Download Lecture 8

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
𝛻2
−
+ 𝑣𝑠 𝒓
2
𝜓𝑛𝑘 = 𝐸𝑛𝑘 𝜓𝑛𝑘
for 𝐸𝑛𝒌 and 𝜓𝑛𝒌 by expanding 𝜓𝑛𝒌 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
PA4311 Quantum Theory of Solids
Band gaps at
the BZ
boundaries
Central equation
⋮
…
…
…
𝒌−𝒈
2
⋮
⋮
𝑣−𝑔
𝑣−2𝑔
2
− 𝐸𝑛𝑘
𝑣𝑔
𝒌2
− 𝐸𝑛𝒌
2
𝑣2𝑔
𝑣𝑔
⋮
⋮
…
⋮
𝑣−𝑔
…
𝒌+𝒈 2
− 𝐸𝑛𝒌
2
⋮
…
𝑐−𝒈
𝑐𝟎
𝑐𝒈
⋮
=0
In a calculation – must cut off the infinite sum at some 𝑮’’ = 𝑮𝑚𝑎𝑥
Supply Fourier components of potential, 𝑣𝑮 , up to 𝑮𝑚𝑎𝑥 then calculate expansion
coefficients 𝑐𝑮 (single particle wavefunctions) and energies 𝐸𝑛𝒌
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
𝑣𝑠 (𝒓) = 𝑣(𝒓) + 𝑣𝐻 [𝑛](𝒓) + 𝑣𝑋𝐶 [𝑛](𝒓)
𝑛𝑡𝑦𝑝𝑒 𝑛𝜅
𝑣 𝜅 (𝒓 − 𝝉𝜅,𝑗 − 𝑻)
𝑣 𝒓 =
𝜅=1 𝑗=1 𝑻
𝑛𝑡𝑦𝑝𝑒
𝑣𝑮 =
𝜅=1
Ω𝑘 𝜅
𝑆 𝑮 𝑣 𝜅 (𝑮)
Ω𝑐𝑒𝑙𝑙
𝑣 𝜅 (𝑮) 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, 𝑎0 = lim 2𝐿
𝐿→∞
If system is large in real space, reciprocal lattice vectors are closely spaced.
So, for a given 𝐸𝑐𝑢𝑡 , 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
Photoemission
𝐸 = 𝑘 2 /2 = primary photoelectron KE
𝐸, Primary
photoelectron
ℏ𝜔
(no scattering –
∴ must originate
close to surface)
𝐸 = ℏ𝜔 − 𝐵 − 𝜙
Vacuum level
ℏω
𝜙
𝐵
𝐸𝐹
⋮
Valence band
Core levels
Photoemission spectrum
from Au, ℏ𝜔 = 1487 eV
Fermi edge,
where 𝐵 =
0
Kinetic energy
PA4311 Quantum Theory of Solids
Angle-resolved photoemission spectroscopy
Surface normal
ℏ𝜔
spectrometer
𝜃
electrons
𝑘⊥
𝑘∥
𝑘∥ = 𝑘 sin 𝜃 = 2𝐸 sin 𝜃 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 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).
PA4311 Quantum Theory of Solids
Linear combination of atomic orbitals
In a crystal, 𝐻 = 𝐻𝑎𝑡 + Δ𝑈 𝑟
𝐻𝑎𝑡 is the single particle hamiltonian for an atom,
𝐻𝑎𝑡 𝜓𝑛 𝒓 = 𝜖𝑛 𝜓𝑛 𝒓
Construct Bloch states of the crystal,
1
𝜙𝑛 𝒌, 𝒓 =
𝑒 𝑖𝒌⋅𝑹 𝜓𝑛 𝒓 − 𝑹 ,
𝑁 𝑹
where 𝐻𝑎𝑡 𝜙𝑛 𝒌, 𝒓 = 𝜖𝑛𝒌 𝜙𝑛 𝒌, 𝒓
Expand crystal wavefunctions (eigenstates of 𝐻 = 𝐻𝑎𝑡 + Δ𝑈 𝑟 ) as
Ψ𝑗 (𝒌, 𝒓) =
𝑐𝑗𝑛 𝒌 𝜙𝑛 𝒌, 𝒓
𝑛
𝑛 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 𝑐𝑗𝑛 𝒌
Minimise 𝐸𝑗𝒌 subject to the constraint that Ψ𝑗 is normalised
𝐸𝑗𝒌 = Ψ𝑗 𝐻 Ψ𝑗 − 𝜖𝑗𝒌 Ψ𝑗 Ψ𝑗 − 1
∗
𝑐𝑗𝑛′
𝑐𝑗𝑛 𝜙𝑛′ 𝐻 𝜙𝑛 − 𝜖𝑗𝒌
𝐸𝑗𝒌 =
𝑛′
𝑛
⋮
(𝐻𝑚𝑛 −𝜖𝑗𝒌 𝛿𝑚𝑛 )𝑐𝑗𝑛 = 0
𝑛
(H − 𝐸I)𝒄 = 0
PA4311 Quantum Theory of Solids
∗
𝑐𝑗𝑛′
𝑐𝑗𝑛 𝜙𝑛′ 𝜙𝑛 − 1
𝑛′
𝑛
s-band from a single s-orbital
Real space lattice – 1 atom basis
𝒂1 = 𝑎(1,0,0)
Reciprocal space lattice
𝒃1 =
2𝜋
(1,0,0)
𝑎
1 atom basis, 1 type of orbital so 𝑛 = 𝑚 = 𝑠, H is a 1 × 1 matrix and
1
𝜖𝒌 = 𝐻𝑠𝑠 =
𝑁
⋮
′
𝑒 𝑖𝒌⋅(𝑹−𝑹 ) 𝜓𝑠 𝒓 − 𝑹′ 𝐻 𝜓𝑠 (𝒓 − 𝑹)
𝑅
𝑅′
𝜖𝑘 = 𝜖𝑠 + 2𝛾1 cos(𝑘𝑎)
PA4311 Quantum Theory of Solids