Optical Properties

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Part III Optical Properties of Materials

Chap. 10 The optical constants

Chap. 11 Atomistic Theory of the Optical Properties

Chap. 12 Quantum Mechanical Treatment of the

Optical Properties

Chap. 13 Applications

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Atomistic Theory of the Optical Properties

11.1 Survey

- Hagen-Rubens equations (Fig 11.1(a))

The validity of equations derived from continuum theory, considering only macroscopic quantities and interrelating experimental data, are often limited to frequencies for which the atomistic structure of solids does not play a major role.

- Drude model (Fig 11.1(a))

In the visible and near IR region, an atomistic model needs to be

considered to explain the optical behavior of metals. Moving electrons collide with certain metal atoms in a nonideal lattice. Absorption band cannot be explained by the Drude theory.

- Lorentz postulations (Fig 11.1(a))

The electrons are considered to be bound to their nuclei, and an external electric field displaces the positive charge of an atomic nuleus against the negative charge of its electron cloud: “harmonic oscillator”

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An oscillator absorbs a maximal amount of

energy when excited near its resonance

frequency. (Fig 11.1b)

11.1 Survey

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11.2 Free Electrons Without Damping

Let’s consider the interaction of a plane-polarized light with the electrons.

The field strength of the plane-polarized light wave is given by

)

0

exp( i t E

E = ω

)

0

exp(

2 2

t i eΕ

dt eE x

m d = = ω

2

4 π

2

ν

m x = − eE

The stationary solution of this vibrational equation is obtained by forming the second derivative of the trial solution

This yields

The vibrating electrons carries an electric dipole moment.

)

0

exp( i t x

x

where ω (= 2πν ) is the angular frequency

ω

=

exΝ

f

P =

^where^N_f is free electrons per cubic centimeter

Atomistic Theory of the Optical Properties

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The dielectric constant

E P

0

1 ε ε = +

2 0

2

f 2

m 1 4

ˆ π ε ν

ε = − e N

Inserting (11.3) and (11.4) into this equation

The dielectric constant equals the square of the index of refraction, n

2 0

2 2 2

m 1 4

ˆ π ε ν

Ν

f

n = − e

We consider two special cases as follows,

(a) For small frequencies, the term is larger than one.

Then is negative and imaginary. An imaginary means that the real part of disappears. Eq.(10.25) becomes, for n = 0

2 0

2 2

m 4 π ε ν

Ν

f

e

ˆ

2

n n ˆ

ˆ

2

n 1 1

1 )

1 (

) 1 (

2 2 2

2

=

+

= + +

+

= −

k k k

n

k

R n i.e., the reflectivity is 100

% (Fig 11.3)

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(b) For large frequency (UV light), the term becomes

smaller than one. Thus is positive and real. The reflectivity for real values of , i.e., for k = 0, becomes

2 0

2 2

m 4

π ε ν

Νf

e

ˆ

2

n n ˆ = n

n ˆ

2 2

) 1 (

+

= − n

R n The material is essentially transparent for these wavelengths (Fig 11.3)

We define a characteristic frequency, ν

₁

often called plasma frequency, which separates the reflective region from the

transparent region. (Fig 11.3) The plasma frequency can also be deduced from (11.6) or (11.7). In these equations must have the unity of the square of a frequency,

which we define to be ν

₁

2 0 2

2

m 4π ε ν

Νf

e

m ε π

Ν

ν e

^f

0 2 2 2

1

= 4

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In Table 11.1, the calculated and the observed values for ν₁ are only identical for sodium : sodium does exactly one free electron per atom contribute to the electron pas.

For other metals “effective number of free electrons” is commonly

introduced, which is defined to be the ratio between the observed and calculated ν₁ values

Ν

eff

) = calculated (

) observed (

2 1

2

ν

1

ν

2

0 2 2

2 2

eff

4 ) 1

(

e

m k

N − n + ν π ε

=

The alkali metals are

transparent near UV and reflect the light in the visible region (Table 11.1) : the s - electrons of the outer shell of the alkali metals can be considered to be free.

N_eff can be obtained by measuring n and k in the red or IR spectrum

Atomistic Theory of the Optical Properties

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11.3 Free Electrons With Damping

)

0

exp(

2 2

t i eE

dt eE γ dx dt

x

m d + = = ω

2

0

2

=

dt x

d = = υ ′

dt dx γ

eE

To take account of the damping, we add to the vibration equation (11.2) a damping term , which is proportional to the velocity

We determine first the damping factor,

The damping is depicted to be a friction force which counteracts the electron motion

) / ( dx dt γ

γ const.

′ = υ

eΝ

f

= j

υ ′ j = σ

₀

E

0 2

σ

f

e γ = N

)

0

exp(

0 2 f 2

2

t i eE

dt eE dx e

Ν dt

x

m d ω

σ = =

+

,

Thus (11.11) becomes,

(11.11)

Atomistic Theory of the Optical Properties

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The stationary solution of eqn.

obtained by differentiating the trial solution by time and inserting the second derivatives into the equation, which yields

)

0

exp(

0 2 f 2

2

t i eE

dt eE dx e

Ν dt

x

m d ω

σ = =

+

)

0

exp( i t x

x = ω

eE x ωω

σ e x Ν

m ω +

^f

=

−

0 2 2

e mω σ

eω Ν

x E

f 2

0

−

=

e i mω σ

eω Ν

E Ρ eΝ

f

2

0

−

=

2 2

0 2

0 0

0

2 4

1 1 ˆ 1

e ν Ν

ε π i m

σ ν E πε

ε ε Ρ

f

− +

= +

=

exΝ

f

P = then

E P

0

1 ε

ε = + then

2 0

2 1 0

2 1

2

0

1 2

2 1 1 ˆ

σ ν ν ν πε

ν ν

ν σ

ν ε πε

− +

=

− +

=

i

i the term

₂ ₁²

0 2

2

to equal m

4 ν

ν ε

π Ν

f

e

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11.3 Free Electrons With Damping

The term in (11.22) has the unity of a frequency. We define a damping frequency⁰

2 1

0 /

2

πε ν σ

0 2 1 0 0

2 1 0

2

2 2 πε ν ρ

σ ν ν = πε =

2 2

2

1

ˆ νν ν

ε ν

+ −

= i

k i nki

n n

2 2

2 2 1

2

2 1

ˆ )

( ν νν

ν

− −

=

−

=

where is identical to

,

Table 11.2 lists values for ν₂which were calculated using experimental ρ₀and ν₁values. Now (11.22) becomes ,

ε ˆ n ˆ

²

Atomistic Theory of the Optical Properties

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Multiplying the numerator and denominator of the fraction in (11.25) by the complex

conjugate of the denominator allows us to equate individually real and imaginary parts.

This provides the Drude equations for the optical constants

2 2 2

2 1 1

2

1

ν ν

ε ν

− +

=

− k n

2 2 2

2 1 2

2

2 2

ν ν

ε σ

= +

=

= nk

m e

0 2

f 2

1 4

π ε

ν

= ^Ν

0 2 1 0 2

2 σ

ν ν = πε

with the characteristic frequencies and

Atomistic Theory of the Optical Properties

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11.4 Special Cases

For the UV, visible, and near IR regions, the frequency varies between 10¹⁴ and 10¹⁵ s^-1. The average damping frequency, ν₂ is 5ⅹ10¹² s^-1

(Table 11.2). Thus, ν² >> ν₂². Equation (11.27) then reduced to

With ν ≈ ν₁ (Table 11.1) We obtain

For very small frequencies ν² << ν₂² , we may neglect ν² in the denominator of (11.27). This yields, with (11.23)

Thus, in the far IR the a.c. conductivity, σ and the d.c. conductivity σ₀ may be considered to be identical

2 2 1 2

2

ν

ν ν ε = ν

ν ε

₂

≈ ν

²

0 0 2

2 1

0

2 4

1

4 πε

σ ν

ν πε

ν = σ = = nk

Atomistic Theory of the Optical Properties

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11.5 Reflectivity

The reflectivity of metal is calculated using (10.29) in conjunction with (11.26) and (11.27).

Atomistic Theory of the Optical Properties

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11.6 Bound Electrons (Classical Electron Theory of Dielectric Materials)

“At higher frequency, the light is absorbed and

reflected by metal as well as by nonmetals in a narrow frequency band” →

It can be interpreted by

Lorentz model: He assumed that under the influence of and external electric field, the positively charged

electron cloud are displaced with respect to each other (Fig 11.7) →

“harmonic oscillator”

Atomistic Theory of the Optical Properties

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11.6 Bound Electrons

Under the influence of and alternating electric field (i.e. by light), the electron is thought to perform forced vibrations

)

0

exp( i t eE

eE = ω

)

0

exp(

2 2

t i eE

dt kx γ dx dt

x

m d + ′ + = ω

[ ( ) ]

exp )

(

₀² ² ² ² ²

2

0

ω φ

ω ω

ω

ε −

+ ′

= − i t

γ m

x e

m

= k

= ₀

0 2

πν

ω

The vibration equation:

kx : restoring force (x is displacement, k is the spring constant), γ’ : damping parameter

The stationary solution for week damping (see Appendix 1)

where is resonance frequency of the oscillator,Φ is the phase

difference between forced vibration and the excitation force of the light wave (see Appendix 1)

) (

2 )

tan (

₂ ₂

0 2

2

0

π ν ν

ν ω

ω φ ω

−

= ′

−

= ′

m γ m

γ

Atomistic Theory of the Optical Properties

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exΝ

a

P = [ ]

2 2 2

2 2

0 2

) (

exp

ω ω

ω

φ ω

γ m

t i E

Ν P e

^a

+ ′

−

= −

[ ( ) ] exp( ) exp( )

exp i ω t − φ = i ω t ⋅ − i φ

) exp(

)

(

₀² ² ² ² ²

2

φ

ω ω

ω γ i

m

E Ν

P e

^a

−

+ ′

= −

) exp(

) (

1 ˆ 2

2 2 2

2 2

0 2 0

2 2

2

φ

ω ω

ω

ε ε i

γ m

E Ν nki e

k

n

^a

−

+ ′ + −

=

−

=

which yield (11.5) and (10.12) Inserting (11.33) yields with

E P

0

1 ε

ε = + n ˆ

²

= n

²

− k

²

− 2 nik ≡ ε ˆ = ε

₁

− i ε

₂

with

φ φ

φ ) cos sin exp( − i = − i

ω φ ω

ω

ε ( ) cos

1

2

2 2 2 2 2

0 2 0

2 2

2

γ m

Ν nki e

k

n

^a

+ ′ + −

=

−

ω φ ω

ω

ε

₀ ²

(

₀² ²

)

² ² ²

sin

2

γ m

Ν

i e

^a

+ ′

− −

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The trigonometric terms in (11.42) are replaced, using (11.35), as follows

2 2 2

2 2

0 2

2 2

0

2

( )

) (

tan 1

cos 1

ω ω

ω

ω ω

φ φ

γ m

m

+ ′

−

= −

= +

2 2 2

2 2

0 2

2

( )

tan 1

sin tan

ω ω

ω

ω φ

φ φ

γ m

γ

+ ′

−

= ′

= +

separating the real and imaginary parts in (11.42) finally provides the optical constants

that is,

] )

( [

)

1

₂

(

₂ ₂ ₂ ₂

0 2 0

2 2

0 2

2 2

1

m γ ν

mN k e

n

^a

+ ′

− + −

=

−

= ε ω ω

ω ε ω

] )

( 4

[

)

1

₂

(

₂ ₂ ₂ ₂

0 2 2 0

2 2

0 2

1

m ν ν γ ν

ν ν

mN

e

_a

+ ′

− + −

= ε π

ε

] )

(

2 [

₂ ₂ ₂ ₂ ₂

0 2 0

2

ε ω ω ω

ε ω

γ m

γ N

nk e

^a

+ ′

−

= ′

= 2

₀

[ 4

² ²

(

₀² ²

)

² ² ²

]

2

m ν ν γ ν

ν γ N e

_a

+ ′

−

= ′

π ε πε

or

Atomistic Theory of the Optical Properties

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11.7* Discussion of the Lorentz Equation for Special Cases Small Damping : γ’ is very small, equation (11.45) reduced to

) (

1 4

₂ ₂

0 0

2 2 2

2

1

m ν ν

N k e

n

^a

+ −

=

−

= π ε

ε

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11.7* Discussion of the Lorentz Equation for Special Cases

Absorption near ν₀: Electrons absorb most energy from light at the

resonance frequency, i.e., ε₂ has a maximum near ν₀. For small damping, the absorption band becomes an absorption line (Fig 11.12)

0 0 2

2

2 γ ν

N e

_a

= ′ ε πε

More than One Oscillator

∑ − − + ′

+

=

−

=

i i i

i a

ν γ ν

ν m

ν ν

f mN

k e

n

₂ ₂ ₂ ₂ ₂

0 2 2

2 2

0 0

2 2

2

1

4 ( )

) 1 (

π ε ε

∑ − ′ + ′

+

=

i i i

i i a

ν γ ν

ν m

γ f N

nk e

₂ ₂ ₂ ₂ ₂

0 2 2 0

2

2 1 2 4 ( )

π

ν ε πε

Atomistic Theory of the Optical Properties

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11.8 Contribution of Free Electrons and Harmonic Oscillators to the Optical Constants

The optical properties of metals may be described by postulating a certain number of free electrons and a certain number of harmonic oscillators.

Both the free electrons and the oscillators contribute to the polarization.

Thus, the equations for the optical constants may be rewritten, by combining (11.26), (11.27),(11.49), and(11.50)

∑ − − + ′

+ +

−

=

i i i

i a

ν γ ν

ν m

ν ν

f mN

e ν

ν ν

2 2 2

2 2

0 2 2

2 2

0 0

2 2

2 1

1

4 ( )

) 1 (

π ε ε

∑ − ′ + ′

+ +

=

i i i

i i a

ν γ ν

ν m

γ ν f N

e ν

ν ν ν

nk ν

₂ ₂ ₂ ₂ ₂

0 2 2 0

2 2

2 1 2

2

2 2 4 ( )

π ε πε

Atomistic Theory of the Optical Properties

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11.8 Contribution of Free Electrons and Harmonic Oscillators to

the Optical Constants