−𝒈 − 𝐸 𝑛𝑘 𝑣 −𝑔 𝑣 −2𝑔 … … 𝑣 𝑔 𝒌 − 𝐸 𝑛𝒌 𝑣 −𝑔 … … 𝑣 2𝑔 𝑣 𝑔 𝒌+𝒈 − 𝐸 𝑛𝒌 … ⋮ ⋮ ⋮ ⋮ 𝑐 −𝒈 𝑐 𝟎 𝑐 𝒈 ⋮ =0In a calculation – must cut off the infinite sum at some 𝑮’’= 𝑮 𝑚𝑎𝑥Supply Fourier components of potential, 𝑣 𝑮 , up to 𝑮 𝑚𝑎𝑥 then calculate expansion coefficients 𝑐 𝑮 (single particle wavefunctions) and energies 𝐸 𝑛𝒌The more terms we include, the better the results will be
5 PseudopotentialsLibraries of ‘standard’ pseudopotentials available for most atoms in the periodic table𝑣 𝑠 (𝒓)=𝑣(𝒓)+ 𝑣 𝐻 [𝑛](𝒓)+ 𝑣 𝑋𝐶 [𝑛](𝒓)𝑣 𝒓 = 𝜅=1 𝑛 𝑡𝑦𝑝𝑒 𝑗=1 𝑛 𝜅 𝑻 𝑣 𝜅 (𝒓− 𝝉 𝜅,𝑗 −𝑻)𝑣 𝑮 = 𝜅=1 𝑛 𝑡𝑦𝑝𝑒 Ω 𝑘 Ω 𝑐𝑒𝑙𝑙 𝑆 𝜅 𝑮 𝑣 𝜅 (𝑮)𝑣 𝜅 (𝑮) is independent of crystal structure- tabulated for each atom typeen.wikipedia.org/wiki/Pseudopotential
6 # 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 cellacell 3*5.53 angstrom # lattice constant =5.53 is the same in all 3 directionsrprim # primitive cell definitionE E E+00 # first primitive cell vector, a_1E E E+00 # a_2E 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.
7 Supercells using plane waves in aperiodic structures Calculate for a periodic structure with repeat length, 𝑎 0 = lim 𝐿→∞ 2𝐿If system is large in real space, reciprocal lattice vectors are closely spaced.So, for a given 𝐸 𝑐𝑢𝑡 , get many more plane waves in the basis
8 ABINIT tutorial Assessed task 14.00 Tuesday November 25th – room G Work through tutorial tasks (based on online abinit tutorial atAssessed taskCalculate GaAs ground state density, band structure, and effective massWrite up results as an ‘internal report’
9 Course Outline Introduction and background The many-electron wavefunction- Introduction to quantum chemistry (Hartree, HF, and CI methods)Introduction to density functional theory (DFT)- Framework (Hohenberg-Kohn, Kohn-Sham)- Periodic solids, plane waves and pseudopotentialsLinear combination of atomic orbitalsEffective mass theoryABINIT computer workshop (LDA DFT for periodic solids)Assessment: 70% final exam30% coursework – mini ‘project’ report for ABINIT calculationSemi-empirical methods
10 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
11 Photoemission 𝐸, Primary photoelectron ℏ𝜔 𝐸 𝐹Core levelsValence bandVacuum level𝐵𝜙𝐸= 𝑘 2 /2 = primary photoelectron KEℏω⋮Photoemission𝐸, Primary photoelectron(no scattering –∴ must originate close to surface)𝐸=ℏ𝜔−𝐵−𝜙ℏ𝜔Photoemission spectrum from Au, ℏ𝜔=1487 eVKinetic energyFermi edge, where 𝐵=0
12 Angle-resolved photoemission spectroscopy Surface normalspectrometerℏ𝜔𝜃𝑘 ⊥electrons𝑘 ∥𝑘 ∥ = 𝑘 sin 𝜃 = 2𝐸 sin 𝜃 is conserved across the boundaryMalterre et al, New J. Phys. 9 (2007) 391
13 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 𝐸(𝒌) for highest valence bands and lowest conduction bandsSolid State Physics, NW Ashcroft, ND MerminPhysical properties of carbon nanotubes, R Saito, G Dresselhaus, MS DresselhausSimplified LCAO Method for the Periodic Potential Problem, JC Slater and GF Koster, Phys. Rev. 94, 1498, (1954).
14 Linear combination of atomic orbitals In a crystal, 𝐻= 𝐻 𝑎𝑡 +Δ𝑈 𝑟𝐻 𝑎𝑡 is the single particle hamiltonian for an atom,𝐻 𝑎𝑡 𝜓 𝑛 𝒓 = 𝜖 𝑛 𝜓 𝑛 𝒓Construct Bloch states of the crystal,𝜙 𝑛 𝒌,𝒓 = 1 𝑁 𝑹 𝑒 𝑖𝒌⋅𝑹 𝜓 𝑛 𝒓−𝑹 ,where 𝐻 𝑎𝑡 𝜙 𝑛 𝒌,𝒓 = 𝜖 𝑛𝒌 𝜙 𝑛 𝒌,𝒓Expand crystal wavefunctions (eigenstates of 𝐻= 𝐻 𝑎𝑡 +Δ𝑈 𝑟 ) asΨ 𝑗 (𝒌,𝒓)= 𝑛 𝑐 𝑗𝑛 𝒌 𝜙 𝑛 𝒌,𝒓𝑛 labels different atomic orbitals and different inequivalent atom positions in the unit cell
15 Expansion coefficients Use the variational method to find the best values of the 𝑐 𝑗𝑛 𝒌Minimise 𝐸 𝑗𝒌 subject to the constraint that Ψ 𝑗 is normalised𝐸 𝑗𝒌 = Ψ 𝑗 𝐻 Ψ 𝑗 − 𝜖 𝑗𝒌 Ψ 𝑗 Ψ 𝑗 −1𝐸 𝑗𝒌 = 𝑛′ 𝑛 𝑐 𝑗𝑛′ ∗ 𝑐 𝑗𝑛 𝜙 𝑛′ 𝐻 𝜙 𝑛 − 𝜖 𝑗𝒌 𝑛′ 𝑛 𝑐 𝑗𝑛′ ∗ 𝑐 𝑗𝑛 𝜙 𝑛 ′ 𝜙 𝑛 −1⋮𝑛 (𝐻 𝑚𝑛 − 𝜖 𝑗𝒌 𝛿 𝑚𝑛 ) 𝑐 𝑗𝑛 =0(H−𝐸I)𝒄=0
16 s-band from a single s-orbital 𝒂 1 =𝑎(1,0,0)Real space lattice – 1 atom basisReciprocal space lattice𝒃 1 = 2𝜋 𝑎 (1,0,0)1 atom basis, 1 type of orbital so 𝑛=𝑚=𝑠, H is a 1×1 matrix and𝜖 𝒌 = 𝐻 𝑠𝑠 = 1 𝑁 𝑅 𝑅′ 𝑒 𝑖𝒌⋅(𝑹− 𝑹 ′ ) 𝜓 𝑠 𝒓− 𝑹 ′ 𝐻 𝜓 𝑠 (𝒓−𝑹)⋮𝜖 𝑘 = 𝜖 𝑠 +2 𝛾 1 cos(𝑘𝑎)
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