Optical Properties of Solids
Claudia Ambrosch-Draxl
Chair of Atomistic Modelling and Design of Materials
University Leoben, Austria
Basics
light scattering
dielectric tensor in the RPA
sumrules
symmetry
the band gap problem
Program
program flow
inputs
outputs
Examples
convergence
results
Outlook
applications
beyond linear optics
beyond RPA
Optics in WIEN2k
Outline
Basics
light scattering
dielectric tensor in the RPA
sumrules
symmetry
the band gap problem
Program
program flow
inputs
outputs
Examples
convergence
results
Outlook
applications
beyond linear optics
beyond RPA
Optics in WIEN2k
Outline
Dielectric function
Optical absorption
Optical gap
Exciton binding energy
Photoemission spectra
Core level spectra
Raman scattering
Compton scattering
Positron annihilation
NMR spectra
Electron spectroscopy
Light emitting diodes
Lasers
Solar cells
Displays
Computer screens
Smart windows
Light bulbs
CDs & DVDs
understand physics
characterize materials
tailor special properties
Excited States
Properties & Applications
Light – Matter Interaction
Polarizability:
Linear approximation:
susceptibility χ
conductivity σ
Fourier transform:
∋
dielectric tensor
Optical Properties
Response to external electric field E
The Dielectric Tensor
Bloch electrons:
Interband contribution:
intraband
interband
independent particle approximation, random phase approximation (RPA)
Optical Properties
Free electrons: Lindhard formula
Light Scattering
hω
E
S
interband transition
ck
EF
hω
intraband transition
vk
wave vector
Optical Properties
Energy
band structure
Optical "Constants"
Optical conductivity:
Complex refractive index:
Reflectivity:
Absorption coefficient:
Loss function:
Kramers-Kronig relations
Optical Properties
Complex dielectric tensor:
Intraband Contributions
Dielectric Tensor:
Drude-like terms
Plasma frequency:
Metals
Optical conductivity:
Optical Properties
Sumrules
Symmetry
monoclinic (α,β=90°)
orthorhombic
tetragonal, hexagonal
cubic
Dielectric Tensor
triclinic
Magneto-optics
without magnetic field, spin-orbit coupling:
cubic
with magnetic field H ‫ װ‬z, spin-orbit coupling: tetragonal
KK
KK
Example: Ni
KK
Be careful ....
Wavefunction vs. Density
Hartree-Fock:
Koopman's theorem
DFT:
Lagrange parameters
Janak's theorem
auxiliary functions
Excited States
ionization energies
Approximations used:
Ground state:
Local Density Approximation (LDA)
Generalized Gradient Approximation (GGA)
Excited state:
Interpretation within one-particle picture
Interpretation of excited states in terms of ground state properties
Electron-hole interaction ignored (RPA)
Where do possible errors come from?
How to treat excited states ab initio?
Excited State Properties
Open Questions
The Band Gap Problem
Ionization energy
Electro-affinity
Band gap
shift of conduction bands: scissors operator
many-body perturbation theory: GW approach
Basics
Program
Examples
Outlook
light scattering
dielectric tensor in the RPA
sumrules
symmetry
the band gap problem
program flow
inputs
outputs
convergence
results
applications
beyond linear optics
beyond RPA
Optics in WIEN2k
Outline
Program Flow
converged potential
kgen
dense mesh
lapw1
eigenstates
lapw2
Fermi distribution
optic
momentum matrix elements
joint
dielectrix tensor components
kram
Re ε
Im ε optical coefficients
broadening
scissors operator
Optics in WIEN2k
SCF cycle
al.inop
2000 1
-5.0 2.2
1
1
OFF
"optic"
number of k-points, first k-point
Emin, Emax: energy window for matrix elements
number of cases (see choices below)
Re <x><x>
unsymmetrized matrix elements written to file?
800 1
-5.0 5.0
3
1
3
7
OFF
number of k-points, first k-point
Emin, Emax: energy window for matrix elements
number of cases (see choices below)
Choices:
Re <x><x>
1......Re <x><x>
Re <z><z>
2......Re <y><y>
Im <x><y>
3......Re <z><z>
4......Re <x><y>
5......Re <x><z>
6......Re <y><z>
7......Im <x><y>
8......Im <x><z>
9......Im <y><z>
Inputs
ni.inop (magneto-optics)
"joint"
al.injoint
lower and upper band index
Emin, dE, Emax [Ry]
output units eV / Ry
switch
number of columns to be considered
broadening for Drude model
choose gamma for each case!
SWITCH
0...JOINT DOS
1...JOINT DOS
2...DOS
3...DOS
4...Im(EPSILON)
5...Im(EPSILON)
6...INTRABAND
7...INTRABAND
for each band combination
sum over all band combinations
for each band
sum over all bands
total
for each band combination
contributions
contributions including band analysis
Inputs
1 18
0.000 0.001 1.000
eV
4
1
0.1 0.2
"kram"
al.inkram
broadening gamma
energy shift (scissors operator)
add intraband contributions 1/0
plasma frequency
gamma(s) for intraband part
0.1
0.0
1
12.6
0.2
as number of colums
as number of colums
70
Silicon
Dielectric function
60
Imε
50
40
si.inkram
Reε
30
20
10
0
-10
Γ=0.05eV
-20
0
1
2
3
Energy [eV]
4
5
6
0.05 broadening gamma
1.00 energy shift (scissors operator)
0
....
Inputs
80
optic
case.symmat
case.mommat
momentum matrix elements, symmetrized
analysis, NLO
joint
Im ε
SWITCH 4
kram
case.epsilon
case.sigmak
case.refraction
case.absorp
case.eloss
complex dielectric tensor
optical conductivity
refractive index
absorption coefficient
loss function
Outputs
case.joint
Basics
light scattering
dielectric tensor in the RPA
sumrules
symmetry
the band gap problem
Program
program flow
inputs
outputs
Examples
convergence
results
Outlook
applications
beyond linear optics
beyond RPA
Optics in WIEN2k
Outline
Results ....
Convergence
Interband Im ε
150
125
100
75
165k
286k
560k
1240k
2456k
3645k
4735k
12.8
12.7
12.6
12.5
12.4
12.3
12.2
12.1
12.0
ωp
0
1000 2000 3000 4000 5000
k-points in IBZ
50
25
0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Energy [eV]
Example: Al
175
Sumrules
5
165 k-points
4735 k-points
Experiment
3
Example: Al
Neff [electrons]
4
2
1
0
0
10 20 30 40 50 60 70 80 90 100
Energy [eV]
Loss Function
120
80
intraband
60
40
total
20
0
Example: Al
Loss function
100
interband
0
5
10
Energy [eV]
15
20
Example: Platinum
Theory - Experiment
K. Glantschnig, and C. Ambrosch-Draxl, (preprint).
C. Ambrosch-Draxl and J. O. Sofo
Comp. Phys. Commun., in print