# Optics in Semiconductors

# Optics in Semiconductors

# Abstract and Keywords

This chapter describes basic optical processes in semiconductor crystals, including interband optical absorption, gain, and emission. The consideration is performed for the two types of crystal structures, zinc-blende and wurtzite, in the framework of a semi-classical approach when the electromagnetic field is treated classically while the electrons are described by the quantum mechanical Hamiltonian and wave functions. Einstein coefficients are introduced in order to define the connection between absorption, stimulated emission, and spontaneous emission. Optical selection rules are obtained in the framework of the k⋅p theory by calculating the interband momentum matrix elements. The concepts of Wannier–Mott excitons and exciton polaritons are discussed in the framework of the effective mass approximation for the case of the direct band-gap semiconductors. The chapter establishes the symmetry classification of excitonic states in the semiconductor crystals in terms of the theory of irreducible group representations.

*Keywords:*
interband absorption, spontaneous emission, stimulated emission, Einstein coefficients, momentum matrix elements effective-mass approximation, Wannier–Mott excitons, exciton polaritons

4.1 Density of states 84

*Plasmonic Effects in Metal–Semiconductor Nanostructures*. Alexey A. Toropov and Tatiana V. Shubina.

© Alexey A. Toropov and Tatiana V. Shubina 2015. Published in 2015 by Oxford University Press.

The aim of this chapter is to discuss the absorption and emission of light by semiconductor crystals. We begin by introducing the basic concept of density of states for systems of different dimensionality. Then we describe the interband absorption and emission processes in semiconductors within a semiclassical approach and introduce the Einstein coefficients in order to define the connection between absorption, stimulated emission, and spontaneous emission. Explicit expressions for the interband transition matrix elements are presented in terms of the $\mathbf{k}\cdot \mathbf{p}$ theory for crystals with both zinc-blende and wurtzite structures. Finally, we briefly discuss the concept of Wannier–Mott excitons in semiconductors in the framework of the effective-mass approximation and establish symmetry classification of the states of excitons and exciton polaritons.

# (p.84) 4.1 Density of states

The concept of *density of states* (DOS) is one of a few basic concepts of quantum physics and plays an important role in any description of light–matter interactions. Let us consider a quantum system described by a Hamiltonian $H$ with allowed energy levels denoted by ${E}_{k}$. The energy spectrum can be either discrete or continuous. The DOS of the system, $\mathrm{\rho}(E)$, is defined as the number of allowed states per unit energy and unit volume lying in the energy range between $E$ and $E+dE$. In this way, the total concentration $N$ of available states in the system between the energies ${E}_{1}$ and ${E}_{2}$ is represented as

$\mathrm{\rho}(E)$ can be conveniently expressed as

where $\mathrm{\delta}(x)$ represents the Dirac delta function, $\mathbf{k}\text{}$ labels all possible states, and $V$ is the system volume. Of course, the delta function in eqn (4.2) makes sense as a mathematical object only when the summation is transformed into integration.

The DOS of a free particle moving in a $D$-dimensional space depends only on a particle dispersion law $E=\mathrm{\hslash}\mathrm{\omega}=f(\mathbf{k})$ and dimensionality $D.$ We are interested in $D=3$ (bulk crystals), $D=2$ (quantum wells), and $D=1$ (quantum wires). To count the particle states, considered as monochromatic plane waves with a frequency $\mathrm{\omega}$ and a wavevector $\mathbf{k}$, we use *the periodic boundary conditions* that the wave function should be periodic in all dimensions with an arbitrary large period $L$. Therefore, for each dimension $\mathrm{\alpha}$ the allowed values of ${k}_{\mathrm{\alpha}}$ are

with integer numbers $j$. These allowed wavevectors make a cubic lattice in the $\mathbf{k}$-space so that the volume of a state is ${\left(\mathrm{\Delta}k\right)}^{D}={\left(2\mathrm{\pi}/L\right)}^{D}$. In other words, the density of the allowed states in the $D$-dimensional $k$-space defined by the periodic boundary conditions (4.3) is ${\left(L/2\mathrm{\pi}\right)}^{D}$. The volume $V={L}^{D}$ represents the *normalization volume*. The periodic boundary conditions are implied everywhere in this chapter.

The sum over the wavevector in eqn (4.2) can be readily reduced to an integral by multiplying and dividing it by $\mathrm{\Delta}\mathbf{k}={\left(2\mathrm{\pi}/L\right)}^{D}$ and assuming that $L$ goes to infinity:

(p.85) The calculation of $\mathrm{\rho}(E)$ simplifies if the integrand is isotropic, e.g. if the particle energy depends only on the wavevector length. This is, for example, the case of the free-electron dispersion ${E}_{k}=\frac{{\mathrm{\hslash}}^{2}{k}^{2}}{2{m}_{0}}$. Evaluating the integral in polar coordinates we get

where ${\mathrm{\Omega}}_{D}$ is the space angle in $D$ dimensions. The involved space angle integral equals $4\mathrm{\pi}$ for $D=3$, $2\mathrm{\pi}$ for $D=2$, and $2$ for $D=1$. Transforming the $k$ integral in eqn (4.5) into an energy integral with

and integrating the delta function we obtain the free-electron density of states

Equation (4.7) shows that the DOS of free electrons strongly depends on their dimensionality:

where $\mathrm{\Theta}(E)$ is the step function, and

For *D* = 0, the DOS is the delta function.

The same technique can be used to calculate the density of states for the photon field, provided that it is described by plane waves with the dispersion

where $c/n$ is the speed of light in the medium with a refractive index $n$. For the most interesting case of $D=3$, substitution in eqn (4.5) of ${E}_{k}=\mathrm{\hslash}{\mathrm{\omega}}_{k}=\frac{\mathrm{\hslash}kc}{n}$ and $d{E}_{k}=\frac{\mathrm{\hslash}c}{n}dk$ gives

# (p.86) 4.2 Interband optical absorption and emission

The *interband absorption process* in semiconductor crystals typically implies a photon-induced excitation of an electron from the filled valence band to the empty conduction band. The excitation of the electron leaves the initial state in the valence band unoccupied. Therefore, the photon absorption creates a hole in the initial state and an electron in the final state that may be considered as the creation of an *electron–hole pair*. Since there is a continuous range of energy states within both bands, these *interband transitions* are possible over a continuous range of frequencies.

The law of energy conservation as applied to the interband absorption process reads

where ${E}_{i}$ is the energy of the electron in the valence band (initial state), ${E}_{f}$ is the energy of the final state in the conduction band, and $\mathrm{\hslash}\mathrm{\omega}$ is the photon energy. Obviously, the interband transitions are not possible unless the photon energy is larger than the crystal band gap ($\mathrm{\hslash}\mathrm{\omega}>{E}_{g}$). Therefore, these transitions give rise to a continuous absorption spectrum above the *fundamental absorption edge* that emerges at the band-gap energy. The shape of the absorption spectrum depends on the band structure and parameters of the semiconductor.

Emission is the inverse process of absorption, when an electron–hole pair inside the semiconductor is destroyed with the emission of electromagnetic radiation. There are, in fact, two essentially different emission processes. In the presence of photons, the interaction of the excited crystal states with the incident photons can stimulate the emission of a photon that is identical to the incident ones. This process, called *stimulated emission*, is the exact opposite to the absorption process. In the absence of any incident photon, the excited state can decay into a lower energy level by itself. This intrinsically stochastic process is called *spontaneous emission*. From a quantum-mechanical viewpoint, the spontaneous emission results from fluctuations of the vacuum state of the electromagnetic field (see, e.g., Yariv 1989; Loudon 2000).

In the stimulated emission process, the number of emitted photons is directly proportional to the number of incident photons. Moreover, all parameters off the emitted photons, like energy, direction, phase, and polarization, are inherited from the incident ones. On the contrary, spontaneous emission may occur at any energy and at any direction, provided there are corresponding allowed transitions. The average time for the spontaneous transitions is a characteristic of the emitting object, while it can be modified by changing its optical environment. The latter property underlies the effect of emission enhancement in plasmonic metal–semiconductor structures, as discussed in detail in Chapter 7. Therefore, just the spontaneous emission will be relevant to most of the content of this book. The stimulated emission will be of particular concern in the discussion of the physics of surface-plasmon lasers in Chapter 8.

(p.87)
Thus, there are three fundamental processes in the interaction of light with matter: absorption, stimulated emission, and spontaneous emission. The probabilities of occurrence for each of these processes are defined by interrelated quantities known as *Einstein coefficients*. Following classical arguments of Einstein, we will treat in this section the basic interaction processes between electromagnetic radiation and matter in the framework of the semiclassical quantum theory. In particular, we derive the interband absorption, gain, and emission spectra for both zinc-blende and wurtzite semiconductors using time-dependent perturbation theory in the form of *Fermi’s golden rule*. This consideration is based on some physically reasonable postulates that can all be justified by consistent quantum-mechanical treatments of the interaction processes (Loudon 2000).

## 4.2.1 Interband optical transitions by Fermi’s golden rule

The description of interband optical absorption and emission is based on applying the quantum mechanical treatment of the light–matter interaction to the band states. We will use *a semiclassical approach* when the electromagnetic field is treated classically while the electrons are described by the quantum mechanical Hamiltonian and wave functions (see, e.g., Schiff 1969; Gasiorowicz 1996). In this approximation, the interaction between the photons and the electrons in the semiconductor can be described by the Hamiltonian

where $\mathbf{A}$ is the vector potential defining the electromagnetic field, $V(\mathbf{r})$ is the periodic crystal potential, and $e$ is the electron charge. This Hamiltonian is obtained from the already introduced unperturbed Hamiltonian (see eqn (3.3) in Chapter 3) by replacing the electron momentum operator $\mathbf{p}$ by $\mathbf{p}+e\mathbf{A}$.

This Hamiltonian can be expanded into

where ${H}_{0}$ is the unperturbed Hamiltonian and ${H}^{\prime}$ describes the interaction between the radiation and electrons. Deriving the expression for ${H}^{\prime}$ we use the Coulomb gauge for the vector potential ($\mathrm{\nabla}\cdot \mathbf{A}=0$ and the scalar potential $\mathrm{\phi}=0$ as $\mathrm{\rho}=0$) such that the operators $\mathbf{p}=\frac{\mathrm{\hslash}}{i}\mathrm{\nabla}$ and $\mathbf{A}$ commute. We also neglect the quadratic term $\frac{{e}^{2}{A}^{2}}{2{m}_{0}}$, which for practical light intensities is much smaller than the term linear in $\mathbf{A}$. This is in line with the assumption that $\mathbf{A}$ is weak enough to validate the use of the time-dependent perturbation theory.

(p.88) Suppose that the electromagnetic field is described by a monochromatic plane wave with the vector potential defined as

where $\text{\xea}$ is a unit vector in the direction of the optical electric field. The optical electric and magnetic fields can be expressed in terms of the vector potential as

where we assume that for a nonmagnetic material the relative permeability of the free space $\mathrm{\mu}=1$ (see, e.g., Jackson 1999). The time-averaged energy density of the optical field is

where $n$ is the refractive index of the material. In deriving eqn (4.20) we have used the fact that $\u3008{\mathrm{sin}}^{2}\left(x\right)\u3009=1/2$. Dividing $W$ by the quantum energy we obtain the photon density

Fermi’s golden rule states that for a time-harmonic perturbation

the transition rate of an electron from an initial state $i$ with energy ${E}_{i}$ to a final state $f$ with energy ${E}_{f}$ is given by

The first term in eqn (4.23) corresponds to the absorption of a photon, since ${E}_{f}={E}_{i}+\mathrm{\hslash}\mathrm{\omega}$, whereas the second term corresponds to the emission of a photon, as ${E}_{f}={E}_{i}-\mathrm{\hslash}\mathrm{\omega}$. Comparing eqns (4.22) and (4.17) we see that the term ${A}_{0}\text{\xea}exp\left(i[\mathrm{\kappa}\mathbf{r}-\mathrm{\omega}t]\right)$ in eqn (4.17) is responsible for the process when the electron in the valence band absorbs the photon energy and is then excited into the conduction band. Another term, ${A}_{0}\text{\xea}exp\left(-i[\mathrm{\kappa}\mathbf{r}-\mathrm{\omega}t]\right)$, describes emission of photons by electrons under the influence of an external optical field that is a stimulated emission process.

(p.89)
To calculate the transition rate of the interband transitions we should evaluate the *interband optical matrix element* of the perturbation operator ${H}^{\prime}$ with the Bloch functions in the valence band ${E}_{v}$ (${\mathrm{\psi}}_{v}$) and the conduction band ${E}_{c}$ (${\mathrm{\psi}}_{c}$), which can be expressed as

where ${u}_{v,{\mathbf{k}}_{\mathbf{v}}}(\mathbf{r})$ and ${u}_{\mathbf{c}\mathbf{,}{\mathbf{k}}_{\mathbf{c}}}(\mathbf{r})$ are the periodic Bloch amplitudes and $N$ is the number of unit cells in a crystal normalization volume $V$ defined by the periodic boundary conditions. For ${H}^{\prime}$ defined by eqns (4.16) and (4.17) the interband matrix element can be written as

Deriving eqn (4.26) we use the periodicity of Bloch amplitudes and replace the integration over the crystal normalization volume by the integration over the unit cell $\mathrm{\Omega}$ and the summation over all crystal lattice vectors ${\mathbf{a}}_{m}$ corresponding to this volume. The latter sum has the form $\sum _{{\mathbf{a}}_{m}}{\text{e}}^{i\mathbf{q}{\mathbf{a}}_{m}}$ ($\mathbf{q}$ is an arbitrary vector from the reciprocal space) and obeys the general summation rule (see, e.g., Anselm 1981):

where

$N$ is the number of unit cells in the normalization volume, and the summation is performed over all lattice vectors in this volume. Applying this rule to (p.90) eqn (4.26) we conclude that the interband optical matrix element is nonzero only when

This relationship is important since it ensures that a Bloch wavevector is conserved in the absorption process on the same footing as a conventional wavevector. It represents the momentum conservation

that is a consequence of the translational symmetry of the crystal.

If we assume that the optical wavevector $\mathrm{\kappa}$ is much smaller that the size of the Brillouin zone, which is true for visible photons, eqn (4.29) transforms to ${\mathbf{k}}_{c}\approx {\mathbf{k}}_{v}$, and the interband transitions are said to be *vertical* or *direct* in the reciprocal space. Since ${\mathbf{k}}_{c}$ and ${\mathbf{k}}_{v}$ are the same, we will drop the subscript $c$ or $v$ to simplify the notation. Equation (4.29) and the smallness of $\mathrm{\kappa}$ allow us to take the exponential term in eqn (4.26) as equal to 1. This approximation is usually referred to as *the electric dipole approximation*. It is equivalent to expanding the term ${e}^{i\mathrm{\kappa}\cdot \mathbf{r}}$ in eqn (4.17) into a Taylor series ($1+i(\mathrm{\kappa}\cdot \mathbf{r})+\dots $) and leaving only the first $\mathrm{\kappa}$-independent term. Taking into account orthogonality of the Bloch amplitudes related to different bands but the same $\mathbf{k}$:

we finally obtain

where

is *the interband momentum matrix element* calculated with periodic Bloch amplitudes.

To calculate *the total upward transition rate per unit volume* we should sum up the first term in eqn (4.23) over all possible $\mathbf{k}$:

The factor 2 appears in eqn (4.34) to take into account two possible spin projections of an electron. Equation (4.34) implies that the valence band is completely occupied and the conduction band is empty. Otherwise we should take into account the probability ${f}_{v}$ that the valence band is occupied and the (p.91) probability $(1\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}{f}_{c})$ that the conduction band is empty, defined by the Fermi–Dirac distributions ${f}_{v}$ and ${f}_{c}$:

${F}_{i}$ are the quasi-Fermi levels for electrons ($i=c$) and holes ($i=v$).

Similarly, the downward transition rate is

and the net upward transition rate $R(\mathrm{\hslash}\mathrm{\omega})={R}_{vc}-{R}_{cv}$ can be written as

Deriving eqn (4.38) we use even parity of the delta function and the fact that ${\left|{M}_{cv}\right|}^{2}={\left|{M}_{cv}^{+}\right|}^{2}$.

Equation (4.38) neglects all mechanisms of spectral broadening due to different scattering processes. A convenient way to include the spectral broadening is to replace the delta function by a Lorentzian function with a proper normalization factor:

where $\mathrm{\gamma}$ is a phenomenological factor taking account of the spectral broadening.

## 4.2.2 Einstein coefficients

Let us consider the interrelation between absorption, stimulated emission, and spontaneous emission for a discrete two-level electron system (level 1 is below level 2 and the inter-level energy gap is ${E}_{21}$) in the presence of an electromagnetic field with a broad spectrum (black body radiation). We assume that the levels are nondegenerate. Energy-conserving processes can occur in which photons of energy $\mathrm{\hslash}\mathrm{\omega}={E}_{21}$ are emitted or absorbed that make transitions between the two states. Making use of Fermi’s golden rule with the perturbation Hamiltonian ${H}^{\prime}$ given by eqn (4.16), we can write the total upward transition rate per unit volume (s^{−1} m^{−3}) as
(p.92)

where ${M}_{21}=\u30082\left|{H}^{\prime}\right|1\u3009$ is the perturbation matrix element,

is the mean number of photons excited in the field mode at temperature *T* at the transition energy ${E}_{21}$, as defined by the Bose–Einstein statistics, and a factor 2 accounts for two polarization states for each photon wavevector. The quantities ${f}_{1}$ and ${f}_{2}$ take into account the occupation probabilities of the levels using Fermi–Dirac distributions (eqn (4.36)).

The summation over the photon wavevector can be reduced to integration by the rule introduced in eqn (4.4) and after integrating the delta function we can express the total transition rate per unit volume as

Here

is the density of photon states, in units J^{−1} m^{−3}, where ${\mathrm{\rho}}_{\text{ph}}(E)$ was defined by eqn (4.12) as a three-dimensional density of states obeying the dispersion law $\mathrm{\hslash}{\mathrm{\omega}}_{k}=\frac{\mathrm{\hslash}kc}{n}$. The quantity ${B}_{12}$ in eqn (4.43), defined as

is one of three Einstein coefficients. It determines the probability of the upward transition rate per incident photon, in (J s^{−1}).

Let us assume that the stimulated emission occurs at a rate (per unit volume) proportional to the incident photon density:

whereas the spontaneous emission rate per unit volume is independent of the photon density:

The quantities ${B}_{21}$ and ${A}_{21}$ in eqns (4.45) and (4.46) represent the two remaining Einstein coefficients.

(p.93) Simple relations between the three Einstein coefficients can be derived for conditions of thermal equilibrium, when the two quasi-Fermi levels, ${F}_{1}$ and ${F}_{2}$, are equal to each other. Using this condition, we find from eqn (4.36) that

Since at thermal equilibrium the average population densities of the two levels are constant with time, we can balance the upward transition rate with the total downward transition rate:

Substitution of eqns (4.41) and (4.47) into eqn (4.48) gives

The left- and right-hand parts of eqn (4.49) are identical at all temperatures *T* only if

and

The relations (4.50) and (4.51) are the essence of the original derivation by Einstein (1917).

The above relations between the Einstein coefficients are derived for conditions of thermal equilibrium. Nevertheless, they are valid for a wide range of conditions since the Einstein coefficients reflect general properties of the medium. It can be seen that the three Einstein coefficients are interrelated and all transition rates can be expressed in terms of a single coefficient. In particular, the ratio of the stimulated and spontaneous emission rates is

Then, the total emission rate from level 2 to level 1 can be written as

Equation (4.53) contains two terms. The first one corresponds to spontaneous emission, while the second term is identified with stimulated emission. It is clear that the stimulated emission rate can be obtained by multiplying the spontaneous emission rate by the occupation number of photons.

## (p.94) 4.2.3 Optical absorption and gain

The optical absorption coefficient $\mathrm{\alpha}$ in a crystal can be defined as the ratio of a number of photons absorbed per unit volume per second to the photon flux density, which is the number of photons injected per second through a unit surface area. In fact, this definition matches the definition represented in Chapter 2 by eqns (2.44)–(2.46).

Transforming the summation in eqn (4.38) into an integration, by the rule introduced in eqn (4.4), and substituting ${M}_{cv}$ (eqn (4.32)) we obtain the net upward transition rate

Deriving eqn (4.54) we assume that the undoped semiconductor is in thermal equilibrium so that the quasi-Fermi levels are equal, ${F}_{v}={F}_{c}={E}_{F}$. Then the valence band is completely occupied, the conduction band is empty, and, hence, ${f}_{v}=1$ and ${f}_{c}=0$.

The photon flux density can be calculated as

where ${N}_{\text{ph}}$ is represented by eqn (4.21), whereas the number of photons absorbed per unit volume per second is given by $R(\mathrm{\hslash}\mathrm{\omega})$. With eqns (4.21) and (4.54) we derive the absorption coefficient at thermal equilibrium as

Equation (4.56) determines the shape of the fundamental absorption edge provided we know the dispersion laws ${E}_{c}(\mathbf{k})$ and ${E}_{v}(\mathbf{k})$ as well as the interband momentum matrix element ${\mathbf{p}}_{cv}(\mathbf{k})$.

In all III–V and II–VI semiconductors with zinc-blende and wurtzite structures, the maximum of the valence band occurs at the $\mathrm{\Gamma}$-point at $k=0$. The minimum of the conduction band, however, can be located either at the $\mathrm{\Gamma}$-point (as in GaAs) or away of it, e.g. in the *X*-point (as in AlAs). Due to the momentum conservation law (eqn (4.30)), optical transitions between two bands with different wavevectors are possible only with the involvement of lattice vibrations (phonons), which facilitate the momentum conservation. Such transitions across an *indirect band gap* are called *indirect transitions*. They are orders of magnitude weaker than direct transitions and, therefore, semiconductors with indirect band gaps have limited application in optoelectronics and photonics.

Let us now consider in more detail the optical transitions at the direct fundamental absorption edge. We assume that the matrix element $\text{\xea}\cdot {\mathbf{p}}_{cv}(\mathbf{k})$ is nonzero (p.95) at the $\mathrm{\Gamma}$-point and is independent of $\mathbf{k}$. The latter condition is justified if we are interested in the absorption coefficient of an isotropic material at the energy not far from the fundamental absorption edge. For simple parabolic bands the dispersion laws are

With these assumptions, the absorption spectrum is defined by

where

and

is the reduced effective mass.

Comparing eqn (4.59) with eqns (4.5) and (4.8), we find that the integral in eqn (4.59) has the meaning of the doubled three-dimensional density of states $2{\mathrm{\rho}}_{3}(\mathrm{\hslash}\mathrm{\omega}-{E}_{g})$ obtained for the quadratic dispersion law with the reduced electron–hole effective mass. This quantity, known as the *reduced* (or *joint*) *3D density of states*, can be written as

The additional factor 2 in eqn (4.62) as compared to eqn (4.8) appears as a result of taking into account the two possible spin projections of an electron. With this definition, the absorption coefficient is expressed as

This absorption spectrum is shown schematically in Fig. 4.1. It can be seen that the square-root behaviour of the absorption coefficient above the band gap is determined by the spectrum of the joint electron–hole density of states.

(p.96) Let us consider a quasi-equilibrium state when there are two distinct quasi-Fermi levels for the electrons and holes, ${F}_{c}$ and ${F}_{v}$, respectively. This situation can be realized under the conditions of strong optical pumping or current injection in a p–n diode structure. In this case, the respective integration over $\mathbf{k}$ for simple parabolic bands gives

where the wavevector

corresponds to the vertical transition with the transition energy $\mathrm{\hslash}\mathrm{\omega}$. We see from eqn (4.64) that $\mathrm{\alpha}(\mathrm{\hslash}\mathrm{\omega})$ becomes negative if

Inequality (4.66) is equivalent to

which leads to

Inequality (4.68) is known as the *Bernard–Duraffourg population inversion condition*. The negative sign of absorption means that there is a positive *gain* in the
(p.97)
medium $g(\mathrm{\hslash}\mathrm{\omega})=-\mathrm{\alpha}(\mathrm{\hslash}\mathrm{\omega})$. The gain exists for the transition energies located in the spectral region

The gain maximum within this interval is determined by the balance between the increasing absorption and decreasing Fermi–Dirac inversion factor, $\left({f}_{v}-{f}_{c}\right)$, with increasing transition energy $\mathrm{\hslash}\mathrm{\omega}$.

Equation (4.63) includes the expression for the reduced density of states ${\mathrm{\rho}}_{r}(\mathrm{\hslash}\mathrm{\omega}-{E}_{g})\sim \sqrt{\mathrm{\hslash}\mathrm{\omega}-{E}_{g}}$, which is derived for simple parabolic bands. The most general expression for the absorption coefficient due to transitions between non-parabolic conduction and valence bands with arbitrary occupation and allowed spectral broadening reads:

## 4.2.4 Spontaneous emission

Let us assume that the photon density is small so that we can neglect the stimulated emission as compared with the spontaneous emission. Then the emission rate for the transition from the conduction band to the valence band is given by

Equation (4.71) says that in order to observe the emission of a photon it is first necessary to excite electrons in the conduction band and holes in the valence band. This can be done optically through absorption of an incident light beam. This process of optical absorption followed by emission is called *photoluminescence* (PL). Alternatively, the electrons and holes can be excited through electrical injection in doped p–n structures. This kind of luminescence process is known as *electroluminescence*.

Anyway, in a luminescence experiment one excites in a semiconductor a non-equilibrium distribution of electron–hole (e–h) pairs. The fastest process after the excitation is usually thermalization towards quasi-equilibrium distributions for electrons ($\phantom{\rule{1.9pt}{0ex}}{f}_{e}$) and holes ($\phantom{\rule{1.9pt}{0ex}}{f}_{h}$), which are often different. In a perfect semiconductor crystal the thermalized electrons and holes are accumulated at the conduction and valence band extrema, where they tend to recombine. From there, several *recombination processes* are possible, both *radiative* and *non-radiative*. The radiative recombination is followed by the spontaneous emission of photons, while in non-radiative processes the energy of the e–h pair is dissipated as heat via excitation of crystal lattice vibrations (phonons).

(p.98)
At high temperatures, the dominant radiative recombination in high-purity and high-quality semiconductors is usually due to *band-to-band transitions* of free electrons and free holes. Let us define ${\mathrm{\tau}}_{r}$ as the average *radiative recombination time* of the e–h pairs. The averaging is necessary since the radiative recombination time may depend on the electron and hole energies. The total decay rate of the excited population of e–h pairs is given by

where ${\mathrm{\tau}}_{\text{nr}}$ defines the average *non-radiative recombination time*. Quantities ${\mathrm{\Gamma}}_{r}=\frac{1}{{\mathrm{\tau}}_{r}}$ and ${\mathrm{\Gamma}}_{\text{nr}}=\frac{1}{{\mathrm{\tau}}_{\text{nr}}}$ are, respectively, the *radiative recombination rate* and the *non-radiative recombination rate*.

The radiative recombination rate of e–h pairs depends on several factors, including the excitation intensity and impurity doping levels. Nevertheless, it is always much higher in direct band-gap semiconductors (like GaAs, InAs, GaN, ZnSe, ZnO, and many others) as compared with indirect band-gap semiconductors (such as Si, Ge, and AlAs) due to the much larger interband optical matrix elements (Pankove 1971).

For a direct band-gap semiconductor, the shape of the band-to-band PL spectra is determined by the spectrum of joint density of states ${\mathrm{\rho}}_{r}(\mathrm{\hslash}\mathrm{\omega})\sim \sqrt{\mathrm{\hslash}\mathrm{\omega}-{E}_{g}}$ (eqn (4.62)) and the quasi-equilibrium distribution functions ${f}_{e}$ and ${f}_{h}$, which can be approximated for low excitation density by Boltzmann distributions. As a result, the PL spectral shape is

for $\mathrm{\hslash}\mathrm{\omega}>{E}_{g}$, and ${I}_{\text{PL}}(\mathrm{\hslash}\mathrm{\omega})=0$ otherwise.

At sufficiently low temperatures, the PL spectrum is determined by *excitonic effects* in pure semiconductors and by localization of carriers at impurities (donors or acceptors) in doped semiconductors with high enough doping levels. The description of excitonic effects is left for Section 4.4. These effects are especially important in wide band-gap semiconductors, such as GaN, AlN, and ZnO, where they survive up to room temperature. As for doped semiconductors, some new types of transitions appear there such as *free-to-bound transitions* and *donor–acceptor pair transitions* (see, e.g., Yu and Cardona 2010). The former transitions involve recombination of a free electron (hole) and a hole (electron) bound to an acceptor (donor) in a p-type (n-type) semiconductor. The donor–acceptor pair transitions occur in *compensated* semiconductors containing both donors and acceptors and imply recombination of the electrons bound to the donors and the holes bound to the acceptors.

# (p.99) 4.3 Interband momentum matrix elements

In the previous sections of this chapter we have shown that all interband optical processes in direct band-gap semiconductors are primarily determined by the values of interband momentum matrix elements near edges of the conduction and valence bands. However, the term involved, ${\left|\text{\xea}\cdot {\mathbf{p}}_{cv}\right|}^{2}$, defines not only the strength of the absorption, emission, and gain, but also the polarization selection rules as determined by crystal symmetry. In this section we calculate the interband momentum matrix elements in terms of the respective Kane parameters for the crystals with zinc-blende and wurtzite symmetries. We show, in particular, that ${\left|\text{\xea}\cdot {\mathbf{p}}_{cv}\right|}^{2}$ in cubic zinc-blende crystals is isotropic, while in wurtzite crystals it is anisotropic owing to the lower hexagonal symmetry.

## 4.3.1 Interband momentum matrix elements in zinc-blende crystals

To evaluate the interband momentum matrix element ${\mathbf{p}}_{cv}(\mathbf{k})$ in a cubic zinc-blende crystal we assume that it is independent of the length of the electron wavevector $\left|\mathbf{k}\right|$, but is dependent on the $\mathbf{k}$ direction. To take account of this dependence, we will average the quantity ${\left|\text{\xea}\cdot {\mathbf{p}}_{cv}\right|}^{2}$ over the solid angle. If the electron wavevector $\mathbf{k}$ has a general direction specified by spherical coordinates $\left(k,\mathrm{\theta},\mathrm{\varphi}\right)$ as

the average over the solid angle reads:

To calculate ${\mathbf{p}}_{cv}$ as a function of $\mathrm{\theta}$ and $\mathrm{\varphi}$ we employ the eight-band Kane’s model presented in Chapter 3. In this model, the wave functions at the band edges (given by eqns (3.34)–(3.41) in Section 3.2.2) are identified as $|iS\uparrow \u3009$ (or $|iS\downarrow \u3009$) for electrons, $|\frac{3}{2},\pm \frac{3}{2}\u3009$ for heavy holes, $|\frac{3}{2},\pm \frac{1}{2}\u3009$ for light holes, and $|\frac{1}{2},\pm \frac{1}{2}\u3009$ for spin split-off holes. Equations (3.34)–(3.41) are obtained for the electron wavevector pointing strictly along the *z*-axis for the involved *p***-**like functions. Averaging of the momentum matrix element over all directions of the electron wavevector requires derivation of the band-edge wave functions in the general coordinate system where the new $z$-axis is set along the $\mathbf{k}$ direction. The explicit expressions for such wave functions, obtained using the respective coordinate transformations for the spatial variables (*X*, *Y*, and *Z*), can be found, for example, in the textbook by Chuang (2009). With these expressions, the momentum matrix elements for conduction to heavy-hole transitions are derived as
(p.100)

For conduction to light-hole transitions, the matrix elements are

Here

and

Consider a zinc-blende film with the growth axis along the *z*-direction. The polarization dependence of the absorption coefficient can be obtained by evaluating $\text{x\u0302}\cdot {\mathbf{p}}_{cv}$, $\text{\u0177}\cdot {\mathbf{p}}_{cv}$, and $\text{\u1e91}\cdot {\mathbf{p}}_{cv}$. For example, let us calculate the average of ${\left|\text{\xea}\cdot {\mathbf{p}}_{cv}(\mathrm{\theta},\mathrm{\varphi})\right|}^{2}$ over the solid angle for the TE polarization (only $\text{x\u0302}\cdot {\mathbf{p}}_{cv}\ne 0$) for the transitions between the conduction band with spin up (${u}_{c}=\u3008iS\uparrow |$) and the heavy hole bands ${u}_{v}=|\frac{3}{2},\pm \frac{3}{2}\u3009$. We find from eqn (4.78) that the matrix element involving
(p.101)
${u}_{v}=|\frac{3}{2},-\frac{3}{2}\u3009$ is equal to zero and does not contribute to absorption. Substituting the nonzero matrix element (4.76) into eqn (4.75) we obtain

The result (4.86) is the same for the other spin in the conduction band. Therefore, a factor of 2 should be additionally used to obtain the total optical absorption. The same result is obtained for the light hole bands ${u}_{v}=|\frac{3}{2},\pm \frac{1}{2}\u3009$. It is easy to check that ${\left|\text{x\u0302}\cdot {\mathbf{p}}_{cv}\right|}^{2}$, ${\left|\text{\u0177}\cdot {\mathbf{p}}_{cv}\right|}^{2}$, and ${\left|\text{\u1e91}\cdot {\mathbf{p}}_{cv}\right|}^{2}$ are equal to each other for all possible transitions, as expected for a cubic crystal, and hence the optical absorption in bulk cubic semiconductors is isotropic.

Comparing eqn (4.57) with the definition of the Kane parameter $\mathbf{P}$ (or ${\mathbf{E}}_{\mathbf{P}}$ in energy units), given by eqns (3.43) and (3.49) in Section 3.2.2, we find

and

Recall that the electron effective mass in Kane’s model is expressed through the parameters ${\mathbf{E}}_{\mathbf{p}}$, ${E}_{g}$, and ${\mathrm{\Delta}}_{so}$ (see eqn (3.50) in Section 3.2.2). Equations (4.86) and (4.88) therefore allow one to relate the strength of the fundamental absorption edge to other experimentally measurable band parameters.

## 4.3.2 Interband momentum matrix elements in wurtzite crystals

In the eight-band Kane’s model, the interband momentum matrix elements in a wurtzite crystal are related to the two Kane parameters, ${\mathbf{P}}_{1}$ and ${\mathbf{P}}_{2}$ (or ${\mathbf{E}}_{pz}$ and ${\mathbf{E}}_{px}$ in energy units), given by eqns (3.84) and (3.85) ((3.90) and 3.91)) in Section 3.2.3. In contrast to zinc-blende crystals, optical absorption in a bulk wurtzite crystal is anisotropic. The polarization selection rules are summarized in Table 4.1, which includes the polarization-dependent band-edge squired momentum matrix elements obtained in terms of ${\mathbf{E}}_{pz}$ and ${\mathbf{E}}_{px}$ (Chuang and Chang 1996). The calculation is performed using the band-edge functions represented by eqns (3.113)–(3.116). The parameters $p$ and $q$ in Table 4.1 are defined by eqns (3.117), (3.118), and (3.125)–(3.127). The reference energy for these definitions has been set at ${E}_{v}=0$, where ${E}_{v}$ is the hypothetical energy of the $ch$ band edge (p.102) in the wurtzite crystal with ignored spin–orbit splitting. The bottom line in Table 4.1 represents the sum rule for each polarization.

Table 4.1 Strain-dependent squared momentum matrix elements ${\left|\u3008{u}_{c}|\text{\xea}\cdot \mathbf{p}|{u}_{i}\u3009\right|}^{2}$ (*i = hh, lh, ch*) for $\text{\xea}$ polarization along the *c*-axis (*z*-direction) and perpendicular to the *c*-axis.

Valence band |
$\text{\xea}\phantom{\rule{thinmathspace}{0ex}}\parallel \phantom{\rule{thinmathspace}{0ex}}c$ |
$\text{\xea}\mathrm{\perp}c$ |
---|---|---|

$hh$ band |
0 |
$\frac{{m}_{0}}{4}}{\mathbf{E}}_{px$ |

$lh$ band |
${q}^{2}\left({\displaystyle \frac{{m}_{0}}{2}}{\mathbf{E}}_{pz}\right)$ |
${p}^{2}\left({\displaystyle \frac{{m}_{0}}{4}}{\mathbf{E}}_{px}\right)$ |

$ch$ band |
${p}^{2}\left({\displaystyle \frac{{m}_{0}}{2}}{\mathbf{E}}_{pz}\right)$ |
${q}^{2}\left({\displaystyle \frac{{m}_{0}}{4}}{\mathbf{E}}_{px}\right)$ |

Sum |
$\frac{{m}_{0}}{2}}{\mathbf{E}}_{pz$ |
$\frac{{m}_{0}}{2}}{\mathbf{E}}_{px$ |

In the absence of elastic strain, the difference between the parameters $p$ and $q$ is determined by the ratio of the $lh$ to $ch$ band edge energies. This ratio is large in crystals where the crystal-field split-off energy (${\mathrm{\Delta}}_{cr}$) is much larger than the spin–orbit split-off energy (${\mathrm{\Delta}}_{so}$). This is, for example, the case of wurtzite-type ZnO, ZnS, and GaN. As a consequence, both $c-hh$ and $c-lh$ interband absorption edges in these crystals are strong for $\text{\xea}\mathrm{\perp}c$, whereas the $c-ch$ absorption edge is stronger for $\text{\xea}\phantom{\rule{thinmathspace}{0ex}}\parallel \phantom{\rule{thinmathspace}{0ex}}c$. The application of elastic strain can drastically modify these selection rules.

# 4.4 Excitons

The concept of *excitons* proposed by Frenkel (1931) is important for semiconductor physics since it describes optical absorption and emission at sufficiently low temperatures near the fundamental band edge of all direct band-gap semiconductors. It became even more significant since wide band-gap semiconductors and low-dimensional structures were invented, where excitonic effects often dominate up to room temperature.

When an e–h pair is excited in a medium, the electron is attracted to the created hole via Coulomb forces. Such a bound electron–hole pair no longer represents two independent particles; instead a new excited state—an exciton—appears. The exciton exhibits the correlated motion of the constituent electron and hole in the medium. This motion does not carry charge, but carries energy. The exciton formation reduces the total energy of the e-h pair by an amount equal to the binding energy. Sharp excitonic resonances are well distinguished in optical spectra below the fundamental absorption edge. The resonances exhibit observable changes at exposure to external electromagnetic fields, elastic stress, and impurity contamination; thus, their study can help to determine material parameters. They are very (p.103) sensitive to emergence nearby of surface plasmon excitations, which opens the route to modifying optical properties via resonant exciton–plasmon interactions.

The type and manifestation of the excitons depend on the medium. In molecular crystals, there is *the Frenkel exciton* of small radius. Its spatial extension is restricted to a region around a single unit cell. An applied electromagnetic field provides electronic polarization of an atom, which in fact is the excitation of its electron to higher states with different parity. After removal of the field, the polarization oscillates with certain frequency. It can resonate with the polarization of neighbouring atoms via interatomic interactions. Thus, the moving Frenkel exciton in the crystal is, in fact, a propagating wave of intra-atomic (intra-molecular) excitation energy (Toyazawa 2003). In ionic crystals, one distinguishes the *charge-transfer exciton* formed always by the electron and hole situating on neighbouring atoms. The *Wannier–Mott exciton* comprises a conduction-band electron and a valence-band hole, which are separated by many interatomic spacings (Wannier 1937; Mott 1938). This type of exciton has been observed in many semiconductors (see the books by Knox (1963) and Rashba and Sturge (1982) for references to early studies). The properties of the Wannier–Mott excitons in semiconductor crystals will be described in this section in the framework of the *effective mass approximation*.

## 4.4.1 Effective mass approximation

Let us assume that the energy dispersion relation for a single band *n* near $\mathbf{k}=0$ is given by

for the Hamiltonian

so that

Suppose that, beside the periodic crystal potential $U(\mathbf{r})$, the electron is subjected to some extra perturbation potential $V(\mathbf{r})$. The perturbation potential can be, for example, an electrostatic potential generated by a charged impurity, a potential defined by an external electrical or magnetic field, or the electron–hole Coulomb potential within an exciton. Then the most important statement of the *effective mass approximation* is that the solution for the Schrödinger equation with the perturbation $V(\mathbf{r})$

(p.104) can be obtained by solving another equation

for the *envelope function* $F(\mathbf{r})$, which can be approximated by

Here ${u}_{n0}(\mathbf{r})$ is the periodic Bloch amplitude. The *effective mass equation* (4.93) contains only the extra perturbation potential $V(\mathbf{r})$, whereas the periodic potential $U(\mathbf{r})$ determines the energy bands and the effective masses. Sometimes, this approach is also referred to as the *envelope function approximation*. The applicability of the model is limited by treatment of the perturbation potentials that are smooth on the scale of the lattice constant. The envelope function approximation can be naturally generalized for degenerate bands, such as the valence bands of zinc-blende- and wurtzite-type semiconductors (Tsidilkovski 1982).

## 4.4.2 Wannier–Mott excitons

In a dielectric crystal characterized by the band gap ${E}_{g}$, the excitation by an electromagnetic field with $\mathrm{\hslash}\mathrm{\omega}>{E}_{g}$ can create an electron in the conduction band and a hole in the valence band, which may make up an exciton. For simple conduction- and valance-band structures, the exciton resembles a hydrogen atom. The hole, which usually possesses the heavier mass, is in centre, and the electron rotates around it, held by the Coulomb attraction (Fig. 4.2).
(p.105)
The attraction force results from the Coulomb potential $U(r)={e}^{2}\phantom{\rule{thinmathspace}{0ex}}/\phantom{\rule{thinmathspace}{0ex}}4\mathrm{\pi}{\mathrm{\epsilon}}_{0}\mathrm{\epsilon}r$, where *r* is the electron–hole distance.

The effective mass approximation can be conveniently used to determine the parameters of the Wannier–Mott exciton. The conventional procedure is to decompose the motion of the electron and hole into two independent parts: (i) the motion of their centre of mass, and (ii) the relative motion of the electron and hole with respect to the centre. The consideration of the centre-of-mass motion is equivalent to introducing a particle with the translational mass $M={m}_{e}^{\ast}+{m}_{h}^{\ast}$ and the centre-of-mass wavevector $\mathbf{K}={\mathbf{K}}_{e}+{\mathbf{K}}_{h}$. The potential in the crystal for such a motion has the translational invariance; therefore **K** is the good quantum number.

The simplest case for consideration is a homogeneous direct band-gap semiconductor, which has parabolic energy dispersion in both conduction and valence bands: ${E}_{e}={E}_{g}+{\mathrm{\hslash}}^{2}{k}_{e}^{2}/2{m}_{e}^{\ast}$ and ${E}_{h}={\mathrm{\hslash}}^{2}{k}_{h}^{2}/2{m}_{h}^{\ast}$. The wave function of the exciton can be expressed via the Bloch functions at the extremum point, i.e. Γ point, and the complete solution for the envelope function is of the form

where ${\mathbf{r}}_{e}$ (${\mathbf{r}}_{h}$) is the radius vector of the electron (hole), $\mathbf{r}={\mathbf{r}}_{e}-{\mathbf{r}}_{h}$, *V* is a normalization crystal volume, and the position of the centre of mass of the exciton is given by the radius vector

$\mathrm{\phi}(\mathbf{r})$ satisfies the two-particle (hydrogen-like) Schrödinger equation

where $\mathrm{\epsilon}$ is the low-frequency dielectric constant (its dispersion is neglected).

By analogy with the hydrogen atom, the relative motion of the electron and the hole within the exciton features both discrete levels (or bound states) and continuum states (see, e.g., Bethe and Salpeter 1957; Landau and Lifshitz 1977). The eigenfunctions of eqn (4.97) can be expressed as

for bound states ($E<0$) and

for continuum states ($E>0$).

(p.106) The spherical harmonics ${Y}_{lm}(\mathrm{\theta},\mathrm{\varphi})$ are presented in Section 2.4.3 (eqn (2.154)) and general expressions for the radial functions ${R}_{nl}(r)$ and ${R}_{El}(r)$ can be found, for example, in Landau and Lifshitz (1977) and Chuang (1995). For bound states, the first few wave functions are:

The exciton Bohr radius ${a}_{B}$ is written as

where $\mathrm{\mu}={m}_{e}^{\ast}{m}_{h}^{\ast}\phantom{\rule{thinmathspace}{0ex}}/\phantom{\rule{thinmathspace}{0ex}}({m}_{e}^{\ast}+{m}_{h}^{\ast})$ is the exciton reduced effective mass. The radius is enlarged compared to that of a hydrogen atom due to the smaller masses of the constituents and the influence of the semiconductor medium $(\epsilon ~10).$

The energy levels are quantized as

where the exciton Rydberg

Such a representation is equivalent to choosing a coordinate system for the relative electron and hole motion, where zero coincides with the exciton centre of mass. This electron–hole pair also makes a translational motion as a whole with the wavevector $\mathbf{K}$. Therefore, the total energy of bound exciton states with respect to the crystal ground state $|0\u3009$, corresponding to the empty conduction band and the completely filled valence band (see Fig. 4.3(a)), is given by

where $M={m}_{e}^{\ast}+{m}_{h}^{\ast}$.

(p.107) The resonant optical creation of the exciton can be realized by photon absorption. Because the photon has a negligibly small momentum, the exciton formation corresponds to the discrete set of energies

These exciton energy levels are expected to be markedly closer to the ionization continuum in comparison with the hydrogen atom.

It was shown by Elliott (1957) that the optical matrix element between the initial state ($|i=|0\u3009$) and the final state $f|\u3009$ corresponding to the electron–hole pair can be expressed in the dipole approximation as

At $\mathbf{r}=0$, the wave functions ${\mathrm{\phi}}_{nlm}(\mathbf{r})$ and ${\mathrm{\phi}}_{Elm}(\mathbf{r})$ represented by eqns (4.98) and (4.99) are not equal to zero only if $l=m=0$. This means that only $s$-states can contribute to the optical absorption and the ground-state exciton wave function is

The energy of the ground-state 1*s* exciton level (with respect to ${E}_{g}$): ${E}_{b}=Ry$, is frequently referred to as the *exciton binding energy*.

(p.108) Substituting the matrix element (4.109) into eqn (4.34) and using eqn (4.21) we calculate the excitonic absorption spectrum as

where the summation over $n$ includes both the bound and continuum exciton states. As for interband optical transitions, the polarization selection rules are governed by the factor ${\left|\text{\xea}\cdot {\mathbf{p}}_{cv}\right|}^{2}$.

For bound states, eqn (4.111) transforms to

The phenomenological exciton linewidth can be introduced by replacing the delta function in eqn (4.112) by a respective Lorentzian function.

When the excitons are ionized in the continuum, the wave functions of the almost free electron and hole states are still sensitive to the Coulomb attraction. As a result, the absorption spectrum at the continuum energy differs from that given by interband transitions (Elliott 1957; Haug and Koch 1990). For the continuum-state contribution, the absorption spectrum can be expressed as

where ${\mathrm{\rho}}_{r}^{3\text{D}}(\mathrm{\hslash}\mathrm{\omega})$ is the reduced 3D density of states (eqn (4.62)) and the factor ${S}_{3\text{D}}$ known as the *3D Sommerfeld enhancement factor* is given by

where

The total excitonic absorption is ${a}_{ex}(\mathrm{\hslash}\mathrm{\omega})={a}_{b}(\mathrm{\hslash}\mathrm{\omega})+{a}_{c}(\mathrm{\hslash}\mathrm{\omega})$. If $\mathrm{\hslash}\mathrm{\omega}\gg {E}_{g}$, ${a}_{ex}(\mathrm{\hslash}\mathrm{\omega})$ approaches the interband optical absorption spectrum caused by free electron–hole transitions (eqn 4.63). A schematic representation of the excitonic spectrum with a finite linewidth is shown in Fig. 4.3(b).

The exciton theory presented above has been developed for crystals with nondegenerate and parabolic valence and conduction bands. However, crystals with the zinc-blende structure have a degenerate band at the $\mathrm{\Gamma}$ point, where the exciton is formed, and therefore the theory for simple bands cannot be directly applied. Baldereschi and Lipari (1970, 1971) investigated the exciton spectrum in the case of degenerate bands using symmetry considerations and second-order (p.109) perturbation theory. They obtained the following simple analytical expression for the binding energy as a function of the band parameters:

where ${R}_{0}$ is the Rydberg (eqn 4.106) relative to an effective mass ${\mathrm{\mu}}_{0}$,

and ${\mathrm{\gamma}}_{1}$, ${\mathrm{\gamma}}_{2}$, and ${\mathrm{\gamma}}_{3}$ are the Luttinger parameters (eqns (3.67)–(3.69)). The effective mass ${\mathrm{\mu}}_{3}$, being generally large in zinc-blende lattice, describes the effects of inversion asymmetry. The first term in eqn (4.116) comes from the isotropic part of the exciton Hamiltonian, the second and the third terms represent the effect of the anisotropy, and the last term represents inversion asymmetry effects. Quantitative estimations performed with eqn (4.116) showed that the accuracy of the simple model in which the degenerate valence bands are replaced by an ‘average’ simple band can be quite satisfactory (Baldereschi and Lipari 1971).

In the direct band-gap semiconductors, a photon with an energy $\mathrm{\hslash}\mathrm{\omega}$ of the order of 1 electronvolt creates an exciton with a momentum $\mathbf{K}\sim 0$. Usually, its kinetic energy can be neglected. The almost vertical transitions provide sharp lines in optical spectra (Fig. 4.3(b)). Their appearance is clear confirmation of the existence of excitons. To allow the optical transitions in indirect band-gap semiconductors (Fig. 4.4), the extra momentum should somehow be lost. Most frequently, this occurs by transferring the momentum to the crystal lattice, by the creation of phonons. If the optical transition leads to the emission of a photon with the wavevector **q**, which is rather small, the momentum conservation law can be expressed as

for the process involving one phonon with the wavevector ${\mathbf{q}}_{\text{ph}}$, and

for the process involving two different phonons. The latter law is more flexible, because additional freedom appears to accommodate the extra momentum (Gross et al. 1971).

(p.110) The peaks of phonon-assisted transitions in the optical spectra are usually smooth; they reflect rather the distribution of the phonon density of states. As a consequence, specific spectral features—the so-called phonon replicas—appear in the optical spectra. They can dominate the luminescence spectra in indirect band-gap semiconductors.

In semiconductors, the formation of different exciton complexes is possible and often energetically favourable. The excitons can be bound by donors or acceptors, which decreases their energy. Other complexes are the *biexcitons* and *charged excitons, or trions*, originally called ‘excitonic molecules’ and ‘excitonic ions’ by analogy with the hydrogen molecule H_{2} and a hydrogen ion (Lampert 1958).

Typical values of the exciton binding energy and Bohr radius in different semiconductors are given for reference in Table 4.2. As can be anticipated, the higher the exciton binding energy, the smaller the exciton radius. In narrow band-gap semiconductors it is possible to observe excitons only at low temperatures or on applying a magnetic field.

Table 4.2 The binding energy and Bohr radius of free excitons shown in the descending ${E}_{B}$ order. The band-gap values at room temperature and the respective material groups are specified.

Semiconductor |
Binding energy (meV) |
Bohr radius (nm) |
Band-gap energy (eV) |
Material groups |
---|---|---|---|---|

CuCl |
190 |
0.7 |
3.2 |
I–VII |

AlN |
50–70 |
1.2 |
6.2 |
III–V |

ZnO |
59 |
1.8 |
3.4 |
II–VI |

CdS |
29 |
2.8 |
2.42 |
II–VI |

GaN |
23 |
2.7 |
3.42 |
III–V |

ZnSe |
19 |
5 |
2.7 |
II–VI |

AlAs |
17 |
4.2 |
2.95 |
III–V |

Si |
15 |
4.9 |
1.12 |
IV |

ZnTe |
13 |
5.6 |
2.26 |
II–VI |

InP |
5.1 |
11.3 |
1.4 |
III–V |

GaAs |
5 |
12 |
1.42 |
III–V |

Ge |
3.6 |
25 |
0.66 |
IV |

InAs |
1 |
34 |
0.4 |
III–V |

## 4.4.3 Classification of excitonic states by symmetry

The ground excitonic state originating from an electron in a conduction band of symmetry ${\mathrm{\Gamma}}_{c}$ and a hole in a valence band of symmetry ${\mathrm{\Gamma}}_{v}$ inherits the symmetries of the constituents via the representation ${\mathrm{\Gamma}}_{\text{ex}}$ obtained as the *direct product* of irreducible representations ${\mathrm{\Gamma}}_{\text{c}}$ and ${\mathrm{\Gamma}}_{\text{v}}$ (Bir and Pikus 1974; Ivchenko and Pikus 1997):

(p.111) Here the direct product of two representations ${\mathrm{\Gamma}}^{\mathrm{\mu}}$ and ${\mathrm{\Gamma}}^{\mathrm{\nu}}$ (with dimensions $\mathrm{\mu}$ and $\mathrm{\nu}$) of the same group of the order $n$,

and

is defined as

In eqns (4.123)–(4.125), ${M}_{i}$ are square matrices of size $\mathrm{\mu}$, ${N}_{i}$ are square matrices of size $\mathrm{\nu}$, and $M\times N$ is the *direct matrix product* so that

(p.112) The double subscript $ij$ ($i=1\dots m$, $j=1\dots n$) in eqn (4.126) enumerates rows of the matrix $M\times N$ in compliance with some arbitrary rule, whereas the matrix columns are numbered by the double index $kl$ following exactly the same rule.

The direct product of the two representations ${\mathrm{\Gamma}}^{\mathrm{\mu}}\times {\mathrm{\Gamma}}^{\mathrm{\nu}}$ is also a representation with dimension $\mathrm{\mu}\cdot \mathrm{\nu}$. However, though ${\mathrm{\Gamma}}_{\text{c}}$ and ${\mathrm{\Gamma}}_{\text{v}}$ are irreducible representations, their direct product ${\mathrm{\Gamma}}_{\text{ex}}$ is, in general, a reducible representation. By means of the theory of group representations (Heine 1960; Knox and Gold 1964), it can be expanded in a certain number $s$ of irreducible representations:

The sign ‘+’ in eqn (4.127) denotes the *direct sum of representations*. For two representations ${\mathrm{\Gamma}}^{\mathrm{\mu}}$ and ${\mathrm{\Gamma}}^{\mathrm{\nu}}$ (eqns (4.123) and (4.124)), this operation is defined as

where the *direct matrix sum* $M+N$ is specified as a square matrix of size $\mathrm{\mu}+\mathrm{\nu}$:

The integer coefficients ${a}_{i}$ in eqn (4.127) show how many times the irreducible representation ${\mathrm{\Gamma}}_{i}$ enters the expansion.

Equations (4.95) and (4.97) describe the states of the so-called *mechanical exciton*, neglecting the exchange interaction between the electron and the hole. As a rule, the excitonic state is multiply degenerate. The short-range and long-range exchange interactions between the electrons and holes partly eliminate the degeneracy of the exciton ground state and split it in accord with the corresponding irreducible representations. The short-range part causes splitting of the exciton states at a zero wavevector, corresponding to different orientations of the spins of the particles. For most semiconductors, this splitting is less than 100 µeV and therefore is not relevant to any observable optical effects. However, as discussed in Chapter 6, the exchange splitting of excitons is considerably enhanced in semiconductor nanostructures, where it can noticeably affect the recombination and relaxation dynamics of excitons. The long-range part is responsible for the longitudinal–transverse splitting of the optically allowed exciton states. Consideration of a Hamiltonian describing this splitting gives the selection rules for the exciton optical transitions (Bir and Pikus 1974; Ivchenko and Pikus 1997). Below we give a brief description of the exciton fine structure and selection rules for two important crystal classes: zinc-blende and wurtzite crystals with direct band gaps at the $\mathrm{\Gamma}$ point.

## (p.113) 4.4.4 Excitons in zinc-blende and wurtzite crystals

According to the T_{d} point symmetry of a zinc-blende-type crystal (e.g. GaAs, InAs, InP), the 1*s* ground state of the exciton described by the ${\mathrm{\Gamma}}_{6}\times {\mathrm{\Gamma}}_{8}$ representation is eightfold degenerate (see Fig. 3.3 and Table 3.1 in Section 3.2.2). The short-range exchange interaction splits the ${\mathrm{\Gamma}}_{6}\times {\mathrm{\Gamma}}_{8}$ state into three levels ${\mathrm{\Gamma}}_{6}\times {\mathrm{\Gamma}}_{8}={\mathrm{\Gamma}}_{3}+{\mathrm{\Gamma}}_{4}+{\mathrm{\Gamma}}_{5}$ (${\mathrm{\Gamma}}_{6}\times {\mathrm{\Gamma}}_{8}={\mathrm{\Gamma}}_{12}+{\mathrm{\Gamma}}_{15}+{\mathrm{\Gamma}}_{25}$ with the notation of Parmenter (1955)). This splitting is generally determined by two constants: ${\mathrm{\Delta}}_{0}$ and ${\mathrm{\Delta}}_{1}$(Bir and Pikus 1974; Ivchenko and Pikus 1997). The constant ${\mathrm{\Delta}}_{1}$, which governs splitting between the ${\mathrm{\Gamma}}_{3}$ and ${\mathrm{\Gamma}}_{5}$ states, is usually small. In the spherical approximation, ${\mathrm{\Delta}}_{1}=0$ and these levels remain degenerate, being shifted relatively to the ${\mathrm{\Gamma}}_{4}$ exciton by ${\mathrm{\Delta}}_{0}$. The triplet dipole-active ${\mathrm{\Gamma}}_{4}$ exciton corresponds to the total angular momentum $J=1$ with projections $M=\pm 1,0$. The ${\mathrm{\Gamma}}_{3}$ and ${\mathrm{\Gamma}}_{5}$ states match the angular momentum $J=2$. In the dipole approximation they are optically inactive.

The ${\mathrm{\Gamma}}_{6}\times {\mathrm{\Gamma}}_{7}$ exciton ground state in a zinc-blende crystal is fourfold degenerate. The short-range exchange interaction splits this state into a dipole-allowed triplet level ${\mathrm{\Gamma}}_{4}$ with $J=1$ and an optically inactive singlet Γ_{2} with $J=0$. If the effective mass of the electron is much less than the effective mass of the hole, the splitting between these states can be approximated as ${{\mathrm{\Delta}}_{0}}^{\prime}=3/4{\mathrm{\Delta}}_{0}$.

The long-range exchange interaction additionally splits the dipole-allowed 1*s* excitons into *longitudinal* and *transverse* states. The terms ‘longitudinal’ and ‘transverse’ define here orientation of exciton polarization with respect to the exciton wavevector $\mathbf{K}$. The frequency difference ${\mathrm{\omega}}_{LT}$ between these states, known as *longitudinal–transverse exciton splitting*, is a measure of light–exciton interaction. It can be expressed via Kane’s parameter $P$ as

In a typical zinc-blende crystal, GaAs, the energy of the longitudinal–transverse splitting $\mathrm{\hslash}{\mathrm{\omega}}_{LT}\approx 0.08$ meV.

Wurtzite-type crystals with ${C}_{6v}$ point symmetry possess three exciton series labelled as A, B, and C according to the valence bands involved. In compounds with the normal ordering of valence bands (e.g. GaN, ZnS, CdS, CdSe) the series are identified as Γ_{7} × Γ_{9} (A), Γ_{7} × Γ_{7} (B), and Γ_{7} × Γ_{7} (C) (see Table 3.2 and Fig. 3.5 in Section 3.2.3). In common opinion, the ordering of the two first series in wurtzite ZnO is likely reversed, whereas in wurtzite AlN the first and the third series are reversed. The ground states in each exciton series are fourfold degenerate. The short-range exchange interaction removes the degeneracy and splits the Γ_{7} × Γ_{9} states into two levels Γ_{5} (angular momentum projection $M=\pm 1$) and two levels Γ_{6} ($M=\pm 2$), while the Γ_{7} × Γ_{7} states are transformed into four levels: Γ_{1} ($M=0$), Γ_{2} ($M=0$), and two Γ_{5} ($M=\pm 1$). The dipole-allowed optical transitions correspond to Γ_{1} for the photon wavevector $\mathbf{k}\phantom{\rule{thinmathspace}{0ex}}\parallel \phantom{\rule{thinmathspace}{0ex}}c$ and Γ_{5} with $\mathbf{k}\mathrm{\perp}c$, where $c$ is the principal axis of the wurtzite crystal.

(p.114)
The structure of the exciton states in the representative wurtzite semiconductor—GaN—is schematically illustrated in Fig. 4.5. Since both the crystal field and spin–orbit interaction in GaN are rather small, the A, B, and C excitons cannot be treated separately. Therefore, splitting between the exciton states with different symmetries has to be calculated by diagonalization of the full 12 × 12 exciton Hamiltonian. For photon wavevectors **k** ⊥ *c*, the long-range exchange interaction splits the spin-singlet states Γ_{5} into a longitudinal (Γ_{5L}) state and a transverse (Γ_{5T}) state. For A excitons, this is about 1 meV in GaN (Monemar et al. 2008) and about 2 meV in ZnO (Klingshirn et al. 2010; Shubina et al. 2011b).

## 4.4.5 Exciton polaritons

The effects of spatial dispersion are usually much weaker than those arising from the frequency dispersion (see Section 2.1), but in a system with exciton
(p.115)
resonances they can lead to qualitatively new phenomena. Because an exciton is a moving excitation with a finite mass, the induced modification of the dielectric susceptibility is not local but spreads through a crystal. The electromagnetic wave propagates in such a medium by way of combined modes called *exciton polaritons* or simply *polaritons*. This phenomenon was first described by Pekar (1958) in terms of additional waves, because at an exciton resonance the polariton splits into several branches. With increasing $\mathbf{k}$, the upper branch tends to the cone of the light dispersion, while the lower one approximates the transverse exciton dispersion curve (Fig. 4.6).

Hopfield and Thomas (1963) have proposed an instructive description of the spatial dispersion, describing each exciton as a harmonic oscillator, whose eigenfrequency ${\mathrm{\omega}}_{0}$ corresponds to the energy of an excitonic transition. The width of the resonance was introduced as a phenomenological damping parameter $\mathrm{\Gamma}$, whose dependence on the wavevector was neglected. The frequency and spatial dispersion of the background dielectric constant ${\mathrm{\epsilon}}_{b}$, characterizing the susceptibility of the medium far from the exciton resonances, was neglected as well. In the simplest case of dipole-active excitonic states with isotropic masses, the link between polarization and electric field is given by the differential equation

(p.116)
Here, $M$ is the exciton translational mass, ${\mathrm{\alpha}}_{0}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}2{d}^{2}/\mathrm{\hslash}{\mathrm{\omega}}_{0}$ is the dimensionless parameter, and *d* is the matrix element of the optical dipole moment of the exciton. For the case of a single excitonic resonance, the dielectric constant is given by

The quantity $4\mathrm{\pi}{\mathrm{\alpha}}_{0}$ is the polarizability related to the oscillator strength of the excitonic transition.

In the vicinity of the fundamental absorption edge of a direct-gap semiconductor when $\left|\mathrm{\omega}-{\mathrm{\omega}}_{0}\right|\ll {\mathrm{\omega}}_{0}$, eqn (4.132) can be simplified (Ivchenko 1982), and the dielectric constant has the form

Here the damping parameter Γ is equal to half of the homogeneous width of the exciton resonance. The quantity $f=2\mathrm{\pi}{\mathrm{\alpha}}_{0}$ is identified as the *oscillator strength*. For the polariton waves, a certain problem is the description of additional boundary conditions at the sample surface. The analysis of this problem can be found in the literature (Hopfield and Thomas 1963; Ivchenko 1982).

In an isotropic crystal with an exciton resonance, both transverse ($\mathbf{P},\mathbf{E}\phantom{\rule{thinmathspace}{0ex}}\mathrm{\perp}\phantom{\rule{thinmathspace}{0ex}}\mathbf{k}$) and longitudinal ($\mathbf{P},\mathbf{E}\phantom{\rule{thinmathspace}{0ex}}\parallel \phantom{\rule{thinmathspace}{0ex}}\mathbf{k}$) waves can be excited. The respective dispersion equations have the forms

and

The longitudinal–transverse splitting, ${\mathrm{\omega}}_{LT}$, is defined as the splitting between the frequencies at which the dielectric constant goes to infinity (${\mathrm{\omega}}_{T}$) and to zero (${\mathrm{\omega}}_{L}$) for $\mathrm{\Gamma}=0$ and $M\to \mathrm{\infty}$. For a single exciton resonance with the central energy ${\mathrm{\omega}}_{0}$, its value equals

In real crystals with several types of excitons, the dielectric constant is written as (Hopfield and Thomas 1963)

(p.117) where each ${X}_{j}$ contribution is similar to the second term in eqn (4.132) or (4.133). This description can be extended to the bound exciton (BX) states, localized by different impurities. The concept of spatial dispersion is meaningless for them; thus, one should assume their masses ${M}_{j,\text{BX}}=\mathrm{\infty}.$

The oscillator strength for exciton transitions can be expressed via the interband momentum matrix element ${p}_{\text{cv}}$ and the wave function of an electron–hole pair at coinciding coordinates (Elliott 1957). In particular, for a free exciton (FX) it is specified by

If excitons are bound to defects with the density *N*, the oscillator strength takes the form (Rashba and Gurgenishvilli 1962)

where $\mathrm{\Psi}(\mathbf{r})$ is the envelope function describing the exciton centre of mass localization at the donor. For an order-of-magnitude estimation one can take ${f}_{\text{BX}}\sim {f}_{\text{FX}}N{R}_{\text{loc}}^{\text{3}}$, where *R*_{loc} is the exciton localization radius, which is of the order of ${a}_{\text{B}}$ (Shubina et al. 2008).

Disorder in any real system causes fluctuations of excitonic parameters, primarily the spread of the resonance frequency *ω*_{0}. One can distinguish between homogeneous broadening in an ideal crystal, when the line of each individual oscillator is broadened in the same way, and inhomogeneous broadening, when the resonance frequencies are distributed through a certain range, although the lines of individual oscillators are not necessarily extra broadened. The mechanisms responsible for homogeneous broadening are radiative damping and the intrinsic interaction with phonons. The latter dominates at elevated temperatures. These mechanisms provide broadening, which is well described by a Lorentzian with the width $\mathrm{\Gamma}$. Frequently, elastic scattering by impurities is also assumed as the homogeneous broadening mechanism (because it is also well described by a Lorentzian), although it is obviously sample dependent. The factors responsible for inhomogeneous broadening are local electric fields, inhomogeneous potentials induced by impurities or localized phonons, and various structural imperfections. The inhomogeneous broadening can be taken into account by convoluting the homogeneously broadened contour with a Gaussian centred at the same frequency. Then the dielectric function is modelled as

Here the spatial dispersion is neglected because such a representation is not consistent with the polaritonic model at the resonance frequency.

This chapter describes basic optical processes in semiconductors, starting from the definition of density of states. As in Chapter 3, a description is provided for two types of crystal structure—zinc-blende and wurtzite. The concept of excitons, including exciton polaritons, is considered, since it frequently determines the optical absorption and emission near the fundamental band edge in direct band-gap semiconductors.