Chapter 5 Optical Properties of Materials

Chapter 5

Optical Properties of Materials Part I Introduction Classification of Optical Processes

Snell’s law

absorption ~ resonance

luminescence Optical medium ~ spontaneous emission a. Specular elastic and • Reflection b. Total internal Inelastic c. Diffused scattering • Propagation nonlinear-optics Optical medium • Transmission Propagation General Optical Process • Incident light is reflected, absorbed, scattered, and/or transmitted

Absorbed: IA Reflected: IR

Transmitted: IT

Incident: I0 Scattered: IS

I 0  IT  IA  IR  IS

Conservation of energy Optical Classification of Materials

Transparent Translucent Opaque Optical Coefficients

If neglecting the scattering process, one has

I0  IT  I A  I R

Coefficient of reflection (reflectivity)

Coefficient of transmission (transmissivity)

Coefficient of absorption (absorbance) Absorption – Beer’s Law

I 0 I(x)

Beer’s law

x 0 l

a is the absorption coefficient (dimensions are m-1). Types of Absorption • Atomic absorption: gas like materials The atoms can be treated as harmonic oscillators, there is a single resonance peak defined by the reduced mass and spring constant.

v v0 Types of Absorption Paschen • Electronic absorption

Due to excitation or relaxation of the electrons in the atoms Molecular Materials Organic (carbon containing) solids or liquids consist of molecules which are relatively weakly connected to other molecules. Hence, the absorption spectrum is dominated by absorptions due to the molecules themselves. Molecular Materials

Absorption Spectrum of Water Molecular Materials • Electronic absorption – molecular orbital transition Solid State Materials Lattice vibration – phonon spectrum Solid State Materials Solid State Materials Electron transition – energy band structures Solid State Materials Electron transition – energy band structures Solid State Materials Electron transition – energy band structures Solid State Materials Conductors Almost any frequency of light can be absorbed Since there is a very high concentration of electrons, practically all the light is absorbed within about 0.1μm of the surface The metal reflects the light very well – about 95% for most metals

The metal appears “silvery” since it acts as a perfect mirror Solid State Materials Conductors Flat

Reflect more red Solid State Materials Semiconductor and dielectrics

Dielectrics and semiconductors behave essentially the same way, the only difference being in the size of the bandgap Solid State Materials Semiconductor and dielectrics Solid State Materials Semiconductor and dielectrics Impurity levels divide up the bandgap to allow transitions with energies less than Eg Recombination can be either radiative (photon) or non-radiative (phonon) depending on the transition probabilities Practical p-n diodes usually contain a small amount of impurity to help recombination because Si has a relatively low recombination “efficiency” Part II

Complex Refractive Index Wave Propagates in Absorption Medium

The incident wave,

The refractive wave, E E(x) k’ wavenumber in the medium. Rewrite the refractive wave in a plane wave format

0 x l The effective wavenumber is Wave Propagates in Absorption Medium The effective wave velocity,

The speed of light in vacuum,

The index of refraction has a complex value

Let

One obtain Wave Propagates in Absorption Medium According to Maxwell’s equations, n   ~   1  i 2 n~2  ~ The relative dielectric constant  is also a complex

2 2 1  n   ,  2  2n

1 1 1 2 2 2 2 n  [1  (1   2 ) ] 2 1 1 1 2 2 2 2   [1  (1   2 ) ] . 2 Part III

Classical Propagation Propagation in a Dense Optical Medium Three types of oscillators:

1. Bound electron (atomic) oscillator

2. Vibrational oscillator;

3. Free electron oscillators Propagation in a Dense Optical Medium Atomic oscillator 1 1 1   ,  m0 mN

KS 0  ,     p  q(r  r ) p(t)  ex(t) Propagation in a Dense Optical Medium Atomic oscillator Propagation in a Dense Optical Medium Atomic oscillator

If  = 0, resonant absorption (Beer’s law)

hv = E2 - E1 re-radiated photon – luminesce radiationless transition

If   0, non-resonant, transparent The oscillators follow the driving wave, but with a phase lag. The phase lag accumulates through the medium and retards the propagation of the wave front, leading to smaller velocity than in free space (v =c / n). -- the origin of n Propagation in a Dense Optical Medium Vibrational oscillators

K S 12 13 0   10 10 Hz 

Infrared spectral region

In a crystalline solid form the Classical model of a polar condensation of polar molecules, molecule (an ionic optical these oscillations are associated medium) with lattice vibrations (phonons). Propagation in a Dense Optical Medium Free electron oscillators

Free electrons, Ks = 0, 0 = 0 Drude-Lorentz model The Lorentz Oscillator

2 d x 2 m0  m00 x  0, mN  m0 dt 2 2 d x dx 2 m0  m0  m00 x  eE(t) E dt 2 dt Light wave will drive oscillations at its own Frequency:  i t E (t )  E 0 e  i t Assuming x (t )  X 0 e

2  it  it 2  it  it  m 0  X 0 e  im 0  X 0 e  m 0  0 X 0 e  eE 0 e

 eE0 / m0 X 0  2 2 0   i The Lorentz Oscillator The electric displacement D:

D   0E  P   0E   0E   0 (1 )E   0E

P is the (macroscopic) density of the permanent and induced electric dipole moments in the material, called the polarization density. The Lorentz Oscillator The macroscopic polarization of medium P:

Presonant  Np  Nexnˆ Ne2 1  2 2 E m0 (0  i) N is the number of atoms per unit volume. The electric displacement D can be expressed as

D  0E  P

  0 E  Pbackground  Presonant Ne2 1   0 E  0E  2 2 E m0 (0   i) The Lorentz Oscillator The dielectric constant  can be obtained

Ne 2 1  ( )  1    2 2  0 m0 ( 0    i ) 2 2 2 Ne  0    1 ( )  1    2 2 2 2  0 m0 ( 0   )  ( ) Ne 2   2 ( )  2 2 2 2  0 m0 ( 0   )  ( ) The Lorentz Oscillator

Frequency dependence of the real and imaginary parts of the complex dielectric constant of a dipole at frequencies close to resonance. Also shown is the real and imaginary part of the refractive index calculated from the dielectric constant. The Lorentz Oscillator Low frequency limit: Ne2  (0)   st 1   2  0m00 High frequency: ()  1 

Ne 2 Thus ( st    )  2  0 m00

2 0  Close to resonance:  1 ( )     ( st    ) 4( ) 2   2

 0  2 ( )  ( st    ) 4( ) 2   2 Multiple Resonance

Take account of all the transitions in the medium (different resonant oscillation frequency)

P  NPr  Nexnˆ Ne2 1   2 2 E m0 j ( j  i j)

The dielectric constant  can be expressed

Ne 2 1  () 1  2 2  0m0 j ( j    i j) Multiple Resonance

Assign a phenomenological oscillator strength fj to each transition: 2 Ne f j  () 1  2 2  0m0 j ( j    i j)

For each atom. where  f j 1. j

Schematic diagram of the frequency dependence of the refractive index and absorption of a hypothetical solid from the infrared to the x-ray spectral region. The solid is assumed to have three resonant frequencies with width of each absorption line has been set to 10% of the center frequency by appropriate choice of the j’s. Multiple Resonance 1. n >>  except near the peaks of the absorption; 2. The transmission range of optical materials is determined by the electronic absorption in UV and the vibrational absorption in IR; 3. IR absorption is caused by the vibrational quanta 13 in SiO2 molecules themselves(1.4  10 Hz (21m) and 3.3  1013 Hz(9.1 m); 4. UV absorption is caused by interband electronic transition(band gap of about 10 eV), threshold at 2  1013 Hz(150 nm)(a ~ 108 m-1); 5. UV absorption departure from Lorentz model; 6. n actually increases with frequency in transparency region, the dispersion originates from wings of two absorption peaks of UV and IR; 7. The phase velocity of light is greater than c in (a) Refractive index and (b) extinction region where n falls below unity; coefficient of fused silica (SiO2) glass from the infrared to the x-ray spectral region. Free Electrons Plasma: A neutral gas of heavy ions and light electrons. Metals and doped semiconductors can be treated as plasmas because they contain equal numbers of fixed positive ions and free electrons. Free electrons in this system experience no restoring force from the medium when they interact with electromagnetic waves. driven by the electric field of a light wave. Free Electrons

Drude-Lorentz model: Considering the oscillations of a free electron induced by AC electric field E(t) of a light wave with polarized along the x direction: 2 d x dx it m0  m0  eE(t)  eE0e . dt 2 at it By substituting x  x0e eE(t) x(t)  2 . m0 (  i) The electric displacement:

D   r  0 E   0 E  P Ne 2 E   0 E  2 m 0 (  i ) Free Electrons Therefore, Ne 2  ( )  1  2  0 m 0 (  i )

Or, 2  p  ( )  1  , ( 2  i ) 1  Ne 2  2 where   .  p    p: plasma frequency   0 m 0  For a lightly damped system,  = 0, so that 2  p  r ( )  1   2

n is imaginary for  < p positive for  > p zero for  = p, Free Electrons 2 n~  1 Later we will know that the reflectivity is R  n~  1

Reflectivity of a less damped free carrier gas as a function of frequency. Free Electrons

Experimental reflectivity of Al as a function of photon energy. The experimental data is compared to predictions of the free electron model with h = 15.8 eV. The dotted line is calculated with no damping. The dashed line with  = 8.010-15 s, which is the value deduced from the DC conductivity. All metals will become transmitting if  > p ( UV transparency of metals) Free Electrons The Drude model: The valence electrons is free. The density N is equal to the density of metal atoms multiplied by their valency; The characteristic scattering time  can be determined by the measurement of .

Free electron density and plasma properties of some metals. The values of N are in the range 1028—1029 m-3. the very large values of N lead to plasma high electrical and thermal conductivities and plasma frequency in the UV region.