Optical Properties

Grundmann, Marius

doi:10.1007/978-3-030-51569-0_9

Marius Grundmann¹¹

Part of the book series: Graduate Texts in Physics ((GTP))

4050 Accesses

Abstract

After introduction of the complex dielectric function, reflection, diffraction are briefly discussed. The focus lies on absorption mechanisms; several transition types (direct and indirect band-band transitions, impurity-related transitions, lattice absorption) are discussed including the effects of excitons, polaritons and high carrier density. Also the various effects of the presence of free carriers are given.

Do not Bodies and Light act mutually upon one another.

Sir I. Newton, 1704 [826]

Access provided by Autonomous University of Puebla. Download chapter PDF

1 Spectral Regions and Overview

The interaction of semiconductors with light is of decisive importance for photonic and optoelectronic devices as well as for the characterization of semiconductor properties. When light hits a semiconductor, reflection, transmission and absorption are considered, as for any dielectric material. The response of the semiconductor largely depends on the photon energy (or wavelength) of the light and various processes contribute to the dielectric function.

An overview of the electromagnetic spectrum in the optical range is given in Table 9.1. The energy and wavelength of a photon are related by^{Footnote 1} $E=h\nu =hc/\lambda $, i.e.

$$\begin{aligned} E\;\mathrm {[eV]}=\frac{1240}{\lambda \;\mathrm {[nm]}} \;. \end{aligned}$$

(9.1)

In the infrared regime, energy is often measured in wave numbers (cm$^{-1}$) for which the conversion 1 meV$=8.056$ cm$^{-1}$ holds.

Table 9.1 Spectral ranges with relevance to semiconductor optical properties

Full size table

2 Complex Dielectric Function

The dielectric function (DF) $\varvec{\epsilon }$ fulfills the relation between the displacement field $\mathbf{D}$, the polarization $\mathbf{P}$ and the electric field $\mathbf{E}$,

$$\begin{aligned} \mathbf{D}=\epsilon _0 \, \mathbf{E}+\mathbf{P}=\epsilon _0 \, \varvec{\epsilon } \, \mathbf{E} \;, \end{aligned}$$

(9.2)

and is generally a tensor of rank 2 since $\mathbf{D}$ and $\mathbf{E}$ must not be collinear. For cubic materials, the DF is isotropic and can be described with a (complex) scalar $\epsilon $. Less symmetric crystals are optically anisotropic and the DF must be used in tensor form. Also, external fields can induce optical anisotropy in an otherwise isotropic material as discussed in Sect. 15.2.2 for magnetic fields or has been observed for mechanical strain fields. The general form of the dielectric function tensor for various crystal symmetries is compiled in Table 9.2.

Table 9.2 General form of the tensor form of the dielectric function for the seven crystallographic systems

Full size table

In most cases in the following, $\epsilon $ will be used as scalar (isotropic case). The dielectric function is frequency dependent $\epsilon (\omega )$ due to the various oscillators playing a role and decreases (non-monotonically) from its static value (for $\omega =0$) to 1 for $\omega \rightarrow \infty $. Major influence on the DF stems from (optical) lattice vibrations (Sect. 9.5) and transitions within the electronic band structure (Sect. 9.6). In some cases also its $\mathbf{k}$-dependence is important, known as ’spatial dispersion’ (cmp. Sect. 9.7.8).

An optic axis in the transparency regime (all tensor elements of $\varvec{\epsilon }\in \mathbb {R}$) is the direction in which the speed of light or the index of refraction is independent of polarization. Uniaxial (biaxial) materials have one (two) of such axes. The anisotropy of the index of refraction and its polarization dependence must be taken into account when light propagation is considered in birefringent semiconductors, e.g. for Raman spectroscopy [827], unless the propagation is along an optic axis.

The dielectric function is generally complex and written as (scalar)

$$\begin{aligned} \epsilon = \epsilon ' + {\mathrm {i}}\, \epsilon '' = \epsilon _1 + {\mathrm {i}}\, \epsilon _2\;. \end{aligned}$$

(9.3)

The real ($\epsilon '$ or $\epsilon _1$) and imaginary ($\epsilon ''$ or $\epsilon _2$) part of the dielectric function are related to each other via the Kramers-Kronig relations (Appendix C).

The complex index of refraction $n^*$ is

$$\begin{aligned} n^* = \sqrt{\epsilon } = n_{\mathrm {r}} + {\mathrm {i}}\, \kappa \;. \end{aligned}$$

(9.4)

From ${n^{*2}}=\epsilon $ follows

$$\begin{aligned} \epsilon '= & {} n_{\mathrm {r}}^2 - \kappa ^2 \end{aligned}$$

(9.5)

$$\begin{aligned} \epsilon ''= & {} 2 \, n_{\mathrm {r}} \, \kappa \;. \end{aligned}$$

(9.6)

From $\epsilon \, \bar{\epsilon }=(n_{\mathrm {r}}^2+\kappa ^2)^2$ and (9.5) follows

$$\begin{aligned} n_{\mathrm {r}}^2= & {} \frac{\epsilon ' + \sqrt{\epsilon '\,^2 + \epsilon ''\,^2}}{2} \end{aligned}$$

(9.7)

$$\begin{aligned} \kappa= & {} \frac{\epsilon ''}{2 \, n_{\mathrm {r}}} \;. \end{aligned}$$

(9.8)

The real part of the complex index of refraction $n_{\mathrm {r}}$ is responsible for the dispersion, the imaginary part $\kappa $ is named extinction coefficient and is related to the absorption coefficient for a plane wave (damping of the intensity $\propto \mathbf{E}^2$) by

$$\begin{aligned} \alpha =2 \, \frac{\omega }{c} \, \kappa = \frac{4 \pi }{\lambda } \, \kappa = 2 \, k\, \kappa \;. \end{aligned}$$

(9.9)

Here, k and $\lambda $ denote the respective values in vacuum. Through the Kramers-Kronig relations (Appendix C), birefringence, i.e. the orientational dependence of the index of refraction, is thus automatically related to dichroism, i.e. the orientational dependence of the absorption coefficient.

As an example, in Fig. 9.1 the dielectric function of GaAs is shown in the vicinity of the band edge and above. Since GaAs is cubic, the dielectric function at each photon energy can represented by a single complex number. The tensor character of the dielectric function is demonstrated in Fig. 9.2a where the four independent tensor elements for (monoclinic) $\beta $-Ga$_2$O$_3$ are depicted [828]. The contributions of various dipole oscillators (strength and orientation) to the dielectric function can be analyzed from these data [829].

In the absorption regime, for biaxial crystals the two optic axes split into four singular optic axes [830] as visualized for $\beta $-Ga$_2$O$_3$ in Fig. 9.2b [831]. It should be noted that optical activity [832] is not considered in the following.

3 Reflection and Diffraction

From Maxwell’s equations and the boundary conditions at a planar interface between two media with different index of refraction for the components of the electric and magnetic fields the laws for reflection and diffraction are derived. We denote the index of refraction as n and also $n_{\mathrm {r}}$ in the following. The interface between two media with refractive indices $n_1$ and $n_2$ is depicted in Fig. 9.3. In the following we assume first that no absorption occurs.

Snell’s law [833] for the angle of diffraction is

$$\begin{aligned} n_1 \sin \phi =n_2 \sin \psi \;. \end{aligned}$$

(9.10)

When the wave enters the denser medium, it is diffracted towards the normal. If the wave propagates into the less-dense medium (reversely to the situation shown in Fig. 9.3), a diffracted wave occurs only up to a critical angle of incidence

$$\begin{aligned} \sin \phi _{\mathrm {TR}}=\frac{n_2}{n_1} \;. \end{aligned}$$

(9.11)

For larger angles of incidence, total internal reflection occurs and the wave remains in the denser medium. Thus, the angle in (9.11) is called the critical angle for total reflection. For GaAs and air the critical angle is rather small, $\phi _{\mathrm {TR}}=17.4^{\circ }$.

The reflectance depends on the polarization (Fresnel formulas [834]). The index ‘p’ (‘s’) denotes parallel polarized/TM (perpendicular polarized/TE) waves.

$$\begin{aligned} R_{\mathrm {p}}= & {} \left( \frac{\tan (\phi -\psi )}{\tan (\phi +\psi )} \right) ^2\end{aligned}$$

(9.12)

$$\begin{aligned} R_{\mathrm {s}}= & {} \left( \frac{\sin (\phi -\psi )}{\sin (\phi +\psi )} \right) ^2 \;. \end{aligned}$$

(9.13)

The situation for GaAs and air is shown for both polarization directions and unpolarized radiation in Fig. 9.4 for a wave going into and out of the GaAs.

When the reflected and the diffracted wave are perpendicular to each other, the reflectance of the p-polarized wave is zero. This angle is the Brewster angle $\phi _{\mathrm {B}}$,

$$\begin{aligned} \tan \phi _{\mathrm {B}}=\frac{n_2}{n_1} \;. \end{aligned}$$

(9.14)

If a wave has vertical incidence from vacuum on a medium with index of refraction $n_{\mathrm {r}}$, the reflectance is given (both polarizations are degenerate) as

$$\begin{aligned} R=\left( \frac{n_{\mathrm {r}}-1}{n_{\mathrm {r}}+1} \right) ^2 \;. \end{aligned}$$

(9.15)

For GaAs, the reflectance for vertical incidence is 29.2%.

4 Absorption

In the absorption process, energy is transferred from the electromagnetic field to the semiconductor. In the case of a linear absorption process, when the probability of light absorption is proportional to the incoming intensity, the decrease of intensity in the absorbing medium is exponential (Lambert–Beer’s law [835, 836]),^{Footnote 2}

$$\begin{aligned} I(x) = I(0) \, \exp (-\alpha \, x) \;. \end{aligned}$$

(9.16)

The quantity $\alpha $ is the absorption coefficient, its reverse the absorption depth.

The spectral dependence $\alpha (E)$, the absorption spectrum, contains the information of the possible absorption processes, their energy, momentum and angular momentum selection rules, and their oscillator strength.

In Fig. 9.5 a schematic absorption spectrum of a semiconductor is depicted. The transition of electrons from the valence to the conduction band begins at the band gap energy. The band gaps of Si, Ge, GaAs, InP, InAs, InSb are in the IR, those of AlAs, GaP, AlP, InN in the VIS, those of GaN and ZnO in the UV, MgO and AlN are in the deep UV. The Coulomb correlation of electrons and holes leads to the formation of excitons that leads to absorption below the band gap. The typical exction binding energy is in the range of 1–100 meV (see Fig. 9.19). Optical transitions from valence-band electrons into donors and from electrons on acceptors into the conduction band lead to band–impurity absorption. In the region from 10–100 meV the interaction with lattice vibrations (phonons) leads to absorption if the phonons are infrared active. Further in the FIR lie transitions from impurities to the closest band edge (donor to conduction and acceptor to valence band). A continuous background is due to free-carrier absorption.

If absorption is considered, the reflectance (9.15) needs to be modified. Using the complex index of refraction $n^*=n_{\mathrm {r}}+{\mathrm {i}}\kappa $, it is given as

$$\begin{aligned} R=\left| \frac{n^*-1}{n^*+1} \right| ^2=\frac{(n_{\mathrm {r}}-1)^2+\kappa ^2}{(n_{\mathrm {r}}+1)^2+\kappa ^2} \;. \end{aligned}$$

(9.17)

5 Dielectric Function due to Optical Phonons

In this section, the dielectric function around the resonance energy of optical phonons is developed. Adjacent atoms oscillate with opposite phase in an optical phonon. If the bond has (partial) ionic character, this leads to a time-dependent polarization and subsequently to a macroscopic electric field. This additional field will influence the phonon frequencies obtained from a purely mechanical approach. We consider in the following the case $\mathbf{k} \approx 0$. The phonon frequency for TO and LO vibrations is given by

$$\begin{aligned} \omega _0=\sqrt{\frac{2 \, C}{M_{\mathrm {r}}}} \;, \end{aligned}$$

(9.18)

where $M_{\mathrm {r}}$ is the reduced mass of the two different atoms (cf. Sect. 5.2.2). $\mathbf{u}$ is the relative displacement $\mathbf{u}_1-\mathbf{u}_2$ of the two atoms in a diatomic base. When the interaction with the electric field $\mathbf{E}$ (which will be calculated self-consistently in the following) is considered, the Hamiltonian for the long-wavelength limit is given by [838]:

$$\begin{aligned} \hat{H}(\mathbf{p},\mathbf{u})=\frac{1}{2} \left( \frac{1}{M_{\mathrm {r}}} \,\mathbf{p}^2 + b_{11} \, \mathbf{u}^2 + 2b_{12} \, \mathbf{u} \cdot \mathbf{E}+b_{22}\, \mathbf{E}^2 \right) \;. \end{aligned}$$

(9.19)

The first term is the kinetic energy ($\mathbf{p}$ stands for the momentum of the relative motion of the atoms 1 and 2 in the base, $\mathbf{p}=M_{\mathrm {r}}\dot{\mathbf{u}}$), the second the potential energy, the third the dipole interaction and the fourth the electric-field energy. The equation of motion for a plane wave $\mathbf{u}=\mathbf{u}_0\exp [-{\mathrm {i}}(\omega t-\mathbf{k}\cdot \mathbf{r})]$ ($\ddot{\mathbf{u}}=-\omega ^2 \mathbf{u}$) yields

$$\begin{aligned} M_{\mathrm {r}} \, \omega ^2 \, \mathbf{u}=b_{11} \, \mathbf{u}+b_{12} \, \mathbf{E} \;. \end{aligned}$$

(9.20)

Thus, the electric field is

$$\begin{aligned} \mathbf{E}=(\omega ^2-\omega _{\mathrm {TO}}^2) \, \frac{M_{\mathrm {r}}}{b_{12}} \, \mathbf{u}\;. \end{aligned}$$

(9.21)

Here, the substitution $\omega _{\mathrm {TO}}^2=b_{11}/M_{\mathrm {r}}$ was introduced that is consistent with (9.18) and $b_{11}=2C$. $\omega _{\mathrm {TO}}$ represents the mechanical oscillation frequency of the atoms undisturbed by any electromagnetic effects. Already now the important point is visible. If $\omega $ approaches $\omega _{\mathrm {TO}}$, the system plus electric field oscillates with the frequency it has without an electric field. Therefore the electric field must be zero. Since the polarization $\mathbf{P}=(\epsilon -1)\epsilon _0 \mathbf{E}$ is finite, the dielectric constant $\epsilon $ thus diverges.

The polarization is

$$\begin{aligned} \mathbf{P}=-\nabla _{\mathbf{E}} \hat{H}=-\left( b_{12}\, \mathbf{u}+b_{22} \, \mathbf{E} \right) \;. \end{aligned}$$

(9.22)

The displacement field is

$$\begin{aligned} \mathbf{D}=\epsilon _0\,\mathbf{E}+\mathbf{P}=\epsilon _0\,\mathbf{E}-\left( b_{22}-\frac{b_{12}^2/M_{\mathrm {r}}}{\omega _{\mathrm {TO}}^2-\omega ^2}\right) \,\mathbf{E}=\epsilon _0\,\epsilon (\omega )\,\mathbf{E} \;. \end{aligned}$$

(9.23)

Therefore, the dielectric function is

$$\begin{aligned} \epsilon (\omega )= \epsilon (\infty )+\frac{\epsilon (0)-\epsilon (\infty )}{1-(\omega /\omega _{\mathrm {TO}})^2} \;. \end{aligned}$$

(9.24)

Here, $\epsilon (\infty )=1-b_{22}/\epsilon _0$ is the high-frequency dielectric constant and $\epsilon (0)=\epsilon (\infty )+b_{12}^2/(b_{11}\epsilon _0)$ the static dielectric constant. The relation (9.24) is shown in Fig. 9.6.

From the Maxwell equation $\nabla \cdot \mathbf{D}=0$ for zero free charge we obtain the relation

$$\begin{aligned} \epsilon _0 \, \epsilon (\omega ) \, \nabla \cdot \mathbf{E}=0 \;. \end{aligned}$$

(9.25)

Thus, either $\epsilon (\omega )=0$ or $\nabla \cdot \mathbf{E}=0$, i.e. $\mathbf{u}$ is perpendicular to $\mathbf{k}$. In the latter case we have a TO phonon and, neglecting retardation effects, using $\nabla \times \mathbf{E}=0$ we find $\mathbf{E}=0$ and therefore $\omega =\omega _{\mathrm {TO}}$, justifying our notation. In the case of $\epsilon (\omega )=0$, we call the related frequency $\omega _{\mathrm {LO}}$ and find the so-called Lyddane–Sachs–Teller (LST) relation [839]

$$\begin{aligned} \frac{\omega _{\mathrm {LO}}^2}{\omega _{\mathrm {TO}}^2}= \frac{\epsilon (0)}{\epsilon (\infty )} \;. \end{aligned}$$

(9.26)

This relation holds reasonably well for optically isotropic, heteropolar materials with two atoms in the basis, such as NaI and also GaAs. Since at high frequencies, i.e. $\omega \gg \omega _{\mathrm {TO}}$, only the individual atoms can be polarized, while for low frequencies the atoms can also be polarized against each other, $\epsilon (0)>\epsilon (\infty )$ and therefore also $\omega _{\mathrm {LO}}>\omega _{\mathrm {TO}}$. For GaAs, the quotient of the two phonon energies is 1.07. Using the LST relation (9.26), we can write for the dielectric function

$$\begin{aligned} \epsilon (\omega )= \epsilon (\infty ) \left( \frac{\omega _{\mathrm {LO}}^2-\omega ^2}{\omega _{\mathrm {TO}}^2-\omega ^2}\right) \;. \end{aligned}$$

(9.27)

The (long-wavelength) TO-phonon does not create a long-range electric field. Using $\nabla \cdot \mathbf{D}=0$ and (9.23) and looking at the longitudinal fields, we have

$$\begin{aligned} \epsilon _0 \, \mathbf{E}=b_{12} \, \mathbf{u} + b_{22} \, \mathbf{E} \;. \end{aligned}$$

(9.28)

This can be rewritten as

$$\begin{aligned} \mathbf{E}=- \, \omega _{\mathrm {LO}} \, \sqrt{\frac{M_{\mathrm {r}}}{\epsilon _0} }\, \sqrt{ \frac{1}{\epsilon (\infty )} - \frac{1}{\epsilon (0)} } \,\, \mathbf{u}\propto -\mathbf{u} \;. \end{aligned}$$

(9.29)

The (long-wavelength) LO-phonon thus creates a long-range electric field acting against the ion displacement and represents an additional restoring force; this is consistent with the fact that $\omega _{\mathrm {LO}}>\omega _{\mathrm {TO}}$.

6 Electron–Photon Interaction

The absorption process within the band structure is quantum mechanically described by the coupling of electrons and photons. The process is described with time-dependent perturbation theory. If $\varvec{H_{\mathrm {em}}}$ is the perturbation operator (electromagnetic field), the transition probability per time $w_{\mathrm {fi}}$ for electrons from (unperturbed) state ‘i’ (initial) to state ‘f’ (final) is given (with certain approximations) by Fermi’s golden rule

$$\begin{aligned} w_{\mathrm {fi}}(\hbar \omega )=\frac{2 \pi }{\hbar } \, \left| H_{\mathrm {fi}}' \right| ^2 \, \delta (E_{\mathrm {f}}-E_{\mathrm {i}}-\hbar \omega ) \;, \end{aligned}$$

(9.30)

where $\hbar \omega $ is the photon energy, $E_{\mathrm {i}}$ ($E_{\mathrm {f}}$) is the energy of the initial (final) state. $H_{\mathrm {fi}}'$ is the matrix element

$$\begin{aligned} H_{\mathrm {fi}}'=\left\langle \Psi _{\mathrm {f}} \left| \varvec{H}' \right| \Psi _{\mathrm {i}} \right\rangle \;, \end{aligned}$$

(9.31)

where $\Psi _{\mathrm {i}}$ ($\Psi _{\mathrm {f}}$) are the wavefunctions of the unperturbed initial (final) state.

$\mathbf{A}$ is the vector potential for the electromagnetic field, i.e. $\mathbf{E}=-\dot{\mathbf{A}}$, $\mu \mathbf{H}=\nabla \times \mathbf{A}$, $\nabla \cdot \mathbf{A}=0$ (Coulomb gauge). The Hamiltonian of an electron in the electromagnetic field is

$$\begin{aligned} \varvec{H}=\frac{1}{2m} \, \left( \hbar \mathbf{k} -q \mathbf{A} \right) ^2 \;. \end{aligned}$$

(9.32)

When terms in $\mathbf{A}^2$ are neglected (i.e. two-photon processes), the perturbation Hamiltonian is thus

$$\begin{aligned} \varvec{H_{\mathrm {em}}}=-\frac{q}{m} \, \mathbf{A \, p} = \frac{{\mathrm {i}}\, \hbar q }{m} \, \mathbf{A} \cdot \nabla \approx q \, \mathbf{r} \cdot \mathbf{E} \;. \end{aligned}$$

(9.33)

The latter approximation is valid for small wavevectors of the electromagnetic wave and is termed the electric dipole approximation.

In order to calculate the dielectric function of the semiconductor from its band structure we assume that $\mathbf{A}$ is weak and we can apply (9.30). The transition probability R for the photon absorption rate at photon energy $\hbar \omega $ is then given by^{Footnote 3}

$$\begin{aligned} R(\hbar \omega )=\frac{2 \pi }{\hbar } \int \limits _{\mathbf{k}_{\mathrm {c}}}\int \limits _{\mathbf{k}_{\mathrm {v}}}\,\left| \langle c |\varvec{H_{\mathrm {em}}} | v\rangle \right| ^2 \delta \left( E_{\mathrm {c}}(\mathbf{k}_{\mathrm {c}}) -E_{\mathrm {v}}(\mathbf{k}_{\mathrm {v}}) - \hbar \omega \right) \mathrm {d}^3\mathbf{k}_{\mathrm {c}} \, \mathrm {d}^3\mathbf{k}_{\mathrm {v}} \;, \end{aligned}$$

(9.34)

with the Bloch functions $|c \rangle $ and $|v\rangle $ of the conduction and valence band, respectively, as given in (6.40b).

The vector potential is written as $\mathbf{A}=A \hat{\mathbf{e}}$ with a unit vector $\hat{\mathbf{e}}$ parallel to $\mathbf{A}$. The amplitude is connected to the electric-field amplitude E via

$$\begin{aligned} A = -\frac{E}{2 \, \omega } \, \left[ \exp \left( {\mathrm {i}}(\mathbf{qr}-\omega t)\right) + \exp \left( -{\mathrm {i}}(\mathbf{qr}-\omega t) \right) \right] \;. \end{aligned}$$

(9.35)

In the electric-dipole approximation the momentum conservation $\mathbf{q}+\mathbf{k}_{\mathrm {v}}=\mathbf{k}_{\mathrm {c}}$, $\mathbf{q}$ being the momentum of the light wave is approximated by $\mathbf{k}_{\mathrm {v}}=\mathbf{k}_{\mathrm {c}}$. The matrix element is then given by

$$\begin{aligned} \left| \langle c |\varvec{H_{\mathrm {em}}} | v \rangle \right| ^2 = \frac{e^2\,|A|^2}{m^2} \, \left| \langle c |\hat{\mathbf{e}} \cdot \mathbf{ p} | v \rangle \right| ^2 \;, \end{aligned}$$

(9.36)

with

$$\begin{aligned} \langle c \left| \hat{\mathbf{e}} \cdot \mathbf{ p} | v \rangle \right| ^2 = \frac{1}{3} \, |\mathbf{p}_{\mathrm {cv}}|^2 = M_{\mathrm {b}}^2 \;, \end{aligned}$$

(9.37)

and the momentum matrix element $\mathbf{p}_{\mathrm {cv}}$ given in (6.39). A $\mathbf{k}$-independent matrix element $|\mathbf{p}_{\mathrm {cv}}|^2$ is often used as an approximation. In Fig. 9.7 the matrix elements for valence to conduction band transitions in GaN are shown as a function of $\mathbf{k}$.

In terms of the electric-field amplitude $E(\omega )$ the transition probability is

$$\begin{aligned} R(\hbar \omega ) = \frac{2 \pi }{\hbar } \left( \frac{e}{m \, \omega } \right) ^2\left| \frac{E(\omega )}{2}\right| ^2 |\mathbf{p}_{\mathrm {cv}}|^2\int \limits _{\mathbf{k}}{ \delta \left( E_{\mathrm {c}}(\mathbf{k}) - E_{\mathrm {v}}(\mathbf{k}) - \hbar \omega \right) \, \mathrm {d}^3\mathbf{k}} \;. \end{aligned}$$

(9.38)

If the integration over $\mathbf{k}$ is restricted to those values allowed in unit volume, the power that is lost from the field in unit volume is given by $R\,\hbar \omega $, leaving a 1/E factor. The dielectric function $\epsilon =\epsilon ' + {\mathrm {i}}\epsilon ''$ is given by

$$\begin{aligned} \epsilon ''= & {} \frac{1}{4 \pi \epsilon _0} \left( \frac{2 \pi \, e}{m \, \omega } \right) ^2|\mathbf{p}_{\mathrm {cv}}|^2 \int \limits _{\mathbf{k}}{ \delta \left( E_{\mathrm {c}}(\mathbf{k}) - E_{\mathrm {v}}(\mathbf{k}) -\hbar \omega \right) \, \mathrm {d}^3\mathbf{k}} \end{aligned}$$

(9.39a)

$$\begin{aligned} \epsilon '= & {} 1+ \int \limits _{\mathbf{k}}{ \frac{e^2}{\epsilon _0 \, m \,\omega _{\mathrm {cv}}^2} \, \frac{2 \, |\mathbf{p}_{\mathrm {cv}}|^2}{m \, \hbar \omega _{\mathrm {cv}}} \, \frac{1}{1-\omega ^2/ \omega _{\mathrm {cv}}^2} \,\mathrm {d}^3\mathbf{k}} \;, \end{aligned}$$

(9.39b)

with $\hbar \omega _{\mathrm {cv}}=E_{\mathrm {c}}(\mathbf{k})-E_{\mathrm {v}}(\mathbf{k})$. Equation (9.39b) has been obtained via the Kramers–Kronig relations^{Footnote 4} (see Appendix C).

Comparison with (D.7) yields that the oscillator strength of the band–band absorption is given by

$$\begin{aligned} f = \frac{e^2}{\epsilon _0\,m\,\omega _{\mathrm {cv}}^2}\,\frac{2\,|\mathbf{p}_{\mathrm {cv}}|^2}{m\,\hbar \omega _{\mathrm {cv}}} = \frac{e^2}{\epsilon _0\,m\,\omega _{\mathrm {cv}}^2} \, N_{\mathrm {cv}}\;, \end{aligned}$$

(9.40)

with the classical ’number’ of oscillators with the frequency $\omega _{\mathrm {cv}}$,

$$\begin{aligned} N_{\mathrm {cv}}=\frac{2 \, |\mathbf{p}_{\mathrm {cv}}|^2}{m \, \hbar \omega _{\mathrm {cv}}} \;. \end{aligned}$$

(9.41)

7 Band–Band Transitions

7.1 Joint Density of States

The strength of an allowed optical transitions between valence and conduction bands is proportional to the joint density of states (JDOS) $D_{\mathrm {j}}(E_{\mathrm {cv}})$ (cf. (6.63), (6.64) and (9.39a))

$$\begin{aligned} D_{\mathrm {j}}(E_{\mathrm {cv}})= 2 \int \limits _{S(\tilde{E})}{\frac{\mathrm {d}^2 S}{(2 \pi /L)^3} \; \frac{1}{|\nabla _{\mathbf{k}}E_{\mathrm {cv}}|}} \;, \end{aligned}$$

(9.42)

where $E_{\mathrm {cv}}$ is an abbreviation for $E_{\mathrm {c}}(\mathbf{k})-E_{\mathrm {v}}(\mathbf{k})$ and $\mathrm {d}^2S$ is a surface element of the constant energy surface with $\tilde{E}=E_{\mathrm {cv}}$. The spin is assumed to generate doubly degenerate bands and accounts for the pre-factor 2. Singularities of the JDOS (van-Hove singularities or critical points) appear where $\nabla _{\mathbf{k}}E_{\mathrm {cv}}$ vanishes. This occurs when the gradient for both bands is zero or when both bands are parallel. The latter generates particularly large JDOS because the condition is valid at many points in $\mathbf{k}$-space.

Generally, the (three-dimensional) energy dispersion $E(\mathbf{k})$ around a three-dimensional critical point (here developed at $\mathbf{k}=0$) can be written as

$$\begin{aligned} E(\mathbf{k})=E(0) + \frac{\hbar ^2 k_{\mathrm {x}}^2}{2 m_{\mathrm {x}}} + \frac{\hbar ^2 k_{\mathrm {y}}^2}{2 m_{\mathrm {y}}} + \frac{\hbar ^2 k_{\mathrm {z}}^2}{2 m_{\mathrm {z}}} \;. \end{aligned}$$

(9.43)

The singularities are classified as $M_0$, $M_1$, $M_2$ and $M_3$ with the index being the number of masses $m_i$ in (9.43) that are negative. $M_0$ ($M_3$) describes a minimum (maximum) of the band separation. $M_1$ and $M_2$ are saddle points. For a two-dimensional $\mathbf{k}$-space there exist $M_0$, $M_1$ and $M_2$ points (minimum, saddle point and maximum, respectively). For a one-dimensional $\mathbf{k}$-space, there exist $M_0$ and $M_1$ points (minimum and maximum, respectively). The functional dependence of the JDOS at the critical points is summarized in Table 9.3. The resulting shape of the dielectric function is visualized in Fig. 9.8.

Table 9.3 Functional dependence of the joint density of states for critical points in 3, 2 and 1 dimensions. $E_0$ denotes the energy (band separation) at the critical point, C stands for a constant value. The type of critical point is given (min.: minimum, saddle: saddle point, max.: maximum)

Full size table

7.2 Direct Transitions

Transitions between states at the band edges at the $\Gamma $ point are possible (Fig. 9.9). The $\mathbf{k}$ conservation requires (almost) vertical transitions in the $E(\mathbf{k})$ diagram because the length of the light $\mathbf{k}$ vector, $k=2\pi /\lambda $, is much smaller than the size of the Brillouin zone $|k|\le \pi /a_0$. The ratio of the lengths of the $\mathbf{k}$ vectors is of the order $a_0/\lambda $ and typically about 10$^{-3}$ for NIR wavelengths.

For isotropic parabolic bands the band-band transition energy versus wavevector relation is

$$\begin{aligned} E_{\mathrm {cv}}(k) = E_{\mathrm {g}} + \frac{\hbar ^2}{2} \, \left( \frac{1}{m_{\mathrm {e}}^*} + \frac{1}{m_{\mathrm {h}}^*} \right) k^2 \;. \end{aligned}$$

(9.44)

When the energy dependence of the matrix element is neglected, the absorption coefficient is determined by the corresponding square-root joint density of states ($M_0$ critical point):

$$\begin{aligned} \alpha (E) \propto \frac{\sqrt{E-E_{\mathrm {g}}}}{E} \approx \, \propto \sqrt{E-E_{\mathrm {g}}} \;. \end{aligned}$$

(9.45)

The approximation is valid if the considered energy interval, e.g. around a band edge, is small.

Absorption spectra of (In$_x$Ga$_{1-x}$)$_2$O$_3$ alloy thin films at room temperature are shown in Fig. 9.10a. The $\alpha ^2$ versus photon energy so-called Tauc plot shows a linear dependence with broadening and additional states at the band edge due to disorder effects. The extrapolation of the linear part yields the absorption edge (Fig. 9.10b).

Absorption spectra of GaAs are shown in Fig. 9.11a for photon energies close to the band gap at various temperatures. The rapid increase, typical for direct semiconductors, is obvious. In particular at low temperatures, however, the absorption lineshape close to the band gap is dominated by an excitonic feature, discussed in Sect. 9.7.6.

Due to the increasing density of states, the absorption increases with the photon energy (Fig. 9.11c). At 1.85 eV there is a step in the absorption spectrum of GaAs due to the beginning of the contribution of transitions between the s-o hole band and the conduction band (see $E_0+\Delta _0$ transition in Fig. 9.1b). When bands run in parallel, i.e. with the same separation, in the $E(\mathbf{k})$ diagram, the absorption processes accumulate at the same transition energy. In this way peaks at higher energy in the complex part of the dielectric function and in the absorption spectrum due to the $E_1$ or $E_0'$ transitions originate as shown in Fig. 9.1.

The selection rules for transitions from valence to conduction band must take into account the angular momentum and spin states of the wavefunctions. The optical transitions for circularly polarized light are shown in Fig. 9.12a, fulfilling the selection rule $\Delta m_j=\pm 1$. A lifting of the energetic degeneracies of these states occurs, e.g. by magnetic fields (cmp. Fig.15.12) or spatial confinement (cmp. Fig. 12.30). For two-photon absorption (Chap.9.7.14), the selection rule is $\Delta m_j=\pm 2$ as shown in Fig. 9.12b [844].

We note that in some materials the direct transition between certain bands is forbidden. An example is SnO$_2$ where the direct transition from the topmost valence band into the lowest conduction band (at $\Gamma $) is forbidden (cmp. Fig. 9.48). If the matrix element increases linearly with $E-E_{\mathrm {g}}$, the absorption coefficient varies like

$$\begin{aligned} \alpha (E) \propto (E-E_{\mathrm {g}})^{3/2} \;. \end{aligned}$$

(9.46)

7.3 Indirect Transitions

In an indirect band structure the missing $\mathbf{k}$ difference (across the Brillouin zone) between valence- and conduction-band state needs to be provided by a second quantum. A phonon can provide the necessary momentum and additionally contributes a small amount of energy $\hbar \omega _{\mathrm {ph}}$. There are several steps in the absorption spectrum due to various involved phonons (or combinations of them). At low temperature ($T=1.6$ K, Fig. 9.13) phonons can only be generated and the absorption starts at energies above the band gap. At higher temperatures (typically above 40 K [845], Fig. 9.13), acoustical phonons assisting the optical absorption transition can also be absorbed from the crystal; in this case due to energy conservation the absorption starts already at an energy $E_{\mathrm {g}}-\hbar \omega _{\mathrm {ph}}$ below the band gap. At even higher temperatures ($>200$ K, Fig. 9.13), also optical phonons can be absorbed.

The perturbation calculation yields an absorption coefficient with a quadratic dependence on energy (9.47a) [846]. Essentially, for the absorption into a specific (empty) conduction band state (with square-root density of states) various initial (filled) valence band states (also with square-root density of states) are possible, making the probability depend on the product of the DOS and thus on the energy to the first power. Integrating over all energy states with energy separation $E\pm \hbar \omega _{\mathrm {ph}}$, yields an $E^2$-dependence.^{Footnote 5} Considering the temperature dependent population of the phonon density of states (Bose statistics, (E.3)) the absorption coefficients for transitions with phonon emission ($\alpha _{\mathrm {e}}$) and phonon absorption ($\alpha _{\mathrm {a}}$) are:

$$\begin{aligned} \alpha _{\mathrm {e}}(E)\propto & {} \frac{(E - (E_{\mathrm {g}} + \hbar \omega _{\mathrm {ph}}))^2}{1-\exp (-\hbar \omega _{\mathrm {ph}}/kT)} \end{aligned}$$

(9.47a)

$$\begin{aligned} \alpha _{\mathrm {a}}(E)\propto & {} \frac{(E - (E_{\mathrm {g}} - \hbar \omega _{\mathrm {ph}}))^2}{\exp (\hbar \omega _{\mathrm {ph}}/kT)-1} \;. \end{aligned}$$

(9.47b)

The two-particle process is less probable than the direct absorption that only involves one photon. The strength of indirect absorption close to the band gap is about 10$^{-3}$ smaller than for the direct transition.

An 11-parameter formula based on terms like (9.47a) can describe the room temperature absorption spectrum of silicon in the visible with a precision of a few percent [847].

The absorption spectra close to the absorption edge are shown for GaP (Fig. 9.13) and Si (Fig. 9.14a). According to (9.47a), the plot of $\sqrt{\alpha }$ versus energy (Macfarlane–Roberts plot [848]) yields a straight line beyond the spectral region of phonon effects. The complicated form close to the (indirect) gap energy is due to the contribution of different phonons. The phonon energies found to contribute to the silicon absorption edge [849] agree with the TA and TO energy at the X minimum [850] (Fig. 9.14b). Also multiple phonons can contribute (Fig. 9.13). The momentum conservation can also be achieved by impurity scattering or electron-electron scattering [851].

We note also that the indirect semiconductors have an optical transition between $\Gamma $ valence- and conduction-band states. However, this transition is at higher energies than the fundamental band gap, e.g. for Si ($E_{\mathrm {g}}=1.12$ eV) at 3.4 eV (see Fig. 6.9a). In Fig. 9.15, the absorption scheme for indirect and direct absorption processes starting with an electron at the top of the valence band is shown together with an experimental absorption spectrum for Ge with the direct transition ($\Gamma _8\rightarrow \Gamma _7$) at 0.89 eV, 0.136 eV above the fundamental band gap.

In Fig. 9.16, the absorption edge of BaTiO$_3$ is shown. An indirect transition with an increase of (weak) absorption $\propto E^2$ and an indirect gap of $E_{\mathrm {i}}=2.66$ eV and a direct transition with an increase of (strong) absorption $\propto E^{1/2}$ and a direct gap of $E_{\mathrm {d}}=3.05$ eV are observed. These transitions could be due to holes at the M (indirect gap) and $\Gamma $ (direct gap) points (cf. Sect. 6.3.11), respectively.

7.4 Urbach Tail

Instead of the ideal $(E-E_{\mathrm {g}})^{1/2}$ dependence of the direct band-edge absorption, often an exponential tail is observed (see Fig. 9.17). This tail is called the Urbach tail [853] and follows the functional dependence (for $E<E_{\mathrm {g}}$)

$$\begin{aligned} \alpha (E) \propto \exp \left( \frac{E-E_{\mathrm {g}}}{E_0}\right) \;, \end{aligned}$$

(9.48)

where $E_0$ is the characteristic width of the absorption edge, the so-called Urbach parameter.

The Urbach tail is attributed to transitions between band tails below the band edges. Such tails can originate from disorder of the perfect crystal, e.g. from defects or doping, and the fluctuation of electronic energy bands due to lattice vibrations. The temperature dependence of the Urbach parameter $E_0$ is thus related to that of the band gap as discussed in [854, 855].

7.5 Amorphous Semiconductors

The sharp features in the dielectric function due to critical points in the band structure of crystalline semiconductors are washed out in amorphous material. As an example the spectra of the imaginary part of the dielectric function for crystalline (trigonal) and amorphous selenium are shown in Fig. 9.18.

7.6 Excitons

An electron in the conduction band and a hole in the valence band form a hydrogen-like state due to the mutual Coulomb interaction. Such a state is called an exciton. The center-of-mass motion is separated and has a dispersion $E=\frac{\hbar ^2}{2M} \mathbf{K}^2$, where $M=m_{\mathrm {e}}+m_{\mathrm {h}}$ is the total mass and $\hbar \mathbf{K}$ is the center-of-mass momentum

$$\begin{aligned} \mathbf{K}=\mathbf{k}_{\mathrm {e}}+\mathbf{k}_{\mathrm {h}} \;. \end{aligned}$$

(9.49)

The relative motion yields hydrogen-like quantized states $E_n\propto n^{-2}$($n\!\ge \!1$):

$$\begin{aligned} E_{\mathrm {X}}^n=-\frac{m_{\mathrm {r}}^*}{m_0} \, \frac{1}{\epsilon _{\mathrm {r}}^2} \, \frac{m_0 e^4}{2(4 \pi \epsilon _0\hbar )^2} \, \frac{1}{n^2} \;, \end{aligned}$$

(9.50)

where $m_{\mathrm {r}}^*$ denotes the reduced effective mass $m_{\mathrm {r}}^{*-1}=m_{\mathrm {e}}^{*-1}+m_{\mathrm {h}}^{*-1}$. The third factor is the atomic Rydberg energy (13.6 eV). The exciton binding energy $E_{\mathrm {X}}^{\mathrm {b}}=-E_{\mathrm {X}}^1$ is scaled by $(m^*/m_0) \, \epsilon _{\mathrm {r}}^{-2} \approx 10^{-3}$. A more detailed theory of excitons beyond the simple hydrogen model presented here, taking into account the valence-band structure, can be found in [858] for direct and [859] for indirect cubic and in [860] for wurtzite semiconductors. The exciton binding energies for various semiconductors are listed in Table 9.4 and shown in Fig. 9.19a versus the band gap.

Table 9.4 Exciton ($E_{\mathrm {X}}^{\mathrm {b}}$) and biexciton ($E_{\mathrm {XX}}^{\mathrm {b}}$, see Sect. 9.7.10) binding energies in various bulk semiconductors. Values for 10 nm GaAs/15 nm Al$_{0.3}$Ga$_{0.7}$As quantum well (QW) are taken from [861]

Full size table

The radius of the exciton is

$$\begin{aligned} r_{\mathrm {X}}^n=n^2 \, \frac{m_0}{m_{\mathrm {r}}^*} \, \epsilon _{\mathrm {r}}\, a_{\mathrm {B}} \;, \end{aligned}$$

(9.51)

where $a_{\mathrm {B}}=0.053\,\mathrm {nm}$ denotes the hydrogen Bohr radius.^{Footnote 6} The Bohr radius of the exciton is $a_{\mathrm {X}}=r_{\mathrm {X}}^1$ (14.6 nm for GaAs, $\sim 2$ nm for ZnO). The exciton moves with the center-of-mass $\mathbf{K}$-vector through the crystal. The complete dispersion is (see Fig. 9.19b)

$$\begin{aligned} E=E_{\mathrm {g}} + E_{\mathrm {X}}^n + \frac{\hbar ^2}{2M} \, \mathbf{K}^2 \;. \end{aligned}$$

(9.52)

The oscillator strength of the exciton states decays $\propto n^{-3}$. The absorption due to excitons is visible in Fig. 9.11a for GaAs at low temperatures. If inhomogeneities are present, typically only the $n=1$ transition is seen. However, under special conditions also higher transitions of the exciton Rydberg series are seen (e.g. $n=2$ and 3 in Fig. 9.11b).

The exciton concept was introduced first for absorption in Cu$_2$O [862]. The $J=1/2$ absorption spectrum (‘yellow series’) is shown in Fig. 9.20. In this particular material both the valence and conduction bands have s character, thus the 1s transition of the exciton is forbidden and the np transitions are observed in normal (one-photon) absorption. With two-photon absorption also the s (and d) transitions can be excited. On a piece of natural Cu$_2$O, the Rydberg series has been measured up to $n=25$ [863] (Fig. 9.21a). The peak energy and the oscillator strength follow the $n^{-2}$ ($E_{\mathrm {X}}^{\mathrm {b}}=92$ meV, $E_{\mathrm {g}}=2.17208$ eV) and $n^{-3}$ laws, respectively, expected from a hydrogen model (Fig. 9.21b). The deviation from the $n^{-3}$-dependence for the oscillator strength at large n is due to interaction effects of excitons with large radius at finite exciton density.

The scattering (unbound) states of the exciton [865] for $E>E_{\mathrm {g}}$ contribute to absorption above the band gap. The factor by which the absorption spectrum is changed is called the Sommerfeld factor. For bulk material it is

$$\begin{aligned} S(\eta )=\eta \, \frac{\exp (\eta )}{\sinh (\eta )} \;, \end{aligned}$$

(9.53)

with $\eta =\pi [E_X^b/(E-E_{\mathrm {g}})]^{1/2}$. The change of the absorption spectrum due to the Coulomb correlation is shown in Fig. 9.22. There is a continuous absorption between the bound and unbound states. At the band gap there is a finite absorption ($S(E\rightarrow E_{\mathrm {g}})\rightarrow \infty $). The detail to which exciton peaks can be resolved depends on the spectral broadening.

In Fig. 9.23 the energy separations of the A-, B-, and C-excitons in GaN are shown [540]. Thus, the ordering of the valence bands depends on the strain state of the semiconductor.

7.7 Phonon Broadening

The scattering with phonons and the related dephasing leads to homogeneous broadening $\Gamma _{\mathrm {hom}}$ of absorption (and recombination) lines. Acoustic and optical phonons contribute to the broadening according to the dependence [867]

$$\begin{aligned} \Gamma _{\mathrm {hom}}(T)=\Gamma _0 + \gamma _{\mathrm {AC}}\,T + \gamma _{\mathrm {LO}} \, \frac{1}{\exp (\hbar \omega _{\mathrm {LO}}/kT)-1} \;, \end{aligned}$$

(9.54)

where $\hbar \omega _{\mathrm {LO}}$ is the optical phonon energy and the last factor is the Bose function (E.24). $\Gamma _0$ is a temperature-independent contribution, $\Gamma _0=\Gamma (T=0)$. The increasing broadening with increasing temperature is obvious, e.g., in absorption spectra (Fig. 9.24a). In Fig. 9.24b experimental data for GaAs, ZnSe and GaN are assembled. The data have been fitted with (9.54); the resulting phonon broadening parameters are listed in Table 9.5.^{Footnote 7} The optical transitions in polar semiconductors exhibit stronger coupling to optical phonons. The phonon coupling parameters from different measurements on GaN are discussed and compared in [870].

Table 9.5 Phonon broadening parameters (FWHM) of various bulk semiconductors. Values from fits with (9.54) to experimental data for GaAs [871], ZnSe [869], GaN [868], ZnO [872] (phonon energy fitted) as shown in Fig. 9.25b

Full size table

7.8 Exciton Polariton

Electrons and holes are particles with spin 1/2. Thus, the exciton can form states with total spin $S=0$ (para-exciton, singlet) and $S=1$ (ortho-exciton, triplet). The exchange interaction leads to a splitting of these states, the singlet being the energetically higher. The singlet state splits into the longitudinal and transverse exciton with respect to the orientation of the polarization carried by the Bloch functions and the center-of-mass motion $\mathbf{K}$ of the exciton. Dipole transitions are only possible for singlet excitons (bright excitons). The triplet excitons couple only weakly to the electromagnetic field and are thus also called dark excitons.

The coupling of these states to the electromagnetic field creates new quasi-particles, the exciton polaritons [873, 874]. The dielectric function of the exciton (with background dielectric constant $\epsilon _{\mathrm {b}}$) is

$$\begin{aligned} \epsilon (\omega )=\epsilon _{\mathrm {b}} \left[ 1+\frac{\beta }{1-(\omega ^2/\omega _{\mathrm {X}})^2} \right] \cong \epsilon _{\mathrm {b}} \left[ 1+\frac{\beta }{1-(\omega ^2/\omega _{\mathrm {T}})^2+\hbar \, K^2/(M\,\omega _{\mathrm {T}})} \right] \;, \end{aligned}$$

(9.55)

where $\beta $ is the oscillator strength and the energy is $\hbar \omega _{\mathrm {X}}=\hbar \omega _{\mathrm {T}}+\hbar ^2\,K^2/2M$. $\hbar \omega _{\mathrm {T}}$ is the energy of the transverse exciton at $K=0$. With this dispersion the wave dispersion must be fulfilled, i.e.

$$\begin{aligned} c^2 \, k^2=\omega ^2 \, \epsilon (\omega ) \;, \end{aligned}$$

(9.56)

where k is the k-vector of the light that needs to be $k=K$ due to momentum conservation. The dependence of the dielectric function on k is called spatial dispersion [875]. Generally, up to terms in $k^2$ it is written as

$$\begin{aligned} \epsilon (\omega )=\epsilon _{\mathrm {b}} \left[ 1+ \frac{\beta }{1-(\omega ^2/\omega _0)^2+D \,k^2} \right] \;. \end{aligned}$$

(9.57)

The term $k^2$ with curvature D (for the exciton polariton $D=\hbar /(M\,\omega _{\mathrm {T}})$) plays a role in particular when $\omega _{\mathrm {T}}^2-\omega ^2=0$. For $\mathbf{k}\ne 0$ even a cubic material is anisotropic. The dimensionless curvature $\hat{D}=Dk^{'2}$ should fulfill $\hat{D}=\hbar /(Mc)\ll 1$ in order to make $k^4$ terms unimportant. For exciton polaritons^{Footnote 8} typically $\hat{D}=\hbar \omega _{\mathrm {T}}/(m\,c^2)\approx 2\times 10^{-5}$ for $\hbar \omega _{\mathrm {T}}=1$ eV and $m^*=0.1$.

From (9.56) together with (9.57) two solutions result:

$$\begin{aligned} 2\omega ^2= & {} c^2 k^2+(1+\beta +D k^2) \, \omega _0^2 \\&\pm \left[ -4 c^2 k^2 (1+D k^2) \, \omega _0^2+(c^2 k^2+(1+\beta +D k^2) \, \omega _0^2)^2\right] ^{1/2} \nonumber \;. \end{aligned}$$

(9.58)

The two branches are shown schematically in Fig. 9.25a. Depending on the k value they have a photonic (linear dispersion) or excitonic (quadratic dispersion) character. The anticrossing behavior at $k'\approx \omega _{\mathrm {T}}/c$ (for $\hbar \omega _{\mathrm {T}}=1$ eV, $k'\approx 0.5\times 10^{-5}$ cm$^{-1}$) creates a bottleneck region in the lower polariton branch. This name stems from the small emission rate of acoustic phonons (i.e. cooling) in that region, as predicted in [876] and experimentally found, e.g. in CdS [877]. The polaritons decay into a photon when they hit the surface. The effect of the oscillator strength of the dispersion is shown in Fig. 9.26 for two-exciton resonance. In the case of several excitons (9.57) reads

$$\begin{aligned} \epsilon (\omega )=\epsilon _{\mathrm {b}} \, \left[ 1+ \sum _{i=1}^{n}\frac{\beta _i}{1-(\omega ^2/\omega _{0,i})^2 + D_i \, k^2} \right] \;. \end{aligned}$$

(9.59)

For $k=0$ either $\omega =0$ (lower polariton branch) or $\epsilon (\omega _{\mathrm {L}})=0$. For the latter we find from (9.57)

$$\begin{aligned} \omega _{\mathrm {L}}=\sqrt{1+\beta } \, \omega _{\mathrm {T}} \;. \end{aligned}$$

(9.60)

Therefore, the energy splitting $\Delta E_{\mathrm {LT}}$, mostly denoted as $\Delta _{\mathrm {LT}}$, between the L- and T-exciton energy given by

$$\begin{aligned} \Delta E_{\mathrm {LT}}= \hbar (\omega _{\mathrm {L}}-\omega _{\mathrm {T}})=\left[ \sqrt{1+\beta }-1 \right] \,\hbar \omega _{\mathrm {T}} \approx \beta \, \hbar \omega _{\mathrm {T}}/2 \end{aligned}$$

(9.61)

is proportional to the exciton oscillator strength (for experimental values see Table 9.6). We note that if (D.9) is used for the dielectric function, $\beta $ in (9.61) needs to be replaced by $\beta /\epsilon _{\mathrm {b}}$.

Table 9.6 Exciton energy (low temperature), LT splitting and exciton polariton oscillator strength for various semiconductors. Values for ZnO from [878], values for GaAs from [879], all other values from [880]

Full size table

The effect of spatial dispersion on the reflection at the fundamental exciton resonance is depicted in Fig. 9.25b. For non-normal incidence an additional feature due to the longitudinal wave is observed for p-polarization [875]. For a detailed discussion additional effects due to anisotropy in wurtzite crystals, an exciton free layer at the semiconductor surface, additional boundary conditions and damping need to be considered [881, 882]. The polariton dispersions of ZnO and GaN are shown in Fig. 9.27.

7.9 Bound-Exciton Absorption

Excitons can localize at impurities or inhomogeneities. Such excitons are called bound excitons. Here, the absorption due to such complexes is discussed. The recombination is discussed in Sect. 10.3.2. In GaP:N excitons are bound to isoelectronic N impurities (substituting P), resulting in the ‘A’ line at 2.3171 eV (at $T=4.2$ K).^{Footnote 9} The absorption due to A excitons is well resolved in the spectrum of Fig. 9.28b. At sufficiently high nitrogen doping, there exist nitrogen pairs, i.e. a complex where a nitrogen impurity has a second nitrogen impurity in the vicinity. The pairs are labeled NN$_n$. It was believed that the second nitrogen atom is in the nth shell around the first one. However, the proper level asignment is probably different in the view of modern theory [544]. Also clusters with more than two nitrogen atoms may exist. NN$_1$ is a prominent level and relates to a N–Ga–N complex having 12 equivalent sites for the second N atom on the next neighbor anion site. The transitions due to excitons bound to NN$_n$, as shown in Fig. 9.28a, give a series of lines (see Table 9.7) that fulfill $\lim _{n\rightarrow \infty }\mathrm {NN}_n=\mathrm {A}$. Although GaP has an indirect band structure, the absorption coefficient of N-related transitions is large, about $10^5$ cm$^{-1}$ for a nitrogen doping level of $10^{19}$ cm$^{-3}$.^{Footnote 10} This is due to the fact that the electron spatially localized at the nitrogen isoelectronic trap (Sect. 7.7.9) has a sizeable $k=0$-component of its wave-function (Fig. 7.40), leading to a large transition probability for $\Gamma $-point holes with an oscillator strength of 0.09 [885].

Table 9.7 Index of nitrogen pairs NN$_n$ and energy separation $\Delta E$ of bound-exciton transitions from the free-exciton line for $n=1 \ldots 10$ and the ‘A’ line

Full size table

7.10 Biexcitons

Similar to two hydrogen atoms forming a hydrogen molecule, two excitons can also form a bound complex, the biexciton involving two electrons and two holes. The biexciton binding energy is defined as

$$\begin{aligned} E_{\mathrm {XX}}^{\mathrm {b}}=2\,E_{\mathrm {X}} - E_{\mathrm {XX}}\;. \end{aligned}$$

(9.62)

Biexcitons are binding in bulk material. Accordingly, the biexciton recombination or absorption occurs at lower energy than that of the exciton. Values of the biexciton binding energy are listed in Table 9.4 for various semiconductors. The ratio of biexciton and exciton binding energies is fairly constant about 0.2. In semiconductors with small exciton binding energy, such as GaAs, biexcitons are hard to observe in bulk material but show up in heterostructures that provide additional carrier confinement (see also Sect. 14.4.4). While the exciton density increases linearly with external excitation, the density of biexcitons increases quadratically.

7.11 Trions

The complexes ‘eeh’ and ‘ehh’ are called trions. Also, the notation X$^{-}$ and X$^{+}$ is common. X$^{-}$ is typically stable in bulk material but hard to observe. In quantum wells or dots, trions are easier to observe. In quantum dots excitons with higher charge, e.g. X$^{2-}$, have also been observed (see Fig. 14.45).

7.12 Band Gap Renormalization

The band structure theory has been developed so far for small carrier densities. If the carrier density is large the interaction of free carriers has to be considered. The first step was exciton formation. However, at high temperatures (ionization) and at large carrier density (screening) the exciton is not stable. Exchange and correlation energy leads to a decrease of the optical absorption edge that is called band gap renormalization (BGR).

An effect due to significant carrier density is to be expected when the density is of the order of the exciton volume, i.e. $n \sim a_{\mathrm {B}}^{-3}$. For $a_{\mathrm {B}} \sim $ 15 nm (GaAs) this means $n \sim 3 \times 10^{17}$cm$^{-3}$. The dimensionless radius $r_{\mathrm {s}}$ is defined via

$$\begin{aligned} \frac{4 \pi }{3} \, r_{\mathrm {s}}^3=\frac{1}{n \, a_{\mathrm {B}}^3} \;. \end{aligned}$$

(9.63)

The sum of exchange and correlation energies $E_{\mathrm {xc}}$ is found to be mostly independent of material parameters [886] (Fig. 9.29a) and follows the form

$$\begin{aligned} E_{\mathrm {xc}}=\frac{a+b \,r_{\mathrm {s}}}{c+d \, r_{\mathrm {s}}+r_{\mathrm {s}}^2} \;, \end{aligned}$$

(9.64)

with $a=-4.8316$, $b=-5.0879$, $c=0.0152$ and $d=3.0426$. Thus the density dependence of the band gap at small carrier density is $\propto n^{1/3}$. Experimental data for a number of II–VI semiconductors roughly follow such a dependence (Fig. 9.29b).

In Fig. 9.30, a theoretical calculation of the absorption spectrum of bulk GaAs for various carrier densities (n=p) [889] is shown. With increasing density, the excitonic resonance broadens and vanishes. The shape approaches the electron–hole plasma shape. The absorption edge shifts to smaller energies. At high carrier density, the absorption becomes negative in a spectral range before absorption sets in. In this spectral region, the material exhibits gain and an incoming light wave is amplified (cmp. Sect. 10.2.6).

7.13 Electron–Hole Droplets

At low temperature and high density, electron–hole pairs in Ge and Si can undergo a phase transition into a liquid state. This electron–hole liquid (EHL) was suggested in [890] and is a Fermi liquid exhibiting the high conductivity of a metal and the surface and density of a liquid. The condensation is due to exchange interaction and correlation. The formation is fostered by the band structure of Ge [891] and the long lifetime of carriers in the indirect band structure. In unstressed Ge typically a cloud of electron–hole droplets with diameter in the $\upmu $m range exists. The phase diagram is shown in Fig. 9.31a. In suitably stressed Ge electron–hole droplets with several hundred $\upmu $m diameter form around the point of maximum shear strain in inhomogeneously strained crystals, as shown in Fig. 9.31b. The pair density in such a liquid is of the order of $10^{17}$ cm$^{-3}$.

We note that the metallic EHL state hinders observation of the Bose–Einstein condensation (BEC) of (bosonic) excitons. The light-exciton mass offers a high condensation temperature in the 1 K range (compared to the mK range for atoms). Recent experiments with spatially indirect excitons in coupled quantum wells lead towards BEC [895, 896]. A sufficiently long lifetime ensures cooling of the excitons close to the lattice temperature. Another potential candidate for BEC are long-living excitons (ms-range) in Cu$_2$O [897]. The condensation of polaritons (cf. Sect. 9.7.8) in microcavities to well-defined regions of $\mathbf{k}$-space has been discussed in [898] and compared to bosonic condensation in bulk.

7.14 Two-Photon Absorption

So far, only absorption processes that involve one photon have been considered. The attenuation of the intensity I of a light beam (of frequency $\omega _0$) along the z direction can be written as

$$\begin{aligned} \frac{\mathrm {d} I}{\mathrm {d} z}= - \alpha \, I - \beta \, I^2 \;, \end{aligned}$$

(9.65)

where $\alpha $ is due to the (linear) absorption coefficient (and possibly scattering) and $\beta $ is the two-photon absorption coefficient. A two-photon process can occur in two steps, e.g. via a midgap level, which is not considered any further here. Here, we consider two-photon absorption (TPA) via the population of a state at $2 \hbar \omega _0$ higher energy than the initial state with a nonlinear optical process. The TPA coefficient is related to the nonlinear third-order electric dipole susceptibility tensor [899] $\chi _{ijkl}$. Within the two-band approximation theory predicts [900]

$$\begin{aligned} \beta \propto \left( 2 \, \hbar \omega _0 - E_{\mathrm {g}} \right) ^{3/2} \;. \end{aligned}$$

(9.66)

The exponent 3/2 is indeed found experimentally, as shown in Fig. 9.32 for GaAs. The strength of absorption depends on the relative orientation of the light polarization with respect to the main crystallographic directions, e.g. TPA for polarization along $\langle 110\rangle $ is about 20% larger than for the $\langle 100\rangle $ orientation.

8 Impurity Absorption

8.1 Shallow Levels

For charge carriers bound to shallow impurities long range Coulomb forces are most important and they exhibit a hydrogen-like term scheme

$$\begin{aligned} E_n=\frac{m^*}{m_0} \frac{1}{\epsilon _{\mathrm {r}}^2} \frac{1}{n^2} \times 13.6\,\mathrm {eV} \;, \end{aligned}$$

(9.67)

with the ionization limit $E_{\infty }$ being the conduction (valence) band edge for donors (acceptors), respectively. They can be excited by light to the nearest band edge. Such absorption is typically in the FIR region and can be used for photodetectors in this wavelength regime. The optical absorption cross section of impurity absorption can be related to the carrier capture cross section [588, 589].

The actual transition energies can deviate from (9.67) due to deviation of the potential close to the impurity from the pure Coulomb potential. Such an effect is known as the chemical shift or central cell correction (cf. Sect. 7.5.5) and is characteristic of the particular impurity. In GaAs such shifts are small ($\sim $100 $\upmu $eV) [902].

The term scheme for P in Si is shown in Fig. 9.33a. The ground state (1s) is split because of a reduction of the tetrahedral symmetry due to intervalley coupling. The anisotropic mass at the X-valley in Si causes the p states (and states with higher orbital angular momentum) to split into p$_0$ and p$_{\pm }$ states. Such an effect is absent in a direct semiconductor with an isotropic conduction-band minimum such as GaAs (Fig. 9.34). Optical transitions between the 1s and various p states can be directly observed in absorption, e.g. for Si:P in [904]. These transitions are also observed in photoconductivity because the missing energy to the ionization into the continuum is supplied by a phonon at finite temperature (photothermal ionization) (Fig. 9.33b) [903]. The splitting of the 2p transition in Fig. 9.34a is the chemical shift due to different donors incorporated in the GaAs (Si, Sn, and Pb). Peak broadening is mostly due to Stark broadening due to neighboring charged impurities. The application of a magnetic field induces Zeeman-like splittings and increases the sharpness of the peaks. The peak width can be further increased by illuminating the sample with light having a higher energy than the band gap. The additional charge carriers neutralize charged impurities and allow higher resolution (Fig. 9.34b).

In Fig. 9.35 absorption spectra of highly doped n-type GaAs are shown. For doping concentrations larger than the critical concentration of ${\sim }1 \times 10^{16}$ cm$^{-3}$ (cf. Table 7.6) significant broadening is observed due to the formation of an impurity band.

8.2 Deep Levels

The absorption of deep levels is typically in the infrared. In Fig. 9.36a the possible optical absorption processes involving the Fe levels in InP (cf. Sect. 7.7.8) during the charge transfer $\mathrm {Fe}^{3+}\rightarrow \mathrm {Fe}^{2+}$ are shown. These transitions and their fine structure (Fig. 9.36b) have been observed in calorimetric absorption spectroscopy (CAS) experiments [682].

In Fig. 9.37 photoproductivity of Si:Mg is shown. The sharp peaks are due to transitions of interstitial, singly ionized Mg, Mg$_{\mathrm {i}}^+$ [907]. Mg in Si is a double donor [653] (see Sect. 7.7.2). Above the ionization limit of about 256 meV, the peaks are replicated, shifted by the LO phonon energy of 59.1 meV. However, now they rather appear as dips. This behavior is typical for a discrete state interacting with a continuum, also called Fano resonance[908, 909] with its characteristic lineshape, going below the continuum level.

The absorption spectra due to various deep acceptors in GaAs are compared in Fig. 9.38. The density of states in the band increases with k (proportional to $\sqrt{E-E_{\mathrm {c}}}$). The carrier on the impurity is strongly localized and described with a wave packet centered around $\Gamma $, its k-components decreasing with increasing k. Thus the maximum absorption will be at an intermediate k-value and an associated energy larger than the ionization energy $E_{\mathrm {i}}$ (lowest transition to continuum at for $k=0$). The lineshapes in Fig. 9.38 fit to a model with a $\delta $-potential (zero range model, neglecting long range Coulomb terms) [910] with maximum absorption close to $2\,E_{\mathrm {i}}$,

$$\begin{aligned} \alpha (E) \propto \frac{E_{\mathrm {i}}^{1/2} \, (E-E_{\mathrm {i}})^{3/2}}{E^3} \;. \end{aligned}$$

(9.68)

9 Absorption in the Presence of Free Charge Carriers

In the presence of charge carriers, various absorption processes can occur. First, the dissipative motion of carriers leads to infrared absorption, termed the free carrier absorption (Sect. 9.9.1). Filling of a band with carriers leads to a shift of the band-band absorption edge, the Burstein-Moss shift (Sect. 9.9.2). Besides the free-carrier absorption, free carriers present in the semiconductor can lead to further absorption processes with transition energies below the band gap. These processes are due to transitions within the band structure and can be

inter-valence band transitions of holes (Sect. 9.9.3),
phonon-assisted inter-valley transitions of electrons (Sect. 9.9.4),
phonon-assisted intra-band transitions of electrons (Sect. 9.9.5).

9.1 Absorption Coefficient, Plasma Frequency

The absorption due to free carriers in the infrared spectral range (away from phonon resonances) can be described with the Drude model [911].

A time-dependent electric field accelerates the charge carriers within a band. The excess energy is subsequently transferred to the lattice via scattering with phonons. A review of the effect of free carriers on optical properties can be found in [912]. In the relaxation-time approximation energy is relaxed with a time constant $\tau $. Thus energy is absorbed from the electromagnetic wave and dissipated. Effectively, this process represents an intra-band excitation.

The complex conductivity (8.37) is given by

$$\begin{aligned} \sigma ^*=\sigma _{\mathrm {r}}+ {\mathrm {i}}\sigma _{\mathrm {i}}=\frac{n \, e^2 \, \tau }{m^*}\left( \frac{1}{1+\omega ^2 \tau ^2} + {\mathrm {i}}\frac{\omega \tau }{1+\omega ^2\tau ^2} \right) \;. \end{aligned}$$

(9.69)

We note that a static magnetic field introduces birefringence as discussed in more detail in Sect. 15.2.2. The wave equation for the electric field is

$$\begin{aligned} \nabla ^2 \mathbf{E} = \epsilon _{\mathrm {r}} \, \epsilon _0 \, \mu _0 \, \ddot{\mathbf{E}} + \sigma ^* \, \mu _0 \, \dot{\mathbf{E}} \;. \end{aligned}$$

(9.70)

For a plane wave $\propto \exp [{\mathrm {i}}(\mathbf{k}\mathbf{r}-\omega t)]$ the wavevector obeys

$$\begin{aligned} k=\frac{\omega }{c} \, \sqrt{ \epsilon _{\mathrm {r}} + {\mathrm {i}}\,\frac{\sigma ^*}{\epsilon _0 \, \omega } } \;, \end{aligned}$$

(9.71)

where $c=(\epsilon _0 \mu _0)^{-1/2}$ is the velocity of light in vacuum, $\epsilon _{\mathrm {r}}$ is the background dielectric constant (for large $\omega $).

The part $\epsilon _{\mathrm {FC}}$ of the dielectric function due to free carriers is

$$\begin{aligned} \epsilon _{\mathrm {FC}} = \frac{{\mathrm {i}}}{\epsilon _0 \, \omega } \, \sigma ^* \;. \end{aligned}$$

(9.72)

The complex index of refraction is

$$\begin{aligned} n^*=n_{\mathrm {r}} + {\mathrm {i}}\, \kappa =\sqrt{ \epsilon _{\mathrm {r}} + {\mathrm {i}}\frac{\sigma ^*}{\epsilon _0 \, \omega } } \;. \end{aligned}$$

(9.73)

Taking the square of this equation yields

$$\begin{aligned} n_{\mathrm {r}}^2 - \kappa ^2= & {} \epsilon _{\mathrm {r}} + {\mathrm {i}}\, \frac{\sigma _{\mathrm {i}}}{\epsilon _0 \, \omega }=\epsilon _{\mathrm {r}}-\frac{n \, e^2}{\epsilon _0 \, m^*} \, \frac{\tau ^2}{1+\omega ^2 \tau ^2} \end{aligned}$$

(9.74a)

$$\begin{aligned} 2 \, n_{\mathrm {r}} \, \kappa= & {} \frac{\sigma _{\mathrm {r}}}{\epsilon _0 \omega } = \frac{n \, e^2}{\epsilon _0 \, \omega \, m^*} \, \frac{\tau }{1+\omega ^2 \tau ^2} \;. \end{aligned}$$

(9.74b)

The absorption coefficient is related to $\kappa $ by (9.9). For the case of higher frequencies, i.e. $\omega \tau \gg 1$, the absorption is

$$\begin{aligned} \alpha =\frac{n \, e^2}{\epsilon _0 \, c \, n_{\mathrm {r}} \, m^* \, \tau } \, \frac{1}{\omega ^2} \propto \lambda ^2\;. \end{aligned}$$

(9.75)

The absorption decreases with increasing frequency like $\omega ^{-r}$. The classical Drude treatment as followed here results in an exponent of $r=2$. This is the case for neutral impurity scattering and also for small frequencies $\hbar \omega \ll E_{\mathrm {F}}$. A more detailed discussion of the energy dependence of free-carrier absorption can be found in [913]. Other exponents have been derived for scattering by acoustical phonons ($r=3/2$), LO phonons ($r=5/2$) and ionized impurities ($r=7/2$). More detailed quantum mechanical treatments of free-carrier absorption in the presence of impurities and phonons can be found in [914–916].

For semiconductors free-carrier absorption is particularly important in the mid- and far-infrared regions when carriers are present due to doping or thermal excitation. In Fig. 9.39a absorption spectra of n-type Ge for various doping concentrations are shown. The absorption coefficient in the transparency regime varies proportionally to $\lambda ^2$ as predicted in (9.75). In Fig. 9.39a, the absorption can be seen to rise for photon energy above 0.7 eV due to absorption in the band structure. Electrons are excited from the valence band across the fundamental band gap into the conduction band (cmp. Sect. 9.7.3), which is an indirect transition in Ge.

In Fig. 9.39b the absorption coefficient due to free carrier absorption at fixed wavelength is shown as a function of dopant concentration.^{Footnote 11} The slope is slightly overlinear, indicating a weak dependence $\tau (n)$. A sub-linear relation has been found for heavily p-doped GaAs [917].

The index of refraction is given by (also for $\omega \tau \gg 1$)

$$\begin{aligned} n_{\mathrm {r}}^2= & {} \epsilon _{\mathrm {r}}-\frac{n e^2}{\epsilon _0 m^* \omega ^2}+\kappa ^2= \epsilon _{\mathrm {r}} \left[ 1-\left( \frac{\omega _{\mathrm {p}}}{\omega } \right) ^2 \right] +\frac{\epsilon _{\mathrm {r}}^2}{4 n_{\mathrm {r}}^2} \left( \frac{\omega _{\mathrm {p}}}{\omega } \right) ^4 \frac{1}{\omega ^2 \tau ^2} \\\approx & {} \epsilon _{\mathrm {r}} \left[ 1-\left( \frac{\omega _{\mathrm {p}}}{\omega } \right) ^2 \right] \nonumber \;, \end{aligned}$$

(9.76)

where

$$\begin{aligned} \omega _{\mathrm {p}}=\sqrt{ \frac{n \, e^2}{\epsilon _{\mathrm {r}} \, \epsilon _0 \, m^*}} \end{aligned}$$

(9.77)

is the plasma frequency. The approximation is valid for small absorption and when $(\omega \tau )^{-2}$ can be neglected. A graphical representation is given in Fig. 9.40a. For coupling to electromagnetic waves (still $\omega \tau \gg 1$)

$$\begin{aligned} \epsilon (\omega )=\epsilon _{\mathrm {r}} \left[ 1-\left( \frac{\omega _{\mathrm {p}}}{\omega } \right) ^2 \right] =\frac{c^2 \, k^2}{\omega ^2} \end{aligned}$$

(9.78)

must be fulfilled. It follows that the dispersion relation in the presence of free carriers (Fig. 9.40b) is

$$\begin{aligned} \omega ^2=\omega _{\mathrm {p}}^2+\frac{c^2 \, k^2}{\epsilon _{\mathrm {r}}} \;. \end{aligned}$$

(9.79)

For $\omega > \omega _{\mathrm {p}}$, $\epsilon >0$, thus waves can propagate. For $\omega <\omega _{\mathrm {p}}$, however, the dielectric constant is negative, i.e. $\epsilon <0$. For such frequencies waves are exponentially damped and cannot propagate or penetrate a layer. This effect can be used in a plasmon waveguide or in metamaterials (cf. Sect. 19.1.10). The expected dependence of the plasmon wavelength on the carrier density $\lambda _{\mathrm {p}}=2\pi c/\omega _{\mathrm {p}}\propto n^{-1/2}$ is depicted in Fig. 9.41 for GaAs. For semiconductors the plasmon frequency is in the mid-or far-infrared spectral region.^{Footnote 12}

9.2 Burstein–Moss Shift

In the discussion so far it has been assumed that all target states in the conduction band are empty. In the presence of free carriers the absorption is modified by the

change of the distribution function
many-body effects (band gap renormalization).

The latter is discussed in the next section. For a degenerate electron distribution all states close to the conduction-band edge are populated. Thus a transition from the valence band cannot take place into such states. This shift of the absorption edge to higher energies is called the Burstein–Moss shift [919, 920]. Originally, the Burstein–Moss shift was evoked to explain the absorption shift in InSb with varying carrier concentration (Fig. 9.42).

$\mathbf{k}$-conserving optical transitions between parabolic hole and electron bands have the dependence

$$\begin{aligned} E=E_{\mathrm {g}}+\frac{\hbar ^2 k^2}{2 \, m_{\mathrm {e}}}+\frac{\hbar ^2 k^2}{2 \, m_{\mathrm {h}}}=E_{\mathrm {g}}+\frac{\hbar ^2 k^2}{2 \,m_{\mathrm {r}}} \;, \end{aligned}$$

(9.80)

where $m_{\mathrm {r}}$ is the reduced mass of electron and hole. About 4kT below the Fermi level all levels in the conduction band are populated (Fig. 9.43). Thus the k value at which the absorption starts is given as

$$\begin{aligned} \hat{k}=\sqrt{\frac{2 \, m_{\mathrm {e}}}{\hbar ^2} \, (E_{\mathrm {F}}-E_{\mathrm {C}}-4kT)} \;. \end{aligned}$$

(9.81)

Besides the energy shift in the conduction band, the corresponding energy shift in the valence band $\hbar {\hat{k}}^2/(2m_{\mathrm {h}})$ must be considered. Thus, the Burstein–Moss shift of the absorption edge is

$$\begin{aligned} \Delta E=\hbar \omega - E_{\mathrm {g}}=(E_{\mathrm {F}}-4kT-E_{\mathrm {C}}) \left( 1+\frac{m_{\mathrm {e}}}{m_{\mathrm {h}}} \right) \;. \end{aligned}$$

(9.82)

The relation between n and the Fermi level is given by (7.6). If $E_{\mathrm {F}}-E_{\mathrm {C}}\gg kT$ the Fermi integral can be approximated by $\frac{2}{\sqrt{\pi }}\frac{2}{3}\left( \frac{E_{\mathrm {F}}-E_{\mathrm {C}}}{kT}\right) ^{3/2}$. Using (7.8) for $N_{\mathrm {C}}$, the Burstein–Moss shift can be written for this case as

$$\begin{aligned} \Delta E=n^{2/3}\,\frac{h^2}{8\,m_{\mathrm {e}}}\,\left( \frac{3}{\pi }\right) ^{2/3}\left( 1+\frac{m_{\mathrm {e}}}{m_{\mathrm {h}}} \right) \approx 0.97 \, \frac{h^2}{8 \, m_{\mathrm {r}}} \, n^{2/3} \;. \end{aligned}$$

(9.83)

The $n^{2/3}$ dependence of the energy shift is found, e.g., for CdO^{Footnote 13} with different carrier concentrations (due to different deposition temperature, no intentional doping) [921] and depicted in Fig. 9.44a. Similar behavior is found for ITO (indium-tin-oxide) thin films, deposited at different sputtering conditions, leading to different carrier concentrations (9.44b).

9.3 Inter-Valenceband Transitions

Transitions within the valence band can occur between three bands, i.e. lh$\rightarrow $hh, so$\rightarrow $hh, and so$\rightarrow $lh, as schematically depicted in Fig. 9.45. Theoretical treatments have been given in [923, 924]. For GaAs, such intravalence-band absorption occurs at photon energies close to $\Delta _0$ as shown in Fig. 9.46a for p-type GaAs:Zn [925]. For p-type GaSb, the absorption coefficient below the fundamental band gap is found almost entirely due to inter-valence band transitions, as shown in Fig. 9.46b for a hole concentration of $p=3.2\times 10^{16}$ cm$^{-3}$ [926].

9.4 Inter-Valley Transitions

Electrons at the conduction band minimum can undergo optical transitions to the same band at a different point of the Brillouin zone. Such intervalley transition, as sketched in Fig. 9.47a, is phonon-assisted to fulfill momentum conservation and occurs around the energy difference $\Delta E$ between the two valleys (cmp. Table 8.4).

For InP with an electron concentration of $n=1.65 \times 10^{18}$ cm$^{-3}$, below the fundamental band edge at 1.4 eV, an additional contribution starting around 0.8–0.9 eV is found besides the free carrier absorption (Fig. 9.47b) [927]. Taking into account the filling of the bottom of the conduction band, an energy separation for the two valleys of $\Delta E=0.90 \pm 0.02$ eV was found for various values of the electron concentrations. This energy corresponds to the energy difference of conduction band minima at $\Gamma $ and X in InP. The lineshape of this absorption processes can be modeled and fits well the difference of measured absorption and extrapolated free-carrier absorption spectra. Transitions to the lower minimum at L ($\Delta E=0.6$ eV) are not observed, possibly masked by the free-carrier absorption.

9.5 Intra-Band Transitions

Phonon-assisted transitions within the lowest conduction band (not to a different valley), as indicated schematically in Fig. 9.48a for the SnO$_2$ band structure [928], can cause absorption at photon energies below the fundamental absorption edge. Actually in SnO$_2$, the optical transition across the fundamental band gap is only weakly dipole-allowed and leads to small absorption coefficient below 100 cm$^{-1}$ directly above the fundamental band gap of about 3.6 eV. The strong dipole-allowed transition with absorption coefficient around $10^5$ cm$^{-1}$ begins at about 4.3 eV and stems from electrons in a lower valence band [929]. The free-carrier absorption due to transitions within the lowest conduction band are calculated to dominate below 2.8 eV (Fig. 9.48b) and thus can also impact transparency in the visible spectral range. The calculated slope is close to $\alpha \propto \lambda ^3$ (cmp. (9.75)), expected from the linear dispersion of the conduction band away from the $\Gamma $-point [928]. A similar effect with the contribution of inter-band and intra-band transitions leading to absorption within the band gap transparency regime has been calculated for Ga$_2$O$_3$ as shown in Fig. 9.49 for various doping levels [930].

10 Lattice Absorption

Due to the lack of a dipole moment of the optical phonons, no first order interaction of optical phonons and (infrared) light exists in the diamond structure for Si and Ge due to crystal structure symmetry [931]. However, higher order processes contribute to lattice absorption in these materials [932, 933]. E.g., two-photon bands are due to a dipole moment that is of second order in the nuclear displacement. Strong absorption effects are present for compound semiconductors. A review can be found in [934].

10.1 Dielectric Constant

The (relative) dielectric constant (with damping parameter $\Gamma $) in the vicinity of the optical phonon energies is given by (cf. (9.27))

$$\begin{aligned} \epsilon (\omega )= \epsilon (\infty ) \left( \frac{\omega _{\mathrm {TO}}^2-\omega ^2- {\mathrm {i}}\omega \Gamma }{\omega _{\mathrm {TO}}^2-\omega ^2- {\mathrm {i}}\omega \Gamma } \right) \;. \end{aligned}$$

(9.84)

The dispersion relation (without damping) can be rewritten as

$$\begin{aligned} \epsilon (\omega )= \epsilon (\infty ) + \frac{\epsilon (0)-\epsilon (\infty )}{1-(\omega /\omega _{\mathrm {TO}})^2}=\epsilon (\infty )\, \left[ 1+ \frac{f}{1-(\omega /\omega _{\mathrm {TO}})^2}\right] \;. \end{aligned}$$

(9.85)

Thus the dimensionless oscillator strength (compare with (D.10)) is $f=(\epsilon (0)/\epsilon (\infty )-1$. With the LST relation (9.26) the oscillator strength is

$$\begin{aligned} f=\frac{\epsilon (0)-\epsilon (\infty )}{\epsilon (\infty )} = \frac{\omega _{\mathrm {LO}}^2-\omega _{\mathrm {TO}}^2}{\omega _{\mathrm {TO}}^2} \approx 2 \, \frac{\omega _{\mathrm {LO}}-\omega _{\mathrm {TO}}}{\omega _{\mathrm {TO}}} \;, \end{aligned}$$

(9.86)

and thus proportional to the splitting $\Delta _{\mathrm {LT}}=\omega _{\mathrm {LO}}-\omega _{\mathrm {TO}}$ between the longitudinal and transverse optical phonon frequency. The approximation in (9.86) is valid for $\Delta _{\mathrm {LT}}\ll \omega _{\mathrm {TO}}$.

The oscillator strength increases with the ionicity, i.e. the electronegativity difference of the atoms in the base (Fig. 9.50). Additionally, the oscillator strength depends on the reduced mass and the high-frequency polarizability; this can be seen, e.g., for the series of the Zn compounds that all have similar ionicity. For the series of the nitrides, the mass effect is small since the reduced mass is dominated by the light N mass. We refer to Fig. 5.23 for the change of phonon oscillator strength in an (Al,Ga)N alloy.

10.2 Reststrahlenbande

The absorption of electromagnetic radiation by optical phonons is governed by the dielectric function that has been derived in (9.84). For small damping, i.e. $\Gamma \ll \Delta _{\mathrm {LT}}$, the dielectric constant is negative between $\omega _{\mathrm {TO}}$ and $\omega _{\mathrm {LO}}$. From $\epsilon _{\mathrm {r}}=n_{\mathrm {r}}^2-\kappa ^2$ it follows that $\kappa ^2$ is much larger than $n_{\mathrm {r}}^2$. Therefore, the reflectance (9.17) will be close to 1. This energy range is the so-called reststrahlenbande. This term stems from multiple reflections in this wavelength regime that suppresses neighboring spectral regions and thus achieves a certain monochromatization in the far-infrared spectral region. Within the semiconductor the absorption is large in the reststrahlenbande (Fig. 9.51).

10.3 Polaritons

The coupled propagation of phonons and electromagnetic radiation is (without phonon damping) related to the dielectric function given in (9.27),

$$\begin{aligned} \epsilon (\omega )=\epsilon (\infty )\left( \frac{\omega _{\mathrm {LO}}^2-\omega ^2}{\omega _{\mathrm {TO}}^2-\omega ^2}\right) =\frac{c^2 \, k^2}{\omega ^2} \;. \end{aligned}$$

(9.87)

There are two branches of propagating waves (real k):

$$\begin{aligned} \omega ^2=\frac{1}{2} \left( \omega _{\mathrm {LO}}^2+\frac{c^2k^2}{\epsilon (\infty )}\right) \pm \sqrt{\frac{1}{4} \left( \omega _{\mathrm {LO}}^2+\frac{c^2k^2}{\epsilon (\infty )}\right) ^2-\left( \frac{c^2 k^2 \omega _{\mathrm {TO}}^2}{\epsilon (\infty )}\right) ^2} \;. \end{aligned}$$

(9.88)

For $k=0$ we find the solutions $\omega =\omega _{\mathrm {LO}}$ and $\omega =k\,c/\sqrt{\epsilon (0)}$. For large k we find $\omega =\omega _{\mathrm {TO}}$ and $\omega =k\,c/\sqrt{\epsilon (\infty )}$. These solutions are shown in Fig. 9.52. Both branches have a phonon- and a photon-like part. The coupled state between the phonon and the photon field is called the (phonon-) polariton.

In the interval $[\omega _{\mathrm {TO}},\omega _{\mathrm {LO}}]$ the wavevector is purely imaginary, i.e. $k={\mathrm {i}}\tilde{k}$ with real $\tilde{k}$. For this case there is only one solution that is also depicted in Fig. 9.52,

$$\begin{aligned} \omega ^2=\frac{1}{2} \left( \omega _{\mathrm {LO}}^2+\frac{c^2 \tilde{k}^2}{\epsilon (\infty )}\right) + \sqrt{ \frac{1}{4} \left( \omega _{\mathrm {LO}}^2+\frac{c^2 \tilde{k}^2}{\epsilon (\infty )}\right) ^2 + \left( \frac{c^2 \tilde{k}^2\omega _{\mathrm {TO}}^2}{\epsilon (\infty )}\right) ^2} \;. \end{aligned}$$

(9.89)

10.4 Phonon–Plasmon Coupling

The coupling of phonons and plasmons in the spectral region of the reststrahlenbande leads to the development of two new branches, the longitudinal phonon plasmon modes (LPP$+$ and LPP−), in the common dispersion. The dielectric function is

$$\begin{aligned} \epsilon (\omega )=\epsilon (\infty )\left( 1+\frac{\omega _{\mathrm {LO}}^2-\omega ^2}{\omega _{\mathrm {TO}}^2-\omega ^2}-\frac{\omega _{\mathrm {p}}^2}{\omega ^2} \right) \;. \end{aligned}$$

(9.90)

For $\epsilon (\omega )=0$ for $k=0$ (coupling to photons) the two solutions $\omega _{\mathrm {LPP+}}$ and $\omega _{\mathrm {LPP-}}$ do not cross as a function of $\omega _{\mathrm {p}}$ (Fig. 9.53),

$$\begin{aligned} \omega _{\mathrm {LPP \pm }}= \frac{1}{2} \left[ \omega _{\mathrm {LO}}^2+\omega _{\mathrm {p}}^2 \pm \sqrt{(\omega _{\mathrm {LO}}^2+\omega _{\mathrm {p}}^2)^2 -4 \, \omega _{\mathrm {TO}}^2 \, \omega _{\mathrm {p}}^2}\right] \;. \end{aligned}$$

(9.91)

For small plasma frequencies $\omega _{\mathrm {LPP+}}=\omega _{\mathrm {LO}}$, i.e. the optical phonons couple to the electromagnetic field without change. Also $\omega _{\mathrm {LPP-}}=\omega _{\mathrm {p}}$. For large carrier density, i.e. $\omega _{\mathrm {p}} \gg \omega _{\mathrm {LO}}$, we find $\omega _{\mathrm {LPP-}}=\omega _{\mathrm {TO}}$ and $\omega _{\mathrm {LPP+}}=\omega _{\mathrm {p}}$. Thus, the carriers have effectively screened the electric field of the phonon that had led to the increase of the TO to the LO frequency.

Notes

1.
The more exact numerical value in (9.1) is 1239.84.
2.
In [836], the absorption coefficient $\mu $ was defined via $I(d)/I(0)=\mu ^d$, i.e. $\mu =\exp -\alpha $.
3.
Here we assume that the valence-band states are filled and the conduction-band states are empty. If the conduction-band states are filled and the valence-band states are empty, the rate is that of stimulated emission.
4.
The real and imaginary parts of the dielectric function are generally related to each other via the Kramers–Kronig relations.
5.
A flat optical phonon dispersion is assumed.
6.
Cf. (7.22); an electron bound to a donor can be considered as an exciton with an infinite hole mass.
7.
Such parameter can be directly determined from spectroscopic broadening (as in [868]) or a time-resolved measurement of the decay of the coherent polarization (four-wave mixing) as in [869]. In the latter, the decay constant of the dephasing $T_2$ is related to the decay constant $\tau $ of the FWM-signal by $T_2=2\tau $ for homogeneous broadening. The Fourier transform of $\exp -t/(2\tau )$ is a Lorentzian of the type $\propto ((E-E_0)^2+\Gamma ^2/4)^{-1}$ with $\Gamma =1/\tau $ being the FWHM.
8.
The dependence of the optical-phonon energies on k is typically too small to make spatial dispersion effects important. According to (5.19) $\hat{D}=-(a_0\omega _{\mathrm {TO}}/4c)^2\approx 4\times 10^{-11}$ for typical material parameters (lattice constant $a_0=0.5$ nm, TO phonon frequency $\omega _{\mathrm {TO}}=15$ THz).
9.
The A line is due to excitons with $J=1$, resulting of coupling of the electron spin 1/2 with the hole angular momentum of 3/2. The B-line is a dipole forbidden line due to ‘dark’ excitons with $J=2$.
10.
Also the recombination (Sect. 10.3.2) is efficient and allows green GaP:N and yellow GaAsP:N light emitting diodes.
11.
Even at low temperature, $n\approx N_{\mathrm {D}}$ since $N_{\mathrm {D}}\gg N_{\mathrm {c}}$ (cf. [594] and Sect. 7.5.7).
12.
The much higher free-electron density in metals shifts the plasma frequency to the UV, explaining the reflectivity of metals in the visible and their UV transparency.
13.
CdO is an indirect semiconductor, the optical band gap is the energy of the direct transition at the $\Gamma $-point, typically obtained from extrapolation in the $\alpha ^2$ versus energy plot. The indirect transitions involve holes from other points in the Brillouin zone (cmp. Fig. 6.13).

Author information

Authors and Affiliations

Felix Bloch Institute for Solid State Physics, Universität Leipzig, Leipzig, Germany
Marius Grundmann

Authors

Marius Grundmann
View author publications
You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marius Grundmann .

Rights and permissions

Reprints and permissions

Copyright information

About this chapter

Cite this chapter

Grundmann, M. (2021). Optical Properties. In: The Physics of Semiconductors. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-51569-0_9

Download citation

DOI: https://doi.org/10.1007/978-3-030-51569-0_9
Published: 07 March 2021
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-51568-3
Online ISBN: 978-3-030-51569-0
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Optical Properties

Abstract

1 Spectral Regions and Overview

2 Complex Dielectric Function

3 Reflection and Diffraction

4 Absorption

5 Dielectric Function due to Optical Phonons

6 Electron–Photon Interaction

7 Band–Band Transitions

7.1 Joint Density of States

7.2 Direct Transitions

7.3 Indirect Transitions

7.4 Urbach Tail

7.5 Amorphous Semiconductors

7.6 Excitons

7.7 Phonon Broadening

7.8 Exciton Polariton

7.9 Bound-Exciton Absorption

7.10 Biexcitons

7.11 Trions

7.12 Band Gap Renormalization

7.13 Electron–Hole Droplets

7.14 Two-Photon Absorption

8 Impurity Absorption

8.1 Shallow Levels

8.2 Deep Levels

9 Absorption in the Presence of Free Charge Carriers

9.1 Absorption Coefficient, Plasma Frequency

9.2 Burstein–Moss Shift

9.3 Inter-Valenceband Transitions

9.4 Inter-Valley Transitions

9.5 Intra-Band Transitions

10 Lattice Absorption

10.1 Dielectric Constant

10.2 Reststrahlenbande

10.3 Polaritons

10.4 Phonon–Plasmon Coupling

Notes

Author information

Authors and Affiliations

Corresponding author

Rights and permissions

Copyright information

About this chapter

Cite this chapter

Download citation

Share this chapter

Publish with us

Search

Navigation