5. Optical Properties

7/28/2019 5. Optical Properties

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Spring Term 12-13 Lecture Notes AP218142

Optical Properties

Metals: when photons having sufficiently large energy impinge on a solid, the

electrons in the crystal get excited into higher energy levels so that unoccupied

higher energy levels are available. Momentum of a photon and its wavevector

(( )pkh

= is much smaller than that of an electron. photk is much smaller than

the diameter of Brillouin zone. Momentum conservation 1 2

1 2

pthk hk hk

k k

+ =

1510

max 2 282 2

pt

10 Energy conservation, ~ ~10 /

( ) ( )3 10

2 2

ptpt

e e

k k ma k kcM M

= =

+ =

k


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1s BZ

E

m-band

Electron transitions at which k remains constant (vertical transitions) are direct

interband transitions.

Optical spectra for metals can be dominated by direct interband transitions for high

photons.

Indirect interband transitions (involving phonons) are much weaker in metals.

ahv

bhv

n-band FE

xk

Smallest photon energy for interband transition: bhv , largest: ahv


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Example: Cu

1L

Ef'

2L P

3L d

|W L

Interband transition: Upper d-band and Fermi energy Transitions ~2.2eV

At higher energies, transitions from EF can also take place.

Intraband (within the band) transitions are also possible for metals (unfilled

electron bands)

Energy

FE

kz

largest photon energy which

can be absorbed by intraband

transition corresponds to an

excitation from lower to upper

band edge. All energies

smaller than maxE are absorbed

continuously.


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At low photon energies, intraband transitions if possible are prevailing mechanism.

(occur in metals only). Above a critical, interband (from band to band) transitions

set in.

Conservation of crystal momentum states that change in wavevector upon

interaction with photon

2 1 0pt

pt

wk k k

c =

(almost direct transition)

Involving phonon, then momentum conservation leads to:

2 1 phk k k =

Optical absorption by semiconductors

1. Transition across bandgapE

e

o

x k


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Valence electron with wavevector 'ek jumps to conduction band, hole with a

wavevector 'h ek k= and a conduction electron with wavevector ek are generated.

22

* *

( )( )

2 2

pt e h

e hpt gd

e h

k k kkk

Em m

= +

= + +

When ,min0e h gd ptk k E == = (onset of direct transition energy gap)

Absorption

transition absorption

photon energy


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vestE photon energy

i pnEg

Direct transition occurs when ,min( 0)vest pt e hE w k k= = =

Optical transition in indirect s.c. requires 3rd entity like phonon or higher energy

photon for direct transition.

Eg:

( )Si i 1.12 eV

Ge(i) 0.66

InAs(d) 0.36

lnP(d) 1.35

(examples of direct and indirectgap semiconductors)


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For indirect gap semiconductor, indirect optical transition involving a phonon

occurs before direct optical transition because the latter requires more energy.

s

p

vestE iEg

o

(absorbed +sign, omitted - sign for phonons)

(if 0)

pt pn e h

e h ph

e ph h

k k k k

k k k

k k k

= +

+

Minimum photon energy required for onset of indirect transition is

,minpt i phEg = when:

'0, .e h e o phk k k k k= = = =

As photon energy increases direct transition eventually becomes possible.

Valence electron with wave

vector 'ek jumps to conduction

band, hole with wavevector'

h ek k= and conduction electron

with ek are generated.


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Excitons: Sometimes optical absorption spectra of a semiconductor show

structures just below the energy gap.

absorption

Co-efficient

EG(band gap) Energyphoton

Small absorption peak just below the gap energy peak is caused by absorption

of photon with creation of bound e h pair called excitons. (Different from free 'e

& holes seen in previous section.) Excitons are observed when electrons or holes

generated are not very mobile such as those generated at low temperatures.

Bound 'e h

pairs can be treated similarly to that of hydrogen atoms;

, ,2

1ex n ex oE E

n

=

4

, 2 2

/

2(4 )

r erex o o

r o r

M mM eE E

= =

excitonlevels

Formation of exciton effectively lowered the

conduction band into the gap; so optical

excitation below band gap energy is possible.


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If photon generates an exciton in a direct gap s.c.

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,* *

( ) ( )

2 2e h

pt gd ex n

e h

k kE E

m m = + + +

Onset of exciton excitation

,min ,1 2

/r ept gd ex gd o

r

M mE E E E

= + =

e.g.: For GaAs,*

*

0.066

0.082

e

h

M me

M me

=

=

Around state energy ~-2.9 meV close to measured value of 3.4 meV.

Many optical transitions are possible in a direct gap s.c.

* excitation of electron across the bandgap

* excitation of electron from valence band to higher lying conduction band

* exciton formation

* excitation of electrons from impurities

* excitation of free carrier (intraband)


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Optical Emission: Reverse process of absorption!

When excited 'e undergoes a transition from a higher energy level to lower energy

level, it gives up the excess energy in the form of photon emission, process is

called optical emission or luminescence which includes:

1. e h recombination across bandgap (and e lying in higher conduction bands)

2. exciton annihilation

3. intraband transition etc.

We need excited electrons first for luminescence.

1. electroluminescence excited e created by electric field

2. photoluminescence created by light

3. black-body radiation: due to recombination of thermally excited 'e s & holes.

Normally, luminescence emits photons in random phase (spontaneous emission is

incoherent). If population of excited electrons can be maintained greater than that

of ground states 'e s (population inversion) then stimulated emission results.

Optical Applications :

- light emission of laser

- light detection of photodetector

- light energy conversion of solar cells- material characterization of Raman

Optical Devices Optical Characterization Spectroscopy Laser Raman XPS Solar Cell


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Raman Scattering:

Inelastic scattering of light due to interaction of photon with phonon is called

Raman scattering.

,1pt ,2ptw ,2ptk ,2ptk

,1pt ,1pt o o

,1ptk ,1ptk

,1pn ,1pn

(Stokes) ,1pnk (Anti-Stokes) ,1pnk

,1 ,2

,1 ,2

(+sign for phonon

emission, stoke proven)

=

pt pt pn

pt pt pnk k k

=

Raman shift defined as:2 1

,2 ,1

1 1 1

2 2 2

pt pt pn

C C C

=

= =

Since photon wavevectors are small phonon wavevectors are very small as well

Raman scattering tends to excite phonons near 0pnk which is on the optical

branch of the diatomic lattice.

1 2

1 1~ 2pnw c M M +

M1and M2 are mass of atoms in the basis.


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Incident photon wavelength is 5145 ( 1

~19400 km-1; Raman shift is about 50 cm-1)

From Kittel


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XPS (Photoelectron Spectroscopy)

Incident X-ray photon ejects a core (or valence e-) outof the solid. If incident

photon has sufficient energy pt to ionize an electron, then:

21

2pt i k eE E mV + = =

One binding energy, EB=Ei (of the core e-) can be deduced by measuring energy of

the electron.

Minimum X-ray photon energy =Binding energy of e-s in solid.

Kaisase et al.J ECS, 152 2005C is core-level XPS spectra of silicon surface annealedunder various conditions.


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Solar Cells:

Light creates electron-hole pairs in a semiconductor using a p-n junction device,

we can connect device to external load (such as a light bulb) and create electricity

direct conversion of light energy into electric energy.

Photons that have energy smaller than s.c. bandgap will not produce e h

pairs. Also photons with energy much larger also will produce e- & holes with

same energy Eg regardless of how large Eg is. Excess energy is dissipated as

heat.

output electrical power

Efficiency: 100outconvin

P

P =

input optical power

Need long carrier lifetime (minority carriers, low series resistance etc. to improve

efficiency)

Can solar cells be used to power an IPod?


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Solar spectral irradiance From Pierret.


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Note that E1 transition

will not give rise to

photoconductivity since e- is

localized.

E2 will give rise to current!

Absorption in ceramics(insulators/wideband-gap materials).

Consider band structure shown below:

Eg E1 E2

(defect state) Absorption E1

E2

Eg Eg

Incident Photon energy (eV)

Color centres: 2 3Al O doped withCr red (Ruby)

2 3Al O doped with Ni blue (Sapphire)

Electronic transitions that cause red color are interlevel transitions between Cr d-

orbitals. Due to influence of crystal field (arising from interaction of O ions with

Cr3+) the d-orbitals are no longer degenerate.

2.2 eV & 3.0 eV absorption leads to major fraction of blue & green end of

visible spectrum being removed, hence ruby appears red.


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Blue Sapphire: doped alumina with Fe and Ti. Fe2+ and Ti4+ can exchange

electrons (when light shines on it)

2 4 0 3 3Fe Ti h Fe Ti+ + + ++ +

When 2 4andFe Ti+ + are nearest neighbors (substitute for nearest neighbor Al3+

ions), broad absorption band is formed at red end of the spectrum crystal

appears blue. This is example of electronic transitions between neighboring ions

rather than single ion (as in previous case). Heat natural stones to enhance their

color important step to get good gems.


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Lasers

Emission of photons:

Spontaneous emission: excited electron reverts back into lower energy state

followed by emission of photon (fluorescence) light is incoherent & often

polychromatic.

Stimulated emission: Consider two energy levels E1 & E2

E2 assume more #of electrons in E2 over E1

E1 (population inversion)

If one electron reverts to lower energy state,

21hv then photon 21hv is emitted. This photon might

cause an electron to descend to E1, thus causing

emission of another photon which vibrates in phase with first one. Similarly, it is

possible to have stimulated emission of light.

Laser is highly monochromatic due to transitions from two narrow energy levels.

Lasing material is contained in a long narrow container known as cavity.

Opposite end of the cavity must be parallel to each other. One face is silvered and

acts as a mirror. Other face is partially silvered and transmits some of the light.


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semi-reflecting mirror

Fraction of the photons escape through the partially transparent mirror; this

constitutes the emitted beam.

How do the electrons reach the higher energy level?

One method is by optical pumping: absorption of light from a

polychromatic light source (Xe flashlamps for pulsed lasers; W-Iodine lamps e.g.)

Pumping efficiency is large if the bandwidth E is large, i.e., an entire frequency

range leads to excited electrons.

E large E2 t large

hv21

pumping hv21

t ~ hE

EnergyRange

Finite spanfor which e- remainsat this level

gt E is large, t is small &vice versa.


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So in a two level configuration, it is difficult to have high pumping efficiency and

population inversion. Three level laser provides improvement over a two level

configuration. Pump band E3is broad which enables good pumping efficiency.

E3

pumping

E2hv4

E1

Even larger population inversion can be achieved using 4 level laser.

Laser examples:

Ruby laser ~ 694 nm(3 level laser; lasingoccurs between (r3+levels)

Nd YAG ~ 1.064 m 4 level laser

He Ne ~ 633

CO2 10,600 nm

He Ne laser. A cavity (about 2mm dia.) filled with 0.1 Torr Ne & 1 Torr He.

Current passes through the gas and produces free electrons. e- excite He gas by

electron-atom collisions: efficient pumping into Ne 2s & 3s levels. Lasing occurs

between Ne s & p levels and produces three characteristics wavelengths.

E3 E2 Through non-radioactive

process E2 E1 since E2 is

sharp, electrons stay longer

(several ms) providing required

population inversion.

5. Optical Properties

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