Optical Properties of Electronic Materials: Fundamentals and Characterization

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.

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In indirect bandgap semiconductors, such as Si and Ge, the photon absorption for photon energies near E_g 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 (E_g − 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 (E_g + 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 f_BE ( 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 k_B 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 α_absorption and the plot of α^1∕2 versus hν has an intercept of ( E_g − hϑ ) .

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

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.

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 m_e 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).

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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 Q_a is known. Using the atomic density of crystalline silicon and four valence electrons per atom, Cody [3.50] estimated Q _a ²  = 0.9 Ų, which gives Q_a ≈  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 Ų, 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 Q_a 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 Q_a 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.

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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 E_g 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].

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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

Excitonic Absorption

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.

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

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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.

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 V_r 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

Normal Incidence Reflectivityand Kramers–Kronig Analyses

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.

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Oblique Incidence Reflectivity R_p and R_s

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 θ_i 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 θ_i 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 θ_i is set close to the principal value (or Brewster angle) where r_p ≈ 0 and r_s and r_p 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).

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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.

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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

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.

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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.

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Swanepoel’s Analysis of Optical Transmission Spectra

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 T_M ( λ )  and T_m ( 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.

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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.

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 E_g 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 E_g, 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.

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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 r_p and r_s 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.

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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.