PHY475

OPTICAL PROPERTIES OF SOLIDS

Prof. Mark Fox

AUTUMN SEMESTER

PHY475: OPTICAL PROPERTIES OF SOLIDS Prof. Mark Fox

Autumn Semester (10 credits)

Course aims and outcomes

• Understand the classical theory of light propagation in solid state dielectric materials; • Understand the quantum theory of absorption and emission in solids; • Appreciate the importance of excitonic effects in solids; • Understand the origin of nonlinear optical effects in crystals. The outcome of the course will be that the student will be familiarised with the optical phenomena that occur in a wide range of solid state materials, based on an understanding of both the classical and quantum theories of how light interacts with dielectric materials.

Course Book: Fox, Optical properties of Solids (Oxford University Press, Second edition 2010) These notes are to be used in conjunction with the course book. A number of hard copies are available in the University Library, as well as an ebook that can be accessed online. Other books that may be useful

• Kittel, Introduction to Solid State Physics (Wiley) • Burns, Solid State Physics (Academic Press) • Ibach and Luth, Solid State Physics (Springer-Verlag)

Assessment: Homework: 15% (3 problem sheets), Exam: 85% (any 3 questions from 5) Course www page: http://www.mark-fox.staff.shef.ac.uk/PHY475/

Lecture Topic Homework Book chapter

1-3 Introduction.

The complex refractive index

1

4-6 Lorentz oscillators.

Dispersion and birefringence

1 2

7-8 Interband absorption 3

9-10 Excitons 2 4

11-12 Interband Luminescence 5

13-14 Quantum confinement 6, 8.5

15-16 Metals. Doped semiconductors 3 7

17-18 Phonons 10

19-20 Nonlinear optics 11

1

Topic 1: Introduction

• Optical coefficients

• Complex dielectric constant

• Complex refractive index

• Introduction to optical materials

Optical coefficients

Figure 1.1: Optical coefficients

incident light

reflected light

transmitted light

propagation through the medium

•  Reflectivity = reflected / incident power

•  Transmissivity = transmitted / incident power

•  T + R =1 if medium is transparent

2

Propagation refraction

absorption and luminescence

scattering

Figure 1.2: Propagation of light through a medium

•  Velocity v= c/n, n is the refractive index •  I(z) = I0 exp(-αz), α is the absorption coefficient •  T = (1-R1) exp(-αL) (1-R2) •  Luminescence : re-emission at lower frequency •  scattering: elastic- change of direction inelastic - change of direction and frequency

Luminescence

Figure 1.3: luminescence mechanisms

abso

rptio

n emission

relaxation

ground state

excited states •  Luminescence comes

out at lower frequency

than absorption due to

internal relaxation

•  The energy shift

between absorption

and luminescence is

called the Stokes shift.

3

Complex optical coefficients Complex relative dielectric constant: εr = ε1 + iε2

Complex refractive index: n = n+ iκAbsorption coefficient: α = 4πκ / λεr = n

2; ε1 = n2 −κ 2; ε2 = 2nκ

n = 1

2ε1 + ε1

2 +ε22( )

1/2"#$

%&'

1/2

κ =1

2−ε1 + ε1

2 +ε22( )

1/2"#$

%&'

1/2

Reflectivity: R = n−1n+1

2

=(n−1)2 +κ 2

(n+1)2 +κ 2

Insulators / semiconductors

Figure 1.4 : transmission spectra of (a) sapphire (Al2O3) (b) CdSe

Al2O3

CdSe

Wavelength (microns)

0.0

1.0

0.1 1 10 0.0

0.8 Tran

smis

sion

infrared UV visible (a)

(b)

• Infrared absorption due to phonons • ultraviolet/visible absorption due to bound electrons • position of fundamental absorption edge depends on the size of band gap • transparency region in between

4

Metals

10 1.0 0.1 0.0 0.2 0.4 0.6 0.8 1.0

Wavelength (µm)

Ref

lect

ivity

infrared visible UV

silver

• Free electrons in the metal absorb ⇒ High reflectivity up to “plasma frequency” in the UV

Figure 1.5: Reflectivity spectrum of silver

Organic materials

Figure 1.6 : Absorption spectrum of polyfluorene

•  Strong absorption in UV/visible spectral region due to

electronic transitions

•  Stokes-shifted emission across the visible spectral region

300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0

Abs

orpt

ion

(a.u

.)

Wavelength (nm)

polyfluorene (F8)

UV/blue band UV visible

5

Doped insulators

200 400 600 800 1000 0.0

0.2

0.4

0.6

0.8

1.0

ruby

sapphire

Wavelength (nm)

Tran

smis

sion

yellow/green band

blue band

Figure 1.7: Transmission spectra of sapphire (Al2O3) and ruby (Al2O3: Cr3+)

•  Sapphire is transparent for visible wavelengths

•  Cr3+ ions doped into sapphire absorb in the blue and yellow/green

spectral regions, hence red colour

Lifting of degeneracies free atom

degenerate magnetic

levels

atom in B field

atom in

crystal

crystal field effect

Zeeman effect

Figure 1.8 : Lifting of degeneracies in the solid state

•  Crystals have directions ⇒ lower symmetry than atoms or molecules

•  new effects due to lower symmetry e.g. lifting of degeneracies

6

Band formation E

Interatomic separation

solid free atom

Figure 1.9: Band formation in the solid state

•  Free atoms have sharp absorption and emission lines

•  Solids have broad absorption and emission bands

•  Two types of bands: electronic and vibronic

7

Topic 2: Dipole oscillators

•  Introduction

•  Lorentz oscillator model

•  Kramers–Kronig relationships

•  Dispersion

•  Optical anisotropy & chirality

Electron oscillators

Figure 2.1: classical model of an atom. Electrons are bound to the nucleus by springs which determine the natural frequencies

Bound electrons (insulators, intrinsic semiconductors) • restoring force for small displacements F = – k x

⇒ natural resonant frequency ω0 = (k/m)1/2 • ω0 lies in the near infrared / visible / UV spectral regions Free electrons (metals, doped insulators) • k = 0 ⇒ ω0 = 0

8

Dipole oscillators

p(t) = – e x(t) P = N p

D = ε0 E + P = εrε0 E

p(t)

t

+ t = 0 t = π

ω0 t = 2π ω0

t

x(t)

x

Refractive index • light ≡ AC electric field at ω (≠ω0) • forced oscillations at ω • atoms lag on the driving field and re-radiate in phase

Light emission • natural oscillations at ω0 cause radiation at ω0 • “Hertzian dipoles”

Absorption • light frequency = ω0 • resonance phenomenon • atoms absorb energy from the light beam

Figure 2.2

Vibrational oscillators

Figure 2.3 polar molecule

+

ε

Molecular physics IR absorption at resonant frequency of the vibrational modes Ionic crystals IR absorption at frequency of optically-active lattice vibrations i.e. the transverse optic (TO) phonons

ω0

9

Lorentz oscillators Dielectric constant Refractive index

Figure 2.4 Lorentz oscillator with ω0 = 1014 rad/s γ = 5×1012 s-1

εst = 12.1 ε∞ = 10

-10

10

30

ε 1

60 100 140 0

20

40

ε 2

ω (1012 rad/s)

γ

ω0

2 4 6

n

60 100 140 0

2

4 κ

ω0

( )γωωωεχωε

i11)( 22

000

2r

−−++=

mNe

•  absorption at ω0 •  γ = 1/τ = damping rate •  FWHM = γ,

Example: atomic absorption line

Figure 2.5 •  sodium gas, N = 1×1017 m–3 (low density ⇒ n0 ≈ 1) •  D2 line at 589.0 nm, hyperfine component with FWHM = 100 MHz

n0 ≈ 1

�200, 0, 200,Δν (MHz)

n

α

0

1700 m�1

3.95 × 10�5

α(ω) =α0γ 2

4Δω 2 +γ 2

n(ω) = n0 −Δn4γΔω

4Δω 2 +γ 2

Δω =ω −ω0

Lorentzian lineshape

10

Multiple resonances Figure 2.6 resonant frequencies: phonons IR valence electrons vis/UV core electrons X-ray

0

1 Ref

ract

ive

inde

x A

bsor

ptio

n

1011 1013 1015 1017 Frequency (Hz)

0

vibrational bands

electronic transitions

IR visible UV X-ray • fj = oscillator strength • classical theory ⇒ fj = 1 • quantum theory ⇒ fj determined by transition probability

( )∑−−

+=j jj

jr

i

fmNe

ωγωωεωε 22

000

21)(

Example: SiO2 glass

1012 1013 1014 1015 1016 1017

10�4,

10�2,

1,

Ext

inct

ion

coef

ficie

nt

κ

Frequency (Hz)

(b)

1

2

3

refr

activ

e in

dex

n

(a) SiO2 glass Figure 2.7 real and imaginary part of the refractive index for SiO2 glass

Transparency region between phonon frequencies in IR and band gap in UV

11

Optical fibre losses

http://nobelprize.org/nobel_prizes/physics/laureates/2009/phyadv09.pdf

Kramers–Kronig relationships

[ ]

2 20

2

2 20

2 ( )( ) 1 P d

( ) 12( ) P d

n

n

ω κ ωω ω

π ω ωω ω

κ ω ωπω ω ω

∞

∞

% %%− =

% −% % −

%= −% −

∫

∫

0 1 2 0

2

4

Ref

ract

ive

inde

x

Band gap wavelength (µm)

• Refractive index (n) and absorption (κ) are the real and imaginary parts of the same function: ñ = n+iκ

• Absorption and refraction are related to each other by the Kramers–Kronig relationships $

n at 10 µm vs band gap

Figure 2.9

12

UV transmission of glass

Add UV absorbers: •  reduces UV transmission •  increases refractive index at

visible frequencies

(at 310 nm) (at 546 nm)

Dispersion Figure 2.10 glass dispersion

0.2 0.4 0.6 0.8 1.0 1.2 1.4

1.45

1.50

1.55

Ref

ract

ive

inde

x, n

Frequency (1015 Hz)

1000 600 400 300 200 Wavelength (nm)

SiO2 glass

2

2d 0dnλ

=

visible ultraviolet IR •  Group velocity dispersion (GVD) ∝ d2n/d!2

•  GVD determines pulse spreading in optical fibres •  GVD = 0 near 1300 nm

13

Dispersive prisms

blue

red white Figure 2.11

Dispersion of light by a prism

•  NORMAL dispersion: n increase with frequency

•  ANOMALOUS dispersion: occurs near resonance lines

For Pink Floyd fans …

normal dispersion anomalous dispersion ?

red red

blue blue

14

Double refraction (Birefringence)

Double refraction in calcite

Figure 2.12 Birefringent crystals such as calcite (Iceland Spar) separate orthogonal light polarizations into the o(rdinary) and e(xtraordinary) rays

109°

71°

6.2°

e-ray

o-ray unpolarized

light

optic axis

Birefringence (optical anisotropy) : refractive index depends on the direction of the polarization vector relative to the crystal axes

15

Optical anisotropy

θ

y

z optic axis

propagation direction

(b) e-ray

ε θ

y

z optic axis

propagation direction

(a) o-ray

ε

Figure 2.13 Birefringence caused by difference of dielectric constants (and hence refractive index) along the different crystal axes.

!!!

"

#

$$$

%

&

!!!

"

#

$$$

%

&=

!!!

"

#

$$$

%

&

z

y

x

z

y

x

EEE

DDD

33

22

11

000

0000

ε

ε

ε

ε

Polarizing beams splitters

e-ray o-ray

unpolarized light "

air gap

optic axis

"

Critical angle: sin "c = 1/n Calcite: no = 1.658, "c = 37.1°; ne = 1.486, "c = 42.3° Hence choose 37.1° ≤ " ≤ 42.3°

Figure 2.14 Glan–Foucault

prism

16

Wave plates

(b) (a) "

optic

axis

d

input output "

o-ray e-ray

input polarization

( )o e2 n n dπ

φλ

Δ = − Half wave plate: !# = $ Quarter wave plate: Δφ = π / 2,

Figure 2.15

Induced birefringence •  Isotropic materials are non-birefringent •  Induce birefringence !n with strain or electric field • Hence photo-elastic and electro-optic effects • Kerr effect (quadratic electro-optic effect) observed in

all materials, including liquids and glass: Δn = ! K E2 ; K = Kerr constant

• Hence Kerr cells (see Fig. 11.8) • Contrast with linear electro-optic effect (Pockels effect)

observed only in anisotropic crystals (See Fig 11.6)

17

Chirality

( )R Ld n nπ

θλ

= −

• Optical properties different for left or right circularly polarized light due to chirality (helicity) of molecules or crystal structure

• Circular dichroism: different absorption for left or right circular light

• Optical activity: different refractive index for left or right circular light.

• Optical activity causes rotation of linear light: Examples: dextrose, laevulose (fructose) [latin dexter, laevus]

amino acid

Magneto-optics •  Induce chirality in non-chiral materials with a magnetic

field • magnetic circular dichrosim in absorbing materials • Faraday effect in transparent materials: rotation of linear

polarization by magnetic field $θ = V B d ; V = Verdet coefficient

"

d

input output

B

Figure 2.16 The Faraday effect

18

Appendix: Local field corrections Figure 2.8 local field ≠ applied field in dense medium

+ + + + + + +

- - - - - - - P

θ

ε

Lorentz correction:,εlocal = ε + P/3ε0 in cubic crystal

Clausius Mossotti relationship

321

r

r aNχεε

=+−

19

Topic 3: Interband absorption

•  Interband transitions: direct and indirect

•  Direct gap materials

•  Optical orientation

•  Indirect gap materials

•  Photodetectors & solar cells

Interband absorption

Ef

Ei

!% Eg

upper band

lower band

Energy •  Photon excites electron from filled valence to empty conduction band

•  Fundamental absorption edge at Eg

•  Process creates an electron–hole pair

Figure 3.1

20

Direct and Indirect absorption

q

valence band

conduction band

(a)  Direct band gap: C.B. minimum at k = 0

(b) Indirect band gap C.B. minimum at k ≠ 0

k

E

k

E

0 0

Eg Eg

•  kphoton = 2π/λ ~ 107 m–1 negligible compared to B.Z. size π/a ~ 1011 m–1 •  Transitions appear as vertical lines on E – k diagrams •  Phonon needed to conserve momentum for indirect gap materials •  Indirect absorption 2nd order process, therefore low probability

Figure 3.2

!% !%

Atomic physics of semiconductors

s

p

s bonding

s antibonding

p bonding

p antibonding

ATOM MOLECULE CRYSTAL

Eg valence band

conduction band

Figure 3.3

•  Four valence electrons per atom: Group IV (C, Si, Ge), •  III-V compounds (GaAs, InAs, InSb, GaN ….) •  II-VI compounds (ZnS, ZnSe, CdSe, HgTe, …. ) •  V.B. → C.B. is p → s, hence allowed transition

21

GaAs band structure Figure 3.4 •  Direct gap at 1.5 eV •  Very important optoelectronic material

•  Strong absorption for !% > Eg -6

-4

-2

0

2

4

Ener

gy (e

V)

Γ X L

Eg !% GaAs

Wave vector k Λ, Δ,

fcc lattice Brillouin zone

Γ,X

L

K W

kz

kx

ky

Symmetry points Γ 000 X 100 010 001 K 110 L 111

Fig D.5

22

Four-band model

k

E

Electron (e) band

Heavy holes (hh)

Light holes (lh)

Split-off holes (so)

Eg

0 �Δ

! "

Figure 3.5 •  Simplified band structure first proposed by Kane (1957)

•  valid near k = 0 ! Heavy hole transition

" Light hole transition

•  Split-off hole transitions also possible

InAs band edge absorption

Figure 3.6 InAs is a direct gap III–V semiconductor with Eg = 0.35 eV !% < Eg : " = 0 !% > Eg : " ∝ (!% – Eg)1/2 0.3 0.4 0.5 0.6

0.0

0.2

0.4

0.6

0.8

1.0

Energy (eV)

α2

(1012

m�2 )

InAs room temperature

23

Magneto absorption in germanium

Figure 3.7 •  direct gap at 0.80 eV •  B-Field quantizes motion in direction perpendicular to B •  Landau level absorption when !% = Eg + (n+½) !%c

%c = eB / me (cyclotron frequency)

0.78 0.80 0.82 0.84 0.86 15 20 25 30 35 40 45

Energy (eV)

Tran

smis

sion

(%) germanium

300 K

B = 3.6 T

B = 0

Spin injection (optical orientation) E

MJ

J = 3/2 J = 1/2

J = 1/2

#1/2 #3/2 +3/2 +1/2

valence band

conduction band

#! 0

Eg

& + & + & # & #

hh hh lh lh

so so Figure 3.8

|M12|2 ∝ MJ ⇒  hh transitions three times stronger than lh

( 1/2) ( 1/2)( 1/2) ( 1/2)N NN N+ ! !" =+ + !

50% spin polarization for σ± excitation

E1 selection rule Δm = ±1 for σ±

24

Direct versus indirect absorption

1.0 1.2 1.4 1.6 1.8 2.0

102 103 104 105 106

Energy (eV)

Abs

orpt

ion

coef

ficie

nt (m

�1 )

GaAs silicon

Figure 3.9

•  Direct absorption is much stronger than indirect absorption

• Silicon has indirect gap at 1.1 eV

•  GaAs has direct gap at 1.4 eV

Germanium band structure

Figure 3.10

-6

-4

-2

0

2

4

Ener

gy (e

V)

Wave vector k

Eg = 0.66eV

direct gap

L Γ,Λ, Δ, X

•  Indirect gap at 0.66 eV •  Direct gap at 0.80 eV

25

Germanium band edge absorption

Figure 3.11

•  indirect absorption: α ∝ (!% – Egind ± !$phonon )2

•  direct absorption: α ∝ (!% – Egdir )1/2

at 300K :

Egind = 0.66 eV

Egdir = 0.80 eV

0.6 0.7 0.8 0.9 0

20

40

60

80

'1/

2 (m#

1/2 )

Energy (eV)

phon

on

abso

rptio

n 291 K

20 K

phonon emission

(a) (b)

0.6 0.7 0.8 0.9 0

2

4

6

8

10

' (1

05 m#

1 )

Energy (eV)

' % (!% # Egdir)1/2

Eg

300 K germanium

Silicon absorption

0 2 4 6 8 10 0

0.5

1.0

1.5

2.0

2.5

Energy (eV)

Abs

orpt

ion

coef

ficie

nt (1

08 m�1 )

silicon 300 K

Eg

E1

E2

Figure 3.12

•  Indirect band gap at 1.1 eV •  Critical points (van Hove singularities) at E1 (3.2 eV) and E2 (4.3 eV)

26

Silicon band structure

Figure 3.13

-12

-8

-4

0

4

Wave vector k

Ener

gy (e

V)

Γ, X L Λ, Δ,

E1 E2

Eg

• density of states g(E) ∝ 2 g(k) (dE/dk)–1

•  Critical points (van Hove singularities) whenever dE/dk = 0 •  occurs when conduction and valence bands are parallel to each other: �Parallel band effect�

Absorption spectroscopy Figures 3.14-15

sample

white-light source

scanning monochromator

transmission detector

reflection detector

vacuum pump

valve

sample in cryostat

collection lenses

spectrograph

entrance slit

computer #

collimation lenses

white-light source

silicon diode array

detector

27

p-i-n diodes (Appendix E) p i n

li

Eg

V0 = 0 z

E

Reverse bias V0 applied

p i n

li

Eg | eV0

|

z

E

i

li

V0 z

p n

• Electric field E = (Vbi – V0) / li

• Vbi = built-in voltage ≈ Eg / e

• V0 is negative in reverse bias

Figs E.1-2

Photodetectors Figure 3.16

•  Embed absorbing region within p-n junction to form p-i-n diode

•  Apply reverse bias V0 (i.e. V0 negative ) •  Electric field ε = (Vbi – V0) / li

•  Vbi = built-in voltage ≈ Eg / e

•  Photons absorbed if !% > Eg

•  Creates electron-hole pairs in i-region

•  Carriers swept out by ε field into external circuit to generate photocurrent Ipc

V0

Ipc

p

i

n

ε + -

!%

li

Examples: •  photodiodes •  solar cells (V0 = 0)

28

Solar cells (photovoltaics) Figure 3.17

V = Ipc R

Ipc

p i n !%

+ –

R

ISC

VOC

V

maximum power

Ipc

• Small gap: large ISC , small VOC • Large gap: small ISC , large VOC • Single junction: max efficiency ~30% • Larger for multi junction

Solar cell efficiency

29

Topic 4: Excitons

•  Introduction

• Wannier excitons

•  Excitonic nonlinearities

•  Frenkel excitons

Excitons Figure 4.1

Free (Wannier) radius >> a small binding energy moves freely through crystal

Tightly-bound (Frenkel) radius ~ a large binding energy localized on one lattice site

e

h

h

e

a

30

Free exciton absorption Figure 4.2

Eg

α,

Photon energy

n = 1

n = 2

(!% – Eg)1/2

•  Hydrogenic series of lines satisfying : !% = Eg – RX / n2 •  enhanced absorption for !% > Eg

•  only observed when T ≤ ( RX / kB)

Excitons in bulk GaAs Figs 4.3-4

α (1

06 m�1 )

0.4

0.8

1.2

Photon energy (eV)

1.514 1.520 0

Eg

n = 1 n = 2

n = 3

α ∝ (!% � 1.425)1/2,

1.42 1.46 1.50 1.54 0

0.4

0.8

1.2

α (

106 m

�1 )

Photon energy (eV)

exciton,

21 K

294 K

•  standard purity sample •  T = 21 K – 294 K

•  ultra pure sample •  T = 1.2 K •  RX = 4.2 meV

RX

31

Field ionization in GaAs Figure 4.5

1.50 1.51 1.52 Photon energy (eV)

Phot

ocur

rent

T = 5 K

V0= +1.44 V$Ε ≈ 0

V0 = +1.00 V, Ε ≈ 5 × 105 V / m

0

GaAs parameters : RX ~ 4.2 meV aX ~ 13 nm li = 1 µm (typical) F ~ 6 × 105 V / m Ε ~ 1.5 × 106 V / m for V0 = 0 !

F aX

F ~ 2 RX / e aX

ionized if ε > F

Ε,!% p i n

V0 li Ε

Nonlinear excitonic absorption Figs 4.6 - 7

(a) Low density Separation » diameter

(b) High density Separation ≈ diameter

1.512 1.516 1.520 0

2

4

6

8

10

Photon energy (eV)

α (1

06 m�1 )

low density 6 × 1022 m�3

2 × 1023 m�3

GaAs, 1.2 K

NMott = [ (4/3)πaX3 ]–1

~ 1.1 × 1023 m�3 in GaAs

32

Frenkel excitons

8 10 12 14 0

1

2

3

α (1

08 m�1 )

Energy (eV)

Eg(eV) RX (eV) NaCl 8.8 0.9 LiF 13.7 0.9

Eg Eg

LiF NaCl

300 K

Figs 4.8 - 9

300 K

3.20 3.30 3.40

Abs

orpt

ion

C16H10 (pyrene)

Energy (eV)

Alkali halides Organic crystals

Also: Rare gas crystals (solid Ne, Ar, Kr, Xe)

33

Topic 5: Luminescence

•  Introduction

•  Photoluminescence

•  Electroluminescence

•  LEDs and lasers

• Cathodoluminescence

Luminescence

• Luminescence spontaneous emission in solids

• Fluorescence fast luminescence electric-dipole allowed, τR ~ ns

• Phosphorescence slow luminescence electric-dipole forbidden, τR ~ µs – ms

• Electroluminescence electrical excitation • Photoluminescence optical excitation • Cathodoluminescence cathode ray (e–beam) excitation

34

Radiative quantum efficiency

!%

inject electrons

inject holes

relaxation

τNR τR

GROUND STATE

EXCITED STATE

Fig. 5.1

•  Radiative transition rate determined by Einstein A-coefficient •  τR = A–1 •  τΝR determined by phonon population, number of traps etc

NRRR /1

1ττ

η+

=

ηR = radiative quantum efficiency = radiative transition rate total transition rate

Direct gap materials

!%

electrons

holes

conduction band

valence band

Eg

E

k k = 0

GaN T = 4 K, Eg = 3.50 eV

3.40 3.50 3.60

Lum

ines

cenc

e

inte

nsity

Energy (eV)

Abs

orpt

ion

•  Strong emission at the band gap •  most III-V and II-VI semiconductors •  linewidth ≥ kBT

Figs 5.2 – 3

35

Indirect gap materials

Figure 5.4

!%

electrons

holes

conduction band

valence band

Eg

E

k

phonon

k = 0

•  Low emission probability (2nd order process) •  Long radiative lifetime ⇒ low radiative quantum efficiency •  diamond, silicon, germanium, AlAs

Photoluminescence

hν

electrons

holes

conduction band

valence band

Eg

E

k

k = 0

0 hνL

Density of states

E

•  Excite using laser with photon energy > Eg •  electrons and holes relax to the bottom of their bands •  thermal distributions formed according to statistical mechanics •  emission from Eg to top of carrier distributions

Figure 5.5

36

Classical (Boltzmann)Statistics

Figure 5.6 GaAs T = 100 K kBT = 8.6 meV Eg = 1.501 eV

1.49 1.50 1.51 1.52 1.53

1.50 1.52 102

103

PL in

tens

ity (a

.u.)

Energy (eV)

Eg kBT

•  Boltzmann statistics: f(E) ∝ exp(–E/kBT) (occupancy factors) •  I(E) ∝ Density of states × fe(E) fh(E) •  PL rises sharply at Eg, then decays exponentially. Linewidth ~ kBT

Fermi’s golden rule Rate ∝ |M|2 ρ(hν)

Degeneracy

electrons

holes

Density of states

E

cFE

vFE

gE hν

0.80 0.90 1.00 Phot

olum

ines

cenc

e (a

.u.) Ga0.47In0.53As, TL = 10 K

24 ps 180 K 250 ps

55 K

Energy (eV)

Eg

6nJ/pulse

Figs 5.7 – 8

•  Degeneracy observed at high density and low temperatures •  Emission from Eg to (Eg + EF

c +EFv )

37

Photoluminescence spectroscopy

sample in cryostat

collection lenses

laser

PL

entrance slit

computer detector

spectrometer

#

mirror

Photoluminescence (PL) spectroscopy • fixed frequency laser, measure spectrum by scanning spectrometer PL excitation spectroscopy (PLE) • detect at peak emission, vary laser frequency • effectively measures absorption Time-resolved PL spectroscopy • short pulse laser + fast detector • measure lifetimes, relaxation processes

Electroluminescence

p n

hν ≈ Eg

current

V0

p-type

n-type

substrate

epitaxial layers

holes

electrons

hν

• Forward-biassed p-n junction • Electrons and holes recombine at the junction • photon energy ~ Eg

• Epitaxial growth of high purity light-emitting layers on substrate crystal • MBE, MOCVD, LPE ….

Figure 5.10

38

Lattice matching

3 4 5 6 0

2

4

6

Ban

d ga

p (e

V)

Lattice constant (Å)

AlN

GaN

InN InAs

AlAs

InP

AlP GaP

SiC GaAs

GaAs

InP

blue red

substrate materials

direct band gap indirect band gap

sapphire

visible spectrum

fibre optics

hexagonal cubic

Figure 5.11

1.2 1.4 1.6 Energy (eV)

1 mA 293 K

Junction electroluminescence

(a) V0 = 0

p n

Eg

holes

electrons

depletion region

(b) V0 ≈ +Eg/e

p n

hν = Eg eV0

EFc EF

v

1.8

GaAs, Eg = 1.42 eV

Elec

trolu

min

esce

nce

•  Emission at Eg

•  Operating voltage ~ Eg / e •  Spectral width ~ kBT

Figs 5.12 – 3

39

Diode lasers

hν

R1 R2

l G

ain γ ν

,

Iin Ith

γth

Out

put

pow

er ,

Iin

p - AlGaAs i - GaAs n - AlGaAs

light output

metal contact oxide

current

metal contact

n- GaAs substrate

•  Mirrors formed by “facets” (i.e. edges) of chip •  Emission wavelength ~ hc / Eg

•  Linewidth determined by cavity modes

Figs 5.14 – 16

Cathodoluminescence

back-scattered electrons

excitation volume

penetration depth (Re) ! 1 µm

electron beam

VACUUM

CRYSTAL cathodoluminescence

•  Commercial use in cathode ray tubes •  Research tool for investigating nanostructures: Observe luminescence from electron microscope

• Focussed spot size ! 100nm • Primary electrons generate

secondary electrons in excitation volume

• Secondary electrons generate e–h pairs, hence luminescence

Primary electrons few keV

Figure 5.17

40

41

Topic 6: Quantum confinement

•  Dimensionality

•  Quantum wells $ Energy levels

$ Optical transitions

$ Quantum confined Stark effect

•  Quantum dots

•  Carbon nanostructures

Dimensionality

z

x y bulk

quantum well

quantum wire

quantum dot

Energy

Den

sity

of s

tate

s

Eg

Dimensionality / Confinement

3 /0

2 / 1

1 / 2

0 / 3

Fig. 6.1

42

Semiconductor quantum wells

d

GaAs quantum wells

C.B.

V.B.

b

AlGaAs

subs

trate

d

AlGaAs

z e-

h+

crystal growth

direction

C.B.

V.B. Eg GaAs Eg AlGaAs

GaAs substrate

GaAs

Single quantum well MQW or superlattice

Fig. 6.2

Eg GaAs Eg AlGaAs

•  Molecular beam epitaxy (MBE) •  Metal-organic chemical vapour deposition (MOCVD)

growth methods

Infinite quantum well Fig. 6.3

0

5

10

0 d/2 �d/2

n = 1

n = 2

n = 3

z

E (h

2/8

m*d

2)

•  kn = nπ/d

•  En = (!kn)2/2m* = (!2π2/2m*d2) n2

•  ψn = (2/d)1/2 sin (knz +nπ/2)

•  symmetry about z = 0 ⇒ wave functions have definite parity

•  ψn has (n–1) nodes

•  En depends on m*, hence heavy and light holes split

43

Finite quantum well Figs 6.4–5

0

4

8

0 2 4 x

y

y = tan(x)

y = 0.85 (13.2-x2)½ / x V0 E2

E1 0

E

n = 1 n = 2

d 2

z 0 d

2 –

•  Wave functions tunnel into the barrier •  wave function still identified by parity and number of nodes •  Confinement energy reduced compared to infinite well •  graphical solution to find En

Example : GaAs/AlGaAs V0 = 0.3 eV, d = 10 nm mw*= 0.067me, mb*= 0.092me E1 = 31.5 meV c.f. infinite well: E1 = 57 meV

Optical transitions

Figs 6.6–7

quantum well

!%

z

n = 2 n = 1

conduction band

valence band

•  Light polarized in x,y plane for normal incidence •  Parity selection rule: Δn = even number •  Infinite well selection rule: Δn = 0

44

2-D absorption

0

Eg

E

!%

kxy

conduction band

valence band 0

Figs 6.8–9

•  Absorption ∝ density of states •  Density of states constant in 2-D: g2D(E) = m / π!2 •  Thresholds whenever !ω exceeds (Eg + Een+ Ehn) •  Band edge shifts to (Eg + Ee1+ Ehh1)

0 5 10 (!%#Eg) in units of (h 2/8d 2µ)

Abs

orpt

ion

coef

ficie

nt

3-D

2-D n = 1 n = 2

n = 3

GaAs quantum wells GaAs/AlGaAs MQW

d = 10 nm

1.4 1.5 1.6 0

5

10

Photon Energy (eV)

α (1

05 m�1 )

0

2

4 bulk

n =1

n = 2 hh lh

300 K

Figs 6.10–11

GaAs/AlAs MQW, d = 7.6 nm

Photon energy (eV) 1.6 1.8 2.0 2.2

Abs

orpt

ion

(au)

0.0

0.5

1.0 hh lh

hh lh

n = 1 n = 2 n = 3

hh lh

T = 6 K

•  Excitonic effects enhanced in quantum wells: strong at room temp •  Pure 2-D: RX

2D = 4 × RX3D

•  Typical GaAs quantum well: RX ~ 10 meV ~ 2.5 × RX (bulk GaAs) •  Splitting of heavy and light hole transitions

45

Spin injection in quantum wells

E

MJ

J = 3/2 J = 1/2

J = 1/2

�1/2,�3/2, +3/2,+1/2,

valence band

conduction band

#!

0

Eg

&+ &+ &# &#

hh hh lh lh

so so

Figure 6.12

Selection rules σ+: Δm = +1,σ�: Δm = �1

Quantum confinement Stark effect (1)

0 –10 10 Position (nm)

0 –10 10

–200

0

1400 1600 1800

Ener

gy (m

eV)

Position (nm)

(a) Ez = 0 (b) Ez = 107 V/m

1455.0 1447.9

–7.4 9.4 1462

.4

1438

.5 e1

hh1

•  10 nm GaAs / Al0.3Ga0.7As quantum well in an electric field •  Eg (GaAs) = 1420 meV

Figure 6.13

46

The quantum confined Stark effect (2)

MQW !% p i n

Ez V0

photocurrent

Energy (eV) 1.4 1.5 1.6 1.7

Phot

ocur

rent

(arb

. uni

ts)

(b) �10 V 1.1×107 V/m

(a) 0 V 1.5×106 V/m

hh1→e2

hh2→e1

hh2→e2 lh1→e1 hh1→e1

GaAs MQW, d = 9.0 nm, 300 K

Figs 6.14–15

•  Red shift of excitons •  Excitons stable to high fields (c.f. Fig 4.5) •  Parity selection rule broken •  used to make modulators

Emission spectrum Figure 6.16

2.4 2.5 2.6 2.7 0

Energy (eV)

PL i

nten

sity

300 K 10 K

Zn0.8Cd0.2Se/ZnSe quantum well

d = 2.5 nm Eg = 2.55eV (10K) Eg = 2.45eV (300K)

•  Emission energy shifted from Eg to (Eg + Ee1 + Ehh1) •  Tune λ by changing d •  Brighter than bulk due to improved electron-hole overlap •  Used in laser diodes and LEDs

47

Intersubband transitions Figure 6.17

n = 1

n = 2 electrons

hν

n-type quantum well

•  Transition energy ~ 0.1 eV (~ 10 µm, infrared) •  Absorption used for infrared detectors •  Emission used for infrared lasers (Quantum cascade lasers)

• Need z polarized light • Parity selection rule: , Δn = odd number

Quantum dots

z

Energy

Den

sity

of s

tate

s

Eg

Bulk (3-D)

Quantum well (2-D)

Quantum wire (1-D)

Quantum dot (1-D)

Figure 6.18

48

Quantum dots

E = 2π 2

2m*nx2

dx2+ny2

dy2+nz2

dz2

!

"

##

$

%

&&

(E�Eg) in units of (h 2/8d 2m*)

Den

sity

of s

tate

s

3-D

0 5 10

d Cuboid dot Spherical dot

R

E = 2

2m*Cnl2 π 2

R2

!

"##

$

%&&

C10 = 1 C11 = 1.43 C12 = 1.83 C20 = 2 "

Colloidal quantum dots

1.5 2.0 2.5 3.0 3.5

Abs

orpt

ion

(arb

. uni

ts)

Energy (eV)

CdSe 10 K

D

C

B

A

(a)

1.6 1.8 2.0 2.2 Energy (eV)

6 nm

5 nm

4 nm

CdTe 300 K

(b)

• Found in semiconductor doped glass (Colour glass filters & stained glass)

• Available commercially

Figure 6.20

49

Self-organized epitaxial dots

InAs quantum dot

200 nm

10 nm GaAs

m

�2, 2,1,�1, 0,

0

1

2

3

Ener

gy in

uni

t of !% 0

0

r V(r)

‘s’ shell

‘p’ shell

‘d’ shell

2-D harmonic oscillator levels

dz ≪ a

z

a

dz

2 2 2

0* 2 ( 1)2

z

z

nE nm dπ

ω≈ + +h h

Figs 6.19, 21

InAs quantum dots Figure 6.22

1.25 1.30 1.35 1.40

200 µm spot

200 nm aperture

Spatially resolved PL Single QD

Far field PL ~107 QDs

T = 10 K

PL In

tens

ity (a

rb. u

nits

)

Energy (eV)

(a)

1.250 1.255 1.260

Phot

ocur

rent

(pA

)

Energy (eV)

PC PL

10 pA

0.6V

PL Intensity (arb. units)

2.6V

Bia

s vol

tage

(b)

T = 10 K

Quantum confined Stark effect

50

Carbon nanostructures

graphene nanotube C60 Bucky ball *

* Buckminster fullerene named after architect R. Buckmintser Fuller

Figure 8.20

Fig. 8.21

Taken from Machon, Phys. Rev B 66, 155410 (2002)

(a) (b)

ky

E

!%

M &

K

(c)

$

$ '

kx

EF

$ &

$ '

& K M –20

–16

–12

– 8

– 4

0

4

Ener

gy (e

V)

&

&

Graphene band structure

Ideal 2-D material

51

Figure 8.22 Graphene transmission

After Nair et al. Science, 320, 1308 (2008).

0 1 2 3 4 5

90

95

100

Tran

smis

sion

(%)

Number of layers

(b)

400 500 600 700 95 96 97 98 99

100

Tran

smis

sion

(%)

Wavelength (nm)

(a)

graphene, single layer

a1

a2

armchair

zigzag

(5,5)

(9,0)

(7,3) "

chiral

Figure 8.23

Nanotube chirality

chiral vector c = n1 a1 + n2 a2

Metallic if n1−n2 = 3m ; otherwise semiconductor

c

Energy

D.O

.S.

Eg

Quantum wire (1-D)

52

Figure 8.24

E E

density of states

kz conduction band

valence band

semiconducting nanotube

E E

kz conduction band

valence band

density of states

metallic nanotube EF

Nanotube transitions

emission

Figure 8.24-5

0 1 2 3 0

1

2

3

Ener

gy g

ap (e

V)

Tube diameter (nm)

E11 m

E11 s

E22 s

900 1000 1100 1200 0.0

1.0

Nor

mal

ized

PL

inte

nsity

(au)

Wavelength (nm)

0.0

1.0

(9,1)

single SWNTs

(7,5)

(8,3)

SWNT ensemble

(9,1) (8,3) (7,5)

Semiconducting Nanotube PL spectra

53

Figure 8.27

0

1

2

3

Ener

gy (e

V)

singlets triplets S0(g)

S1(g)

S2(u)

T1(g) 1.2 ns

1.8 µs

(a) (b)

Energy (eV) 2 3 4 5

0 1 2 3 4

Abs

orpt

ion

(104 c

m–1

)

1.6 1.8 2.0 2.2

Absorption

Energy (eV)

PL (a

.u.)

S1 S2 C60

Carbon bucky ball: C60

54

55

Topic 7: Free electrons

•  Free carrier reflectivity

•  Metals

•  Doped semiconductors

•  Plasmons (bulk & surface)

•  Negative refraction

Free electron refelctivity Fig 7.1

ω / ωp

Ref

lect

ivity

0 1 2 0.0

0.5

1.0

Free electron effects observed in: •  metals •  doped semicondcutors

• Key parameter: Plasma frequency , ωp = (Ne2/ε0m0)1/2 • Lightly-damped system, εr(ω) = 1 � ωp

2/ω2,

• R = 100% below ωp

• transmitting above ωp ⇒ ultraviolet transparency of metals

56

The Drude model Fig 7.2

( )

2/1

00

2p

2

2p

r i

1

!!"

#$$%

&=

+−=

mNeε

ω

γωω

ωωε

• τ = 1/γ = momentum scattering time = m0σ/Ne2, σ = DC conductivity

• Reflectivity < 100% below ωp • Discrepancies caused by interband absorption transitions

0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0

Energy (eV)

Ref

lect

ivity

experimental data

τ = 8.0 fs γ = 0

!%p Aluminium !%p = 15.8 eV, σ = 3.6 × 107 Ω–1 m–1

Interband transitions in Aluminium

Fig 7.3

• Transitions from filled states below EF to empty states above EF • Very high density of states for parallel bands • “parallel band effect” at the critical points • Hence reflectivity dip at 1.5 eV,

Transition energy ~ 1.5 eV

Ener

gy (e

V)

Γ, X W L Γ, K X 0

4

8

12

16

Wave vector k

EF

57

Transition metals

Density of states

E

3d band

4s band

EF

optical transitions

0 2 4 6 8 10 12 14 0.0 0.2 0.4 0.6 0.8 1.0

copper

Energy (eV) R

efle

ctiv

ity

!%p

visi

ble

Fig. 7.4, 7.6

•  Inner d orbitals form narrow bands below the Fermi energy •  threshold for optical transitions to outer half-full s band gives a sharp absorption edge •  hence characteristic colours of metals, eg copper, gold

Copper band structure

W L Γ, X W K

-8 -6 -4 -2 0 2

Ener

gy (e

V)

0 2 4 6

Energy (eV)

Density of states (states eV�1)

0 2 4 6 8 10

-8 -6 -4 -2 0 2

Integrated density of states

Wave vector k

EF

copper 3d and 4s bands

3d bands

minimum transition energy ~ 2.2 eV (560 nm)

Fig 7.5

Electronic configuration: 3d10 4s1

58

Doped semiconductors

10 20 30 0

20

40

60

80

100

Wavelength (µm)

Ref

lect

ivity

(%) 4.0 × 1024

2.8 × 1024 1.2 × 1024 6.2 × 1023 3.5 × 1023

n-type InSb at 300 K

Plasma frequency ,ωp = (Ne2/ε0εrm*)1/2

•  Control plasma frequency by varying the doping density •  Plasma edge in the infrared

Fig 7.7

Free carrier absorption

k

E

EF

scattering process

!%

E = !k 2

2me *

• Transitions from filled states below EF to empty ones above EF • α ∝ λ2 / τ • Scattering process required to conserve momentum • Continuous infrared absorption in doped semiconductors

20

2carrier free

* τωεα

ncmNe

=

Fig 7.8

59

Intervalence band absorption

k

E

EF !!"! #$

hh band

lh band SO band

Δ,

•  Infrared absorption in p-type semiconductors

Fig 7.9

Impurity transitions Fig. 7.10

conduction band

valence band

1 2

n donor levels

conduction band

valence band

1 2 n

(a) low temperature: infrared absorption

(b) high temperature: band tail below Eg

• Impurity levels: En = –(m*/m0)(1/εr2) RH / n2

n-type material

60

Donor absorption in n-type silicon

32 34 36 38 40 42 44 0

1.0

Photon energy (meV)

Abs

orpt

ion

(103

m�1 )

2p0

4p0 3p0

2p± 3p±

4p± 5p±

Phosphorous doped Silicon

H2r0

*e*

2* 1 ,

n11 R

mmRRh

εν =#

$

%&'

( −=

Fig. 7.11

N = 1.2 × 10 20 m–3 T = 4.2 K me* ~ 0.5 m0 εr = 12 R* ~ 45 meV

•  simple model:

•  complications due to effective mass anisotropy

k

Plasmons

• Quantized plasma waves: i.e. longitudinal oscillations of the electron plasma at ωp

Fig. 7.12

61

Observation of bulk plasmons

-400 -200 0 200 400 Energy shift (cm�1)

plasmon absorption

plasmon emission

Ram

an sc

atte

ring

rate

Metals: !%p ~ 3–20 eV observe by electron energy loss spectroscopy : Eout = Ein − n!%p

Ein ~ keV

Doped semiconductors: !%p ~ 10 meV Measure by Raman scattering:

!%out = !%in ± n!%p

n-type GaAs, T = 300 K N = 1.8 × 10 23 m–3 !%p = 19 meV (150 cm–1)

Ein Eout !%p

sample

!%p

Figs 7.12-13

Surface plasmons

0

z

dielectric: (d

x

metal: (m

z = 0

z

k

•  Surface waves at boundary between metal and insulator

•  Plasmonics: propagate energy as surface plasmons in nano-circuits

•  Cannot couple directly to free-space photons due to wave vector mismatch: hence need for couplers (e.g. gratings)

+++ ### +++ ###

metal

dielectric

Figs 7.14 - 15

Ezd (z)

Ezm(z)

Ez

62

Surface plasmon polaritons

0 1 2 0 1 2 ck /%p ckx /%p

%p %p

0 0

% %

% = ck

(a) (b)

%sp

% = ckx /(εd

bulk surface

psp

d1+!

!"

= Hence ωsp = ωp/√2 if εd = 1 (i.e. air)

Fig. 7.16

Localized surface plasmons

300! 500! 700! 900!0.0!

0.4!

0.8!

1.2!

Abs

orpt

ion

(a.u

.)!

Wavelength (nm)!

Gold nanoparticles 6-7nm diameter

Courtesy M. Sugden T. Richardson

•  Colloidal gold is not gold coloured ! •  Localized plasma oscillations at surface of metal nanoparticles •  ωsp ~ ωp/√((d+2) i.e. ωp/√3 for air-metal boundary, but varies

somewhat with size and shape

MesoGold® • pure gold

nanoparticles suspended in water Fig. 7.17

63

Negative refraction

)r

µr

n > 0 ) = 0

n < 0 ) = 0

ñ = i) ) > 0

I II

III IV

ñ = i) ) > 0

"i "r

n < 0 n = 1

perfect lens

metamaterial

• Need both (r and µr to be negative i.e. region III

• Not possible in natural materials: hence need metamaterial

Figs 7.18 - 19

64

65

Topic 8: Phonons (Chapter 10)

•  Infrared active phonons

•  Reststrahlen

•  Inelastic light scattering

$  Raman scattering

•  Phonon lifetimes

Lattice absorption Figs 10.1–2

ω, k Ω, q

Wave vector (q)

Freq

uenc

y

acoustic branch

optical branch

π a

0

phot

on

•  Resonant absorption when photon frequency and wave-vector match the phonon Ω, q

•  Only possible for optical phonons of ionic or polar (partly ionic) crystals

•  Phonon must be transverse: hence TO phonons involved

•  Resonance occurs in the infrared ~ 10 – 100 µm

66

TO phonon resonance

ε = ε0eiqz

x+

x� z

Figure 10.3

•  TO phonon modes of ionic crystals: positive and negative ions move in opposite transverse directions

•  This generates transverse electric field waves

•  Light resonates with these modes when the wave vectors and frequencies match

Reststrahlen

6 8 10 12 14 -20 0

20 40 (a)

ε r

Frequency (THz) 6 8 10 12 14

0.0

1.0 (b)

Frequency (THz)

Ref

lect

ivity

Figure 10.4

22

2)()(

νν

νεεενε

−−+= ∞∞

TO

TOstr

Example: νTO = 10 THz εst = 12.1 νLO = 11 THz ε∞ = 10

•  100% reflectivity between νTO and νLO

•  Lyddane-Sachs-Teller relationship: νlO / νTO = (εst / ε∞ )1/2

67

Infrared reflectivity

200 240 280 0.0

1.0

GaAs InAs

Ref

lect

ivity

Wave number (cm�1)

Figure 10.5

•  Reststrahlen band observed between νTO and νLO •  reflectivity less than 100% due to damping •  damping caused by anharmonic phonon decay. Typical lifetime ~ 10 ps

GaAs : νTO = 273.3 cm–1

νLO = 297.3 cm–1

InAs : νTO = 218.9 cm–1

νLO = 243.3 cm–1

1 cm–1 = 2.998 × 1010 Hz

Polaritons

0.0 1.0 2.0 0

10

20

Wave vector q (106 m�1)

ω /

2π

(TH

z)

ΩTO / 2π

ΩLO / 2π ω = vst q ω = v∞ q

Figs 10.6 – 7

Raman scattering data on GaP

!$LO

lower branch

upper branch

0 1 2 3 36

40

44

48

52 Energy (meV

)

Wave vector (106 m�1)

Calculated: ΩTO / 2π = 10 THz, εst = 12.1, ε∞ = 10

•  Polariton = coupled TO phonon polarization wave & photon wave •  anticrossing of photon and TO phonon modes when ω ~ ΩTO •  modifies the dispersion of both photons and TO phonons

68

Polarons Fig. 10.8

e�

•  free electron in a polar solid produces a local lattice distortion through the electron-phonon coupling αep

•  modifies the effective mass to : m** = m*(1 – αep/6)–1

•  Polaron effects are small in III-V�s (e.g. ~ 1% in GaAs) but increase as the crystal becomes more ionic.

•  Self-trapping leads to hopping conduction instead of band conduction

•  Organic conductors show strong polaron effects

Inelastic light scattering Fig. 10.9

ω1 , k1 ω2 , k2

Ω , q

• Conservation laws: ,ω1 = ω2 ± Ω ; k1 = k2 ± Q

• Stokes scattering: phonon emission (+ sign)

• Anti-Stokes scattering: phonon absorption (– sign)

• Ianti-Stokes / IStokes = exp(–!$/kBT)

•  Raman scattering: scattering from optical phonons •  Brillouin scattering: scattering from acoustic phonons •  Qmax = 2k ~ 106 m–1 for back-scattering geometry ∴ can only probe small wave vector phonons •  Scattering can also be observed from plasmons, magnons etc

69

Raman scattering arrangement Fig. 10.10

sample in cryostat

collection lenses

laser beam

reflected laser

scattered light

CCD array detector

scanning double

spectrometer

entrance slit

computer

#

Raman scattering data Fig. 10.11

200 300 400

GaP

AlSb

InP

GaAs

300 K Ram

an si

gnal

(a.u

.)

Energy shift (cm�1)

•  Raman scattering from TO and LO phonons in various III-V semiconductors using a Nd:YAG laser at 1.06 µm •  1cm–1 = 3 × 1010 Hz = 0.124 meV

TO LO

70

Phonon lifetimes

Ω1 , q1

Ω2 , q2

Ω3 , q3

Ω1 , q1

Ω2 , q2 Ω3 , q3

Figs 10.12 – 13

•  3 phonon processes caused by anharmonicity in the crystal

Wave vector (q)

Freq

uenc

y (Ω

)

acoustic branch

optical branch

0 π a

ΩTO

ΩTO

2

0 π a

•  TO phonon can decay into two acoustic phonons •  Lifetime typically ~ 10 ps.

“Klemens channel” (see Klemens 1966)

71

Topic 9: Nonlinear optics (Chapter 11)

•  Definitions

•  Resonant vs non-resonant nonlinearities

•  Second-order nonlinearities

$ frequency doubling

$ electro-optics

•  third-order nonlinearities

Nonlinear susceptibility D = ε0 E + P ≡ ε0 εr E

P = ε0 χ(1) E + ε0 χ(2) E2 + ε0 χ(3) E3 + ….

% εr = 1 + χ(1) + χ(2) E + χ(3) E2 + ….

• χ(n) = nth order nonlinear susceptibility,

• χ(2) = 0 in materials with inversion symmetry (centrosymmetric)

ie: all isotropic materials (gases, liquids, glasses)

+ some crystals (eg NaCl)

• optical properties depend on the light intensity

• needs large electric fields (ie high power lasers)

72

Nonresonant nonlinear response P

ε,

ε (t),

t,

t

P(t) P

ε,

ε (t),

t,

t

P(t) (a) (b) Figure 11.1

distorted output contains terms at 2ω, 3ω etc

small amplitude sine wave input

large amplitude sine wave input

sine wave output

•  laser is tuned to the transparency region of crystal •  far from resonance with any atomic transitions

Resonant nonlinear response

Figs. 11.2-3

N2

N1

2

1 uν A

z dz

I(z) I – dI

•  laser is tuned to resonance with an atomic transition •  stimulated emission reduces net absorption rate at high intensity •  equivalent to “Pauli blocking” of upper state •  saturable absorption: α(I) = α0 / (1 + I / Is) ; Is = saturation intensity

•  α(I) ≈ α0 – α2 I at small I ; α2 = α0 / Is

•  Δεr ∝ I ∝ E2 ∴third-order nonlinearity

73

Second order nonlinear effects

ω2

ω1

ω1+ω2

Process Input Ouptut Frequency mixing ω1, ω2 , , ω1 + ω2 Frequency doubling ω, ω 2ω,Down conversion ω1 ω2, ω1 � ω2,

�ω2

ω1

ω1�ω2 (b)

Fig. 11.4

(a)

Pi(2) = ε0 Σjk χ(2)

ijk Ej Ek χ(2) = 0 in centrosymmetric crystals

Phase matching

• Phase matching condition: 2kω = k2ω (conservation of momentum) ⇒ 2× n(ω) ω / c = n(2ω) 2ω / c ⇒ n(ω) = n(2ω) • not usually possible due to dispersion. • use a birefringent crystal with optic axis at angle θ :$

kω 2ω

χ(2) ω

k2ω

2o

2

2e

2

2o )2(

cos

)2(

sin

)(

1

ω

θ

ω

θ

ω nnn+=

θ

74

2ω,

Frequency doubling / tripling etc

Figure 11.5

Nd:YAG laser 1064 nm

nonlinear crystal

532 nm

355 nm

266 nm

2ω,

4ω,

ω,

ω,

2ω, 3ω,

nonlinear crystal

nonlinear crystal

• use a 2nd order nonlinear crystal

Electro-optic modulators

z x

y

x'

y' 45°

E polarizer

crossed polarizer E-O crystal

V input

output

• Linear electro-optic (Pockels) effect: Field-induced birefringence: Δn ∝ rij E

• rij = electro-optic coefficient ≠ 0 for non centro-symmetric crystal

• Consider as χ(2) process with DC field • Use to make intensity modulators Figure 11.6

75

Third-order nonlinear processes

Figure 11.7

(b) Frequency tripling

ω,ω,ω,

3ω,

(c) Optical Kerr effect

ω ω �ω

ω

(a) Four-wave mixing

ω1 ω2 ω3

ω1+ω2+ω3

(d) Raman effect

ω �ω ωs

ωs

Kerr cells

E

polarizer

crossed polarizer Kerr

medium

input

output

V

polarizer

crossed polarizer Kerr

medium

input

output

laser pulse

E

Figure 11.8

• Quadratic electro-optic effect: Field-induced birefringence: Δn ∝ χ(3) E 2

• Occurs even in isotropic materials (eg glass, liquids) • DC Kerr effect: rotate polarization with DC field • Optical Kerr effect: induce birefringence with light pulse

76

Nonlinear refractive index

Figure 11.9

Δεr = χ(3) E2

Δn = n2 I ⇒ n2 = χ(3) / n0

2 c ε0

n(I) = n0 + n2 I

• Third order nonlinear effects found in all materials including isotropic media e.g. glasses, liquids, gases • self-phase modulation: Δφ = 2 π Δn L / λ = 2 π n2 I L / λ,

• soliton: balance dispersion and self-phase modulation • propagation of high intensity short pulses along optical fibres

Semiconductor nonlinearities Figs. 11.10-11

k

E

Eg

10 102 103 104 105 0 2 4 6 8

α (

103 c

m�1 )

I (W cm�2)

α( )/

II

=+

+6700

1 2902000

E = 0.77 eV

0.7 0.9 1.1 0

10

20

30

40

α (

103 c

m�1 )

E (eV)

GaInAs/InP MQW

exciton

300 K

•  strong Pauli blocking when !% ~ Eg

⇒ saturable absorption •  third order nonlinearity •  excitonic enhancements •  see also Fig. 4.7

77

Saturable absorption in quantum dots Fig. 11.12

0.1 1 10 100 1000 10–4

10–3

10–2

Power (nW)

Pea

k ab

sorb

ance

1.26545 1.26550 1.26555 0

5

10

15

Energy (eV)

Abs

orba

nce

(uni

ts o

f 10–3

)

3.5 µeV

•  Tune laser to QD exciton •  Behaves as saturable two-level atom

N2

N1

2

1 uν