3.1  Introduction

The devices introduced in this book involve light–matter interaction. Light appears to be a trivial phenomenon known to everyone; however, its mathematical description is quite intricate and has been a major point of discussion in the physics community. Historically, this was manifested in a longstanding dispute on the nature of light, featuring Isaac Newton ^† as the advocate of a corpuscular model and Christiaan Huygens ^‡ as the proponent of its wave nature.

Today, more than 300 years later, both perceptions have been reconciled and the description of light in terms of a wave–particle dualism is well established in the physics community. A full quantum-electrodynamic description, relieving the apparent contradiction between wave and particle nature, is often not required, i.e., in many cases, light can be well described as either a wave or a particle. This will be mirrored in this chapter, where both the classical and the quantum perspectives are introduced.

As often stated, the general guideline is that light may be treated as a wave when propagating in vacuum, yet it has to be quantized in the case of the interaction with matter. However, this is not always true, as, e.g., the behavior of light in a waveguide can well be understood in terms of the wave picture. In fact, quantum-optical effects only started to be discussed intensively in the 1950s and 1960s. Their investigation is closely related to the development of the laser and the requirement of solid theoretical foundations in this field. Simply speaking, the corpuscular theory of light states that it is distributed in quantized packages of energy

, where h is Planck's constant, f is the light frequency, , and ω is the angular frequency . A particle nature is thus attributed to light, where the light particle is known as a photon. The photon may be described quantum mechanically in the framework of second quantization, a formalism in which the light field and other physical field quantities are written as operators. This method was first developed by Paul Dirac ^§ in 1927 in analogy to the first quantization where physical quantities such as position and momentum become operators; see Ref. [1]. Further information may be found in textbooks on quantum mechanics.

In this chapter, we consider electron–photon interaction, i.e., light–matter coupling. There are different electron–photon interaction mechanisms, e.g., the photoelectric effect, Compton scattering, and pair production. In the energy range of interest in optoelectronics, the photoelectric effect is the only relevant effect and will thus be treated exclusively.

The chapter briefly introduces the theoretical treatment of electron–photon interaction for different and mostly strongly simplified cases. A large part is devoted to the two-level system in different formalisms. This allows us to concentrate on the differences and commonalities of the various approaches. We look at the two-level system in the Schrödinger and the Heisenberg representation (Section 3.2) and then move on to a solid-state system (Section 3.3). Section 3.4 deals with the quantum-optical treatment of the two-level system, while Section 3.5 is devoted to the treatment of photonic structures.

Before we enter the detailed mathematics of a quantum mechanical system, some basics will be dis- cussed. In any ensemble of atoms or molecules in interaction with an electromagnetic field, there are three processes occurring: spontaneous emission, absorption, and stimulated emission. Although spontaneous emission is quite common in everyday life—e.g., most light sources like light bulbs or light-emitting diodes (LEDs) operate based on spontaneous emission—it is difficult to treat theoretically. The existence of spontaneous emission, however, was already postulated by Albert Einstein in 1916/1917 with his argument based on thermodynamic equilibrium. For a handwaving explanation, consider an ensemble of atoms placed inside a closed cavity whose walls are held at a constant temperature. The walls, furthermore, are assumed to be perfect absorbers and emitters of radiation. After the system has reached thermal equilibrium, the cavity will be filled with the so-called “blackbody radiation” according to the temperature T of the system. The condition of thermal equilibrium then requires that for each transition, the spontaneous emission matches the absorption of blackbody radiation, necessitating the existence of spontaneous emission.

3.2  Two-Level System

3.3  The Solid-State System

We now move on to a much more complex system, the semiconductor. Not only do we encounter a system of many atoms forming an energy landscape known as band structure here (see also Chapter 1), but, in addition, we also need to take into account a large number of electrons. However, a semiconductor is what the diodes, amplifiers, laser diodes, optical modulators, and solar cells discussed in this book are made out of. In order to get a flavor of a many-body treatment of electron–photon interaction, the derivation of the semiconductor Bloch equations will be discussed briefly here. For an in-depth treatment, the reader is referred to Ref. [7], the references therein, and a wealth of original literature. Since a many-body problem can only be solved approximately, for every case the equations have to be examined separately and the proper approximations made. This is far beyond the scope of this chapter.

In optoelectronics, we customarily deal with direct-bandgap semiconductors. Here, we consider the simplest possible case, a direct-gap two-band semiconductor and derive the relevant equations of motion. It comprises a valence and a conduction band, where in the ground state, all the electrons are in the valence band and fill up its states completely. Similar to the two-level system, they may be raised to the conduction band by optical excitation. Figure 3.2 shows a simplified schematic of the direct-gap two-band semiconductor. Equations of motion for more complicated systems may be obtained by expanding the indices in the derivation below to all the existing bands; a derivation for a multiband system is thus straightforward but more cumbersome.

Looking at the semiconductor–light interaction, we need to keep in mind that the size of an atom is in the range of a single Ångström, while the wavelength of visible light is approximately 400–700 nm, i.e., 4000–7000 Ångström. The electric field is thus approximately constant over the extension of a single atom, and all quantities in Maxwell's equations ^† are averaged over a large number of atoms.

The Hamiltonian of the solid-state system is made up of three contributions,

where is the Hamiltonian of the electronic system, consisting of the sum of the individual single particle contributions, Ĥ ₀, and the Coulomb interaction between the electrons, . As before, denotes the interaction with the light field.

The structure of an inorganic semiconductor is characterized by its crystallinity, i.e., periodicity. In other words, the environment of an electron is the same in position r and position

, where is any lattice vector, i.e., a vector linking two identical sites in an infinite lattice. The periodicity is taken into account in the Bloch theorem ^† for the wave functions, Consequently, the Hamiltonian is expanded in electronic Bloch wave functions, is the volume of the crystal and λ denotes the band index, comprising the conduction band and the valence band . Bloch functions take into account the lattice periodicity of the system by including a lattice periodic function , The introduction of the crystal volume necessitates further explanation, as Equation 3.39 is only valid in an infinite system. Customary crystals are so large that the assumption of infinity is quite fitting. Since for a proper normalization of the wave functions, one needs a finite crystal volume, we consider the (approximately infinite) crystal to be made up of mesoscopic cubes of size and demand periodic boundary conditions of the wave function, i.e., the wave function has to have the same value on either side of the cube.

For the single particle contribution, we derive

( ) is the annihilation (creation) operator of an electron in state .

Since the energies

are rather difficult to assess, parabolic bands are often assumed, corresponding to the effective mass approximation. At the extrema of the band structure, the linear term in the Taylor expansion becomes zero, such that a parabolic approximation may be made. Formally, a constant called the “effective mass,” , may then be introduced, Here, denotes higher-order terms. For the Bloch functions, this implies that the lattice-periodic part becomes momentum independent, i.e., . This approximation is always valid in a certain range around an extremum of the band structure but is used extensively in calculations even far away from these points. In the effective-mass approximation, the electrons behave like free particles with the effective mass rather than the free-particle mass. may be positive or negative, examples being the conduction and valence band in the case of the two-band semiconductor we are looking at. When the sign is negative, this is often included into the energy by transferring to the electron–hole picture. Furthermore, the effective mass, in general, is not independent of direction and must be written as a tensor, The direction dependence of the effective mass is discussed, e.g., in Refs. [9, 10, 11].

Returning to light–matter coupling in semiconductors, we have seen that light couples to electrical dipoles,

Expanding the interaction Hamiltonian in these wave functions yields

where is the optical interband polarization. are the band and momentum-dependent optical transition dipole matrix elements with in-plane momentum k and subband labels , corresponding to the transition dipole matrix elements , in the two-level system.

To obtain Equation 3.46, the following approximations are made: (1) the photon momentum is neglected and (2) intraband transitions are not considered. Rather than retracing the entire derivation, ^† let us discuss these approximations:

1. Neglect of the photon momentum: In this book, we concentrate on light in and around the visible range, i.e., with wavelengths roughly between 400 and 700 nm. For a central wavelength of 550 nm, the wave number is
   $k=2\mathrm{\pi }/\mathrm{\lambda }=11\times 10^6{\mathrm{m}}^{-1}$
   , which is very small compared to typical wave vectors in a semiconductor band structure. Photonic transitions are consequently vertical in a band diagram.
2. Neglect of intraband transitions: Here, only interband transitions will be considered, assuming an excitation at an energy near the band gap. Suppression of intraband transitions in a regular structure becomes obvious from the vanishing photon momentum as discussed earlier.

Looking at the system Hamiltonian, Equation 3.38, we still need to calculate the term

describing Coulomb interaction between the carriers. is specified using the so-called jellium model, which substitutes the ionic background of the lattice with a smooth background. The derivation is not included here, but may be found in Ref. [7]. The final expression of the Hamiltonian is where the second line spells out the interaction for our two-band model. thus describes the repulsion between electrons in the valence band, electrons in the conduction band and between valence and conduction band. is the Fourier transform of the Coulomb potential, where for the calculation a convergence generating factor needs to be used. and are the vacuum and relative material permittivity, respectively. The sum in Equation 3.47 omits the case where obviously the Coulomb matrix element has a singularity; this term is canceled by the ionic background charge in the jellium model.

The complete Hamiltonian is thus

This Hamiltonian is now going to be used for the calculation of the optical polarization

, where are the microscopic interband coherences or polarizations and h.c. denotes the Hermitian conjugate. The optical polarization is a quantity that describes the reaction of matter to a light field, coupling light (electric field ) to matter in the wave equation with the index of refraction n and the vacuum permeability μ₀. The optical polarization is a fundamental parameter from which numerous optical variables can be calculated, e.g., the absorption α or laser gain g being given as see Ref. [8]. In Equation 3.53, the index of refraction n was assumed to be constant in the region of interest. Calculation of the optical polarization thus gives us deep insight into the optical properties of matter and allows us to simulate optical devices such as the ones discussed in this book.

Returning to the actual calculation of the optical polarization, the next step is the insertion of the many-body Hamiltonian into the Heisenberg equation of motion for the microscopic polarizations

Calculating the appropriate commutators and taking the expectation values on both sides renders where and are the carrier densities in the conduction and valence band, respectively. The transition dipole matrix element was assumed to be real, .

Obviously, the polarization as a two-operator quantity on the left-hand side of Equation 3.55 couples to four-operator quantities on the right side via the Coulomb interaction. Computing the equation of motion for these four-operator terms will in turn lead to a coupling to six-operator quantities, which will then couple to eight-operator quantities and so on, with ever-rising computational cost. This so-called infinite hierarchy problem, which is an expression of the many-body character of the electron system, necessitates a truncation at some level. A relatively simple approximation at two-operator level is the dynamic Hartree–Fock approximation. ^† Roughly speaking, it assumes that any particle experiences the impact of all other particles as a mean field created by them. Technically, in Hartree–Fock approximation, we split four-operator quantities into all possible combinations of two-operator quantities. For example, the first Coulomb term of Equation 3.55 yields

using . In the second and third lines, we have taken into account that only terms that are diagonal in momentum are excited in a spatially homogeneous system. Approximating the other terms likewise, the resulting equation of motion for is with renormalized energies where is the exchange self-energy.

Equation 3.56 is not yet closed, but couples to the carrier densities

, . To close the system of equations, the equations of motion for these quantities are derived in analogy to the prior derivation of Equation 3.56, and the following equations are obtained: We now have a closed system of equations for , , and , which constitute the semiconductor Bloch equations in Hartree–Fock approximation. For a given electromagnetic light field, these equations may be solved numerically (starting with the unpolarized system before excitation and calculating the temporal evolution).

Let us now look at the interaction between the carriers, i.e., the Coulomb effects. These are marked by the Coulomb potential

and occur in the renormalized energies and in a renormalization to the Rabi frequency, A characteristic of the Hartree–Fock approximation is thus that Coulomb effects may be represented as renormalization terms.

Neglecting the Coulomb interaction, we regain the optical Bloch equations:

Here, electrons at different k-values do not couple to each other, i.e., the semiconductor is described as an ensemble of uncoupled two-level systems. Only the Coulomb interaction couples the different wave vectors and makes the semiconductor a true many-body system.

Using the optical Bloch equations and an approach similar to the derivation of Equation 3.36 in Section 3.2 and Equation 3.53, one obtains an often-used formula for the semiconductor absorption or gain [12],

is the density of states in a three-dimensional parabolic band structure, which occurs when transferring from an integral over equally spaced k-values to an energy integral [8, 12]. The effective-mass approximation has been used, and is the reduced mass originating from the effective masses of hole and electron. is the unit vector pointing in the direction of the light field, and is the free electron mass. The carrier densities , have been replaced by Fermi–Dirac distributions , , assuming that scattering is fast enough to keep the carrier distributions in thermal equilibrium. This is in general a good approximation especially in a high-density (laser) system; although hole burning or other deviations from an equilibrium distribution may occur [13, 14], kinetic holes in the carrier distribution are often smoothed out by the energy integral in Equation 3.62.

Equation 3.62 is thus obtained making, apart from common assumptions such as an infinite crystal lattice, the following approximations:

• Two-band approximation
• Effective-mass approximation, including the assumption of direction-independence of the effective mass
• Neglect of Coulomb interaction between the carriers; many body effects are only included as a phenomenological broadening
• Assumption of a carrier distribution in thermal equilibrium
• Dephasing time
  $T_2=1/\mathrm{\Gamma }$
  for the decline of the polarization

Looking at laser gain, the spectrum is rather unstructured and deviations between theory and experiment are often compensated by fitting the free parameters of the theory, such as carrier density and broadening; usually, the bandgap is also obtained by fits to experiment. Many gain spectra may thus be fitted with Equation 3.62. A predictive theory, however, needs to take into account many-body effects and closely look at the impact of the other simplifications mentioned above for the structure and experimental conditions it simulates and has to go far beyond the approximations made in Equation 3.62. An excellent overview of the effects of different approximations is given in Ref. [8].

The Hartree–Fock approximation that was introduced above is a widespread method to include many-body effects while keeping computation time reasonable, yet there are many other possible approaches. These include, e.g., (1) the linear case, where the carrier densities are assumed to be zero. The polarization may then be calculated in Hartree–Fock approximation. (2) Calculations beyond Hartree–Fock, where the Coulomb terms are treated in a more elaborate way. In particular, the Coulomb terms also include carrier scattering, which is not mirrored in the Hartree–Fock approximation. Therefore, Hartree–Fock equations require an artificial relaxation to be included. Instead, scattering can be included in the second Born approximation ^† by deriving the equations of motion for four-operator terms and factorizing at this level. This avoids having to include an ad hoc-scattering time, which can only be determined experimentally, and additional changes in temperature, carrier density and other experimental parameters.

Another fact that has not been discussed so far is that realistic semiconductor structures are never perfect. There has been incredible progress in the growth of semiconductor structures since Wolfgang Pauli termed solid-state physics as “dirt physics” in 1931 [15], but there still remain some inhomogeneities and disturbances of the crystal lattice. As was mentioned previously (see Section 3.2.3), disorder introduces a different kind of broadening termed inhomogeneous, where the lineshape is often assumed to be Gaussian.

Thus, different effects are manifest in the lineshape and linewidth of resonances; in other words, the lineshape says a lot about the processes determining it. To illustrate this, Figure 3.3 [16] shows a calculation of a single absorption resonance including homogeneous and inhomogeneous broadening in differing amounts. Both lineshapes include a homogeneous phonon broadening corresponding to a temperature of 30 K; for the calculation of the solid line, an inhomogeneous broadening of 0.287 meV has been added, while for the dotted line only half the disorder strength was included and an additional phenomenological homogeneous broadening of 0.066 meV was introduced via a T ₂-time approach. It becomes obvious that homogeneous broadening shifts oscillator strength to the tails of the resonance. In fact, the tails of a Lorentzian lineshape become so strong that unphysical effects may appear, e.g., absorption below the gain resonance of a laser spectrum.