Transcript 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 โ mini โprojectโ report for ABINIT calculation
www.abinit.org
PA4311 Quantum Theory of Solids
Last timeโฆ
Solved the single-electron Schrödinger equation
๐ป2
โ
+ ๐ฃ๐ ๐
2
๐๐๐ = ๐ธ๐๐ ๐๐๐
for ๐ธ๐๐ and ๐๐๐ by expanding ๐๐๐ in a basis of plane waves
Derived the central equation โ an infinite set of coupled simultaneous equations
Examined solutions when the potential was zero, or weak and periodic
Reduced
zone scheme,
๐โ = ๐ โ ๐ฎ
PA4311 Quantum Theory of Solids
Band gaps at
the BZ
boundaries
Central equation
โฆ
โฆ
โฆ
๐โ๐
2
โฎ
โฎ
โฎ
๐ฃโ๐
๐ฃโ2๐
โฆ
๐2
โ ๐ธ๐๐
2
๐ฃโ๐
โฆ
๐+๐ 2
โ ๐ธ๐๐
2
โฎ
โฆ
2
โ ๐ธ๐๐
๐ฃ๐
๐ฃ2๐
๐ฃ๐
โฎ
โฎ
โฎ
๐โ๐
๐๐
๐๐
โฎ
=0
In a calculation โ must cut off the infinite sum at some ๐ฎโโ = ๐ฎ๐๐๐ฅ
Supply Fourier components of potential, ๐ฃ๐ฎ , up to ๐ฎ๐๐๐ฅ then calculate expansion
coefficients ๐๐ฎ (single particle wavefunctions) and energies ๐ธ๐๐
The more terms we include, the better the results will be
PA4311 Quantum Theory of Solids
Pseudopotentials
Libraries of โstandardโ pseudopotentials
available for most atoms in the periodic table
๐ฃ๐ (๐) = ๐ฃ(๐) + ๐ฃ๐ป [๐](๐) + ๐ฃ๐๐ถ [๐](๐)
๐๐ก๐ฆ๐๐ ๐๐
๐ฃ ๐
(๐ โ ๐๐
,๐ โ ๐ป)
๐ฃ ๐ =
๐
=1 ๐=1 ๐ป
๐๐ก๐ฆ๐๐
๐ฃ๐ฎ =
๐
=1
ฮฉ๐ ๐
๐ ๐ฎ ๐ฃ ๐
(๐ฎ)
ฮฉ๐๐๐๐
๐ฃ ๐
(๐ฎ) 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
# โโฆ an enormous effect on the quality of a calculation; โฆthe larger ecut is, the better converged the
calculation is. For fixed geometry, the total energy MUST always decrease as ecut is raisedโฆโ
# 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
โข using plane waves in aperiodic structures
Calculate for a periodic
structure with repeat
length, ๐0 = lim 2๐ฟ
๐ฟโโ
If system is large in real space, reciprocal lattice vectors are closely spaced.
So, for a given ๐ธ๐๐ข๐ก , get many more plane waves in the basis
PA4311 Quantum Theory of Solids
ABINIT tutorial
โข 14.00 Tuesday November 25th โ room G
โข Work through tutorial tasks (based on online abinit tutorial at
www.abinit.org)
Assessed task
โข Calculate GaAs ground state density, band structure, and effective mass
โข Write up results as an โinternal reportโ
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 โ mini โprojectโ 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
๐ธ = ๐ 2 /2 = primary photoelectron KE
Photoemission
๐ธ, Primary
photoelectron
โ๐
(no scattering โ
โด must originate
close to surface)
๐ธ = โ๐ โ ๐ต โ ๐
Vacuum level
โฯ
๐
๐ต
๐ธ๐น
โฎ
Valence band
Core levels
Photoemission spectrum
from Au, โ๐ = 1487 eV
Fermi edge,
where ๐ต =
0
Kinetic energy
PA4311 Quantum Theory of Solids
Angle-resolved photoemission spectroscopy
Surface normal
โ๐
spectrometer
๐
electrons
๐โฅ
๐โฅ
๐โฅ = ๐ sin ๐ = 2๐ธ sin ๐ is conserved across the boundary
Malterre et al, New J.
Phys. 9 (2007) 391
PA4311 Quantum Theory of Solids
Tight binding or LCAO method
โข Plane wave basis good when the potential is weak and electrons are nearly free
(e.g simple metals)
โข But many situations where electrons are highly localised (e.g. insulators,
transition metal d-bands etc.)
โข Describe the single electron wavefunctions in the crystal in terms of atomic
orbitals (linear combination of atomic orbitals)
โข Calculate ๐ธ(๐) for highest valence bands and lowest conduction bands
โข Solid State Physics, NW Ashcroft, ND Mermin
โข Physical properties of carbon nanotubes, R Saito, G Dresselhaus, MS
Dresselhaus
โข 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, ๐ป = ๐ป๐๐ก + ฮ๐ ๐
๐ป๐๐ก is the single particle hamiltonian for an atom,
๐ป๐๐ก ๐๐ ๐ = ๐๐ ๐๐ ๐
Construct Bloch states of the crystal,
1
๐๐ ๐, ๐ =
๐ ๐๐โ
๐น ๐๐ ๐ โ ๐น ,
๐ ๐น
where ๐ป๐๐ก ๐๐ ๐, ๐ = ๐๐๐ ๐๐ ๐, ๐
Expand crystal wavefunctions (eigenstates of ๐ป = ๐ป๐๐ก + ฮ๐ ๐ ) as
ฮจ๐ (๐, ๐) =
๐๐๐ ๐ ๐๐ ๐, ๐
๐
๐ 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 ๐๐๐ ๐
Minimise ๐ธ๐๐ subject to the constraint that ฮจ๐ is normalised
๐ธ๐๐ = ฮจ๐ ๐ป ฮจ๐ โ ๐๐๐ ฮจ๐ ฮจ๐ โ 1
โ
๐๐๐โฒ
๐๐๐ ๐๐โฒ ๐ป ๐๐ โ ๐๐๐
๐ธ๐๐ =
โฎ
๐โฒ
๐
(๐ป๐๐ โ๐๐๐ ๐ฟ๐๐ )๐๐๐ = 0
๐
(H โ ๐ธI)๐ = 0
PA4311 Quantum Theory of Solids
โ
๐๐๐โฒ
๐๐๐ ๐๐โฒ ๐๐ โ 1
๐โฒ
๐
s-band from a single s-orbital
Real space lattice โ 1 atom basis
๐1 = ๐(1,0,0)
๐1 =
Reciprocal space lattice
2๐
(1,0,0)
๐
1 atom basis, 1 type of orbital so ๐ = ๐ = ๐ , H is a 1 × 1 matrix and
1
๐๐ = ๐ป๐ ๐ =
๐
โฎ
โฒ
๐ ๐๐โ
(๐นโ๐น ) ๐๐ ๐ โ ๐นโฒ ๐ป ๐๐ (๐ โ ๐น)
๐
๐
โฒ
๐๐ = ๐๐ + 2๐พ1 cos(๐๐)
PA4311 Quantum Theory of Solids