Prof.P. Ravindran,folk.uio.no/ravi/cutn/pmat/7.optical_prop.pdf · 2015-12-21 · P.Ravindran,...
Transcript of Prof.P. Ravindran,folk.uio.no/ravi/cutn/pmat/7.optical_prop.pdf · 2015-12-21 · P.Ravindran,...
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
http://folk.uio.no/ravi/PMAT2013
Prof.P. Ravindran, Department of Physics, Central University of Tamil
Nadu, India
Optical Properties of Materials
1
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
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Light – Matter InteractionResponse to external electric field E
Linear approximation: susceptibility c
conductivity s
dielectric tensor
Fourier transform:
Polarizability:
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
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We want to develop a set of equations to describe the absorption of a photon in
semiconductor material.
The electromagnetic field is a quantized system (with a set of modes, each of
which is a harmonic oscillator).
In absorption, a photon is absorbed by the crystal and the energy of the
electromagnetic field is transferred to the crystal.
The initial state in the region of interest in the crystal is Ei,while the final state is
Ef.
Optical absorption in Semiconductors
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
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What are the assumptions and approximations we must
consider?
The electromagnetic field is perturbed by the electronic crystal.
If the wavelength associated with a mono-energetic field is larger than theperturbing charge (like in an atom or quantum dot), then we can make the dipoleapproximation and assume there is no position dependence to the field (and solvejust using the time-dependent field E(t)=E0sin( t)).
Otherwise, we assume, Bloch waves.
This means we neglect the action of the charge back on to the field (back
action).
We can assume the intensity of the field is large,so that changes in the photonnumber in each mode is small. Called semiclassical approximation (which wewill make most of the time).
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
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Fermi’s Golden Rule
The probability of absorption or emission will depend on the overlap
and energy difference of the initial and final state, and the density of
these states.
In order to determine the probability or amplitude of the absorption we must find
the overlap of the initial and final wavefunctions.
Instead of single initial and final states in single-particle picture, we
have in principle a large density of final states - (k)
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
6In quantum structures case
Choice of the wavefunctions for the initial and final states
Two different kinds of possibilities in quantum structure
Transitions between the valence and conduction bands
Transitions between the quantum-confined states within a given band,
so-called "intersubband“ transitions
Pump
E2
E1
HH1
LH1
K||
Eg
E
V.B
C.B
Probe
LH1
HH1
E2
E1
Eg
Barrier QW Barrier
Emission
LHx HHx
e1-e2 ISBT
Resonant optical transition
Pump
E2
E1
HH1
LH1
K||
Eg
E
V.B
C.B
Probe
LH1
HH1
E2
E1
Eg
Barrier QW BarrierBarrier QW Barrier
Emission
LHx HHx
e1-e2 ISBT
Resonant optical transition
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
Light Scattering: Interband transition
E
S
intraband transitioninterband transition
En
erg
y
wave vector
EF
band structure
kc
kv
7
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
Linear optical parameters
Kramers-Kronig relations
Complex dielectric tensor:
Optical conductivity:
Loss function:
Absorption coefficient:
Reflectivity:
Complex refractive index:
8
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
Intraband Contributions: Metals
Drude-like termsDielectric Tensor:
Optical conductivity:
Plasma frequency:
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
Optical Sum rules10
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
11Form of wavefunctions for "non-excitonic“ quantum
well absorption (quantum well)
Start by neglecting any excitonic effects (and other Coulombeffects –many particle effects)
Treat the initial state as being some electron statecorresponding to an electron in the valence band or somelower subband
Treat the final state as an electron in the conduction band or ahigher subband
The absorption process is an interaction between the matter and
the electromagnetic field.
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
General aspects
Optical absorption and luminescence occur by transition of
electrons and holes between electronic states (bands, tail
states, gap states). If electron-phonon coupling is strong
enough self-trapping occurs.
Choose valence band wavefunction as initial state.
Conduction band wavefunction as the final state.
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
Optical Absorption
Absorption coefficient α is defined by I(z) = Io exp {- α z}
where I(z) is the flux density if incident light is Io, z is the
distance measured from the incident surface. Hence
α = - (1/I(z)) dI(z)/dz
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
Tauc law (Tauc plot, A region)
The absorption coefficient, α, due to interband transition near
the band-gap is well described:
αħω = B (ħ ω – Eg)2
ħω is photon energy, Eg is optical gap.
This Tauc plot defines the optical gap in semiconductors.
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
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Urbach Tail in Absorption
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
Urbach tail (B region)
The absorption coefficient at the photon energy below the
optical gap (tail absorption) depends exponentially on the
photon energy:
α(ħ ω) ~ exp (ħ ω/Eu)
where Eu is called Urbach energy.
In addition, optical absorption by defects also appears at energy lower
than optical gap (C region). Likewise α is written as another
exponential function of photon energy:
α(ħω) ~ exp (ħω/Ed),
Ed belongs to the width of the defect states. C region is rather sensitive to
the structural properties of materials.
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
Photoluminescence
Photoluminescence occurs as a result of the transition of electrons and
holes from excited states to ground state.
After interband excitation, electrons (holes) relax to the bottom (top) of
the conduction (valence) band by emitting phonons much more quickly
than the radiative transition.
In the case of crystalline semiconductors (without defects, there is no
localized state) photoluminescence occurs by transition between the
bottom of the conduction band and the top of the valence band. k
selection rule must be satisfied: kphoton = ki – kf . (kphoton, ki and, kf are
the wave numbers of photons, electron of initial and final states.
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
Direct/indirect transition
Since kphoton is much smaller than ki and kf, we can rewrite the selection
rule:
ki = kf.
The semiconductors satisfying this condition is called direct-gap
semiconductors. c-Si is not satisfying k-selection rule (indirect-gap
semiconductor). Transition is allowed by either absorption of phonons or
their emission.
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
19Microscopic Theory of Linear Optical Properties of Semiconductors
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Semic-classical Theory of Interband Transitions
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
26Optical Transitions
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
27Optical Properties
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
28Beer – Lambert Law
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Absorption in Semiconductors : processes
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
30Absorption in semiconductors: processes cont.
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Optical Properties: Semiconductors & Insulators
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Optical Properties : Impurities
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Absorption in semiconductors: band-to-band
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
34Direct band gap and Indirect band gap
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
35Indirect Band Gap
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
36Interband absorption above the band gap
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Dielectric Function and Critical Points in Ge
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
38Comparing Direct and Indirect Bandgap Absorption
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
39Optical absorptions in Si
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40
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
41RPA Approximation
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
42
One can predict optical properties from
DFT calculations
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
43Silicon –Optical Absorption
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
44Joint Density of States
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
45Band edge absorption in direct gap semiconductors
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
46External Electric and Magnetic Field Effects
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
47Radiative and Non-radiative Recombination
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
48
Feasible Recombination Processes
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
49
Interband absorption
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
50Interband absorption ….
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
51
Direct versus indirect absorption
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
52
Silicon band structure
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
53Summary of Indirect optical transitions
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
54Phonon Assisted Optical Transition
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
55Excitonic Effect : Two particle (e-h) interaction
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
56
Absorption via Excitons
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
57
Electron-Hole interaction: Excitons
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
58Experimental Absorption Edges with exciton
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
59Exciton Effect above the bandgap
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
60Plasma reflectivity : metals
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
61Drude Model
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
62
Interband transitions in metals
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
63
Noble Metals : Copper
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
64Band structure and DOS in Copper
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
65Doped Semiconductors
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
66Optical transitions in semiconductors: Impurities
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
67
Donor absorption in n-type silicon
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
68Optical Anisotropy
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
Symmetry of Dielectric Tensor
cubic
monoclinic (a,b=90°) orthorhombic
tetragonal, hexagonal
triclinic
69
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00
25
50
75
100
125
150
175
0 1000 2000 3000 4000 500012.0
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
p
k-points in IBZ
165k
286k
560k
1240k
2456k
3645k
4735k
Inte
rband I
m
Energy [eV]
70
Convergence : Al
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
4
5
165 k-points
4735 k-points
Experiment
Neff [ele
ctr
ons]
Energy [eV]
Sumrules : Al71
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
Exa
mp
le:
Al
0 5 10 15 200
20
40
60
80
100
120
total
intraband
interband
Loss fu
nction
Energy [eV]
72
Loss Function
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
73
Sum Rules.
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
74Optical Properties of Metals.
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
75
Optical Properties of Metals: Al and Pd
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
76
Optical Properties of Metals: Cu and Cd
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
77
Joint Density of States (JDOS)
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
78
Dielectric Function (Real and Imaginary parts)
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
79
Comparison of theory vs. Experiment: ε2(ω) for Ge
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
80
Absorption Coefficient : α(ε)
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
81
Index of Refraction: n(ω)
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
82Optical Properties: Reflectance & Dielectric Function : Si
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
83Optical Properties: Reflectance & Dielectric Function : GaAs
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
84
Sensitivity of Reflectivity to Surface Contamination
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
85
Crystalline vs. Amorphous (Exp & Theory)
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
86
Origin of strong change in absorption
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
87
Band structure of Au: relativistic effects
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
88
DOS and Joint DOS for Au: relativistic effect
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
89Dielectric function for Au: relativistic effect
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
90
Optical Spectra : Impact on Solar Cells
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
91Optical Spectra : Impact on Solar Cells
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
92Current Developments
Gradient Corrections (GGA)
LDA + U
Exact Exchange (EXX)
Self-interaction correction (SIC)
Non-local exchange / screened exchange
Kohn-Sham theory
Generalized Kohn-Sham theory
Time dependent DFT
band gap problem
excitonic effects
non-local effects
correlation effects
band gap problem
Many-body perturbation theory
GW + Bethe-Salpeter equation
response to
time-dependet perturbation
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
93Theory of Optical Properties
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P.Ravindran, PHY085 – Properties of Materials, Autum 2013 Optical Properties of Materials
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