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**Quantum Theory of Solids**

Mervyn Roy (S6) www2.le.ac.uk/departments/physics/people/mervynroy

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**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 pseudopotentials Linear combination of atomic orbitals Effective mass theory ABINIT computer workshop (LDA DFT for periodic solids) Assessment: 70% final exam 30% coursework – mini ‘project’ report for ABINIT calculation

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**Last time… Solved the single-electron Schrödinger equation**

− 𝛻 𝑣 𝑠 𝒓 𝜓 𝑛𝑘 = 𝐸 𝑛𝑘 𝜓 𝑛𝑘 for 𝐸 𝑛𝒌 and 𝜓 𝑛𝒌 by expanding 𝜓 𝑛𝒌 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, 𝒌’=𝒌−𝑮 Band gaps at the BZ boundaries

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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 ⋮ =0 In a calculation – must cut off the infinite sum at some 𝑮’’= 𝑮 𝑚𝑎𝑥 Supply Fourier components of potential, 𝑣 𝑮 , up to 𝑮 𝑚𝑎𝑥 then calculate expansion coefficients 𝑐 𝑮 (single particle wavefunctions) and energies 𝐸 𝑛𝒌 The more terms we include, the better the results will be

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Pseudopotentials Libraries of ‘standard’ pseudopotentials available for most atoms in the periodic table 𝑣 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 type en.wikipedia.org/wiki/Pseudopotential

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**# 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_1 E E E+00 # a_2 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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**Supercells using plane waves in aperiodic structures**

Calculate for a periodic structure with repeat length, 𝑎 0 = lim 𝐿→∞ 2𝐿 If 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

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**ABINIT tutorial Assessed task 14.00 Tuesday November 25th – 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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**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 pseudopotentials Linear combination of atomic orbitals Effective mass theory ABINIT computer workshop (LDA DFT for periodic solids) Assessment: 70% final exam 30% coursework – mini ‘project’ report for ABINIT calculation Semi-empirical methods

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**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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**Photoemission 𝐸, Primary photoelectron ℏ𝜔**

𝐸 𝐹 Core levels Valence band Vacuum level 𝐵 𝜙 𝐸= 𝑘 2 /2 = primary photoelectron KE ℏω ⋮ Photoemission 𝐸, Primary photoelectron (no scattering – ∴ must originate close to surface) 𝐸=ℏ𝜔−𝐵−𝜙 ℏ𝜔 Photoemission spectrum from Au, ℏ𝜔=1487 eV Kinetic energy Fermi edge, where 𝐵=0

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**Angle-resolved photoemission spectroscopy**

Surface normal spectrometer ℏ𝜔 𝜃 𝑘 ⊥ electrons 𝑘 ∥ 𝑘 ∥ = 𝑘 sin 𝜃 = 2𝐸 sin 𝜃 is conserved across the boundary Malterre et al, New J. Phys. 9 (2007) 391

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**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 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).

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**Linear combination of atomic orbitals**

In a crystal, 𝐻= 𝐻 𝑎𝑡 +Δ𝑈 𝑟 𝐻 𝑎𝑡 is the single particle hamiltonian for an atom, 𝐻 𝑎𝑡 𝜓 𝑛 𝒓 = 𝜖 𝑛 𝜓 𝑛 𝒓 Construct 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,𝒓 Expand crystal wavefunctions (eigenstates of 𝐻= 𝐻 𝑎𝑡 +Δ𝑈 𝑟 ) as Ψ 𝑗 (𝒌,𝒓)= 𝑛 𝑐 𝑗𝑛 𝒌 𝜙 𝑛 𝒌,𝒓 𝑛 labels different atomic orbitals and different inequivalent atom positions in the unit cell

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**Expansion coefficients**

Use the variational method to find the best values of the 𝑐 𝑗𝑛 𝒌 Minimise 𝐸 𝑗𝒌 subject to the constraint that Ψ 𝑗 is normalised 𝐸 𝑗𝒌 = Ψ 𝑗 𝐻 Ψ 𝑗 − 𝜖 𝑗𝒌 Ψ 𝑗 Ψ 𝑗 −1 𝐸 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

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**s-band from a single s-orbital**

𝒂 1 =𝑎(1,0,0) Real space lattice – 1 atom basis Reciprocal space lattice 𝒃 1 = 2𝜋 𝑎 (1,0,0) 1 atom basis, 1 type of orbital so 𝑛=𝑚=𝑠, H is a 1×1 matrix and 𝜖 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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