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Optical Properties of Electronic Materials: Fundamentals and Characterization

  • Jan Mistrik
  • Safa Kasap
  • Harry E. Ruda
  • Cyril Koughia
  • Jai Singh
Part of the Springer Handbooks book series (SPRINGERHAND)

Abstract

Light interacts with materials in a variety of ways; this chapter focuses on refraction and absorption. Refraction is characterized by a material’s refractive index. We discuss some of the most useful models for the frequency dependence of the refractive index, such as those due to Cauchy, Sellmeier, Gladstone–Dale, and Wemple–DiDominico. Examples are given of the applicability of the models to actual materials. We present various mechanisms of light absorption, including absorption by free carriers, phonons, excitons and impurities. Special attention is paid to fundamental and excitonic absorption in disordered semiconductors and to absorption by rare earth, trivalent ions due to their importance in modern photonics. We also discuss the effect of an external electric field on absorption, and the Faraday effect. Practical techniques for determining the optical parameters of thin films are outlined. Finally, we present a short technical classification of optical glasses and materials.

3.1 Optical Constants

The changes that light undergoes upon interacting with a particular substance are known as the optical properties of that substance. These optical properties are influenced by the macroscopic and microscopic properties of the substance, such as the nature of its surface and its electronic structure. Since it is usually far easier to detect the way a substance modifies light than to investigate its macroscopic and microscopic properties directly, the optical properties of a substance are often used to probe other properties of the material. There are many optical properties, including the most well known: reflection, refraction, transmission and absorption. Many of these optical properties are associated with important optical constants, such as the refractive index and the extinction coefficient. In this section we review these optical constants, such as the refractive index and the extinction coefficient. Books by Adachi [3.1], Fox [3.2] and Simmons and Potter [3.3] are highly recommended. In addition, Adachi also discusses the optical properties of III–V compounds in this handbook in Chap.  30.

3.1.1 Refractive Indexand Extinction Coefficient

The refractive index n of an optical or dielectric medium is the ratio of the velocity of light c in a vacuum to its velocity v in the medium; \(n=c/v\). Using this and Maxwell’s equations, one obtains the well-known formula for the refractive index of a substance as \(\smash{n=\sqrt{\varepsilon_{\mathrm{r}}\mu_{\mathrm{r}}}}\), where εr is the relative permittivity (dielectric constant) and μr is the relative magnetic permeability. As μr = 1 for nonmagnetic substances, one gets \(n=\sqrt{\varepsilon_{\mathrm{r}}}\), which is very useful for relating the dielectric properties of a material to its optical properties at any particular frequency of interest. Since εr depends on the wavelength of the light, the refractive index depends on it too, which is called dispersion. In addition to dispersion, an electromagnetic wave propagating through a lossy medium (one that absorbs or scatters radiation passing through it) experiences attenuation, which means that it loses its energy due to various loss mechanisms such as the generation of phonons (lattice waves), photogeneration, free-carrier absorption and scattering. In such materials, the refractive index is a complex function of the frequency of the light wave.

The complex refractive index N, with real part n and imaginary part K (called the extinction coefficient), is related to the complex relative permittivity, \(\varepsilon_{\mathrm{r}}=\varepsilon_{\mathrm{r}}^{\prime}-\mathrm{i}\varepsilon_{\mathrm{r}}^{\prime\prime}\) by
$$N=n-\mathrm{i}K=\sqrt{\varepsilon_{\mathrm{r}}}=\sqrt{\varepsilon_{\mathrm{r}}^{\prime}-\mathrm{i}\varepsilon_{\mathrm{r}}^{\prime\prime}}$$
(3.1)
where ε r and \(\varepsilon_{\mathrm{r}}^{\prime\prime}\) are, respectively, the real and imaginary parts of εr. Equation (3.1 ) gives
$$n^{2}-K^{2}=\varepsilon_{\mathrm{r}}^{\prime}\quad\text{and}\quad 2nK=\varepsilon_{\mathrm{r}}^{\prime\prime}\;.$$
(3.2)
The optical constants 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. For normal incidence, the reflection coefficient r is obtained as
$$r=\frac{1-N}{1+N}=\frac{1-n+\mathrm{i}K}{1+n-\mathrm{i}K}\;.$$
(3.3)
The reflectance R is then defined by
$$R=|r|^{2}=\left|{\frac{1-n+\mathrm{i}K}{1+n-\mathrm{i}K}}\right|^{2}=\frac{(1-n)^{2}+K^{2}}{(1+n)^{2}+K^{2}}\;.$$
(3.4)
Notice that whenever K is large, for example over a range of wavelengths, the absorption is strong, and the reflectance is almost unity. The light is then reflected, and any light in the medium is highly attenuated.
Optical properties of materials are typically presented by showing either the frequency dependence (dispersion relation) of n and K or ε r and \(\varepsilon_{\mathrm{r}}^{\prime\prime}\). An intuitive guide to explaining dispersion in insulators is based on a single oscillator model in which the electric field in the light induces forced dipole oscillations in the material (displaces the electron shells to oscillate about the nucleus) with a single resonant frequency ω0 [3.1]. The frequency dependences of ε r and \(\varepsilon_{\mathrm{r}}^{\prime\prime}\) are then obtained as
$$\varepsilon_{\mathrm{r}}^{\prime}=1+\frac{N_{\mathrm{at}}}{\varepsilon_{0}}\alpha_{\mathrm{e}}^{\prime}\quad\text{and}\quad\varepsilon_{\mathrm{r}}^{\prime\prime}=1+\frac{N_{\mathrm{at}}}{\varepsilon_{0}}\alpha_{\mathrm{e}}^{\prime\prime}\;,$$
(3.5)
where Nat is the number of atoms per unit volume, ε0 is the vacuum permittivity, and α e and \(\alpha_{\mathrm{e}}^{\prime\prime}\) are the real and imaginary parts of the electronic polarizability, given respectively by
$$\alpha_{\mathrm{e}}^{\prime}=\alpha_{\mathrm{e}0}\frac{1-(\omega/\omega_{0})^{2}}{[1-(\omega/\omega_{0})^{2}]^{2}+(\gamma/\omega_{0})^{2}(\omega/\omega_{0})^{2}}\;,$$
(3.6a)
and
$$\alpha_{\mathrm{e}}^{\prime\prime}=\alpha_{\mathrm{e}0}\frac{(\gamma/\omega_{0})(\omega/\omega_{0})}{[1-(\omega/\omega_{0})^{2}]^{2}+(\gamma/\omega_{0})^{2}(\omega/\omega_{0})^{2}}\;,$$
(3.6b)
whereαe0 is the direct current (DC) polarizability corresponding to ω = 0 and γ is the loss coefficient. Using (3.1), (3.2) and (3.5) with (3.1.1), it is then possible to study the frequency dependence of n and K. Figure 3.1a shows the dependence of n and K on the normalized frequency ω ∕ ω0 for a simple single electronic dipole oscillator of resonance frequency ω0. It is seen that n and K peak close to ω = ω0. If a material has a \(\varepsilon_{\mathrm{r}}^{\prime\prime}\gg\varepsilon_{\mathrm{r}}^{\prime}\), then \(\varepsilon_{\mathrm{r}}\approx-\mathrm{i}\varepsilon_{\mathrm{r}}^{\prime\prime}\) and \(n=K\approx\sqrt{{\varepsilon_{\mathrm{r}}^{\prime\prime}/2}}\) is obtained from (3.2). Figure 3.1b shows the dependence of the reflectance R on the frequency. It is observed that R reaches its maximum value at a frequency slightly above ω = ω0, and then remains high until ω reaches nearly 3ω0; so the reflectance is substantial while absorption is strong. The normal dispersion region is the frequency range below ω0 where n falls as the frequency decreases; that is, n decreases as the wavelength λ increases. The anomalous dispersion region is the frequency range above ω0, where n decreases as ω increases. Below ω0, K is small and if εDC is εr ( 0 ) , the refractive index becomes (approximately)
$$n^{2}\approx 1+(\varepsilon_{\mathrm{DC}}-1)\frac{\omega_{0}^{2}}{\omega_{0}^{2}-\omega^{2}}\;;\quad\omega<\omega_{0}\;.$$
(3.7)
Since \(\lambda=2\uppi c/\omega\), defining \(\lambda_{0}=2\uppi c/\omega_{0}\) as the resonance wavelength, one gets
$$n^{2}\approx 1+(\varepsilon_{\mathrm{DC}}-1)\frac{\lambda^{2}}{\lambda^{2}-\lambda_{0}^{2}}\;;\quad\lambda> \lambda_{0}\;.$$
(3.8)
While intuitively useful, the dispersion relation in (3.8) is far too simple. More rigorously, we have to consider the dipole oscillator quantum-mechanically, which means that a photon excites the oscillator to a higher energy level; see, for example, [3.2, 3.3]. The result is that we have a series of \(\lambda^{2}/(\lambda^{2}-\lambda_{i}^{2})\) terms with various weighting factors A i that add to unity, where the λ i represent different resonance wavelengths. The weighting factors A i involve quantum-mechanical matrix elements.
Fig. 3.1a,b

The dipole oscillator model. (a) Refractive index and extinction coefficient versus normalized frequency. (b) Reflectance versus normalized frequency

Figure 3.2 shows the complex relative permittivity and the complex refractive index of crystalline silicon in terms of photon energy hν. For photon energies below the bandgap energy (1.1 eV), both \(\varepsilon_{\mathrm{r}}^{\prime\prime}\) and K are negligible and n is close to 3.7. Both \(\varepsilon_{\mathrm{r}}^{\prime\prime}\) and K increase and change strongly as the photon energy becomes greater than 3 eV, far beyond the bandgap energy. Notice that both ε r and n peak at hν ≈ 3.5 eV, which corresponds to a direct photoexcitation process – the excitation of electrons from the valence band to the conduction band – as discussed later.
Fig. 3.2

(a) Complex relative permittivity of a silicon crystal as a function of photon energy plotted in terms of real (ε r ) and imaginary (\(\varepsilon_{\mathrm{r}}^{\prime\prime}\)) parts. (b) Optical properties of a silicon crystal versus photon energy in terms of real ( n )  and imaginary ( K )  parts of the complex refractive index. (After [3.4])

3.1.2 Kramers–Kronig Relations

If we know the frequency dependence of the real part, ε r , of the relative permittivity of a material, then we can use the Kramers–Kronig relations to determine the frequency dependence of the imaginary part \(\varepsilon_{\mathrm{r}}^{\prime\prime}\), and vice versa. The transform requires that we know the frequency dependence of either the real or imaginary part over a frequency range that is as wide as possible, ideally from DC to infinity, and that the material is linear; in other words it has a relative permittivity that is independent of the applied field. The Kramers–Kronig relations for \(\varepsilon_{\mathrm{r}}=\varepsilon_{\mathrm{r}}^{\prime}+\mathrm{i}\varepsilon_{\mathrm{r}}^{\prime\prime}\) are given by [3.4]
$$\varepsilon_{\mathrm{r}}^{\prime}(\omega)=1+\frac{2}{\uppi}P\int_{0}^{\infty}{\frac{{\omega}^{\prime}{\varepsilon}_{\mathrm{r}}^{\prime\prime}({\omega}^{\prime})}{{\omega}^{\prime 2}-\omega^{2}}}\,\mathrm{d}{\omega}^{\prime}\;,$$
(3.9a)
and
$${\varepsilon}_{\mathrm{r}}^{\prime\prime}(\omega)=-\frac{2\omega}{\uppi}P\int_{0}^{\infty}{\frac{{\varepsilon}_{\mathrm{r}}^{\prime}({\omega}^{\prime})}{{\omega}^{\prime 2}-\omega^{2}}}\,\mathrm{d}{\omega}^{\prime}\;,$$
(3.9b)

whereω is the integration variable, P represents the Cauchy principal value of the integral and the singularity at \(\omega=\omega^{\prime}\) is avoided.

Similarly, one can relate the real and imaginary parts of the polarizability, α ( ω )  and \(\alpha^{\prime\prime}(\omega)\), and those of the complex refractive index, n ( ω )  and K ( ω )  as well. For α ( ω )  and \(\alpha^{\prime\prime}(\omega)\), one can analogously write
$$\alpha^{\prime}(\omega)=\frac{2}{\uppi}P\int_{0}^{\infty}{\frac{{\omega}^{\prime}{\alpha}^{\prime\prime}({\omega}^{\prime})}{{\omega}^{\prime 2}-\omega^{2}}}\,\mathrm{d}{\omega}^{\prime}$$
(3.10a)
and
$${\alpha}^{\prime\prime}({\omega}^{\prime})=-\frac{2\omega}{\uppi}P\int_{0}^{\infty}{\frac{{\alpha}^{\prime}({\omega}^{\prime})}{{\omega}^{\prime 2}-\omega^{2}}}\,\mathrm{d}{\omega}^{\prime}\;.$$
(3.10b)

3.2 Refractive Index

There are several simplified models describing the spectral dependence of the refractive index n.

3.2.1 Cauchy Dispersion Equation

The dispersion relationship for the refractive index ( n )  versus the wavelength of light (λ) is stated in the following form
$$n=A+\frac{B}{\lambda^{2}}+\frac{C}{\lambda^{4}}\;,$$
(3.11)
where A, B and C are material-dependent specific constants. The Cauchy equation (3.11 ) is typically used in the visible region of the spectrum for various optical glasses, and is applied to normal dispersion. The third term is sometimes dropped for a simpler representation of n versus λ behavior. The original expression was a series in terms of the wavelength, λ, or frequency, ω, of light as
$$n=a_{0}+a_{2}\lambda^{-2}+a_{4}\lambda^{-4}+a_{6}\lambda^{-6}+{\ldots}\;{\lambda}> \lambda_{\mathrm{h}}\;,$$
(3.12a)
or
$$n=n_{0}+n_{2}\omega^{2}+n_{4}\omega^{4}+n_{6}\omega^{6}+{\ldots}\;\omega<\omega_{\mathrm{h}}\;,$$
(3.12b)
whereℏω is the photon energy, \(\hbar\omega_{\mathrm{h}}=hc/\lambda_{\mathrm{h}}\) is the optical excitation threshold (the bandgap energy), while a0, a2 …  ,  and n0, n2 , … are constants. A Cauchy relation of the following form
$$n=n_{-2}(\hbar\omega)^{-2}+n_{0}+n_{2}(\hbar\omega)^{2}+n_{4}(\hbar\omega)^{4}\;,$$
(3.13)
can be used satisfactorily over a wide range of photon energies. The dispersion parameters, calculated from (3.13), of a few materials are listed in Table 3.1.
Table 3.1

Cauchy’s dispersion parameters (obtained from (3.11)) for a few materials. (Data compiled from [3.5])

Material

ω ( eV ) 

n−2 ( eV2 ) 

n 0

n2 ( eV−2 ) 

n4 ( eV−4 ) 

Diamond

0.05–5.47

\(\mathrm{-1.07\times 10^{-5}}\)

2.378

\(\mathrm{8.01\times 10^{-3}}\)

\(\mathrm{1.04\times 10^{-4}}\)

Si

0.002–1.08

\(\mathrm{-2.04\times 10^{-8}}\)

3.4189

\(\mathrm{8.15\times 10^{-2}}\)

\(\mathrm{1.25\times 10^{-2}}\)

Ge

0.002–0.75

\(\mathrm{-1.00\times 10^{-8}}\)

4.0030

\(\mathrm{2.20\times 10^{-1}}\)

\(\mathrm{1.40\times 10^{-1}}\)

Cauchy’s dispersion relation, given in (3.13), was originally called the elastic-ether theory of the refractive index [3.5, 3.6, 3.7]. It has been widely used for many materials, although in recent years it has been largely replaced by the Sellmeier equation, which we consider next.

3.2.2 Sellmeier Dispersion Equation

The dispersion relationship can be quite complicated in practice. An example of this is the Sellmeier equation, which is an empirical relation between the refractive index n of a substance and the wavelength λ of light in the form of a series of \(\lambda^{2}/(\lambda^{2}-\lambda_{i}^{2})\) terms, given by
$$n^{2}=1+\frac{A_{1}\lambda^{2}}{\lambda^{2}-\lambda_{1}^{2}}+\frac{A_{2}\lambda^{2}}{\lambda^{2}-\lambda_{2}^{2}}+\frac{A_{3}\lambda^{2}}{\lambda^{2}-\lambda_{3}^{2}}+\cdots\;,$$
(3.14)
where λ i is a constant and A1, A2, A3, λ1, λ2 and λ3 are called Sellmeier coefficients, which are determined by fitting this expression to the experimental data. The full Sellmeier formula has more terms of similar form, such as \(A_{i}\lambda^{2}/(\lambda^{2}-\lambda_{i}^{2})\), where \(i=4,5,\ldots\) but these can generally be neglected when considering n versus λ behavior over typical wavelengths of interest and by ensuring that the three terms included in the Sellmeier equation correspond to the most important or relevant terms in the summation. Examples of Sellmeier coefficients for some materials, including pure silica ( SiO2 )  and 86.5 mol% SiO2-13.5 mol% GeO2, are given in Table 3.2 . Two methods are used to find the refractive index of silica-germania glass ( SiO2)1−x ( GeO2) x : (a) a simple, but approximate, linear interpolation of the refractive index between known compositions, for example \(n(x)-n(0.135)=(x-0.135)[n(0.135)-n(0)]/0.135\) for ( SiO2)1−x ( GeO2) x , so n ( 0.135 )  is used for 86.5 mol% SiO2-13.5 mol% GeO2 and n ( 0 )  is used for SiO2; (b) an interpolation for the coefficients A i and λ i between SiO2 and GeO2
$$\begin{aligned}\displaystyle n^{2}-1&\displaystyle=\frac{\{A_{1}(S)+X[A_{1}(G)-A_{1}(S)]\}\lambda^{2}}{\lambda^{2}-\{\lambda_{1}(S)+X[\lambda_{1}(G)-\lambda_{1}(S)]\}^{2}}\\ \displaystyle&\displaystyle\quad+\cdots\;,\end{aligned}$$
(3.15)
where X is the atomic fraction of germania, S and G in parentheses refer to silica and germania [3.8]. The theoretical basis of the Sellmeier equation is that the solid is represented as a sum of N lossless (frictionless) Lorentz oscillators such that each takes the form of \(\lambda^{2}/(\lambda^{2}-\lambda_{i}^{2})\) with different λ i , and each has different strengths, with weighting factors (A i , \(i=1,2,\ldots N\)) [3.10, 3.9]. Knowledge of appropriate dispersion relationships is essential when designing photonic devices, such as waveguides.
Table 3.2

Sellmeier coefficients of a few materials (λ1, λ2, λ3 are in μm)

Material

A 1

A 2

A 3

λ 1

λ 2

λ 3

SiO2 (fused silica)

0.696749

0.408218

0.890815

0.0690660

0.115662

9.900559

86.5%SiO2-13.5%GeO2

0.711040

0.451885

0.704048

0.0642700

0.129408

9.425478

GeO2

0.80686642

0.71815848

0.85416831

0.068972606

0.15396605

11.841931

Barium fluoride

0.63356

0.506762

3.8261

0.057789

0.109681

46.38642

Sapphire

1.023798

1.058264

5.280792

0.0614482

0.110700

17.92656

Diamond

0.3306

4.3356

 

0.175

0.106

 

Quartz, n0

1.35400

0.010

0.9994

0.092612

10.700

9.8500

Quartz, ne

1.38100

0.0100

0.9992

0.093505

11.310

9.5280

KDP, n0

1.2540

0.0100

0.0992

0.09646

6.9777

5.9848

KDP, ne

1.13000

0.0001

0.9999

0.09351

7.6710

12.170

There are other dispersion relationships that inherently take account of various contributions to optical properties, such as the electronic and ionic polarization and the interactions of photons with free electrons. For example, for many semiconductors and ionic crystals, two useful dispersion relations are
$$n^{2}=A+\frac{B\lambda^{2}}{\lambda^{2}-C}+\frac{D\lambda^{2}}{\lambda^{2}-E}\;,$$
(3.16)
and
$$n^{2}=A+\frac{B}{\lambda^{2}-\lambda_{0}^{2}}+\frac{C}{\left(\lambda^{2}-\lambda_{0}^{2}\right)^{2}}+D\lambda^{2}+E\lambda^{4}\;,$$
(3.17)
where A, B, C, D, E and λ0 are constants particular to a given material. Table 3.3 provides a few examples.
Table 3.3

Parameters from (3.16) and (3.17) for some selected materials (Si data from [3.12]; others from [3.13])

Material

λ0 ( μm ) 

A

B ( μm)2

C ( μm)−4

D ( μm)−2

E ( μm)−4

Silicon

0.167

3.41983

0.159906

−0.123109

\(\mathrm{1.269\times 10^{-6}}\)

\(\mathrm{-1.951\times 10^{-9}}\)

MgO

0.11951

2.95636

0.021958

0

\(\mathrm{-1.0624\times 10^{-2}}\)

\(\mathrm{-2.05\times 10^{-5}}\)

LiF

0.16733

1.38761

0.001796

\(\mathrm{-4.1\times 10^{-3}}\)

\(\mathrm{-2.3045\times 10^{-3}}\)

\(\mathrm{-5.57\times 10^{-6}}\)

AgCl

0.21413

4.00804

0.079009

0

\(\mathrm{-8.5111\times 10^{-4}}\)

\(\mathrm{-1.976\times 10^{-7}}\)

The refractive index of a semiconductor material typically decreases with increasing bandgap energy Eg. There are various empirical and semi-empirical rules and expressions that relate n to Eg. In Moss’ rule, n and Eg are related by \(n^{4}E_{\mathrm{g}}=K=\) constant (≈ 100 eV). In the Hervé–Vandamme relationship [3.11],
$$n^{2}=1+\left({\frac{A}{E_{\mathrm{g}}+B}}\right)^{2}\;,$$
(3.18)
where A and B are constants (A ≈ 13.6 eV and B ≈ 3.4 eV and \(\mathrm{d}B/\,\mathrm{d}T\approx{\mathrm{2.5\times 10^{-5}}}\,{\mathrm{e{\mskip-2.0mu}V/K}}\)). The refractive index typically increases with increasing temperature. The temperature coefficient of the refractive index (TCRI ) of a semiconductor can be found from the Hervé–Vandamme relationship [3.11]
$$\text{TCRI}=\frac{1}{n}\frac{\mathrm{d}n}{\mathrm{d}T}=\frac{(n^{2}-1)^{3/2}}{13.6n^{2}}\left({\frac{\mathrm{d}E_{\mathrm{g}}}{\mathrm{d}T}+\frac{\mathrm{d}B}{\mathrm{d}T}}\right)\;.$$
(3.19)
TCRI is typically in the range \({\mathrm{10^{-6}}}{-}{\mathrm{10^{-4}}}\).

3.2.3 Gladstone–Dale Formula

The Gladstone–Dale formula is an empirical equation that allows the average refractive index n of an oxide glass to be calculated from its density ρ and its constituents as
$$\frac{n-1}{\rho}=p_{1}k_{1}+p_{2}k_{2}+\cdots=\sum_{i=1}^{N}{p_{i}k_{i}}=C_{\mathrm{GD}}\;,$$
(3.20)
where the summation is for various oxide components (each a simple oxide), p i is the weight fraction of the i-th oxide in the compound, and k i is the refraction coefficient that represents the polarizability of the i-th oxide. The right-hand side of (3.20 ) is called the Gladstone–Dale coefficient CGD. In more general terms, as a mixture rule for the overall refractive index, the Gladstone–Dale formula is frequently written as
$$\frac{n-1}{\rho}=\frac{n_{1}-1}{\rho_{1}}w_{1}+\frac{n_{2}-1}{\rho_{2}}w_{2}+\cdots\;,$$
(3.21)
where n and ρ are the effective refractive index and effective density of the whole mixture respectively, n1, n2, … are the refractive indices of the constituents, and ρ1, ρ2, … represent the densities of each constituent. Gladstone–Dale equations for the polymorphs of SiO2 and TiO2 give the average n values as [3.14]
$$n\mathrm{(SiO_{2})}=1+0.21\rho\quad\text{and}\quad n\mathrm{(TiO_{2})}=1+0.40\rho\;.$$
(3.22)

3.2.4 Wemple–DiDominico Dispersion Relation

The Wemple–DiDominico dispersion relation is a semi-empirical relationship used to find the refractive indices of a variety of materials for photon energies below the interband absorption edge, given by [3.15]
$$\begin{aligned}\displaystyle n^{2}=1+\frac{E_{0}E_{\mathrm{d}}}{E_{0}^{2}-(h\nu)^{2}}\;,\end{aligned}$$
(3.23)
where ν is the frequency, h is the Planck constant, E0 is the single oscillator energy, Ed is the dispersion energy, which is a measure of the average strength of interband optical transitions; Ed = βNcZaNe (eV), where Nc is the effective coordination number of the cation nearest-neighbor to the anion (Nc = 6 in NaCl, Nc = 4 in Ge), Za is the formal chemical valency of the anion (Za = 1 in NaCl; 2 in Te; 3 in GaP), Ne is the effective number of valence electrons per anion excluding the cores (Ne = 8 in NaCl, Ge; 10 in TlCl; 12 in Te; \(\smash{9\tfrac{1}{3}}\) in As2Se3), and β is a constant that depends on whether the interatomic bond is ionic (βi) or covalent (βc): \(\beta_{\mathrm{i}}=0.26\pm{\mathrm{0.04}}\,{\mathrm{e{\mskip-2.0mu}V}}\) (this applies to halides such as NaCl and ThBr and most oxides, including Al2O3, for example), while \(\beta_{\mathrm{c}}=0.37\pm{\mathrm{0.05}}\,{\mathrm{e{\mskip-2.0mu}V}}\) (applies to tetrahedrally bonded A N B8−N zinc blende- and diamond-type structures such as GaP, ZnS, for instance). (Note that wurtzite crystals have a β that is intermediate between βi and βc.) Also, empirically, E0 = CEg(D), where Eg(D) is the lowest direct bandgap and C is a constant, typically ≈ 1.5. E0 has been associated with the main peak in the \(\varepsilon_{\mathrm{r}}^{\prime\prime}({h\nu})\) versus hν spectrum. The parameters required to calculate n from (3.23) are listed in Table 3.4 [3.15].
Table 3.4

Examples of parameters for the Wemple–DiDomenico dispersion relationship (3.23), for various materials

Material

N c

Z a

N e

E0 ( eV ) 

Ed ( eV ) 

β ( eV ) 

β

Comment

NaCl

6

1

8

10.3

13.6

0.28

β i

Halides, LiF, NaF, etc.

CsCl

8

1

8

10.6

17.1

0.27

β i

CsBr, CsI, etc.

TlCl

8

1

10

5.8

20.6

0.26

β i

TlBr

CaF2

8

1

8

15.7

15.9

0.25

β i

BaF2, etc.

CaO

6

2

8

9.9

22.6

0.24

β i

Oxides, MgO, TeO2, etc.

Al2O3

6

2

8

13.4

27.5

0.29

β i

 

LiNbO3

6

2

8

6.65

25.9

0.27

β i

 

TiO2

6

2

8

5.24

25.7

0.27

β i

 

ZnO

4

2

8

6.4

17.1

0.27

β i

 

ZnSe

4

2

8

5.54

27

0.42

β c

II–VI, Zinc blende, ZnS, ZnTe, CdTe

GaAs

4

3

8

3.55

33.5

0.35

β c

III–V, Zinc blende, GaP, etc.

Si (crystal)

4

4

8

4.0

44.4

0.35

β c

Diamond, covalent bonding; C (diamond), Ge, β-SiC etc.

SiO2 (crystal)

4

2

8

13.33

18.10

0.28

β i

Average crystalline form

SiO2 (amorphous)

4

2

8

13.38

14.71

0.23

β i

Fused silica

CdSe

4

2

8

4.0

20.6

0.32

βi − βc

Wurtzite

3.2.5 Group Index ( Ng ) 

The group index represents the factor by which the group velocity of a group of waves in a dielectric medium is reduced with respect to propagation in free space, \(N_{\mathrm{g}}=c/v_{\mathrm{g}}\), where vg is the group velocity. The group index can be determined from the ordinary refractive index n via
$$N_{\mathrm{g}}=n-\lambda\frac{\mathrm{d}n}{\,\mathrm{d}\lambda}\;,$$
(3.24)
where λ is the wavelength of light. The relation between Ng and n for SiO2 is illustrated in Fig. 3.3.
Fig. 3.3

Refractive index n and the group index Ng of pure SiO2 (silica) glass as a function of wavelength

3.3 Optical Absorption

The main optical properties of a semiconductor are typically its refractive index n and its extinction coefficient K or absorption coefficient α (or equivalently the real and imaginary parts of the relative permittivity), as well as their dispersion relations (their dependence on the electromagnetic radiation wavelength λ or photon energy hν) and the changes in the dispersion relations with temperature, pressure, alloying, impurities, and so on. A typical relationship between the absorption coefficient and the photon energy observed in a crystalline semiconductor is shown in Fig. 3.4, where various possible absorption processes are illustrated. The important features of the behavior of the α versus hν can be summarized as follows:
  1. 1.

    Free-carrier absorption due to the presence of free electrons and holes, an effect that decreases with increasing photon energy

     
  2. 2.

    An impurity absorption band (usually narrow) due the various dopants

     
  3. 3.

    Reststrahlen or lattice absorption in which the radiation is absorbed by vibrations of the crystal ions

     
  4. 4.

    Exciton absorption peaks that are usually observed at low temperatures and are close the fundamental absorption edge; and

     
  5. 5.

    Band-to-band or fundamental absorption of photons, which excites an electron from the valence to the conduction band.

     
Type 5 absorption has a large absorption coefficient and occurs when the photon energy reaches the bandgap energy Eg. It is probably the most important absorption effect; its characteristics for hν > Eg can be predicted using the results of Sect. 3.3.3 . The values of Eg, and its temperature shift, dEg ∕ dT, are therefore important factors in semiconductor-based optoelectronic devices. In nearly all semiconductors Eg decreases with temperature, hence shifting the fundamental absorption to longer wavelengths. The refractive index n also changes with temperature. dn ∕ dT depends on the wavelength, but for many semiconductors \((\mathrm{d}n/\mathrm{d}T)/n\approx{\mathrm{5\times 10^{-5}}}\,{\mathrm{K^{-1}}}\); for example, for GaAs, \((\mathrm{d}n/\mathrm{d}T)/n={\mathrm{4\times 10^{-4}}}\,{\mathrm{K^{-1}}}\) at λ = 2 μm. There is a good correlation between the refractive indices and the bandgaps of semiconductors in which, typically, n decreases as Eg increases; semiconductors with wider bandgaps have lower refractive indices. The refractive index n and the extinction coefficient K (or α) are related by the Kramers–Kronig relations. Thus, large increases in the absorption coefficient for hν near and above the bandgap energy Eg also result in increases in the refractive index n versus hν in this region. Optical (and some structural) properties of various semiconductors are listed in Table 3.5.
Fig. 3.4

Absorption coefficient plotted as a function of the photon energy in a typical semiconductor, illustrating various possible absorption processes

Table 3.5

Crystal structure, lattice parameter a, bandgap energy Eg at 300 K, type of bandgap (D=direct and I=indirect), change in Eg per unit temperature change ( dEg ∕ dT )  at 300 K, bandgap wavelength λg and refractive index n close to λg (A=amorphous, D=diamond, W=wurtzite, ZB=zinc blende). Approximate data from various sources

Semiconductors

Crystal

a

( nm ) 

E g

( eV ) 

Type

dE g  ∕ dT

( meV ∕ K ) 

λ g

( μm ) 

n ( λg ) 

dn ∕ dT

\(({\mathrm{10^{-5}}}\,{\mathrm{K^{-1}}})\)

Group IV

Diamond

D

0.3567

5.48

I

−0.05

0.226

2.74

1.1

Ge

D

0.5658

0.66

I

−0.37

1.87

4

27.6

42.4 (4 μm)

Si

D

0.5431

1.12

I

−0.25

1.11

3.45

13.8

16 (5 μm)

a-Si:H

A

 

1.7–1.8

  

0.73

  

SiC( α ) 

W

0.3081 a

1.5120 c

2.9

I

−0.33

0.42

2.7

9

III–V Compounds

AlAs

ZB

0.5661

2.16

I

−0.50

0.57

3.2

15

AlP

ZB

0.5451

2.45

I

−0.35

0.52

3

11

AlSb

ZB

0.6135

1.58

I

−0.3

0.75

3.7

 

GaAs

ZB

0.5653

1.42

D

−0.45

0.87

3.6

15

GaAs0.88Sb0.12

ZB

 

1.15

D

 

1.08

  

GaN

W

0.3190 a

0.5190 c

3.44

D

−0.45

0.36

2.6

6.8

GaP

ZB

0.5451

2.26

I

−0.54

0.40

3.4

 

GaSb

ZB

0.6096

0.73

D

−0.35

1.7

4

33

In0.53Ga0.47As

on InP

ZB

0.5869

0.75

D

 

1.65

  

In0.58Ga0.42As0.9P0.1

on InP

ZB

0.5870

0.80

D

 

1.55

  

In0.72Ga0.28As0.62P0.38

on InP

ZB

0.5870

0.95

D

 

1.3

  

InP

ZB

0.5869

1.35

D

−0.40

0.91

3.4–3.5

9.5

InAs

ZB

0.6058

0.36

D

−0.28

3.5

3.8

2.7

InSb

ZB

0.6479

0.18

D

−0.3

7

4.2

29

II–VI Compounds

ZnSe

ZB

0.5668

2.7

D

−0.50

0.46

2.3

6.3

ZnTe

ZB

0.6101

2.3

D

−0.45

0.55

2.7

 

3.3.1 Lattice or Reststrahlen Absorption and Infrared Reflection

In the infrared wavelength region, ionic crystals reflect and absorb light strongly due to the resonance interaction of the electromagnetic (EM) wave field with the transverse optical (TO ) phonons. The dipole oscillator model based on ions driven by an EM wave results in
$$\varepsilon_{\mathrm{r}}=\varepsilon_{\mathrm{r}}^{\prime}-\mathrm{i}\varepsilon_{\mathrm{r}}^{\prime\prime}=\varepsilon_{\mathrm{r}\infty}+\frac{\varepsilon_{\mathrm{r}\infty}-\varepsilon_{\mathrm{r}0}}{\left({\frac{\omega}{\omega_{\mathrm{T}}}}\right)^{2}-1+\mathrm{i}\frac{\gamma}{\omega_{\mathrm{T}}}\left({\frac{\omega}{\omega_{\mathrm{T}}}}\right)}\;,$$
(3.25)
where εr0 and εr∞ are the relative permittivity at ω = 0 (very low frequencies) and ω = ∞ (very high frequencies) respectively, γ is the loss coefficient per unit reduced mass, representing the rate of energy transfer from the EM wave to optical phonons, and ω0 is a resonance frequency that is related to the spring constant between the ions. By definition, the frequency ωT is
$$\omega_{\mathrm{T}}^{2}=\omega_{0}^{2}\left({\frac{\varepsilon_{\mathrm{r}\infty}+2}{\varepsilon_{\mathrm{r}0}+2}}\right)\;.$$
(3.26)
The loss \(\varepsilon_{\mathrm{r}}^{\prime\prime}\) and the absorption are maxima when ω = ωT, and the wave is attenuated by the transfer of energy to the transverse optical phonons, so the EM wave couples to the transverse optical phonons. At ω = ωL, the wave couples to the longitudinal optical (LO ) phonons. Figure 3.5 shows the optical properties of AlSb [3.16] in terms of n, K and R versus wavelength. The peaks in the extinction coefficient K and reflectance R occur over roughly the same wavelength region, corresponding to the coupling of the EM wave to the transverse optical phonons. At wavelengths close to \(\lambda_{\mathrm{T}}=2\uppi/\omega_{\mathrm{T}}\), n and K peak, and there is strong absorption of light (which corresponds to the EM wave resonating with the TO lattice vibrations), and then R rises sharply.
Fig. 3.5a,b

Infrared refractive index n, extinction coefficient K (a), and reflectance R (b) of AlSb. Note that the wavelength axes are not identical, and the wavelengths λT and λL, corresponding to ωT and ωL respectively, are shown as dashed vertical lines. (After [3.16])

3.3.2 Free-Carrier Absorption (FCA)

An electromagnetic wave with sufficiently low frequency oscillations can interact with free carriers in a material and thereby drift the carriers. This interaction results in an energy loss from the EM wave to the lattice vibrations through the carrier scattering processes. Based on the Drude model, the relative permittivity εr ( ω )  due to N free electrons per unit volume is given by
$$\begin{aligned}\displaystyle\varepsilon_{\mathrm{r}}&\displaystyle=\varepsilon_{\mathrm{r}}^{\prime}-\mathrm{i}\varepsilon_{\mathrm{r}}^{\prime\prime}=1-\frac{\omega_{\mathrm{p}}^{2}}{\omega^{2}-\mathrm{i}\omega/\tau}\;,\\ \displaystyle\omega_{\mathrm{p}}^{2}&\displaystyle=\frac{Ne^{2}}{\varepsilon_{0}m_{\mathrm{e}}}\;,\end{aligned}$$
(3.27)
where ωp is a plasma frequency that depends on the electron concentration, while τ is the relaxation time of the scattering process (the mean scattering time). For metals where the electron concentration is very large, ωp is of the order of \(\approx{\mathrm{10^{16}}}\,{\mathrm{rad/s}}\), at ultraviolet (UV) frequencies, and for ω > ωp, εr ≈ 1, and the reflectance becomes very small. Metals lose their free-electron reflectance in the UV range, thus becoming UV-transparent. The reflectance does not fall to zero because there are other absorption processes such as interband electron excitations or excitations from core levels to energy bands. Plasma edge transparency, where the reflectance almost vanishes, can also be observed in doped semiconductors. For example, the reflectance of doped InSb has a plasma edge wavelength that decreases with increasing free-carrier concentration [3.17]. Equation (3.27 ) can be written in terms of the conductivity σ0 at low frequencies (DC) as
$$\begin{aligned}\displaystyle\varepsilon_{\mathrm{r}}=\varepsilon_{\mathrm{r}}^{\prime}-\mathrm{i}\varepsilon_{\mathrm{r}}^{\prime\prime}=1&\displaystyle-\frac{\tau\sigma_{0}}{\varepsilon_{0}[(\omega\tau)^{2}+1]}\\ \displaystyle&\displaystyle-\mathrm{i}\frac{\sigma_{0}}{\varepsilon_{0}\omega[(\omega\tau)^{2}+1]}\;.\end{aligned}$$
(3.28)
In metals, σ0 is high. At frequencies where \(\omega<1/\tau\), the imaginary part \(\varepsilon_{\mathrm{r}}^{\prime\prime}=\sigma_{0}/(\varepsilon_{0}\omega)\) is normally much more than 1, and \(\smash{n=K\approx\sqrt{\varepsilon_{\mathrm{r}}^{\prime\prime}/2}}\), so that the free-carrier attenuation coefficient α is then given by
$$\alpha=2k_{0}K\approx\frac{2\omega}{c}\left({\frac{\varepsilon_{\mathrm{r}}^{\prime\prime}}{2}}\right)^{1/2}\approx\left({2\sigma_{0}\mu_{0}}\right)^{1/2}\omega^{1/2}\;.$$
(3.29)
Furthermore, the reflectance can also be calculated using \(n=K\approx\sqrt{\varepsilon_{\mathrm{r}}^{\prime\prime}/2}\), which leads to the well-known Hagen–Rubens relationship [3.18]
$$R\approx 1-2\left({\frac{2\omega\varepsilon_{0}}{\sigma_{0}}}\right)^{1/2}\;.$$
(3.30)
In semiconductors, one typically encounters \(\sigma_{0}/(\varepsilon_{0}\omega)<1\), since the free-electron concentration is small, and we can treat n as constant due to various other polarization mechanisms such as electronic polarization. Since \(2nK=\varepsilon_{\mathrm{r}}^{\prime\prime}\), the absorption coefficient becomes [3.19]
$$\alpha=2k_{0}K\approx\frac{2\omega}{c}\left({\frac{\varepsilon_{\mathrm{r}}^{\prime\prime}}{2n}}\right)=\frac{\sigma_{0}}{nc\varepsilon_{0}[(\omega\tau)^{2}+1]}\;.$$
(3.31)
At low frequencies where \(\omega<1/\tau\), we have \(\alpha(\lambda)\propto\sigma_{0}/n(\lambda)\) so that α should be controlled by the DC conductivity, and hence the amount of doping. Furthermore, α will exhibit the frequency dependence of the refractive index n (\(\alpha(\lambda)\propto 1/n(\lambda)\)), in which n will typically be determined by the electronic polarization of the crystal.
At high frequencies where \(\omega> 1/\tau\),
$$\alpha\propto\frac{\sigma_{0}}{\omega^{2}}\propto N\lambda^{2}\;,$$
(3.32)
where α is proportional to N, the free-carrier concentration, and λ2. Experimental observations on FCA in doped semiconductors are in general agreement with these predictions. For example, α increases with N, whether N is increased by doping or by carrier injection [3.23]. However, not all semiconductors show the simple α ∝ λ2 behavior. A proper account of the field-driven electron motion and scattering must consider the fact that τ will depend on the electron energy. The correct approach is to use the Boltzmann transport equation [3.24] with the appropriate scattering mechanism. FCA can be calculated using a quantum-mechanical approach based on second-order time-dependent perturbation theory with Fermi–Dirac statistics [3.25].
Absorption due to free carriers is commonly written as α ∝ λ p , where the index p depends primarily on the scattering mechanism, although it is also influenced by the compensation doping ratio, if the semiconductor has been doped by compensation, and the free-carrier concentration. In the case of lattice scattering, one must consider scattering from acoustic and optical phonons. For acoustic phonon scattering p ≈ 1.5, for optical phonon scattering p ≈ 2.5, and for impurity scattering p ≈ 3.5. The observed free-carrier absorption coefficient will then have all three contributions
$$\alpha=A_{\text{acoustic}}\lambda^{1.5}+A_{\text{optical}}\lambda^{2.5}+A_{\text{impurity}}\lambda^{3.5}\;.$$
(3.33)
Inasmuch as α for FCA depends on the free-carrier concentration N, it is possible to evaluate the latter from the experimentally measured α, given its wavelength dependence and p as discussed by [3.26]. The free-carrier absorption coefficients α (mm−1) for GaP, n-type PbTe and n-type ZnO are shown in Fig. 3.6.
Fig. 3.6

Free-carrier absorption in n-GaP at 300 K [3.20], p- and n-type PbTe [3.21] at 77 K and In doped n-type ZnO at room temperature. (After [3.22])

Free-carrier absorption in p-type Ge demonstrates how the FCA coefficient α can be dramatically different than what is expected from (3.33). Figure 3.7a shows the wavelength dependence of the absorption coefficient for p-Ge over the wavelength range from about 2 to 30 μm [3.27]. The observed absorption is due to excitations of electrons from the spin-off band to the heavy hole band, from the spin-off band to the light hole band, and from the light hole band to the heavy hole band, as marked in the Fig. 3.7b.
Fig. 3.7

(a) Free-carrier absorption due to holes in p-Ge [3.27]. (b) The valence band of Ge has three bands; heavy hole, light hole and spin-off bands

3.3.3 Band-to-Bandor Fundamental Absorption

Crystalline Solids

Band-to-band absorption or fundamental absorption of radiation is due to the photoexcitation of an electron from the valence band to the conduction band. There are two types of band-to-band absorptions, corresponding to direct and indirect transitions.

A direct transition is a photoexcitation process in which no phonons are involved. The photon momentum is negligible compared with the electron momentum, so that when the photon is absorbed, exciting an electron from the valence band (VB ) to the conduction band (CB ); the electron’s k-vector does not change. A direct transition on an E − k diagram is a vertical transition from an initial energy E and wavevector k in the VB to a final energy E and a wavevector k in the CB where \(k^{\prime}=k\), as shown in Fig. 3.8a. The energy \((E^{\prime}-E_{\mathrm{c}})\) is the kinetic energy \((\hbar k)^{2}/(2m_{\mathrm{e}}^{\ast})\) of the electron with an effective mass m e , and ( E v  − E )  is the kinetic energy \((\hbar k)^{2}/(2m_{\mathrm{h}}^{\ast})\) of the hole left behind in the VB. The ratio of the kinetic energies of the photogenerated electron and hole depends inversely on the ratio of their effective masses.
Fig. 3.8

(a) A direct transition from the valence band (VB) to the conduction band (CB) through the absorption of a photon. Absorption behavior represented as ( αhυ)2 versus photon energy hυ near the band edge for single crystals of (b) p-type GaAs, from [3.28] and (c) CdTe. (After [3.29])

The absorption coefficient α is derived from the quantum-mechanical probability of transition from E to E, the occupied density of states at E in the VB from which electrons are excited, and the unoccupied density of states in the CB at E + hν. Thus, α depends on the joint density of states at E and E + hν, and we have to suitably integrate this joint density of states. Near the band edges, the density of states can be approximated by a parabolic band, and α rises with the photon energy following
$$\alpha h\nu=A(h\nu-E_{\mathrm{g}})^{1/2}\;,$$
(3.34)
where the constant \(A\approx[e^{2}/({nch}^{2}m_{\mathrm{e}}^{\ast})](2\mu^{\ast})^{3/2}\) in which μ is a reduced electron and hole effective mass, n is the refractive index, and Eg is the direct bandgap, with minimum Ec − E v at the same k value (see [3.3] for the derivation). Experiments indeed show this type of behavior for photon energies above Eg and close to Eg, as shown in Fig. 3.8b for a GaAs crystal [3.28] and in Fig. 3.8c for a CdTe crystal [3.29]. The extrapolation to zero photon energy gives the direct bandgap Eg, which is about 1.40 eV for GaAs and 1.46–1.49 eV for CdTe. For photon energies very close to the bandgap energy, the absorption is usually due to exciton absorption, especially at low temperatures, and is discussed later in this chapter.
In indirect bandgap semiconductors, such as Si and Ge, the photon absorption for photon energies near Eg requires the absorption and emission of phonons during the absorption process, as illustrated in Fig. 3.9 a. The absorption onset corresponds to a photon energy of (Eg − hϑ ) , which represents the absorption of a phonon with energy hϑ. For the latter, α is proportional to \([h\nu-(E_{\mathrm{g}}-h\vartheta)]^{2}\). Once the photon energy reaches (Eg + hϑ ) , then the photon absorption process can also occur by phonon emission, for which the absorption coefficient is larger than that for phonon absorption. The absorption coefficients for the phonon absorption and emission processes are given by [3.30]
$$\begin{aligned}\displaystyle\alpha_{\mathrm{absorption}}&\displaystyle=A[f_{\mathrm{BE}}(h\vartheta)][h\nu-(E_{\mathrm{g}}-h\vartheta)]^{2}\;;\\ \displaystyle h\nu&\displaystyle> (E_{\mathrm{g}}-h\vartheta)\;,\end{aligned}$$
(3.35)
and
$$\begin{aligned}\displaystyle\alpha_{\mathrm{emission}}&\displaystyle=A[(1-f_{\mathrm{BE}}(h\vartheta)][h\nu-(E_{\mathrm{g}}+h\vartheta)]^{2}\;;\\ \displaystyle h\nu&\displaystyle> (E_{\mathrm{g}}+h\vartheta)\;,\end{aligned}$$
(3.36)
where A is a constant and fBE ( hϑ )  is the Bose–Einstein distribution function at the phonon energy hϑ, \(f_{\mathrm{BE}}(h\vartheta)=[\exp(h\vartheta/(k_{\mathrm{B}}T))-1]^{-1}\), where kB is the Boltzmann constant and T is the temperature. As we increase the photon energy in the range \((E_{\mathrm{g}}-h\vartheta)<h\nu<(E_{\mathrm{g}}+h\vartheta)\), the absorption is controlled by αabsorption and the plot of α1∕2 versus hν has an intercept of ( Eg − hϑ ) .
Fig. 3.9

(a) Indirect transitions across the bandgap involve phonons. Direct transitions in which dE ∕ dk in the CB is parallel to dE ∕ dk in the VB lead to peaks in the absorption coefficient. (b) Fundamental absorption in Si at two temperatures. The overall behavior is well described by (3.35) and (3.36)

On the other hand, for \(h\nu> (E_{\mathrm{g}}+h\vartheta)\), the overall absorption coefficient is αabsorption + αemission, but at slightly higher photon energies than ( Eg + hϑ ) , αemission quickly dominates over αabsorption since \(f_{\mathrm{BE}}(h\vartheta)\gg 1-f_{\mathrm{BE}}(h\vartheta)\). Figure 3.9b shows the behavior of α1∕2 versus photon energy for Si at two temperatures for hν near band edge absorption. At low temperatures, fBE ( hϑ )  is small and αabsorption decreases with decreasing temperature, as apparent from Fig. 3.9b. Equations (3.35) and (3.36) intersect the photon energy axis at ( Eg − hϑ )  and ( Eg + hϑ ) , which can be used to obtain Eg.

An examination of the extinction coefficient K or \(\varepsilon_{\mathrm{r}}^{\prime\prime}\) versus photon energy for Si in Fig. 3.2 shows that absorption peaks at certain photon energies, hν ≈ 3.5 and 4.3 eV. These peaks are due to the fact that the joint density of states function peaks at these energies. The absorption coefficient peaks whenever there is a direct transition in which the E versus k curve in the VB is parallel to the E versus k curve in the CB, as schematically illustrated in Fig. 3.9 a, where a photon of energy hν12 excites an electron from state 1 in the VB to state 2 in the CB in a direct transition, i. e., k1 = k2. Such transitions where E versus k curves are parallel at a photon energy hν12 result in a peak in the absorption versus photon energy behavior, and can be represented by the condition that
$$(\nabla_{\boldsymbol{k}}E)_{\mathrm{CB}}-(\nabla_{\boldsymbol{k}}E)_{\mathrm{VB}}=0\;.$$
(3.37)
The above condition is normally interpreted as the joint density of states reaching a peak value at certain points in the Brillouin zone called van Hove singularities. Identification of peaks in K versus hν involves the examination of all E versus k curves of a given crystal that can participate in a direct transition. The silicon \(\varepsilon_{\mathrm{r}}^{\prime\prime}\) peaks at hν ≈ 3.5 eV and 4.3 eV correspond to (3.37) being satisfied at points L, along ⟨ 111 ⟩  in k-space, and X along ⟨ 100 ⟩  in k-space, at the edges of the Brillouin zone.
In degenerate semiconductors, the Fermi level EF is in a band; for example, in the CB for a degenerate n-type semiconductor. Electrons in the VB can only be excited to states above EF in the CB rather than to the bottom of the CB. The absorption coefficient then depends on the free-carrier concentration since the latter determines EF. Fundamental absorption is then said to depend on band filling, and there is an apparent shift in the absorption edge, called the Burstein–Moss shift. Furthermore, in degenerate indirect semiconductors, the indirect transition may involve a nonphonon scattering process, such as impurity or electron-electron scattering, which can change the electron’s wavevector k. Thus, in degenerate indirect bandgap semiconductors, absorption can occur without phonon assistance and the absorption coefficient becomes
$$\alpha\propto[h\nu-(E_{\mathrm{g}}+\Updelta E_{\mathrm{F}})]^{2}\;,$$
(3.38)
where ΔEF is the energy depth of EF into the band measured from the band edge.
Heavy doping of degenerate semiconductors normally leads to a phenomenon called bandgap narrowing and bandtailing. Bandtailing means that the band edges at E v and Ec are no longer well-defined cut-off energies, and there are electronic states above E v and below Ec where the density of states falls sharply away with energy from the band edges. Consider a degenerate direct bandgap p-type semiconductor. One can excite electrons from states below EF in the VB, where the band is nearly parabolic, to tail states below Ec, where the density of states decreases exponentially with energy into the bandgap, away from Ec. Such excitations lead to α depending exponentially on hν, a dependence that is called the Urbach rule [3.31, 3.32], given by
$$\alpha=\alpha_{0}\exp\left(\frac{h\nu-E_{0}}{\Updelta E}\right)\;,$$
(3.39)
where α0 and E0 are material-dependent constants, and ΔE, called the Urbach width, is also a material-dependent constant. The Urbach rule was originally reported for alkali halides. It has been observed for many ionic crystals, degenerately doped crystalline semiconductors, and almost all amorphous semiconductors. While exponential bandtailing can explain the observed Urbach tail of the absorption coefficient versus photon energy, it is also possible to attribute the absorption tail behavior to strong internal fields arising, for example, from ionized dopants or defects.

Amorphous Solids

In a defect-free crystalline semiconductor, a well-defined energy gap exists between the valence and conduction bands. In contrast, in an amorphous semiconductor, the distributions of conduction band and valence band electronic states do not terminate abruptly at the band edges. Instead, some electronic states called the tail states encroach into the gap region [3.33]. In addition to tail states, there are also other localized states deep within the gap region [3.34]. These localized tail states in amorphous semiconductors are contributed by defects. The defects in amorphous semiconductors are considered to be all cases of departure from the normal nearest-neighbor coordination (or normal valence requirement). Examples of defects are: broken and dangling bonds (typical for amorphous silicon); over- and under-coordinated atoms (such as ‘valence alternation pairs’ in chalcogenide glasses); and voids, pores, cracks and other macroscopic defects. Mobility edges exist, which separate these localized states from their extended counterparts; tail and deep defect states are localized [3.35, 3.36, 3.37]. These localized tail and deep defect states are responsible for many of the unique properties exhibited by amorphous semiconductors.

Despite years of intensive investigation, the exact form of the distribution of electronic states associated with amorphous semiconductors remains a subject of some debate. While there are still some unresolved theoretical issues, there is general consensus that the tail states arise as a consequence of the disorder present within amorphous networks, and that the width of these tails reflects the amount of disorder present [3.38]. Experimental results (from, for example, [3.39, 3.40]) suggest exponential distributions for the valence and conduction band tail states in a-Si:H, although other possible functional forms [3.41] cannot be ruled out. Singh and Shimakawa [3.37] have derived separate effective masses of charge carriers in their extended and tail states. That means the density of states (DOS ) of extended and tail states can be represented in two different parabolic forms. The relationship between the absorption coefficient and the distribution of electronic states for the case of a-Si:H may be found in [3.37, 3.42, 3.43, 3.44].

The existence of tail states in amorphous solids has a profound impact upon the band-to-band optical absorption. Unlike in a crystalline solid, the absorption of photons in an intrinsic amorphous solid can also occur for photon energies ℏω ≤ E0 due to the presence of tail states in the forbidden gap. E0 is the energy of the optical gap, which is usually close to the mobility gap – the energy difference between the conduction band and valence band mobility edges.

In a crystalline semiconductor, the energy and crystal momentum of an electron involved in an optical transition must be conserved. In an amorphous semiconductor, however, only the energy needs to be conserved [3.36, 3.37]. As a result, for optical transitions caused by photons of energy ℏω ≥ E0 in amorphous semiconductors, the approach of a joint density of states is not applicable [3.37, 3.45]. One has to consider the product of the densities of both conduction and valence electronic states when calculating the absorption coefficient [3.37, 3.46]. Assuming that both the valence band and conduction band DOS functions have square-root dependencies on energy, one can derive the absorption coefficient α as [3.37]
$$(\alpha\hbar\omega)^{1/2}=B^{1/2}(\hbar\omega-E_{0})\;,$$
(3.40)
where, if one assumes that the transition matrix element is constant,
$$B=\frac{1}{nc\varepsilon_{0}}\left(\frac{e}{m_{\mathrm{e}}^{\ast}}\right)^{2}\left(\frac{L(m_{\mathrm{e}}^{\ast}m_{\mathrm{h}}^{\ast})^{3/2}}{2^{2}\hbar^{3}}\right)\;,$$
(3.41)
where m e and m h are the effective masses of an electron and a hole respectively, L denotes the average interatomic separation in the sample, and n is the refractive index. Plotting ( αω)1∕2 as a function of ℏω yields a straight line that is usually referred to as Tauc’s plot, from which one can determine the optical gap energy E0 experimentally. Equation (3.40) is also known as Tauc’s relation and B as Tauc’s constant [3.47]. Experimental data from many, but not all, amorphous semiconductors and chalcogenides fit to (3.40) very well. Deviations from Tauc’s relation have been observed. For instance, some experimental data in a-Si:H fit much better to a cubic root relation given by Mott and Davis [3.36]
$$(\alpha\hbar\omega)^{1/3}=C(\hbar\omega-E_{0})$$
(3.42)
and therefore the cubic root has been used to determine the optical gap E0. Here C is another constant.
If one considers that the optical transition matrix element is independent of photon energy [3.37, 3.48], one finds that the cubic root dependence on photon energy can be obtained only when the valence band and conduction band DOS depend linearly on energy. Using such DOS functions, the cubic root dependence has been explained by Mott and Davis [3.36]. Another way to obtain the cubic root dependence has been suggested in [3.49]. Using (3.40 ), Sokolov et al. [3.49] have modeled the cubic root dependence on photon energy by considering the fluctuations in the optical bandgap due to structural disorder. Although their approach gives a way of getting the cubic root dependence, Sokolov et al.’s model is hardly different from the linear density of states model suggested by Mott and Davis [3.36]. Cody [3.50] has shown an alternative approach using a photon energy-dependent transition matrix element. Thus, the absorption coefficient is obtained as [3.37]
$$(\alpha\hbar\omega)={B}^{\prime}(\hbar\omega)^{2}(\hbar\omega-E_{0})^{2}\;,$$
(3.43)
where
$${B}^{\prime}=\frac{e^{2}}{nc\varepsilon_{0}}\left(\frac{\left(m_{\mathrm{e}}^{\ast}m_{\mathrm{h}}^{\ast}\right)^{3/2}}{2\uppi^{2}\hbar^{7}\nu\rho_{\mathrm{A}}}\right)Q_{\mathrm{a}}^{2}\;,$$
(3.44)
and Qa is the average separation between the excited electron and hole pair in an amorphous semiconductor, ν denotes the number of valence electrons per atom and ρA represents the atomic density per unit volume.
Equation (3.43) suggests that ( αω )  depends on the photon energy in the form of a polynomial of order 4. Then, depending on which term of the polynomial may be more significant in which material, one can get square, cubic or fourth root dependences of ( αω )  on the photon energy. In this case, (3.43) may be expressed as
$$(\alpha\hbar\omega)^{x}\propto(\hbar\omega-E_{0})\;,$$
(3.45)
where \(x\leq 1/2\). Thus, in a way, any deviation from the square root or Tauc’s plot may be attributed to the energy-dependent matrix element [3.37, 3.46]. Another possible explanation has been recently discussed by Shimakawa and coworkers on the basis of fractal theory [3.51].
Another problem is how to determine the constants B (3.41) and B (3.44), which involve the effective masses of an electron and a hole. In other words, how do we determine the effective masses of charge carriers in amorphous solids? Recently, a simple approach [3.37, 3.46] has been developed to calculate the effective masses of charge carriers in amorphous solids. Different effective masses of charge carriers are obtained in the extended and tail states. The approach applies the concepts of tunneling and effective medium, and one obtains the effective mass of an electron in the conduction extended states, denoted by m ex , and in the tail states, denoted by m et , as
$$m_{\mathrm{ex}}^{\ast}\approx\frac{E_{L}}{2(E_{2}-E_{\mathrm{c}})a^{1/3}}m_{\mathrm{e}}\;,$$
(3.46)
and
$$m_{\mathrm{et}}^{\ast}\approx\frac{E_{L}}{(E_{\mathrm{c}}-E_{\mathrm{ct}})b^{1/3}}m_{\mathrm{e}}\;,$$
(3.47)
where
$$E_{L}=\frac{\hbar^{2}}{m_{\mathrm{e}}L^{2}}\;.$$
(3.48)
Here \(a=N_{1}/N<1\), N1 is the number of atoms contributing to the extended states, \(b=N_{2}/N<1\), N2 is the number of atoms contributing to the tail states such that \(a+b=1\) \((N=N_{1}+N_{2})\), and me is the free-electron mass. The energy E2 in (3.46) corresponds to the energy of the middle of the extended conduction states, at which the imaginary part of the dielectric constant becomes maximum (Fig. 3.10; see also 3.2).
Fig. 3.10

Schematic illustration of the electronic energy states E2, Ec, Ect, Evt, Ev and Ev2 in amorphous semiconductors. The shaded region represents the extended states. Energies E2 and Ev2 correspond to the centers of the conduction and valence extended states and Ect and Evt represent the ends of the conduction and valence tail states respectively

Likewise, the effective masses of the hole m hx and m ht in the valence extended and tail states are obtained, respectively, as
$$m_{\mathrm{hx}}^{\ast}\approx\frac{E_{L}}{2(E_{\mathrm{v}}-E_{\mathrm{v}2})a^{1/3}}m_{\mathrm{e}}\;,$$
(3.49)
and
$$m_{\mathrm{ht}}^{\ast}\approx\frac{E_{L}}{(E_{\mathrm{vt}}-E_{\mathrm{v}})b^{1/3}}m_{\mathrm{e}}\;,$$
(3.50)
where Ev2 and Evt are energies corresponding to the half-width of the valence extended states and the end of the valence tail states respectively; see Fig. 3.10.
Using (3.46) and (3.47 ) and the values of the parameters involved, different effective masses of an electron are obtained in the extended and tail states. Taking, for example, the density of weak bonds contributing to the tail states as 1 at.%, so b = 0.01 and a = 0.99, the effective masses and energies E L calculated for hydrogenated amorphous silicon (a-Si:H) and germanium (a-Ge:H) are given in Table 3.6.
Table 3.6

Effective mass of electrons in the extended and tail states of a-Si:H and a-Ge:H calculated using (3.46) and (3.47) for a = 0.99, b = 0.01 and \(E_{\mathrm{ct}}=E_{\mathrm{vt}}=E_{\mathrm{c}}/2\). E L is calculated from (3.48). All energies are given in eV. Note that since the absorption coefficient is measured in cm−1, the value used for the speed of light is in cm ∕ s (a [3.52]; b [3.53]; c [3.33]; d [3.54])

 

L ( nm ) 

E 2

E c

E L

Ec − Ect

m ex

m et

a-Si:H

0.235a

3.6b

1.80c

1.23

0.9

0.34 me

6.3 me

a-Ge:H

0.245a

3.6

1.05d

1.14

0.53

0.22 me

10.0 me

According to (3.46), (3.47), (3.49) and (3.50), for sp3 hybrid amorphous semiconductors such as a-Si:H and a-Ge:H, effective masses of the electron and hole are expected to be the same. In these semiconductors, since the conduction and valence bands are two equal halves of the same electronic band, their widths are the same and that gives equal effective masses for the electron and the hole [3.37, 3.55]. This is one of the reasons for using \(E_{\mathrm{ct}}=E_{\mathrm{vt}}=E_{\mathrm{c}}/2\), which gives equal effective masses for electrons and holes in the tail states as well. This is different from crystalline solids where m e and m h are usually not the same. This difference between amorphous and crystalline solids is similar to, for example, having direct and indirect crystalline semiconductors but only direct amorphous semiconductors.

Using the effective masses from Table 3.6 and (3.41), B can be calculated for a-Si:H and a-Ge:H. The values thus obtained with the refractive index n = 4 for a-Si:H and a-Ge:H are \(B={\mathrm{6.0\times 10^{6}}}\,{\mathrm{cm^{-1}{\,}e{\mskip-2.0mu}V^{-1}}}\) for a-Si:H and \(B={\mathrm{4.1\times 10^{6}}}\,{\mathrm{cm^{-1}{\,}e{\mskip-2.0mu}V^{-1}}}\) for a-Ge:H, which are an order of magnitude higher than those estimated from experiments [3.36]. However, considering the quantities involved in B (3.41), this can be regarded as a reasonable agreement.

In a recent paper, Malik and O’Leary [3.56] studied the distributions of conduction and valence band electronic states associated with a-Si:H. They noted that the effective masses associated with a-Si:H are material parameters that are yet to be experimentally determined. In order to remedy this deficiency, they fitted square-root DOS functions to experimental DOS data and found that \(m_{\mathrm{h}}^{\ast}=2.34m_{\mathrm{e}}\) and \(m_{\mathrm{e}}^{\ast}=2.78m_{\mathrm{e}}\).

The value of the constant B in (3.44) can also be calculated theoretically, provided that Qa is known. Using the atomic density of crystalline silicon and four valence electrons per atom, Cody [3.50] estimated Q a 2  = 0.9 Å2, which gives Qa ≈  0.095 nm, less than half the interatomic separation of 0.235 nm in a-Si:H, but of the same order of magnitude. Using ν = 4, \(\rho_{\mathrm{A}}={\mathrm{5\times 10^{28}}}\,{\mathrm{m^{-3}}}\), Q a 2  = 0.9 Å2, and extended state effective masses, we get \({B}^{\prime}={\mathrm{4.6\times 10^{3}}}\,{\mathrm{cm^{-1}{\,}e{\mskip-2.0mu}V^{-3}}}\) for a-Si:H and \({\mathrm{1.3\times 10^{3}}}\,{\mathrm{cm^{-1}{\,}e{\mskip-2.0mu}V^{-3}}}\) for a-Ge:H. Cody estimated an optical gap, E0 = 1.64 eV, for a-Si:H, which, using (3.43), gives \(\alpha={\mathrm{1.2\times 10^{3}}}\,{\mathrm{cm^{-1}}}\) at a photon energy of ℏω = 2 eV. This agrees reasonably well with the \(\alpha={\mathrm{6.0\times 10^{2}}}\,{\mathrm{cm^{-1}}}\) used by Cody. If we use interatomic spacing in place of Qa in (3.44), we get \({B}^{\prime}={\mathrm{2.8\times 10^{4}}}\,{\mathrm{cm^{-1}{\,}e{\mskip-2.0mu}V^{-3}}}\), and then the corresponding absorption coefficient becomes \({\mathrm{3.3\times 10^{3}}}\,{\mathrm{cm^{-1}}}\). This suggests that, in order to get an estimate, one may use the interatomic spacing in place of Qa if the latter is unknown. Thus, both B and B can be determined theoretically, a task not possible before due to a lack of knowledge of the effective masses in amorphous semiconductors.

The absorption of photons of energy less than the bandgap energy ℏω < E0 in amorphous solids involves the localized tail states and hence follows neither (3.40) nor (3.42). Instead, the absorption coefficient depends on the photon energy exponentially, as given in (3.39 ), giving rise to Urbach’s tail. Abe and Toyozawa [3.57] have calculated the interband absorption spectra in crystalline solids, introducing the Gaussian site diagonal disorder and applying the coherent potential approximation. They have shown that Urbach’s tail occurs due to static disorder (structural disorder). However, the current stage of understanding is that Urbach’s tail in amorphous solids occurs due to both thermal effects and static disorder [3.50]. More recent issues in this area have been addressed by Orapunt and O’Leary [3.48].

Keeping the above discussion in mind, the optical absorption in amorphous semiconductors near the absorption edge is usually characterized by three types of optical transitions corresponding to transitions between tail and tail states, tail and extended states, and extended and extended states. The first two types correspond to ℏω ≤ E0, and the third one corresponds to ℏω ≥ E0. Thus, the plot of absorption coefficient versus photon energy (α versus ℏω) has three different regions, A, B and C respectively, that correspond to these three characteristic optical transitions shown in Fig. 3.11a,b.
Fig. 3.11a,b

Typical spectral dependence of the optical absorption coefficient in amorphous semiconductors. (a) In the A and B regions, the absorption is controlled by optical transitions between tail and tail states and tail and extended states respectively, and in the C region it is dominated by transitions from extended to extended states. In domain B, the absorption coefficient follows the Urbach rule (3.39). In region C, the absorption coefficient follows Tauc’s relation (3.40) in a-Si:H, as shown in (b)

In the small absorption coefficient range A (also called the weak absorption tail (WAT ), where \(\alpha<{\mathrm{10^{-1}}}\,{\mathrm{cm^{-1}}}\), the absorption is controlled by optical transitions from tail-to-tail states. As stated above, the localized tail states in amorphous semiconductors are contributed to by defects. To some extent, the absolute value of absorption in region A may be used to estimate the density of defects in the material. In region B, where typically \(10^{-1}<\alpha<{\mathrm{10^{4}}}\,{\mathrm{cm^{-1}}}\), the absorption is related to transitions from the localized tail states above the valence band edge to extended states in the conduction band and/or from extended states in the valence band to localized tail states below the conduction band. The spectral dependence of α usually follows the so-called Urbach rule, given in (3.39). For many amorphous semiconductors, ΔE has been related to the width of the valence (or conduction) band tail states, and may be used to compare the widths of such localized tail states in different materials; ΔE is typically 0.05–0.1 eV. In region C, the absorption is controlled by transitions from extended to extended states. For many amorphous semiconductors, the α versus ℏω behavior follows the Tauc relation given in (3.40). The optical bandgap, E0, determined for a given material from the α versus ℏω relations obtained in (3.40), (3.42) and (3.43), can vary as shown in Table 3.7 for a-Si:H alloys.
Table 3.7

Optical bandgaps of a-Si1−xC x :H films obtained from Tauc’s (3.40), Sokolov et al.’s (3.42) and Cody’s (3.43) relations. (After [3.58])

 

Egat

\(\alpha={\mathrm{10^{3}}}\,{\mathrm{cm^{-1}}}\)

Egat

\(\alpha={\mathrm{10^{4}}}\,{\mathrm{cm^{-1}}}\)

Eg = E0

(Tauc)

Eg = E0

(Cody)

Eg = E0

(Sokolov)

ΔE ( meV ) 

a-Si:H

1.76

1.96

1.73

1.68

1.60

46

a-Si0.88C0.18:H

2.02

2.27

2.07

2.03

1.86

89

3.3.4 Exciton Absorption

Excitons in Crystalline Semiconductors

Optical absorption in crystalline semiconductors and insulators can create an exciton, which is an electron and hole pair excited by a photon and bound together through their attractive Coulomb interaction. This means that the absorbed optical energy remains held within the solid for the lifetime of the exciton. There are two types of excitons that can be formed in nonmetallic solids: Wannier or Wannier–Mott excitons and Frenkel excitons. The concept of Wannier–Mott excitons is valid for inorganic semiconductors such as Si, Ge and GaAs, because in these materials the large overlap of interatomic electronic wavefunctions enables the electrons and holes to be far apart but bound in an excitonic state. For this reason, these excitons are also called large-radii orbital excitons. Excitons formed in organic crystals are called Frenkel excitons. In organic semiconductors/insulators or molecular crystals, the intermolecular separation is large and hence the overlap of intermolecular electronic wavefunctions is very small and electrons remain tightly bound to individual molecules. Therefore, the electronic energy bands are very narrow and closely related to individual molecular electronic energy levels. In these solids, the absorption of photons occurs close to the individual molecular electronic states and excitons are also formed within the molecular energy levels [3.59]. Such excitons are therefore also called molecular excitons. For a more detailed look at the theory of Wannier and Frenkel excitons, readers may like to refer to Singh [3.59].

In Wannier–Mott excitons, the Coulomb interaction between the hole and electron can be viewed as an effective hydrogen atom with, for example, the hole establishing the coordinate reference frame about which the reduced mass electron moves. If the effective masses of the isolated electron and hole are m e and m h respectively, their reduced mass μ x is given by
$$\mu_{x}^{-1}=\left(m_{\mathrm{e}}^{\ast}\right)^{-1}+\left(m_{\mathrm{h}}^{\ast}\right)^{-1}\;.$$
(3.51)
Note that in the case of so-called hydrogenic impurities in semiconductors (both shallow donor and acceptor impurities), the reduced mass of the nucleus takes the place of one of the terms in (3.51) and hence the reduced mass is given to good approximation by the effective mass of the appropriate carrier. When the exciton is the carrier, the effective masses are comparable and hence the reduced mass is markedly lower; accordingly, the exciton binding energy is markedly lower than that for hydrogenic impurities. The energy of a Wannier–Mott exciton is given by (e. g., [3.59])
$$E_{x}=E_{\mathrm{g}}+\frac{\hbar^{2}K^{2}}{2M}-E_{n}\;,$$
(3.52)
where Eg is the bandgap energy, ℏK is the linear momentum, M \((=m_{\mathrm{e}}^{\ast}+m_{\mathrm{h}}^{\ast})\) is the effective mass associated with the center of mass of an exciton, and E n is the exciton binding energy given by
$$E_{n}=\frac{\mu_{x}e^{4}\kappa^{2}}{2\hbar^{2}\varepsilon^{2}}\frac{1}{n^{2}}=\frac{\text{Ry}^{x}}{n^{2}}\;,$$
(3.53)
where e is the electronic charge, \(\kappa={1}/{(4\uppi\varepsilon_{0})}\), ε is the static dielectric constant of the solid, and n is the principal quantum number associated with the internal excitonic states n = 1 ( s ) , 2 ( p ) , … (Fig. 3.12). According to (3.53), the excitonic states are formed within the bandgap below the conduction band edge. However, as the exciton binding energy is very small (a few meV in bulk Si and Ge crystals), exciton absorption peaks can only be observed at very low temperatures. Ry x is the so-called effective Rydberg of the exciton given by \(\text{Ry}(\mu_{x}/m_{\mathrm{e}})/\varepsilon^{2}\), where Ry = 13.6 eV. For bulk GaAs, the exciton binding energy (n = 1) corresponds to about 5 meV. Extrapolating from the hydrogen atom model, the extension of the excitonic wavefunction can be found from an effective Bohr radius a B given in terms of the Bohr radius as \(a_{\mathrm{B}}=h^{2}/(\uppi e^{2})\); that is, \(a_{\mathrm{B}}^{\ast}=a_{\mathrm{B}}(\varepsilon/\mu_{x})\). For GaAs this corresponds to about 12 nm or about 21 lattice constants – in other words the spherical volume of the exciton radius contains \(\smash{\approx(a_{\mathrm{B}}^{\ast}/a_{0})^{3}}\) or ≈ 9000 unit cells, where a0 is the lattice constant of GaAs. As \(\text{Ry}^{\ast}\ll E_{\mathrm{g}}\) and \(a_{\mathrm{B}}^{\ast}\gg a_{0}\), it is clear from this example that the excitons in GaAs are large-radii orbital excitons, as stated above. It should be noted that the binding energies of excitons in semiconductors tend to be a strong function of the bandgap. The dependences of Ry x (exciton binding energy) and \(a_{\mathrm{B}}^{\ast}/a_{0}\) on the semiconductor bandgap are shown in Fig. 3.13a,b a and b respectively. For excitons with large binding energies and correspondingly small radii (approaching the size of a lattice parameter), the excitons become localized on a lattice site, as observed in most organic semiconductors. As stated above, such excitons are commonly referred to as Frenkel excitons or molecular excitons. Unlike Wannier-based excitons, which are typically dissociated at room temperature, these excitations are stable at room temperature. For the binding energies of Frenkel excitons, one can refer to Singh for example [3.59].
Fig. 3.12

Schematic illustration of excitonic bands for n = 1 and 2 in semiconductors. Eg represents the energy gap

Fig. 3.13a,b

Dependence of the (a) exciton binding energy (R y x ) (3.29) and (b) size (in terms of the ratio of the excitonic Bohr radius to lattice constant [\(a_{\mathrm{B}}^{\ast}/a_{0}\)]) as a function of the semiconductor bandgap. Exciton binding energy increases along with a marked drop in exciton spreading as bandgap increases. The Wannier-based description is not appropriate above a bandgap of about 2 eV

Excitons can recombine radiatively, emitting a series of hydrogen-like spectral lines, as described by (3.52 ). In bulk (3-D) semiconductors such as GaAs, exciton lines can only be observed at low temperatures: they are easily dissociated by thermal fluctuations. On the other hand, in quantum wells and other structures of reduced dimensionality, the spatial confinement of both the electron and hole wavefunctions in the same layer ensures strong excitonic transitions of a few meV below the bandgap, even at room temperature. Excitonic absorption is well located spectrally and very sensitive to optical saturation. For this reason, it plays an important role in nonlinear semiconductor devices (nonlinear Fabry–Perot resonators, nonlinear mirrors, saturable absorbers, and so on). If the valence hole is a heavy hole, the exciton is called a heavy exciton; conversely, if the valence hole is light, the exciton is a light exciton. For practical purposes, the excitonic contribution to the overall susceptibility around the resonance frequency νex can be written as
$$\chi_{\mathrm{exc}}=-A_{0}\frac{(\nu-\nu_{\mathrm{ex}})+\mathrm{i}\Gamma_{\mathrm{ex}}}{(\nu-\nu_{\mathrm{ex}})^{2}+\Gamma_{\mathrm{ex}}^{2}(1+S)}\;,$$
(3.54)
where Γex is the linewidth and \(S=I/I_{\mathrm{S}}\) the saturation parameter of the transition. For instance, in GaAs MQW, the saturation intensity IS is as low as 1 kW ∕ cm2, and Γex (≈ 3.55 meV at room temperature) varies with the temperature according to
$$\Gamma_{\mathrm{ex}}=\Gamma_{0}+\frac{\Gamma_{1}}{\exp(\hbar\omega_{\mathrm{LO}}/kT)-1}\;,$$
(3.55)
where ℏΓ0 is the inhomogeneous broadening (≈ 2 meV), ℏΓ1 is the homogeneous broadening (≈ 5 meV), and ℏωLO is the longitudinal optical phonon energy (≈ 36 meV). At high carrier concentrations (provided either by electrical pumping or by optical injection), the screening of the Coulomb attractive potential by free electrons and holes provides an efficient mechanism for saturating the excitonic line.

The above discussion refers to so-called free excitons formed between conduction-band electrons and valence-band holes. According to (3.52), such an excitation is able to move throughout a material with a given center-of-mass kinetic energy (second term on the right-hand side). It should be noted, however, that free electrons and holes move with a velocity ℏ ( dE ∕ dk )  where the derivative is taken for the appropriate band edge. To move through a crystal, both the electron and the hole must have identical translational velocities, restricting the regions in k-space where these excitations can occur to those with \((\mathrm{d}E/\mathrm{d}k)_{\mathrm{electron}}=(\mathrm{d}E/\mathrm{d}k)_{\mathrm{hole}}\), commonly referred to as critical points.

A number of more complex pairings of carriers can also occur, which may also include fixed charges or ions. For example, for the case of three charged entities with one being an ionized donor impurity (D+), the following possibilities can occur: (D+)(+)(−), (D+)(−)(−) and (+)(+)(−) as excitonic ions, and (+)(+)(−)(−) and (D+)(+)(−)(−) as biexcitons or even bigger excitonic molecules [3.60]. Complexity abounds in these systems, as each electronic level possesses a fine structure corresponding to allowed rotational and vibrational levels. Moreover, the effective mass is often anisotropic. Note that when the exciton or exciton complex is bound to a fixed charge, such as an ionized donor or acceptor center in the material, the exciton or exciton complex is referred to as a bound exciton. Indeed, bound excitons may also involve neutral fixed impurities. It is usual to relate the exciton in these cases to the center binding them; thus, if an exciton is bound to a donor impurity, it is usually termed a donor-bound exciton.

Excitons in Amorphous Semiconductors

The concept of excitons is traditionally valid only for crystalline solids. However, several observations in the photoluminescence spectra of amorphous semiconductors have revealed the occurrence of photoluminescence associated with singlet and triplet excitons [3.37]. Applying the effective mass approach, a theory for the Wannier–Mott excitons in amorphous semiconductors has recently been developed in real coordinate space [3.37, 3.46, 3.55, 3.61]. The energy of an exciton thus derived is obtained as
$$W_{x}=E_{0}+\frac{P^{2}}{2M}-E_{n}(S)\;,$$
(3.56)
where P is the linear momentum associated with the exciton’s center of motion and E n  ( S )  is the binding energy of the exciton, given by
$$E_{n}(S)=\frac{\mu_{x}e^{4}\kappa^{2}}{2\hbar^{2}{\varepsilon}^{\prime}(S)^{2}n^{2}}\;,$$
(3.57)
where
$$\varepsilon^{\prime}(S)=\varepsilon\left[{1-\frac{(1-S)}{A}}\right]^{-1}\;,$$
(3.58)
where S is the spin (S being = 0 for singlet and = 1 for triplet) of an exciton and A is a material-dependent constant representing the ratio of the magnitude of the Coulomb and exchange interactions between the electron and the hole of an exciton. Equation (3.57) is analogous to (3.53) obtained for excitons in crystalline solids for S = 1. This is because (3.53) is derived within the large-radii orbital approximation, which neglects the exchange interaction and hence is valid only for triplet excitons [3.59, 3.62]. As amorphous solids lack long-range order, the exciton binding energy is found to be larger in amorphous solids than in their crystalline counterparts; for example, the binding energy is higher in hydrogenated amorphous silicon (a-Si:H) than in crystalline silicon (c-Si). This is the reason that it is possible to observe both singlet and triplet excitons in a-Si:H [3.63] but not in c-Si.

Excitonic Absorption

Since exciton states lie below the edge of the conduction band in a crystalline solid, absorption to excitonic states is observed below this edge. According to (3.53 ), the difference in energy in the bandgap and the excitonic absorption gives the binding energy. As the exciton-photon interaction operator and excited electron and hole pair and photon interaction operator depend only on their relative motion, then these interactions take the same form for band-to-band and excitonic absorption. Therefore, to calculate the excitonic absorption coefficient, one can use the same form of interaction as that used for band-to-band absorption, but one must use the joint density of states. Using the joint density of states, the absorption coefficient associated with the excitonic states in crystalline semiconductors is obtained as [3.37]
$$\alpha\hbar\omega=A_{x}(\hbar\omega-E_{x})^{1/2}$$
(3.59)
with the constant
$$A_{x}=\frac{4\sqrt{2}\,e^{2}|p_{\textit{xv}}|^{2}}{nc\sqrt{\mu_{x}}\,\hbar^{2}}\;,$$
where p xv is the transition matrix element between the valence and excitonic bands. Equation (3.59) is similar to that seen for direct band-to-band transitions, discussed above (3.60), and is only valid for the photon energies ℏω ≥ E x . There is no absorption below the excitonic ground state in pure crystalline solids. Absorption of photons to excitonic energy levels is possible through either the excitation of electrons to higher energy levels in the conduction band and then nonradiative relaxation to the excitonic energy level, or through the excitation of an electron directly to the exciton energy level. Excitonic absorption occurs in both direct and indirect semiconductors.
In amorphous semiconductors, the excitonic absorption and photoluminescence can be quite complicated. According to (3.56), the excitonic energy level is below the optical bandgap by an energy equal to the binding energy given in (3.57). However, there are four transition possibilities:
  1. 1.

    Extended valence to extended conduction states

     
  2. 2.

    Valence to extended conduction states

     
  3. 3.

    Valence extended to conduction tail states and

     
  4. 4.

    Valence tail to conduction tail states.

     
These possibilities will have different optical gap energies, E0, and different binding energies. Transition 1 will give rise to absorption in the free exciton states, transitions 2 and 3 will give absorption in the bound exciton states, because one of the charge carriers is localized in the tail states, and absorption through transition 4 will create localized excitons, which are also called geminate pairs. This can be visualized as follows: if an electron-hole pair is excited by a high-energy photon through transition 1 and forms an exciton, initially its excitonic energy level and the corresponding Bohr radius will have a reduced mass corresponding to both charge carriers being in extended states. As such an exciton relaxes downward nonradiatively, its binding energy and excitonic Bohr radius will change because its effective mass changes in the tail states. When both charge carriers reach the tail states (transition 4), their excitonic Bohr radius will be maintained although they are localized.

The excitonic absorption coefficient in amorphous semiconductors can be calculated using the same approach as presented in Sect. 3.3.3, and similar expressions to (3.40) and (3.43) are obtained. This is because the concept of the joint density of states is not applicable in amorphous solids. Therefore, by replacing the effective masses of the charge carriers by the excitonic reduced mass and the distance between the excited electron and hole by the excitonic Bohr radius, one can use (3.40) and (3.43) to calculate the excitonic absorption coefficients for the four possible transitions above in amorphous semiconductors. However, such a detailed calculation of the excitonic transitions in amorphous semiconductors is yet to be performed.

3.3.5 Impurity Absorption

Impurity absorption can be observed as the absorption coefficient peaks lying below fundamental (band-to-band) and excitonic absorption. It is usually related to the presence of ionized impurities or, simply, ions. The peaks occur due to electronic transitions between ionic electronic states and the conduction/valence band or due to intra-ionic transitions (within d or f shells, between s and d shells, and so on). The first case leads to intense and broad lines, while the characteristics of the features arising from the latter case depend on whether or not these transitions are allowed by parity selection rules. For allowed transitions, the absorption peaks are quite intense and broad, while forbidden transitions produce weak and narrow peaks. General reviews of this topic may be found in Blasse and Grabmaier [3.64], Henderson and Imbusch [3.65] and DiBartolo [3.66]. In the following section, we concentrate primarily on the properties of rare earth ions, which are of great importance in modern optoelectronics.

Optical Absorption of Trivalent Rare Earth Ions: Judd–Ofelt Analysis

Rare earths (RE s) is the common name used for the elements from Lanthanum (La) to Lutetium (Lu). They have atomic numbers of 57 to 71 and form a separate group in the periodic table. The most notable feature of these elements is an incompletely filled 4f shell. The electronic configurations of REs are listed in Table 3.8. The RE may be embedded in different host materials in the form of divalent or trivalent ions. As divalent ions, REs exhibit broad absorption-emission lines related to allowed 4f→5d transitions. In trivalent form, REs lose two 6 s electrons and one 4f or 5d electron. The Coulomb interaction of a 4f electron with a positively charged core means that the 4f level gets split into a complicated set of manifolds with energies, to a first approximation, that are virtually independent of the host matrix because the 4f level is well screened from external influences by the 5s and 5p shells [3.67]. Figure 3.14 shows an energy level diagram for the low-lying 4f N states of the trivalent ions embedded in LaCl3. To a second approximation, the exact construction and precise energies of the manifolds depend on the host material, via crystal field and via covalent interactions with the ligands surrounding the RE ion. A ligand is an atom (or molecule or radical or ion) with one or more unshared pairs of electrons that can attach to a central metallic ion (or atom) to form a coordination complex. Examples of ligands include ions (F, Cl, Br, I, S2−, CN, NCS, OH, NH 2 ) and molecules (NH3, H2O, NO, CO) that donate a pair of electrons to a metal atom or ion. Some ligands that share electrons with metals form very stable complexes.
Table 3.8

Occupation of outer electronic shells for rare earth elements

57

La

4s2

4p2

4d10

5s2

5p6

5 d 1

6s2

58

Ce

4s2

4p2

4d10

4 f 1

5s2

5p6

5 d 1

6s2

59

Pr

4s2

4p2

4d10

4 f 3

5s2

5p6

6s2

60

Nd

4s2

4p2

4d10

4 f 4

5s2

5p6

6s2

         

68

Er

4s2

4p2

4d10

4 f 12

5s2

5p6

6s2

         

70

Yb

4s2

4p2

4d10

4 f 14

5s2

5p6

6s2

71

Lu

4s2

4p2

4d10

4 f 14

5s2

5p6

5 d 1

6s2

Fig. 3.14

Energy level diagram of the low-lying 4f N states of trivalent ions doped in LaCl3. The pendant semicircles indicate fluorescent levels. (After [3.65, 3.68, 3.69])

Optical transitions between 4f manifold levels are forbidden by a parity selection rule that states that the wavefunctions of the initial and final states of an atomic (ionic) transition must have different parities for them to be permitted. Parity is a property of any function (or quantum mechanical state) that describes the function after mirror reflection. Even functions (states) are symmetric (identical after reflection, for example a cosine function), while odd functions (states) are antisymmetric (for example a sine function). The parity selection rule may be partially removed for an ion (or atom) embedded in host material due to the action of the crystal field, which gives rise to forbidden lines. The crystal field is the electric field created by a host material at the position of the ion.

The parity selection rule is weakened by the admixture of 5d states with 4f states and by the disturbed RE ion symmetry due to the influence of the host, which increases with the covalency. Higher covalency implies stronger sharing of electrons between the RE ion and the ligands. This effect is known as the nephelauxetic effect. The resulting absorption-emission lines are characteristic of individual RE ions and quite narrow because they are related to forbidden inner shell 4f transitions.

Judd–Ofelt (JO) analysis allows the oscillator strength of an electric dipole (ED) transition between two states of a trivalent rare earth (RE) ion embedded in a particular host lattice to be calculated. The possible states of an RE ion are often referred to as 2S+1L J , where \(L=0,1,2,3,4,5,6\;\ldots\) determines the electron’s total angular momentum, and is conventionally represented by the letters S, P, D, F, G, I. The term ( 2S + 1 )  is called the spin multiplicity and represents the number of spin configurations, while J is the total angular momentum, which is the vector sum of the overall (total) angular momentum and the overall spin \((J=L+S)\). The value (2J + 1) is called the multiplicity and corresponds to the number of possible combinations of overall angular momentum and overall spin that yield the same J. Thus, the notation 4I15∕2 for the ground state of Er3+ corresponds to the term \((J,L,S)=(15/2,6,3/2)\), which has a multiplicity of \(2J+1=16\) and a spin multiplicity of \(2S+1=4\). If the wavefunctions | ψ i  ⟩  and | ψ f  ⟩  correspond to the initial (2S+1L J ) and final (\({}^{2S^{\prime}+1}L_{J^{\prime}}\)) states of an electric dipole transition of an RE ion, the line strength of this transition, according to JO theory, can be calculated using
$$\begin{aligned}\displaystyle S_{\mathrm{ed}}&\displaystyle=\left|\langle\psi_{f}\left|H_{\mathrm{ed}}\right|\psi_{i}\rangle\right|^{2}\\ \displaystyle&\displaystyle=\sum_{k=2,4,6}{\Omega_{k}\left|{\left\langle{f_{\gamma}^{N}S^{\prime}L^{\prime}J^{\prime}\left|{U^{(k)}}\right|f_{\gamma}^{N}SLJ}\right\rangle}\right|^{2}}\;,\end{aligned}$$
(3.60)
where Hed is the ED interaction Hamiltonian, Ω k are coefficients reflecting the influence of the host material, and U(k) are reduced tensor operator components, which are virtually independent of the host material, and their values are calculated using the so-called intermediate coupling approximation [3.70]. The theoretical values of Sed calculated from this are compared with the values derived from experimental data using
$$S_{\exp}=\frac{3hcn}{8\uppi^{3}e^{2}\left\langle\lambda\right\rangle}\frac{2J+1}{\chi_{\mathrm{ed}}}\int_{\mathrm{band}}{\frac{\alpha(\lambda)}{\rho}}\,\mathrm{d}\lambda\;,$$
(3.61)
where ⟨λ⟩ is the mean wavelength of the transition, h is the Plank constant, c is the speed of light, e is the elementary electronic charge, α ( λ )  is the absorption coefficient, ρ is the RE ion concentration, n is the refraction index and the factor \(\chi_{\mathrm{ed}}=(n^{2}+2)^{2}/9\) is the so-called local field correction. The key idea of JO analysis is to minimize the discrepancy between experimental and calculated values of line strength, Sed and Sexp, by choosing the coefficients Ω k , which are used to characterize and compare materials, appropriately. The complete analysis should also include the magnetic dipole transitions [3.71]. The value of Ω2 is of prime importance because it is the most sensitive to the local structure and material composition and is correlated with the degree of covalence. The values of Ω k are used to calculate radiative transition probabilities and appropriate radiative lifetimes of excited states, which are very useful for numerous optical applications. More detailed analysis may be found in, for example, [3.71]. Ω k values for different ions and host materials can be found in Gschneidner Jr. and Eyring [3.72].

3.3.6 Effects of External Fields

Electroabsorptionand the Franz–Keldysh Effect

Electroabsorption is the absorption of light in a device where the absorption is induced by an applied (or changing) electric field within the device. Such a device is an electroabsorption modulator. There are three fundamental types of electroabsorption processes. In the Franz–Keldysh process, a strong applied field modifies the photon-assisted probability of an electron tunneling from the valence band to the conduction band, and thus it corresponds to an effective reduction in the bandgap energy, inducing the absorption of light with photon energies of slightly less than the bandgap. It was first observed for CdS, in which the absorption edge was observed to shift to lower energies with the applied field; that is, photon absorption shifts to longer wavelengths with the applied field. The effect is normally quite small but is nonetheless observable. In this type of electroabsorption modulation, the wavelength is typically chosen to be slightly smaller than the bandgap wavelength so that absorption is negligible. When a field is applied, the absorption is enhanced by the Franz–Keldysh effect. In free-carrier absorption, the concentration of free carriers N in a given band is changed (modulated), for example, by an applied voltage, changing the extent of photon absorption. The absorption coefficient is proportional to N and to the wavelength λ of the light raised to some power, typically 2–3. In the confined Stark effect, the applied electric field modifies the energy levels in a quantum well. The energy levels are reduced by the field by an amount proportional to the square of the applied field. A multiple quantum well (MQW ) pin-type device has MQWs in its intrinsic layer. Without any applied bias, light with photon energy just less than the quantum well (QW ) exciton excitation energy will not be significantly absorbed. When a field is applied, the energy levels are lowered and the incident photon energy is then sufficient to excite an electron and hole pair in the QWs. The relative transmission decreases with the reverse bias Vr applied to the pin device. Such MQW (pin) devices are usually not very useful in the transmission mode because the substrate material often absorbs the light (for example a GaAs/AlGaAs MQW pin would be grown on a GaAs substrate, which would absorb the radiation that excites the QWs). Thus, a reflector would be needed to reflect the light back before it reaches the substrate; such devices have indeed been fabricated.

The Faraday Effect

The Faraday effect, originally observed by Michael Faraday in 1845, is the rotation of the plane of polarization of a light wave as it propagates through a medium subjected to a magnetic field parallel to the direction of propagation of the light. When an optically inactive material such as glass is placed in a strong magnetic field and plane-polarized light is sent along the direction of the magnetic field, the emerging light’s plane of polarization is rotated. The magnetic field can be applied, for example, by inserting the material into the core of a magnetic coil – a solenoid. The specific rotatory power induced, given by θ ∕ L, has been found to be proportional to the magnitude of the applied magnetic field B, which gives the amount of rotation as
$$\theta=\vartheta\boldsymbol{B}L\;,$$
(3.62)
where L is the length of the medium, and ϑ is the so-called Verdet constant, which depends on the material and the wavelength of the light. The Faraday effect is typically small. For example, a magnetic field of ≈ 0.1 T causes a rotation of about 1 through a glass rod of length 20 mm. It appears that optical activity is induced by the application of a strong magnetic field to an otherwise optically inactive material. There is, however, an important distinction between natural optical activity and the Faraday effect. The sense of rotation θ observed in the Faraday effect for a given material (Verdet constant) depends only on the direction of the magnetic field B. If ϑ is positive, for light propagating parallel to B, the optical field E rotates in the same sense as an advancing right-handed screw pointing in the direction of B. The direction of light propagation does not change the absolute sense of rotation of θ. If we reflect the wave to pass through the medium again, the rotation increases to 2θ. The Verdet constant depends not only on the wavelength λ but also on the charge-to-mass ratio of the electron and the refractive index n ( λ )  of the medium through
$$\vartheta=-\frac{(e/m_{\mathrm{e}})}{2c}\lambda\frac{\mathrm{d}n}{\,\mathrm{d}\lambda}\;.$$
(3.63)
Verdet constants for some materials are listed in Table 3.9. Terbium gallium garnet is commercially used in optical isolators.
Table 3.9

Verdet constants for some materials

Material

Quartz

Flint glass

Tb-Ga garnet

Tb-Ga garnet

ZnSe

Crown glass

NaCl

λ (nm)

589.3

632

632

1064

633

589.3

589.3

ϑ (\(\mathrm{rad{\,}m^{-1}{\,}T^{-1}}\))

4.0

4.0

−134

−40

118

6.4

10

3.4 Optical Characterization

Various methods have been developed for the determination of optical constants of solids. They differ with respect to sample geometry (bulks, thin films, multilayers, powders etc.) and optical properties (absorbing and nonabsorbing). In this section we are mainly concerned with bulk samples and thin films with flat surfaces. From available experimental techniques we select those based on two fundamental properties of light (used as a probe) – its intensity and polarization. Generally, both the light intensity and polarization change upon light interaction with the sample and if measured in suitable configurations they can be used for the determination of the material refractive index and the extinction coefficient. Optical transmittance, reflectance and ellipsometry parameters are convenient measurable quantities.

Normal incidence optical transmittance is defined as a ratio of light intensity of transmitted and incident waves, \(T=I^{\mathrm{t}}/I^{\mathrm{i}}\). Reflectance is usually measured for oblique incidence and therefore depends on light polarization and angle of incidence. It is convenient to define optical reflectance for two linear polarizations: p represents the electric intensity vector oscillations in parallel direction with respect to the plane of incidence and s represents the electric intensity vector oscillations in perpendicular direction with respect to the plane of incidence. Reflectance is then defined by the relation \({R}_{\text{p,s}}=I_{\text{p,s}}^{\mathrm{r}}/I_{\text{p,s}}^{\mathrm{i}}\) as the ratio of reflected and incident beam intensities for specific polarizations. Experimentally, easily accessible transmittance and reflectance can be related to transmission and reflection coefficients of a sample
$$T =|{t}|^{2}\;, {t} =\frac{\hat{{E}}^{\mathrm{t}}}{\hat{{E}}^{\mathrm{i}}}$$
(3.64)
$${R}_{\text{p,s}} =|{r}_{\text{p,s}}|^{2}\;, {r}_{\text{p,s}} =\frac{\hat{{E}}_{\text{p,s}}^{\mathrm{r}}}{\hat{{E}}_{\text{p,s}}^{\mathrm{i}}}\;,$$
(3.65)
where amplitudes of electric fields of incident, reflected and transmitted waves are complex amplitudes \(\hat{{E}}={E}_{0}\mathrm{e}^{{\mathrm{i}\phi}}\) of a plane electromagnetic wave
$${E}={E}_{0}\mathrm{e}^{\mathrm{i}\phi}\mathrm{e}^{\mathrm{i}(\omega t-kr)}=\hat{{E}}\mathrm{e}^{\mathrm{i}(\omega t-kr)}\;.$$
(3.66)
All discussed cases of transmission and reflection are presented in Figs. 3.153.18 together with explicit expressions of incident, reflected and transmitted waves. It should be noted here that transmission and reflection coefficients are generally complex numbers providing information not only on the amplitude change but also on the phase change of the wave. On the other hand, transmittance and reflectance are real quantities that do not carry phase change information as they measure the ratio of light intensities
$$I=\frac{1}{2}c\varepsilon_{0}{E}_{0}^{2}\;.$$
(3.67)
Fig. 3.15

Incident and transmitted waves in the case of normal incidence transmittance measurement

Fig. 3.16

Incident and reflected s-polarized waves in the case of oblique incidence reflectance measurement

Fig. 3.17

Incident and reflected p-polarized waves in the case of oblique incidence reflectance measurement

Fig. 3.18

Linearly polarized incident wave with equal contributions of s- and p-components reflected from the sample surface. The geometrical meaning of ellipsometric angles Ψ and Δ is indicated

An elegant way to get access to a phase change is by measuring the polarization state of the reflected or transmitted wave with respect to the polarization state of the incident wave. The change of polarization is often represented by two angles Ψ, Δ that are, in the case of reflection, defined by the relation
$$\tan\Psi\mathrm{e}^{\mathrm{i}\Delta}=\frac{r_{\mathrm{p}}}{r_{\mathrm{s}}}\;.$$
(3.68)
The polarization state of a reflected wave – that is analyzed – generally takes an elliptical form, therefore this experimental method is called ellipsometry . The geometrical meaning of angles Ψ, Δ follows from their definition. Consider a linearly polarized incident wave with equal contributions (equal amplitudes) of p- and s- components that are in phase (Fig. 3.18). In the reflected wave, due to different reflection coefficients rp and rs, the p- and s- components are generally mutually shifted by an angle difference φrp − φrs that equals the ellipsometric angle Δ. Furthermore, the amplitudes of p- and s- components in a reflected wave are no more equal and their ratio E 0p r  ∕ E 0s r equals tanΨ. This interpretation is consistent with following mathematical treatment of relation (3.68)
$$\frac{{r}_{\mathrm{p}}}{{r}_{\mathrm{s}}}=\frac{{E}_{0\text{p}}^{\mathrm{r}}\mathrm{e}^{\mathrm{i}\phi_{\text{rp}}}/{E}_{0\text{p}}^{\mathrm{i}}\mathrm{e}^{\mathrm{i}\phi_{\text{ip}}}}{{E}_{0\text{s}}^{\mathrm{r}}\mathrm{e}^{\mathrm{i}\phi_{\text{rs}}}/{E}_{0\text{s}}^{\mathrm{i}}\mathrm{e}^{\mathrm{i}\phi_{\text{is}}}}=\frac{{E}_{0\text{p}}^{\mathrm{r}}}{{E}_{0\text{s}}^{\mathrm{r}}}\mathrm{e}^{\mathrm{i}(\phi_{\text{rp}}-\phi_{\text{rs}})}=\tan{\Psi}\mathrm{e}^{\mathrm{i}\Delta}\;.$$
(3.69)

For selected sample geometries, transmission and reflection coefficients can be expressed analytically in the frame of phenomenological electromagnetic theory [3.73]. Optical constants are explicitly covered in these expressions and therefore the measurable quantities of optical reflectance , transmittance and ellipsometry angles can be used for determination of the material refractive index and the extinction coefficient. This will be discussed in more detail for the case of single interface (bulk) and thin film materials in the following sections.

A large number of books and articles cover instrumental aspects of reflectance and transmittance measurements in different spectral ranges. For a broad overview of this field the reader is referred to [3.74] and references therein. Instrumentation of spectroscopic ellipsometers is discussed in detail by monographs devoted to spectroscopic ellipsometry [3.75, 3.76, 3.77]. Historically, the first ellipsometers that were developed were null ellipsometers. In this configuration, the orientation of the polarizer and compensator are adjusted such that the light reflected from the sample is linearly polarized. The analyzer is then rotated to the position that the light intensity on the detector is extinguished or nulled. From the positions of the polarizer, compensator and analyzer, the ellipsometric angles Ψ and Δ can be calculated. This configuration, if performed in four equivalent zones, can be very accurate and with low systematic errors. However, even when automated, this approach is relatively slow and measurements are time consuming. In order to speed up measurements, rotating analyzer/polarizer ellipsometers were developed. In these systems, either the analyzer or polarizer is continuously rotated at a constant angular velocity (typically about 10–100 Hz) about the optical axis. The operating characteristics of both of these configurations are similar. However, the rotating polarizer system requires the light source to be totally unpolarized. Any residual polarization in the source results in a source of measurement error unless corrected. Similarly, a rotating analyzer system is susceptible to the polarization sensitivity of the detector. However, solid-state semiconductor photodetectors have extremely high polarization sensitivities. Thus, commercial systems tend to use rotating analyzer systems where residual polarization in the source is not an issue. Fourier analysis of the variation of the amplitude of the detector signal provides values for Ψ and Δ. Such systems can provide high-speed and accurate measurements. Spectroscopic ellipsometers extend the concepts developed for measurements at a single wavelength to measurements at multiple wavelengths. Being able to measure the dispersion in optical constants with wavelength adds another dimension to the analysis, permitting unambiguous determination of material and structure parameters.

3.4.1 Bulk Samples

The term bulk is used here for samples with (at least locally) flat surfaces and for samples that are made from:
  1. 1.

    Absorbing materials thick enough with respect to penetration depth of light – used as a probe – that it senses only the sample surface (and not the backside of the sample or its deeper interfaces).

     
  2. 2.

    Nonabsorbing materials thick enough with respect to coherence length of the light – used as a probe – or shaped in such a geometry that light interference in the sample can be neglected.

     

For optical characterization of samples of type 1, reflection configuration is the most convenient, whereas for samples of type 2, refraction (possibly combined with reflection) is usually used.

First we will consider the determination of optical constants of absorbing bulk materials. In this case light senses only the sample surface, hence reflection coefficients for p- and s-polarized waves get the form of Fresnel equations derived for single interface
$$\begin{aligned}\displaystyle&\displaystyle{r}_{\text{p}}=\frac{{N}_{\text{t}}\cos{\theta}_{\text{i}}-{N}_{\text{i}}\cos{\theta}_{\text{t}}}{{N}_{\text{t}}\cos{\theta}_{\text{i}}+{N}_{\text{i}}\cos{\theta}_{\text{t}}}\;,\\ \displaystyle&\displaystyle{r}_{\mathrm{s}}=\frac{{N}_{\text{i}}\cos{\theta}_{\text{i}}-{N}_{\text{t}}\cos{\theta}_{\text{t}}}{{N}_{\text{i}}\cos{\theta}_{\text{i}}+{N}_{\text{t}}\cos{\theta}_{\text{t}}}\;,\end{aligned}$$
(3.70)
where Ni and Nt are (complex) refractive indices of the ambient and sample respectively [3.73]. It is useful to shortly discuss the dependence of reflection coefficients on the incidence angle θi (angle of refraction is denoted by θt and can be calculated by Snell’s law ). Figure 3.19a,b represents incident angle dependence of absolute values of p- and s-reflection coefficients and their phases. Two cases are distinguished: (a) the ambient is optically rarer; and (b) the ambient is optically denser with respect to the sample. The angle of incidence where the rp coefficient becomes zero is called the Brewster angle θB. However, for optically absorbing samples, the rp coefficient does not reach a zero value. Instead, its angle dependence forms a nonzero minimum there and its position is called the principal angle. When the ambient is optically denser then the critical angle θc can be distinguished. For incidence angles greater than the critical, both reflection coefficients are equal in absolute value to unity. Total reflection is observed under these conditions. Calculations of reflectance and ellipsometry parameters for a specific angle of incidence is straightforward using Fresnel equations (3.70) and definition relations (3.65) and (3.69).
Fig. 3.19a,b

Incident angle dependence of single interface reflection coefficient (its absolute value and phase) for a p- and s-polarized wave. Fresnel relations with ni = 1; nt = 1.5 (a) and ni = 1.5; nt = 1 (b) were used. Brewster θB and critical θC angles are also indicated

However, precaution must be used when approximating a surface of a bulk material with a single interface. A single interface is an ideal structure that presents an abrupt and sharp change of refractive index across a planar interface between two semi-infinitive spaces – in our case, the ambient and sample. In practice it is rather difficult to prepare such surfaces. Mechanical polishing of a sample surface usually causes a so-called Beilby overlayer that accumulates various defects due to the mechanism of polishing [3.78]. This damaged layer can be reduced in thickness if a proper polishing process is selected or removed by electropolishing. The Beilby overlayer may present different optical properties than the bulk and, if not carefully considered, erroneous optical constants are determined. Alternatively, sample cutting or cleaving in a vacuum or the deposition of an optically thick film is possible. Ellipsometry as a phase sensitive method is extremely sensitive to the surface quality; and it is made use of for ultrathin film or overlayer characterization. The following sections cover selected approaches used to derive optical constants from reflectance and ellipsometry measurements carried out on sample surfaces that can be successfully approximated by the single interface model. For further use we provide here also the Fresnel formulae of transmission coefficients for a single interface
$$\begin{aligned}\displaystyle&\displaystyle{t}_{\text{p}}=\frac{2{N}_{\text{i}}\cos{\theta}_{\text{i}}}{{N}_{\text{t}}\cos{\theta}_{\text{i}}+{N}_{\text{i}}\cos{\theta}_{\text{t}}}\;,\\ \displaystyle&\displaystyle{t}_{\text{s}}=\frac{2{N}_{\text{i}}\cos{\theta}_{\text{i}}}{{N}_{\text{i}}\cos{\theta}_{\text{i}}+{N}_{\text{t}}\cos{\theta}_{\text{t}}}\;.\end{aligned}$$
(3.71)

Normal Incidence Reflectivityand Kramers–Kronig Analyses

For normal incidence, p- and s-polarization cannot be distinguished. The reflection coefficient is then polarization-independent and if air is considered as an ambient we get
$${r}=\frac{1-{n}+{\mathrm{i}K}}{1+{n}-{\mathrm{i}K}}\;.$$
(3.72)
Hence, determination of the refractive index n and extinction coefficient K of the studied material requires knowledge of both the absolute value and phase of the normal incidence reflection coefficient
$${r}=\left|{r}\right|\mathrm{e}^{\mathrm{i}\phi}=\sqrt{{R}}\mathrm{e}^{{\mathrm{i}\phi}}\;.$$
(3.73)
Experimentally accessible normal incidence reflectivity R provides only the absolute value of the reflection coefficient. However when reflectivity is known in the whole spectral range then the phase of the reflection coefficient can be calculated with the help of the Kramers–Kronig relation  [3.79]
$${\phi}(\omega)=-\frac{2{\omega}}{{\uppi}}{P}\int_{0}^{{\infty}}\frac{{\ln}\sqrt{{R}\left({\omega}^{\prime}\right)}}{{\omega}^{\prime 2}-{\omega}^{2}}{\mathrm{d}\omega^{\prime}}\;.$$
(3.74)
The final relations for the refractive index and extinction coefficient of the probed material are obtained from
$$\begin{aligned}\displaystyle&\displaystyle{n}=\frac{1-{R}}{1+{R}+2\sqrt{{R}}\cos{\phi}}\;,\\ \displaystyle&\displaystyle{K}=\frac{2\sqrt{{R}}\sin{\phi}}{1+{R}+2\sqrt{{R}}\cos{\phi}}\;.\end{aligned}$$
(3.75)
However, in reality, reflectivity cannot be measured in the whole spectral range covering all frequencies. Nevertheless, the Kramers–Kronig relation (3.74) can still be used when the extrapolation of the experimentally available spectrum of reflectivity is carefully done in both spectral limits. Often this method combines visible (VIS ), ultraviolet (UV) and vacuum ultraviolet (VUV ) spectrophotometry or synchrotron facility to cover as wide a spectral range as possible. For the high energy of photons, the surface quality (mainly surface roughness) becomes extremely important; for nonnegligible roughness, a correction on light scattering lost in the reflected beam must be considered [3.80]. As an example of an application of the Kramers–Kronig approach we provide a normal incidence reflectivity spectrum recorded on PZT and the corresponding calculated spectra of the PZT refractive index and extinction coefficient in Fig. 3.20a,b.
Fig. 3.20a,b

Normal incidence reflectivity of PZT (a) and its complex refractive index (b) determined by Kramers–Kronig relations. Experimental data recorded on a synchrotron facility UVSOR, Japan. Comparison with ellipsometry is also presented

Oblique Incidence Reflectivity Rp and Rs

Even if a wide spectral range is not experimentally available for a normal incidence reflectivity measurement, we can still make use of reflectance measurements – carried out for oblique incidence – and determine optical constants of material from these spectra. Fresnel relations (3.70) inserted into definitions (3.65) make a link between reflectance for p- and s-waves and optical constants of the material
$$\begin{aligned}\displaystyle&\displaystyle{R}_{\mathrm{s}}=\frac{({Y}-{u})^{2}+{v}^{2}}{({Y}+{u})^{2}+{v}^{2}}\;,\\ \displaystyle&\displaystyle{R}_{\mathrm{p}}=\frac{(\varepsilon_{1}-{Z}{u})^{2}+(\varepsilon_{2}-{Z}{v})^{2}}{(\varepsilon_{1}+{Z}{u})^{2}+(\varepsilon_{2}+{Z}{v})^{2}}{\ }\;,\end{aligned}$$
(3.76)
where
$$\begin{aligned}\displaystyle&\displaystyle{Y}={n}_{\text{i}}\cos{\theta}_{\text{i}}\;,\quad{Z}=\frac{{n}_{\text{i}}}{\cos{\theta}_{\text{i}}}\;,\\ \displaystyle&\displaystyle\varepsilon_{1}={n}_{\text{t}}^{2}-{K}_{\text{t}}^{2}\;,\quad{\varepsilon}_{2}=2{n}_{\text{t}}{K}_{\text{t}}\;,\quad{\varepsilon}_{3}=\left({n}_{\text{i}}\cos{\theta}_{\text{i}}\right)^{2}\;,\\ \displaystyle&\displaystyle 2{u}^{2}={\varepsilon}_{1}-{\varepsilon}_{3}+\sqrt{\left({\varepsilon}_{1}-{\varepsilon}_{3}\right)^{2}+{\varepsilon}_{2}^{2}}\;,\\ \displaystyle&\displaystyle 2{v}^{2}={\varepsilon}_{3}-{\varepsilon}_{1}+\sqrt{\left({\varepsilon}_{1}-{\varepsilon}_{3}\right)^{2}+{\varepsilon}_{2}^{2}}\;.\end{aligned}$$
These equations are highly nonlinear and moreover transcendental, therefore only approximate inverse solutions exist [3.81]. Presently, numerical methods and nonlinear fitting of experimental spectra of Rs and Rp can be routinely performed to adjust searched optical constants.

Ellipsometry

The determination of optical constants of a material from ellipsometry parameters Ψ and Δ recorded on a bulk sample is straightforward. It can be shown [3.82] that the complex refractive index of a material can be expressed as
$$n-\mathrm{i}K=\sqrt{\sin^{2}\theta_{\mathrm{i}}\left(1+\tan^{2}\theta_{\mathrm{i}}\left(\frac{1-\rho}{1+\rho}\right)^{2}\right)}\;,$$
(3.77)
where ρ represents the ratio of Fresnel reflection coefficients for p- and s-polarized waves; \(\rho=r_{\mathrm{p}}/r_{\mathrm{s}}\) and the ambient is assumed to be air. Since the angle of incidence θi is set in the experiment, the two parameters measured from the experiment (Ψ and Δ) for a given wavelength can be used to deduce the complex parameter ρ and further, two remaining unknown variables in the equation above – namely, n and K. Alternatively, we can reformulate relation (3.77) and express the real and imaginary part of the material dielectric permittivity as an explicit function of ellipsometry parameters
$$\begin{aligned}\displaystyle&\displaystyle\varepsilon_{1}=n^{2}-K^{2}=\\ \displaystyle&\displaystyle\quad\sin^{2}\theta_{\mathrm{i}}\left[1+\left(\vphantom{\frac{1}{2}}\tan^{2}\theta_{\mathrm{i}}\right.\right.\\ \displaystyle&\displaystyle\quad\left.\left.\times\frac{\cos^{2}(2\Psi)-\sin^{2}(2\Psi)\sin^{2}\Delta}{[1+\sin(2\Psi)\cos\Delta]^{2}}\right)\right]\;,\\ \displaystyle&\displaystyle\varepsilon_{2}=2nK=\frac{\sin^{2}\theta_{\mathrm{i}}\tan^{2}\theta_{\mathrm{i}}\sin(4\Psi)\sin\Delta}{[1+\sin(2\Psi)\cos\Delta]^{2}}\;.\end{aligned}$$
(3.78)
The single interface is the only case where the inverse solution is analytically available. For other structures such as thin films, multilayers, and so on, we have to use numerical methods to adjust searched optical and geometrical parameters of a sample. Even if the incidence angle θi appears explicitly in relation (3.77), the value of the complex refractive index does not depend on its selection. However, the most precise results are obtained when θi is set close to the principal value (or Brewster angle) where rp ≈ 0 and rs and rp coefficients have significantly different values (Fig. 3.19a,b). Figure 3.21a-c represents an example of Ψ and Δ measured for incidence angles 55 and 65 on the surface of a pellet pressed from blue organic pigment and the corresponding optical constants determined by relation (3.77). For both incidence angles, the obtained optical constants are nearly identical. On the other hand, if determined n and K are incidence angle-dependent then the approximation of a single interface is no longer valid. In this case, the sample surface may have a more complex structure (for example the presence of an overlayer).
Fig. 3.21a–c

Ellipsometric parameters recorded on pressed pellets of blue organic pigment for incidence angle of 55 (a) and 65 (b) together with refractive index (c) determined by analytical formula (3.70)

Alternatively, the searched spectral dependence of a complex refractive index may be parameterized by an appropriate function (model dielectric function). Its parameters may be then determined by nonlinear fitting of experimental data Ψ and Δ. For this purpose, ellipsometric spectra for several incidence angles are usually recorded and fitted simultaneously to increase the number of experimental data and hence increase the reliability of the results. Several examples of model dielectric function are provided above in Sect. 3.2 . Other examples for specific materials and spectral ranges are available for example in monographs focused on spectroscopic ellipsometry [3.75, 3.76]. Figure 3.22a,b presents a model dielectric function of polycrystalline ZnSe defined as a sum of five oscillators whose parameters were fitted against experimental ellipsometric spectra recorded for several incidence angles.
Fig. 3.22a,b

Real (a) and imaginary (b) part of dielectric permittivity of ZnSe determined by spectroscopic ellipsometry. Its deconvolution to five oscillators is also indicated

Ellipsometry is a very precise and sensitive spectroscopic tool but when operated solely in reflection configuration then its sensitivity to low values of material absorption is limited. This is due to the fact that reflected light senses only the sample surface and its close vicinity. It is argued that ellipsometry can provide reliable results for absorption coefficient \({\alpha}> {\mathrm{10^{5}}}\,{\mathrm{cm^{-1}}}\) [3.83]. If α is in the range 103–105 cm−1 usually transmittance spectrum is added and treated simultaneously with ellipsometry data to ensure accurate determination of the absorption coefficient. In the region of low absorption \(\alpha<{\mathrm{10^{3}}}\,{\mathrm{cm^{-1}}}\) other techniques such as photocurrent measurement or photothermal deflection spectroscopy are more appropriate [3.84].

Combined Measurement of T and R

The combined measurement of reflectance and transmittance also enables material absorption determination. This is often done for the study of absorption edge onset of amorphous semiconductors (Fig. 3.11a,b) and glasses where the absorption coefficient is relatively weak. The determination of the absorption coefficient in a spectral range where the penetration depth decreases as the absorption increases is usually realized by measurements of a series of samples with various thicknesses. Samples are usually in the form of plane-parallel plates (both sides polished to defined thickness), hand blown glass bubbles or films deposited on substrates. The absorption coefficient is then expressed by following relation
$$\alpha=\frac{1}{d}\ln\left(\frac{\left(1-{R}\right)^{2}+\sqrt{\left(1-{R}\right)^{4}+4{R}^{2}T^{2}}}{2T}\right)\;,$$
(3.79)
where d stands for sample thickness.

There is a wide class of materials that in some limited spectral range are nonabsorbing and their optical transparency is extensively used in various applications (optical elements, integrated optics, etc.). In this case, the refractive index has a real value, because its imaginary part (extinction) can be neglected. Of course, the refractive index of these nonabsorbing materials can be determined by most methods mentioned above that were based on light reflection but much more precise are methods based on light refraction, i. e., refractometry .

Minimal Deviation

The refractive indices of dielectrics and semiconductors in a spectral range of their transparency can be determined from a prism of the material of an apex angle, A, by measuring the angle of minimal deviation, D, of radiation passing through the prism as in Fig. 3.23. The refractive index, n, is then given by the relation
$$n=\frac{{\sin}\frac{1}{2}(A+D)}{\sin\frac{1}{2}A}\;.$$
(3.80)
It should be noted here that high quality samples (prisms) free of any volume imperfections and defects are required, otherwise scattered radiation in the transmitted beam does not enable a precise determination of the angle of minimal deviation. This technique can give a refraction index in absolute precision of about 0.001.
Fig. 3.23

Angle of deviation technique for measuring the refactive index

To complete this section we present one more convenient method for determination of refractive index of nonabsorbing materials based on the critical angle measurement.

Critical Angle

Consider a bulk material with an unknown refractive index, optically coupled to the prism with a known index of refraction (Fig. 3.24). If the angle of incidence is scanned, then an abrupt change of intensity is detected when the angle of incidence reaches the critical angle (onset of total reflection on prism/sample interface) (Fig. 3.19a,b). From the condition of critical angle,
$${n}={n}_{\text{p}}{\sin}{\theta}_{\text{c}}$$
(3.81)
we can determine the refractive index of the bulk sample with a high precision of about 0.0005. The condition of total reflection requires that the refractive index of the sample is higher than that of the prism. This technique can also be used for refractive index and thickness determination of transparent films deposited on a substrate. In this case, for some discrete values of incidence angle, the light is coupled as a guided wave to the film. From these angle values, film parameters of interest can be determined and the method itself is referred to as a prism coupler method.
Fig. 3.24

Experimental setup for measuring the refractive index by the critical angle technique

3.4.2 Thin Film Optics

Thin film optics involves multiple reflections of light entering a thin film dielectric (typically on a substrate) so that the reflection and transmission coefficients are determined by multiple wave interference phenomena, as shown in Fig. 3.25.
Fig. 3.25

Thin film coated on a substrate and multiple reflections of incident light, where n1 , n2 and n3 are the refractive indices of the medium above the thin film, the thin film, and the substrate respectively

Consider a thin film coated on a substrate. Assuming that the incident wave has an amplitude of E0, then there are various transmitted and reflected waves, as shown in Fig. 3.25. We then have the following amplitudes based on the definitions of the reflection and transmission coefficients
$$\begin{aligned}\displaystyle A_{1}&\displaystyle=E_{0}r_{12}\mathrm{e}^{\mathrm{i}\phi_{\text{A1}}}\\ \displaystyle A_{2}&\displaystyle=E_{0}t_{12}r_{23}t_{21}\mathrm{e}^{\mathrm{i}\phi_{\text{A2}}}\\ \displaystyle A_{3}&\displaystyle=E_{0}t_{12}r_{23}r_{21}r_{23}t_{21}\mathrm{e}^{\mathrm{i}\phi_{\text{A3}}}\\ \displaystyle B_{1}&\displaystyle=E_{0}t_{12}\mathrm{e}^{\mathrm{i}\phi_{\text{B1}}}\\ \displaystyle B_{2}&\displaystyle=E_{0}t_{12}r_{23}\mathrm{e}^{\mathrm{i}\phi_{\text{B2}}}\\ \displaystyle B_{3}&\displaystyle=E_{0}t_{12}r_{23}r_{21}\mathrm{e}^{\mathrm{i}\phi_{\text{B3}}}\\ \displaystyle B_{4}&\displaystyle=E_{0}t_{12}r_{23}r_{21}r_{23}\mathrm{e}^{\mathrm{i}\phi_{\text{B4}}}\\ \displaystyle B_{5}&\displaystyle=E_{0}t_{12}r_{23}r_{21}r_{23}r_{21}\mathrm{e}^{\mathrm{i}\phi_{\text{B5}}}\\ \displaystyle B_{6}&\displaystyle=E_{0}t_{12}r_{23}r_{21}r_{23}r_{21}r_{23}\mathrm{e}^{\mathrm{i}\phi_{\text{B6}}}\\ \displaystyle C_{1}&\displaystyle=E_{0}t_{12}t_{23}\mathrm{e}^{\mathrm{i}\phi_{\text{C1}}}\\ \displaystyle C_{2}&\displaystyle=E_{0}t_{12}r_{23}r_{21}t_{23}\mathrm{e}^{\mathrm{i}\phi_{\text{C2}}}\\ \displaystyle C_{3}&\displaystyle=E_{0}t_{12}r_{23}r_{21}r_{23}r_{21}t_{23}\mathrm{e}^{\mathrm{i}\phi_{\text{C3}}}\end{aligned}$$
(3.82)
and so on, where r12 is the reflection coefficient of a wave in medium 1 incident on medium 2, and t12 is the transmission coefficient from medium 1 into 2. For simplicity, we will assume normal incidence. The phase change upon traversing the thin film thickness d is \(\phi_{\text{C}1}=-(2\uppi/\lambda)n_{2}d\), where λ is the free space wavelength. In the following relations the phase φ is defined as \(\phi=-\phi_{\text{C}1}\).
The reflection and transmission coefficients under normal incidence are given by
$$\begin{aligned}\displaystyle r_{1}=r_{12}&\displaystyle=\frac{n_{1}-n_{2}}{n_{1}+n_{2}}=-r_{21}\;,\\ \displaystyle r_{2}=r_{23}&\displaystyle=\frac{n_{2}-n_{3}}{n_{2}+n_{3}}\;,\end{aligned}$$
(3.83a)
and
$$\begin{aligned}\displaystyle t_{1}&\displaystyle=t_{12}=\frac{2n_{1}}{n_{1}+n_{2}}\;,\quad t_{2}=t_{21}=\frac{2n_{2}}{n_{1}+n_{2}}\;,\\ \displaystyle t_{3}&\displaystyle=t_{23}=\frac{2n_{3}}{n_{2}+n_{3}}\;,\end{aligned}$$
(3.83b)
where
$$1-t_{1}t_{2}=r_{1}^{2}\;.$$
(3.84)
The total reflection coefficient is then obtained as
$$r=\frac{A_{\text{reflected}}}{E_{0}}=r_{1}-\frac{t_{1}t_{2}}{r_{1}}\sum_{k=1}^{\infty}\left(-r_{1}r_{2}\mathrm{e}^{-\mathrm{i}2\phi}\right)^{k}\;,$$
(3.85)
which can be summed to be
$$r=\frac{r_{1}+r_{2}\mathrm{e}^{-\mathrm{i}2\phi}}{1+r_{1}r_{2}\mathrm{e}^{-\mathrm{i}2\phi}}\;.$$
(3.86)
The total transmission coefficient is obtained as
$$\begin{aligned}\displaystyle t=\frac{C_{\text{transmitted}}}{E_{0}}&\displaystyle=-\frac{t_{1}t_{3}\mathrm{e}^{\mathrm{i}\phi}}{r_{1}r_{2}}\sum_{k=1}^{\infty}\left(-r_{1}r_{2}\mathrm{e}^{-\mathrm{i}2\phi}\right)^{k}\\ \displaystyle&\displaystyle=\left(\frac{t_{1}t_{3}\mathrm{e}^{\mathrm{i}\phi}}{r_{1}r_{2}}\right)\frac{r_{1}r_{2}\mathrm{e}^{-\mathrm{i}2\phi}}{1+r_{1}r_{2}\mathrm{e}^{-\mathrm{i}2\phi}}\;,\end{aligned}$$
(3.87)
which can be summed to be
$$t=\frac{t_{1}t_{3}\mathrm{e}^{-\mathrm{i}\phi}}{1+r_{1}r_{2}\mathrm{e}^{-\mathrm{i}2\phi}}\;.$$
(3.88)
Equations (3.86) and (3.88) are very useful when studying the optical properties of thin films coated on a substrate as discussed by Kasap, Ray and Gould in Chap.  28 on Thin Films in this Handbook. Reflection and transmission coefficients of thin film systems for oblique incidence are also expressed by (3.86) and (3.88) respectively. However, single interface reflection and transmission coefficients (\(r_{1},r_{2},t_{1},t_{3}\)) have to be replaced by Fresnel relations for s- and p-polarized waves (Eqs. (3.70) and (3.71)). Moreover, phase change has to be modified to \({\phi}=(2{\uppi}/\lambda)d(n_{2}^{2}-n_{1}^{2}\sin^{2}\theta)^{1/2}\), where θ is the incidence angle, to account for oblique propagation of light in the layer. Reflectance of the thin film system is equal to the square of the absolute value of the reflection coefficient of the thin film (3.65). On the other hand, when considering transmittance of the film deposited on a substrate, it must be remembered that expression (3.88) relates the amplitudes of the waves in the media bounding the film (ambient and substrate), but transmitted wave intensity is not measured in the substrate but in free space with a refractive index n1. Therefore, normal incidence transmittance is related to the transmission coefficient by relation
$$T=\frac{{n}_{3}}{{n}_{1}}\left|{t}\right|^{2}\;.$$
(3.89)
In practice, the two most popular approaches for thin film optical characterization are either to analyze optical transmission spectra, which may be observed using a standard spectrophotometer, or ellipsometric investigations in reflection. Both approaches are briefly explained below.

Swanepoel’s Analysis of Optical Transmission Spectra

One of the simplest and most practically realizable approaches to the problem was developed by Swanepoel [3.85]. He showed that for a uniform film, with a thickness d, a refractive index n and an absorption coefficient α, deposited on the substrate with a refractive index s, the transmittance can be expressed as
$$T(\lambda)=\frac{Ax}{B-Cx\cos\varphi+Dx^{2}}\;,$$
(3.90a)
where
$$A =16n^{2}s\;,$$
(3.90b)
$$B =(n+1)^{3}\left(n+s^{2}\right)\;,$$
(3.90c)
$$C =2\left(n^{2}-1\right)\left(n^{2}-s^{2}\right)\;,$$
(3.90d)
$$D =(n-1)^{3}\left(n-s^{2}\right)\;,$$
(3.90e)
$$\varphi =4\uppi nd/\lambda\;,$$
(3.90f)
$$x =\exp(-\alpha d)\;,$$
(3.90g)

andn is a function of λ.

For a nonuniform film with a wedge-like cross-section, (3.90a ) must be integrated over the thickness of the film, giving
$$T_{\Updelta d}\left(\lambda\right)=\frac{1}{\varphi_{2}-\varphi_{1}}\int_{\varphi_{1}}^{\varphi_{2}}\frac{Ax}{B-Cx\cos\varphi+Dx^{2}}\mathrm{d}x\;,$$
(3.91)
where
$$\varphi_{1}=4\uppi n\left(d-\Updelta d\right)/\lambda$$
and
$$\varphi_{2}=4\uppi n\left(d+\Updelta d\right)/\lambda\;,$$
(3.92)
where d is the average thickness of the film and Δd is the variation of the thickness throughout the illumination (testing) area. The concept behind Swanepoel’s method is to construct two envelopes TM ( λ )  and Tm ( λ )  that pass through the maxima and minima of T ( λ )  and to split the entire spectral range into three regions: negligible absorption, weak absorption and strong absorption regions, as shown in Fig. 3.26.
Fig. 3.26

Optical transmission of an a-Se thin film. Calculations are done using (3.4.2.1) with the n ( λ )  and α ( λ )  relations shown in Fig. 3.27. The film was prepared by the thermal evaporation of photoreceptor-grade selenium pellets. Film thickness was 2 μm. Tentative regions of strong, weak and negligible absorption are also shown

In the region of negligible absorption, (3.91) yields
$$\begin{aligned}\displaystyle T_{\mathrm{M}/\mathrm{m}}&\displaystyle=\frac{\lambda}{2\uppi n\Updelta d}\frac{a}{\sqrt{1-b^{2}}}\\ \displaystyle&\displaystyle\quad\;\tan^{-1}\left[\frac{1\pm b}{\sqrt{1-b^{2}}}\tan\left(\frac{2\uppi n\Updelta d}{\lambda}\right)\right]\;,\end{aligned}$$
(3.93)
where \(a=A/(B+D)\) and \(b=C/(B+D)\). The plus sign in (3.93) corresponds to transmission maxima and minus to minima. Equation (3.74) is used to find Δd and n.
In the region of weak absorption,
$$\begin{aligned}\displaystyle T_{\mathrm{M}/\mathrm{m}}&\displaystyle=\frac{\lambda}{2\uppi n\Updelta d}\frac{a_{x}}{\sqrt{1-b_{x}^{2}}}\\ \displaystyle&\displaystyle\quad\;\tan^{-1}\left[\frac{1\pm b_{x}}{\sqrt{1-b_{x}^{2}}}\tan\left(\frac{2\uppi n\Updelta d}{\lambda}\right)\right]\;,\end{aligned}$$
(3.94)
where \(a_{x}=Ax/(B+Dx^{2})\) and \(b_{x}=Cx/(B+Dx^{2})\). Equation (3.94) allows us to find x and n using the previously found Δd.

The previous results are used to construct n ( λ )  and to create approximations using the Cauchy, Sellmeier or Wemple–DiDominico dispersion equations. An example of a Cauchy approximation is shown in Fig. 3.27.

The positions of the extrema are given by the equation
$$2nd=m\lambda\;,$$
(3.95)
where \(m=1,2,3,{\ldots}\) for maxima and \(m=1/2,3/2,{\ldots}\) for minima. Therefore, d can be found from the slope of (n ∕ λ) versus m.
In the region of strong absorption, the absorption coefficient is calculated as
$$x=\frac{A-\sqrt{A^{2}-4T_{i}^{2}BD}}{2T_{i}D}\;,$$
(3.96)
where
$$T_{i}=\frac{2T_{\mathrm{M}}T_{\mathrm{m}}}{T_{\mathrm{M}}+T_{\mathrm{m}}}$$
(3.97)
and A, B and D are calculated using the above-mentioned n ( λ )  approximation.

It is worth noting that the division into negligible absorption, weak absorption and strong absorption regions is quite arbitrary and should be checked using trial-and-error methods.

Ellipsometry

An example of ellipsometric spectra recorded in reflection configuration on a thin film is presented in Fig. 3.28a-d. For photon energies lower than bandgap Eg of thin film material, we can distinguish interference phenomena (interference fringes) due to multiple reflections in the film. When photon energy in the spectrum increases and has a higher value than Eg, the onset of light absorption reduces the amplitude of fringes; and for even higher absorption values (higher photon energies) interference fringes completely disappear. In this spectral range, the incident light probes only the film surface, hence the approach derived for a single interface in the previous section can be used. This qualitative interpretation is valid also for the transmittance spectra treated previously by the Swanepoel method.
Fig. 3.27

(a) The spectral dependence of the refractive index of the a-Se thin film from Fig. 3.26. The line corresponds to the Cauchy approximation with the parameters shown on the figure. (b) The spectral dependence of the absorption coefficient of the same a-Se thin film

Fig. 3.28a–d

Ellipsometric (a,b) transmittance (c) and reflectance (d) spectra recorded on As50Se50 amorphous thin film deposited on float glass substrate (experimental data – black solid lines) compared with best-fit theoretical spectra (broken lines)

It should be noted here that in optics term thin film is usually used for films with thicknesses significantly smaller than coherence length of interacting (probe) light. In this case, the formalism of totally coherent light can be applied to the study of interference phenomena in the film. Quantitative analyses of ellipsometric spectra can provide information on the geometrical (film thickness) and optical (film complex refractive index) properties of a sample. However, desired optical and geometrical parameters cannot be expressed analytically as explicit functions of experimental Ψ and Δ values as was done previously in the case of a single interface (bulk samples). The solution of this inverse problem requires construction of a sample model. Selected parameters are then adjusted by a numerical procedure where differences between theoretically calculated and experimental spectra are iteratively minimized (nonlinear fitting). Theoretical spectra of Ψ and Δ are easily obtained when rp and rs reflection coefficients – determined for the thin film (see (3.86) and related remarks) – are inserted into the definition relation of ellipsometric angles (3.68). Often, it is helpful to add transmittance and reflectance experimental spectra and treat them simultaneously with ellipsometry spectra (recorded for several incidence angles). In this way, we can reduce possible correlations between searched parameters.

As an example of ellipsometry data treatment we present in Fig. 3.28a-d experimental ellipsometric, transmittance and reflectance spectra compared with best-fit theoretical ones. Spectra were recorded on a chalcogenide As50Se50 amorphous film deposited onto float glass. The back side of the substrate was grounded before ellipsometry and reflectance measurements to suppress spurious reflections from this interface. Sample model structure was then designed as a homogenous film sandwiched between two semi-infinitive media: ambient and substrate. In the case of transmittance, the effect of finite substrate was accounted for by equation (3.89). The refractive index of float glass was determined formerly by spectroscopic ellipsometry carried out on a naked substrate considering it as a bulk material (see previous section). Optical constants of the chalcogenide film were parameterized by a Tauc–Lorentz formula (that is, a combination of the Lorentz oscillator and Tauc absorption edge models) that is the appropriate parameterization for amorphous semiconductors. The imaginary part of electric permittivity \(\varepsilon_{r}^{\prime\prime}\) at a photon energy E is then expressed as
$$\begin{aligned}\displaystyle&\displaystyle\varepsilon_{\mathrm{r}}^{\prime\prime}=\frac{AE_{0}C\left(E-E_{\mathrm{g}}\right)^{2}}{\left(E^{2}-E_{0}^{2}\right)^{2}+CE^{2}}\frac{1}{E}&\displaystyle\quad&\displaystyle E> E_{\mathrm{g}}\\ \displaystyle&\displaystyle\varepsilon_{\mathrm{r}}^{\prime\prime}=0&\displaystyle\quad&\displaystyle E<E_{\mathrm{g}}\end{aligned}$$
and the real part ε r can be calculated with the help of Kramers–Kronig relations ((3.9a) and (3.9b)) or by an analytic solution [3.86, 3.87]. Parameters of the Tauc–Lorentz formula together with the film thickness were free parameters adjusted by the fitting procedure. Figure 3.29 represents the determined spectral dependence of the relative permittivity of As50Se50 with adjusted values of searched parameters.
Fig. 3.29

Determined electrical permittivity spectra of As50Se50 film with thickness of 1085.5 ± 1.1 nm. Adjusted parameters of Tauc–Lorentz parameterization are \(A={\mathrm{155.16}}\pm{\mathrm{1.39}}\,{\mathrm{eV}}\); \(E_{0}={\mathrm{4.55}}\pm{\mathrm{0.03}}\,{\mathrm{eV}}\); \(C=6.09\pm{\mathrm{0.09}}\,{\mathrm{eV}}\); \(E_{\mathrm{g}}={\mathrm{2.351}}\pm{\mathrm{0.001}}\,{\mathrm{eV}}\); \(\varepsilon_{\infty}=1.05\pm 0.01\)

Of course, thin films usually have various defects and nonidealities, for example nonuniform thickness, surface roughness, internal nonuniformities, refractive index gradient, and so on. Ellipsometry is a sensitive method capable of detecting most of the mentioned defects and, moreover, theoretical approaches have been developed to account for them and analyze them. Monographs focused on spectroscopic ellipsometry [3.75, 3.76, 3.77] and related articles treat these topics.

3.5 Optical Materials

3.5.1 Abbe Number or Constringence

In an optical medium, the Abbe number is defined as the inverse of its dispersive power; that is, it represents the relative importance of refraction and dispersion. There are two common definitions based on the use of different standard wavelengths. The Abbe number νd is defined by
$$\nu_{\mathrm{d}}=(n_{\mathrm{d}}-1)/(n_{\mathrm{F}}-n_{\mathrm{C}})\;,$$
(3.98)
where nF, nd and nC are the refractive indices of the medium at the Fraunhofer standard wavelengths corresponding to the helium d-line (λd = 587.6 nm, yellow), the hydrogen F-line (λF = 486.1 nm, blue) and the hydrogen C-line (λC = 656.3 nm, red) respectively. The Abbe number of a few glasses are listed in Table 3.10 . The Abbe number νe, on the other hand, is defined by
$$\nu_{\mathrm{e}}=(n_{\mathrm{e}}-1)/(n_{\mathrm{F}^{\prime}}-n_{\mathrm{C}^{\prime}})\;,$$
(3.99)
where ne, \(n_{\mathrm{F}^{\prime}}\) and \(n_{\mathrm{C}^{\prime}}\) are the refractive indices at wavelengths of the e-line (546.07 nm), the F-line (479.99 nm) and the C-line (643.85 nm) respectively.
Table 3.10

Abbe numbers for a few glasses. PC denotes polycarbonate, PMMA is polymethylmethacrylate, and PS represents polystyrene

Optical glass →

SF11

F2

BaK1

Crown glass

Fused silica

PC

PMMA

PS

νd → 

25.76

36.37

57.55

58.55

67.80

34

57

31

The relationship between the refractive index nd and the Abbe number νd is given by the Abbe diagram, where nd is plotted on the y-axis against its corresponding νd value for different glasses, usually with the Abbe numbers decreasing on the x-axis, as shown in Fig. 3.30.
Fig. 3.30

The Abbe diagram is a diagram in which the refractive indices nd of glasses are plotted against their Abbe numbers in a linear nd versus νd plot and, usually, with the Abbe number decreasing along the x-axis, rather than increasing. A last letter of F or K represents flint or crown glass. Other symbols are as follows: S, dense; L, light; LL, extra light; B, borosilicate; P, phosphate; LA or La, lanthanum; BA or Ba, barium. Examples: BK, dense flint; LF, light flint; LLF, extra light flint; SSK, extra dense crown; PK, phosphate crown; BAK, barium crown; LAF, lanthanum flint

3.5.2 Optical Materials

Optical materials are those crystalline solids and glasses that are commonly used in the construction of optical components, such as lenses, prisms, windows, mirrors and polarizers. In addition to having the required optical properties, such as a well-defined refractive index n with a known dependence on wavelength n = n ( λ )  and temperature, optical materials should also have various other desirable material properties, such as good homogeneity (including negligible macroscopic variations in the refractive index), negligible thermal expansion (small αL, thermal expansion coefficient), resistance to mechanical damage (resistance to scratching), and resistance to chemical degradation (chemical corrosion, staining). Also, in the case of optical glasses, it is desirable to have negligible air (or gas) bubbles incorporated into the glass structure during fabrication (during shaping for example), negligible stress-induced birefringence, negligible nonlinear properties (unless used specifically due to its nonlinearity), and negligible fluorescence. In all of these cases, negligible implies less than a tolerable quantity within the context of the particular optical application. For example, at a given wavelength and for many glass materials used in optics, n should not vary by more than 10−5, while 10−6 is required for certain optics applications in astronomy. In addition, they should have good or reliable manufacturability at an affordable cost. There are various useful optical materials that encompass not only single crystals (such as CaF2, MgF2, quartz, sapphire) but also a vast range of glasses (which are supercooled liquids with high viscosity, such as flint and crown glasses as well as fused silica). Higher refractive index materials have more refractive power and allow lens designs that need less curvature to focus light, and hence tend to give fewer aberrations. Flint glasses have a larger refractive index than crown glasses. On the other hand, crown glasses are chemically more stable, and can be produced more to specification. While most optical materials are used for their optical properties (such as in optical transmission), certain optical materials (auxiliary materials) are used in optical applications such as mirror substrates and optical spacers for their nonoptical properties, such as their negligible thermal expansion coefficients. Some optical properties of selected optical materials and their applications are listed in Table 3.11.
Table 3.11

The refractive indices, nd, and Abbe numbers, vd, (3.98 ) of selected optical materials (compiled from the websites of Oriel, Newport and Melles-Griot); nd at λd = 587.6 nm, αL is the linear thermal expansion coefficient

Glasses

Transmission

(typical, nm)

n d

υ d

Applications

Comment

Fused silica

175–2000

1.45846

67.8

Lenses, windows, prisms, interferometric FT-IR components. UV lithography

Synthetic. Has UV properties; transmittance and excellent thermal low αL. Resistant to scratching

SF 11, flint

380–2350

1.78472

25.76

Lenses, prisms

Flint glasses have vd < 50

LaSFN9, flint

420–2300

1.85025

32.17

Lenses, prisms

High refractive index. More lens power for less curvature

BK7, borosilicate crown

380–2100

1.51680

64.17

Visible and near-IR optics. Lenses, windows, prisms, interferometric components

All around excellent optical lens material. Not recommended for temperature-sensitive applications

BaK1, barium crown

380–2100

1.57250

57.55

Visible and near-IR optics. Lenses, windows, prisms, interferometric components

All around excellent optical lens material. Not recommended for temperature-sensitive applications

Optical crown

380–2100

1.52288

58.5

Lenses, windows, prisms, interferometric components

Lower quality than BK7

Pyrex, borosilicate glass

 

1.43385

66

Mirrors

Low thermal expansion

Crystals

CaF2 crystal

170–7000

1.43385

94.96

Lenses, windows for UV optics, especially for excimer laser optics

Sensitive to thermal shock

MgF2 crystal

150–7000

n0 = 1.37774

ne = 1.38956

 

Lenses, windows, polarizers, UV transmittance

Positive birefringent crystal. Resistant to thermal and mechanical shock

Quartz, SiO2 crystal

150–2500

n0 = 1.54431

ne = 1.55343

 

UV optics. Wave plates. Polarizers

Positive uniaxial birefringent crystal

Sapphire, Al2O3 crystal

150–6000

1.7708 (546.1 nm)

 

UV–far-IR windows, high power laser optics

High surface hardness, scratch resistant. Chemically inert

Auxiliary optical materials

ULE

SiO2-TiO2glass

   

Optical spacers

Very small thermal expansion

Zerodur, glass ceramic composite

 

1.5424

56–66

Mirror substrates. Not suitable for transmission optics due to internal scattering

Ultra-low αL. Fine mixture of glass and ceramic crystals (very small size)

3.5.3 Optical Glasses

Optical glasses are a range of noncrystalline transparent solids used to fabricate various optical components, such as lenses, prisms, light pipes and windows. Most (but not all) optical glasses are either crown (K) types or flint (F) types. K-glasses are usually soda-lime-silica glasses, whereas flint glasses contain substantial lead oxide; hence F-glasses are denser and have higher refractive powers and dispersions. Barium glasses contain barium oxide instead of lead oxide and, like lead glasses, have high refractive indices, but lower dispersions. There are other high refractive index glasses, such as lanthanum- and rare earth-containing glasses. Optical glasses can also be made from various other glass formers, such as boron oxide, phosphorus oxide and germanium oxide. The Schott glass code or number is a special number designation (511604.253 for Schott glass K7) in which the first three numbers (511) represent the three decimal places in the refractive index (nd = 1.511), the next three numbers (604) represent the Abbe number (νd = 60.4), and the three numbers after the decimal (253) represent the density (\(\rho={\mathrm{2.53}}\,{\mathrm{g/cm^{3}}}\)). A different numbering system is also used, where a colon is used to separate nd and νd; for example, 517:645 for a particular borosilicate crown means nd = 1.517, νd = 64.5 (Sect. 3.5.1).

In the Schott glass coding system, optical glasses are represented by letters in which a last letter of K refers to crown, and F to flint. The first letters usually represent the most important component in the glass, such as P in the case of phosphate. The letters Kz (Kurz), L (leicht) and S (schwer) before K or F represent short, light and dense (heavy) respectively (from German). S after K or F means special. Examples include: BK, borosilicate crown; FK, fluor crown; PK, phosphate crown; PSK, dense phosphate crown; BaLK, light barium crown; BaK, barium crown; BaSK, dense barium crown; SSK, extra dense barium crown; ZnK, zinc crown; LaK, lanthanum crown, LaSK, dense lanthanum crown; KF, crown flint; SF, dense flint; SFS, special dense flint; BaF, barium flint; BaLF, barium light flint; BaSF, dense barium flint; LLF, extra light flint; LaF, lanthanum flint.

Notes

Acknowledgements

Authors are grateful to Stephen Karrer O’Leary (University of British Columbia, Okanagan Campus) for many insightful discussions on the subject and his invaluable help in the first edition chapter. This work was supported by the grants LM2015082 and CZ.1.05/4.1.00/11.0251 from the Ministry of Education, Youth and Sports of the Czech Republic.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Jan Mistrik
    • 1
  • Safa Kasap
    • 2
  • Harry E. Ruda
    • 3
  • Cyril Koughia
    • 4
  • Jai Singh
    • 5
  1. 1.Center of Materials and NanotechnologyUniversity of PardubicePardubiceCzech Republic
  2. 2.University of SaskatchewanSaskatoonCanada
  3. 3.Dept. of Materials Science and Engineering, and Edward Rogers Dept. of Electrical and Computer EngineeringUniversity of TorontoTorontoCanada
  4. 4.Dept. of Electrical and Computer EngineeringUniversity of SaskatchewanSaskatoonCanada
  5. 5.School of Engineerng and Information TechnologyCharles Darwin UniversityDarwinAustralia

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