14 Optical Properties,Lecture 3

K Asokan

Learning Objectives:

From this module students may get to know about the following:

Propagation of an electromagnetic wave in solid.
Lattice absorption.
Band to band absorption
Direct and indirect semiconductor.

COMPLEX REFRACTIVE INDEX AND LIGHT ABSORPTION

Generally when light propagates through a material, it becomes attenuated in the direction of propagation as illustrated in Figure 1. We distinguish between absorption and scattering both of which give rises to a loss of intensity in the regular direction of propagation. In absorption, the loss in the power in the propagating EM wave is due to the conversion of light energy to other forms of energy, e.g., lattice vibrations (heat) during the polarization of the molecules of the medium, local vibrations of impurity ions, and excitation of electrons from the valence band to the conduction band.

It is instructive to consider what happens when a monochromatic light wave such as

= 0 ( − ) ……………….(1

is propagating in a dielectric medium. The electric field E in Equation (1) is either parallel to x or y since propagation is along z. As the wave travels through the medium, the molecules become polarized. This polarization effect is represented by the relative permittivity of the medium. If there were no losses in the polarization process, then the relative permittivity would be a real number and the corresponding refractive index = √ would also be a real number. However, we know that there are always some losses in all polarization processes. For example, when the ions of an ionic crystal are displaced from their equilibrium positions by an alternating electric field and made to oscillate, some of the energy from the electric field is coupled and converted to lattice vibrations (intuitively, “sound” and heat). These losses are generally accounted for by describing the whole medium in terms of a complex relative permittivity (or dielectric constant) , that is,

= ′ − ′′ ………………(2)

Where the real part ′ determines the polarization of the medium with losses ignored and the imaginary part ′′ describes the losses in the medium. For a lossless medium, obviously = ′ . The loss ′′ depends on the frequency of the wave and usually peaks at certain natural (resonant) frequencies. If the medium has a finite conductivity (e.g., due to a small number of conduction electrons), then there will be a Joule loss due to the electric field in the wave driving these conduction electrons. This type of light attenuation is called free carrier absorption. In such cases, ′′ and are related by

′′ =		…………….(3)
0

Where 0 is the absolute permittivity and is the conductivity at the frequency of the EM wave.

Since is a complex quantity, we should also expect to have a complex refractive index.

An EM wave that is traveling in a medium and experiencing attenuation due to absorption can be generally described by a complex propagation constant k that is,

= ′ − ′′ …………….(4)

Where k’ and k” are the real and imaginary parts. If we put Equation 4 into Equation 1, we will find the following,

= 0 exp(− ′′ ) exp ( − ′ ) …………….(5)

The amplitude decays exponentially while the wave propagates along z. The real ′ part of the complex propagation constant (wave vector) describes the propagation characteristics, e.g., phase

velocity = ′. The imaginary k” part describes the rate of attenuation along z. The intensity at any point along z is

∝ | |2 ∝ exp(−2 ′′ )

So the rate of change in the intensity with distance is

		= −2 ′′	……………..(6)
Where the negative sign represents attenuation.

Suppose that 0 is the propagation constant in a vacuum. This is a real quantity as a plane wave suffers no loss in free space. The complex refractive index N with real part n and imaginary part K is defined as the ratio of the complex propagation constant in a medium to propagation constant in free space,

= − =	= (	1	) [ ′ − ′′]	.…………..(7)
0
	0

i.e.

=	′	and=	′′	……………(8)
0	0

The real part n is simply and generally called the refractive index and K is called the extinction coefficient. In the absence of attenuation,

′′ = 0 , = ′	and	= =		=	′
	0
			0

We know that in the absence of loss, the relationship between the refractive index n and the relative permittivity is = √ . This relationship is also valid in the presence of loss except that we must use complex refractive index and complex relative permittivity, that is,

= − = √ = √ − ′′ ……………(9)

By squaring both sides we can relate n and K directly to ′ and ′′. The final result is

Optical properties of materials are typically reported either by showing the frequency dependences of n and K or ′ and ′′. Clearly we can use Equation 10 to obtain one set of properties from the other. Figure 2 shows the real (n) and imaginary (K) parts of the complex refractive index of amorphous silicon (noncrystalline form of Si) as a function of photon energy (h v). For photon energies below the band gap energy, K is negligible and n is close to 3.5. Both n and K change strongly as the photon energy increases far beyond the band gap energy.

Figure 2: Optical properties of an amorphous silicon film in terms of real (n) and imaginary (K) parts of the complex refractive index.

If we know the frequency dependence of the real part ′ of the relative permittivity of a material, we can also determine the frequency dependence of the imaginary part ′′, and vice versa. This may seem remarkable, but it is true provided that we know the frequency dependence of either the real or imaginary part over as wide a range of frequencies as possible (ideally from dc to infinity) and the material is linear, i.e., it has a relative permittivity that is independent of the applied field; the polarization response must be linearly proportional to the applied field. The relationships that relate the real and imaginary parts of the relative permittivity are called

Kramers-Kronig relations. If ′ ( ) and ′′( ) represent the frequency dependences of the real and imaginary parts, respectively, then one can be determined from the other as depicted schematically in Figure 3.

Figure 3: Kramers-Kronig relations allow frequency dependences of the real and imaginary parts of the relative permittivity to be related to each other. The material must be a linear system.

The optical properties n and K can be determined by measuring the reflectance from the surface of a material as a function of polarization and the angle of incidence (based on Fresnel’s equations).

It is instructive to mention that the reflection and transmission coefficients that we derived in previous lecture were based on using a real refractive index that is, neglecting losses. We can still use the reflection and transmission coefficients if we simply use the complex refractive index N instead of n. For example, consider a light wave traveling in free space incident on a material at normal incidence ( = 90°). The reflection coefficient is now

= −	−1	= −	− −1	……………(11)

+1	− +1

The reflectance is then

	− −1 2		( −1)2	+ 2	……………(12)
= \|		\|	=
− +1	( +1)2	+ 2

Which reduce to the usual forms when the extinction coefficient K = 0.

LATTICE ABSORPTION:

In optical absorption, some of the energy from the propagating EM wave is converted to other forms of energy, for example, to heat by the generation of lattice vibrations. There are a number of absorption processes that dissipate the energy from the wave. One important mechanism is called lattice absorption (Reststrahlen absorption) and involves the vibrations of the lattice atoms as illustrated in Figure 4. The crystal in this example consists of ions, and as an EM wave propagates it displaces the oppositely charged ions in opposite directions and forces them to vibrate at the frequency of the wave. In other words, the medium experiences ionic polarization. It is the displacements of these ions that give rise to ionic polarization and its contribution to the relative permittivity . As the ions and hence the lattice is made to vibrate by the passing EM wave, as shown in Figure 4, some energy is coupled into the natural lattice vibrations of the solid. This energy peaks when the frequency of the wave is close to the natural lattice vibration frequencies. Typically these frequencies are in the infrared region. Most of the energy is then absorbed from the EM wave and converted to lattice vibrational energy (heat). We associate this absorption with the resonance peak or relaxation peak of ionic polarization loss (imaginary part of the relative permittivity ′′).

Figure 4: Lattice absorption through a crystal. The field in the EM wave oscillates the ions which consequently generate “mechanical” waves in the crystal; energy is thereby transferred from the wave to lattice vibrations.

Figure 5 shows the infrared resonance absorption peaks in the extinction coefficient K versus wavelength characteristics of GaAs and CdTe; both crystals have substantial ionic bonding. These absorption peaks in Figure (5) are usually called Reststrahlen bands because absorption occurs over a band of frequencies (even though the band may be narrow), and in some cases may even have identifiable features.

Figure 5: Lattice or Reststrahlen absorption in CdTe and GaAs in terms of the extinction coefficient versus wavelength. For reference, n versus k for CdTe is also shown.

Although Figure 4 depicts an ionic solid to visualize absorption due to lattice waves, energy from a passing EM wave can also be absorbed by various ionic impurities in a medium as these charges can couple to the electric field and oscillate. Bonding between an oscillating ion and the neighboring atoms causes the mechanical oscillations of the ion to be coupled to neighboring atoms. This leads to a generation of lattice waves which takes away energy from the EM wave.

BAND-TO-BAND ABSORPTION:

The photon absorption process for photo generation, that is, the creation of electron-hole pairs (EHPs), requires the photon energy to be at least equal to the band gap energy of the semiconductor material to excite an electron from the valence band (VB) to the conduction band (CB). The upper cut-off wavelength (or the threshold wavelength) for photogenerative

absorption is therefore determined by the band gap energy of the semiconductor, so ℎ ( ) =

( ) = ( )

……………(13)

For example, for Si, = 1.12 eV and is 1.11 whereas for Ge, = 0.66 eV and the corresponding = 1.87 . It is clear that Si photodiodes cannot be used for optical communications at 1.3 and 1.55 , whereas Ge photodiodes are commercially available for use at these wavelengths. Table1 lists some typical band gap energies and the corresponding cut-off wavelengths of various photodiode semiconductor materials.

Table 1: Band gap energy at 300 K, cut-off wavelength , and type of bandgap (D = direct and I = indirect) for some photo detector materials.

Semiconductor	( )	( )	Type

InP	1.35	0.91	D

GaAs0.88Sb0.12	1.15	1.08	D

Si	1.12	1.11	I

In0.7Ga0.3As0.64P0.36	0.89	1.4	D

In0.53Ga0.47As	0.75	1.65	D

Ge	0.66	1.87	I

InAS	0.35	3.5	D

InSb	0.18	7	D

Incident photons with wavelengths shorter than become absorbed as they travel in the semiconductor, and the light intensity, which is proportional to the number of photons, decays exponentially with distance into the semiconductor. The light intensity I at a distance x from the semiconductor surface is given by

( ) = 0 exp(− ) ……………..(14)

Where 0 is the intensity of the incident radiation and is the absorption coefficient that depends on the photon energy or wavelength . The absorption coefficient is a material property. Most of the photon absorption (63%) occurs over a distance 1, and 1 is called the penetration depth . Figure (6) shows the versus k characteristics of various semiconductors where it is apparent that the behavior of with the wavelength depends on the semiconductor material.

Figure 6: Absorption coefficient versus wavelength for various semiconductors.

Absorption in semiconductors can be understood in terms of the behavior of the electron energy

(E) with the electron momentum (ℏ ) in the crystal, called the crystal momentum. If is the wavevector of the electron’s wave function in the crystal, then the momentum of the electron within the crystal is ℏ . E versus ℏ behaviors for electrons in the conduction and valence bands of direct and indirect band gap semiconductors are shown in Figure 7a and b, respectively. In direct band gap semiconductors such as III-V semiconductors (e.g., GaAs, In As, InP, GaP) and in many of their alloys (e.g., InGaAs, GaAsSb) the photon absorption process is a direct process which requires no assistance from lattice vibrations. The photon is absorbed and the electron is excited directly from the valence band to the conduction band without a change in its -vector, or its crystal momentum ℏ , inasmuch as the photon momentum is very small. The change in the electron momentum from the valence to the conduction band is

Figure7: Electron energy E versus crystal momentum ℏ and photon absorption.

(a) Photon absorption in a direct band gap semiconductor.

(b) Photon absorption in an indirect band gap semiconductor (VB = valence band; CB = conduction band).

This process corresponds to a vertical transition on the electron energy (E) versus electron momentum (ℏ ) diagram as shown in Figure 9.24a. The absorption coefficient of these semiconductors rises sharply with decreasing wavelength from as apparent for GaAs and InP in Figure 6.

In indirect band gap semiconductors such as Si and Ge, the photon absorption for photon energies near requires the absorption and emission of lattice vibrations, that is, phonons, during the absorption process as shown in Figure 7. If K is the wave vector of a lattice wave (lattice vibrations travel in the crystal), then ℏ represents the momentum associated with such a lattice vibration; that is,ℏ is a phonon momentum. When an electron in the valence band is excited to the conduction band, there is a change in its momentum in the crystal, and this change in the momentum cannot be supplied by the momentum of the incident photon which is very small. Thus, the momentum difference must be balanced by a phonon momentum,

ℏ − ℏ = ℎ = ℏ

The absorption process is said to be indirect as it depends on lattice vibrations which in turn depend on the temperature. Since the interaction of a photon with a valence electron needs a third body, a lattice vibration, the probability of photon absorption is not as high as in a direct transition. Furthermore, the cut-off wavelength is not as sharp as for direct band gap semiconductors. During the absorption process, a phonon may be absorbed or emitted. If is the frequency of the lattice vibrations, then the phonon energy is ℎ . The photon energy is ℎ where the photon frequency is. Conservation of energy requires that

ℎ = ± ℎ

Thus, the onset of absorption does not exactly coincide with but typically it is very close to in as much as ℎ is small (< 0.1 eV). The absorption coefficient initially rises slowly with decreasing wavelength from about as apparent in Figure 6 for Si and Ge.

Summary:

(1) Effect of a solid on a propagating EM wave in a solid is explained.

(2) The dependence of the relative permittivity with the loss is described.

(3) The relationship between real and imaginary part of the relative permittivity is briefly described

(4) Lattice absorption of a EM wave is explained.

(5) Band to band absorption is also explained.

References:

A.J. Dekker (1957). Solid state physics, Prentce-Hall, Inc .
Leonid A. Azaroff (1960), Introduction to solids, McGRAW-HILL BOOK COMPANY INC.
3 Kasap, S. (2006). Principles of electronic materials and devices (3rd ed.). Boston: McGraw-Hill
Frederick Wooten (1972) ACADEMIC PRESS INC
Harald Ibach, Hans Luth (2009) Springer series on material science
Ashcroft & Mermin (1976) Harcourt College Publisher.
Philip Hofmann (2008) , Solid State Physics An Introduction, WILEY-VCH Verlag GmbH & KGaA.

References and Suggestive Readings

James Patterson, Bernard Bailey (2010) Solid State Physics, Second Edition , Springier series on material science.
S.L. Kakani (2004), Material Science , New age international (P) Limited, publishers
Harald lbach & Hans Luth (1995) , Solid state physics- An introduction to material science , Springier
Donald A. Neamen (2003), Semiconductor physics and devices, Third edition, Mc-Graw Hill Higher Education

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