Prof.P. Ravindran,folk.uio.no/ravi/cutn/pmat/7.optical_prop.pdf · 2015-12-21 · P.Ravindran,...

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

Light – Matter InteractionResponse to external electric field E

Linear approximation: susceptibility c

conductivity s

dielectric tensor

Fourier transform:

Polarizability:

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

Optical absorption in Semiconductors

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).

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)

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

Barrier QW Barrier

Emission

LHx HHx

e1-e2 ISBT

Resonant optical transition

Barrier QW BarrierBarrier QW Barrier

Emission

LHx HHx

e1-e2 ISBT

Resonant optical transition

Light Scattering: Interband transition

intraband transitioninterband transition

wave vector

band structure

Linear optical parameters

Kramers-Kronig relations

Complex dielectric tensor:

Optical conductivity:

Loss function:

Absorption coefficient:

Reflectivity:

Complex refractive index:

Intraband Contributions: Metals

Drude-like termsDielectric Tensor:

Optical conductivity:

Plasma frequency:

Optical Sum rules10

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.

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.

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

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.

Urbach Tail in Absorption

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.

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.

Direct/indirect transition

Since kphoton is much smaller than ki and kf, we can rewrite the selection

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.

19Microscopic Theory of Linear Optical Properties of Semiconductors

Semic-classical Theory of Interband Transitions

26Optical Transitions

27Optical Properties

28Beer – Lambert Law

Absorption in Semiconductors : processes

30Absorption in semiconductors: processes cont.

Optical Properties: Semiconductors & Insulators

Optical Properties : Impurities

Absorption in semiconductors: band-to-band

34Direct band gap and Indirect band gap

35Indirect Band Gap

36Interband absorption above the band gap

Dielectric Function and Critical Points in Ge

38Comparing Direct and Indirect Bandgap Absorption

39Optical absorptions in Si

41RPA Approximation

One can predict optical properties from

DFT calculations

43Silicon –Optical Absorption

44Joint Density of States

45Band edge absorption in direct gap semiconductors

46External Electric and Magnetic Field Effects

47Radiative and Non-radiative Recombination

Feasible Recombination Processes

Interband absorption

50Interband absorption ….

Direct versus indirect absorption

Silicon band structure

53Summary of Indirect optical transitions

54Phonon Assisted Optical Transition

55Excitonic Effect : Two particle (e-h) interaction

Absorption via Excitons

Electron-Hole interaction: Excitons

58Experimental Absorption Edges with exciton

59Exciton Effect above the bandgap

60Plasma reflectivity : metals

61Drude Model

Interband transitions in metals

Noble Metals : Copper

64Band structure and DOS in Copper

65Doped Semiconductors

66Optical transitions in semiconductors: Impurities

Donor absorption in n-type silicon

68Optical Anisotropy

Symmetry of Dielectric Tensor

monoclinic (a,b=90°) orthorhombic

tetragonal, hexagonal

triclinic

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00

0 1000 2000 3000 4000 500012.0

k-points in IBZ

rband I

Energy [eV]

Convergence : Al

0 10 20 30 40 50 60 70 80 90 1000

165 k-points

4735 k-points

Experiment

Neff [ele

Energy [eV]

Sumrules : Al71

0 5 10 15 200

intraband

interband

Loss fu

nction

Energy [eV]

Loss Function

Sum Rules.

74Optical Properties of Metals.

Optical Properties of Metals: Al and Pd

Optical Properties of Metals: Cu and Cd

Joint Density of States (JDOS)

Dielectric Function (Real and Imaginary parts)

Comparison of theory vs. Experiment: ε2(ω) for Ge

Absorption Coefficient : α(ε)

Index of Refraction: n(ω)

82Optical Properties: Reflectance & Dielectric Function : Si

83Optical Properties: Reflectance & Dielectric Function : GaAs

Sensitivity of Reflectivity to Surface Contamination

Crystalline vs. Amorphous (Exp & Theory)

Origin of strong change in absorption

Band structure of Au: relativistic effects

DOS and Joint DOS for Au: relativistic effect

89Dielectric function for Au: relativistic effect

Optical Spectra : Impact on Solar Cells

91Optical Spectra : Impact on Solar Cells

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

93Theory of Optical Properties

Prof.P. Ravindran,folk.uio.no/ravi/cutn/pmat/7.optical_prop.pdf · 2015-12-21 · P.Ravindran,...

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