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Solid State Optics

Optical properties of solids

Pok.132A GF, bacewicz@if.pw.edu.pl www.if.pw.edu.pl/~bacewicz

Password: wf.sso

Rajmund Bacewicz

Literature:

M. Fox „Optical Properties of Solids” Oxford University Press, (2010);

C.F. Klingshirn, Semiconductor Optics, 4th ed., Graduate Texts in Physics, DOI 10.1007/978-3-642-28362-8 6, © Springer-Verlag Berlin Heidelberg (2012);

J.Garcia Sole, L.E.Bausa and D.Jaque, „Optical Spectroscopy of Inorganic Solids” Wiley (2005);

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Solid State Optics

30 h lecture + examination

15 h lab

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Aim of the Lecture:

• To explain the optical properties of materials in terms of interaction

mechanisms between light and matter

• To provide physical basics of photonic

devices

(4)

Prerequisites

• Electrodynamics – basic course

• Introduction to Solid State Physics

• Quantum Mechanics - basics

(5)

Syllabus

1. Electric permittivity of solids in classical electrodynamics.

Optical constants;

2. Lorentz-Drude theory of dispersion;

3. Interband transitions in solids. Fundamental absorption edge

4. Excitons and role of the electron-hole interaction in optical properties of crystals

5. Absorption of light by defects. Configurational diagram 6. Light – free carriers interaction

7. Surface plasmons – plasmonics.

8. Emission of light from solids

9. Interaction of light with atom vibrations 10. Inelastic light scattering in solids

(6)

What is the wavelength range of optics?

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Wavelength Frequency (Hz) Photon energy

Quantum transition

Gamma -ray 0.005 – 0.1Å 3×1019 - 6×1020 ∼ MeV Nuclear X-ray 0.1 – 100 Å 3×1016 - 3×1019 ∼ keV Inner core

electrons Ultraviolet 10 – 400 nm 7×1014 - 3×1016 3 – 120 eV Bonding

electrons Visible 400 – 700 nm 4×1014 - 7.5×1014 1.8 - 3 eV Bonding

electrons Near-

infrared

800 – 1000 nm 3×1014 - 4×1014 ∼1 eV Rotationa/

vibrational Infrared 1 – 100 µm 3×1012 - 3×1014 0.01 – 1 eV Rotationa/

vibrational Terahertz 100 – 1000 µm 3×1011 - 3×1012 0.01 – 0.001 eV Rotational

Microwave 1 – 4 mm <3×1011 Rotational

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THz radiation

THz scanners

(9)

Interaction of Light with Matter

Light – classical Matter - classical

Light – classical Matter - quantum Different theoretical approaches

+ Quantum optics

(10)

Maxwell equations

E ε P ε

E D ε

r r

r r

0

0

+ =

=

Macroscopic description of light – solid interaction

Electric displacement field, or

Electric Flux Density

t E B

− ∂

=

×

r r

t H D

= ∂

×

r r

0 0

=

=

B D r

r

 

 

= ∂

, z

, y

x

(11)

[ ( ) ]

exp )

,

( r t E i q r t

E r r = r

o

r r − ω

λ π / q = 2

λ πν π

ω = 2 = 2 c

Plane wave – a Fourier component of any electric field

wavevector

angular frequency

λ π /

q = 2 q

Is a small due to a large wavelength of light as compared to interatomic distances

(12)

Assume, that we have external electric field of the electromagnetic wave propagating in a medium.

E induces dipoles in a medium

Optical response is given by the polarization vector P(r, t)

representing induced dipole moment of the unit volume of a solid.

Optical response

p N P r r

=

d q p

r r

=

E = 0

dipole moment

(13)

Most general linear, nonlocal relation between P and E:

t 0

-

P (t, r ) = ε dt ' d r 'f (t, t ', r, r ') E (t ', r ') ˆ

∫ ∫

for an arbitrar

y

E field

Nonlocality in time

Nonlocality in space causality

(14)

t 0

-

P (t, r ) = ε dt ' d r 'f (t, t ', r, r ') E (t ', r ') ˆ

∫ ∫

t

i (q r - t) i (q r ' - t ')

0 -

P ( , q ) ω e

ω

= ε dt ' d r 'f (t - t ', r - r ') ˆ e

ω

E ( , q) ω

∫ ∫

assume

' r r

R r r r

=

't t −

τ =

Let’s substitute

(plane wave)

[ ( ) ]

exp )

,

( r t E i q r t

E r r = r

o

r r − ω

For a single Fourier component:

(15)

0

ˆ

P ( , q ) = ( , q) E ( , q) ω ε χ ω ω

-i (q R - ) 0

ˆ ( , q ) = d d R f ( , R ) e ˆ

ωτ

χ ω τ τ

∫ ∫

-

electric susceptibility

χ ˆ

Electric susceptibility

(16)

E ε P ε

E D ε

r r

r r

0

0

+ =

=

-i (q R - ) 0

ˆ ( , q) = 1 + ( , q ) = 1 + d ˆ d R f ( , R ) e ˆ

ωτ

ε ω χ ω τ τ

∫ ∫

i

d e

0

( ) = 1 + f ( )

ωτ

ε ω τ τ

Time dispersion and space dispersion

ε

- electric permittivity

usually

Electric permittivity

Electric

displacement field

(17)

) (

i )

( )

( ω ε ω ε ω

ε = 1 + 2

t H D

rot

= ∂ r r

) ( ω ε 2

) ( ω

ε 1

- dispersion

- energy dissipation, absorption

(18)

Kramers – Kronig Relations

d '

2

1 2 2

0

2 ' ( ')

( ) - 1 = _ -

'

ω ε ω

ω ω

ε π ω ω

P

d '

1

2 2 2

0

2 ( ') - 1

( ) = - _

- '

ω ε ω

ω ω

ε π ω ω

Dyspersion and absorption are interrelated!

P

(19)

t E B

− ∂

=

×

r r

t H D

= ∂

×

r r 0

0

=

=

B D r

r

E ε P ε

E D ε

r r

r r

0

0

+ =

=

×

Optical Constants

Wave equation 2

0

2

2

=

− ∂

×

×

t

E E c

r ε r

(20)

2 2

2

( )

=

q c

ε ω ω

= 0 for a transverse wave

0 )

( 2

2 2

=

− +

E

E c q

E q q

r r r

r εω

[ ( ) ]

exp )

,

( r t E i q r t

E r r = r

o

r r − ω

2 0

2

=

×

× E

E c q

q

r r r

r εω

2

0

2

2

=

− ∂

×

×

t

E E c

r ε r

For a plane wave:

Dispersion relation

(21)

)

2

( ω = n

c

ε

n

n c = + n

c

- complex refractive index

f

c

v

n c )

n

Re( = =

κ -

extinction coefficient

Relation between ε and optical constants

v

f phase velocity Let’s define

κ

=

)

n

Im(

c

(22)

2

n c

) ( ω =

ε

n κ ω

ε

n κ ω

ε

2 )

(

) (

2

2 2

1

=

=

2 2

2

1

ε 2 κ κ

ε + i = n + n i

Relation between ε and optical constants

(23)

) (

0

t r ω q

e

i

E

E =

r

r

r

r

Plane wave

s i κ c n

q r ω r

( + )

=

) (

0 )

( 0

t ω c

z nω c i

z ω t κ

ω c

z n ω

i

E e e

e E

E =

z

=

r r

) r

1 , 0 , 0 (

=

s

(24)

) (

0

t r ω q

e

i

E

E =

r

r

r

r

Plain wave

s i κ c n

q r ω r

( + )

=

) c t

n z ( c i

) z c t

n z (

i

E e e

e E

E

c ω

ω κω

ω ω

=

=

0 0

r r

) r

1 , 0 , 0 (

= s

Intensity of the light wave (energy flow) c αz

z ω κ c

z ω κ

e e

e E

I

2

=

2 0

- Absorption coefficient

nc ωε c

ωκ

α 2

2

=

=

(25)

n = 1 n = 2

q0 nq0

vacum mediuml

Absorption depth = 1/α

λ0

λ0/n Shorter wavelength

I (z) = I(0) exp(- α z )

)]

t qz

( i exp[

) 0 ( E ) t , z (

E = 0 ω

r

r Er =Er0 exp

(

κq0z

)

exp

[

i

(

nq0z ωt

) ]

κ

i n

n

c

= +

(26)

Water absorption

VIS

Absorption depth(1/α)

λ

1 km 1 m 1 mm 1 µm 1 nm

UV

X-ray

Radio Microwave

IR

1 mm 1 km

1 µm 1 m

(27)

φ

κ

ω κ

i

c

c

R e

i n

i n n

) n (

r =

+ +

= + +

= −

1 1 1

1

Light reflection

I0 R I0

0 0

E r = E

r

Fresnel formula

2 2

2 2 2

) 1 (

) 1 (

n κ n κ r

R + +

+

= −

=

Normal incidence (θ = 0) θ

Amplitude reflection coefficient

Intensity reflection coefficient

(28)

Oblique incidence - ellipsometry

tg e

i p

s

= r =

ρ r ψ

(29)

Oblique incidence - ellipsometry

(30)

Ellipsometry

A method of probing surfaces with light

Obraz

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