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.
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.
For a more detailed treatment, we now move to the simplest existing system, the two-level system. It includes one electron that can occupy either one of two nondegenerate states. The two levels of the model may represent, e.g., the highest occupied and lowest unoccupied level in an atom, molecule, or quantum structure. We devote this entire section to the two-level system, including a discussion of the Einstein coefficients as well as calculations in the Schrödinger (Section 3.2.2) and the Heisenberg formalism (Section 3.2.3). In Section 3.4, we again return to the two-level system for a quantum-optical discussion. For an even more detailed treatment of two-level systems, the reader is referred to the book by Allen and Eberly [2].
A schematic of the two-level system is shown in Figure 3.1. The system consists of the ground state, level 1 and one excited state, level 2, with the respective energies ϵ_{1} and ϵ_{2}. One of the levels is occupied by an electron, described by the eigenstates
or . We consider the interaction with an incident light field with photon density , where ω_{0} refers to the angular light frequency corresponding to the energy difference of the two-level system, . Let us now think of an ensemble of identical two-level systems, where the number of electrons in state 1 is N _{1} and the number of electrons in state 2 is N _{2}. The following processes indicated in the schematic may occur in any of them:Figure 3.1 Relevant electron–photon interaction processes in a two-level system.
A,
, and are the Einstein coefficients, named after Albert Einstein (1879–1955), who first introduced them in 1916; see Ref. [3]. Considering all three processes, we get an expression for the overall rate of change of the occupancies of the two levels:We now proceed to a quantum-mechanical treatment of the electron in the two-level system. At this point, we stick to a semiclassical treatment, i.e., a classical treatment of light. As we see, this takes us a long way toward understanding absorption and stimulated emission as well as deriving the optical Bloch equations. The reader is referred to the quantum-optical treatment presented in Section 3.4 to deal with spontaneous emission as it cannot be understood semiclassically.
The two-level system is described by the Hamiltonian of the undisturbed system, H _{0}, and the Hamiltonian of an optical field coupling to the electronic transition dipole moment,
:The wave function
may be expanded in the eigenfunctions of the system,Applying a monochromatic incident field,
, we deriveConsidering an optical field near the resonance frequency of the system ω_{0}, one of the addends in either equation is almost time independent and the other is oscillating quickly. In the rotating wave approximation (RWA), the latter term is ignored due to averaging out and we are left with
For resonance conditions
, this system of coupled equations may easily be solved by time-differentiating the first equation and inserting the second, yieldingStarting with
, i.e., an electron in the excited state, we observe that the electron oscillates between ground and excited states with a cosine dependence and frequency Ω, describing the interplay between absorption and stimulated emission. These so-called Rabi flops are typical for any two-level system driven by an oscillatory field, including atoms in a light field as discussed here, but also particles in an oscillating magnetic field as investigated in nuclear magnetic resonance.Looking at the optical processes possible in a two-level system with or without the incidence of light, we note the following: In the absence of the external perturbation represented by the laser pulse, the system is described by the Hamiltonian Ĥ _{0}. If, at an initial time
, it is in an eigenstate, it will remain in this state for any length of time. This applies to the ground state as well as to the excited state . It is, however, obviously not a realistic description, the reason for this being the absence of spontaneous emission in the semiclassical picture we used here. We see later that spontaneous emission enters the model with a quantization of light.Now let us briefly consider the example of a laser system. Laser operation is basically as follows: An initial photon, usually produced by spontaneous emission, is confined to an active medium using a resonator and amplified by stimulated emission. Obviously, this will only work if stimulated emission exceeds absorption. Going back to Equation 3.10, we know that the Einstein coefficients for absorption and stimulated emission are equal and thus the occupancies of the ground and excited states govern the relative strength. Therefore, inversion has to be achieved for laser operation, i.e., the number of carriers in the excited state has to exceed the number of carriers in the ground state. Under certain conditions, quantum theory also allows lasing without inversion; since this requires intensive tailoring of the system, it will not be discussed here, but the reader is referred to the original literature, see, e.g., Refs. [5,6].
Unfortunately, inversion can, in general, not be achieved in a two-level system as there is no efficient pumping mechanism. Thus, a laser requires at least a three-level system for operation. The relevant equations for the three-level system may be derived in analogy to the case of the two-level system discussed here.
We now introduce yet another description of the two-level system, changing from the Schrödinger picture to the Heisenberg picture. The passage from the Schrödinger picture to the Heisenberg picture is made by transferring the time dependence from the wave functions to the operator; for details, see any see any textbook on quantum mechanics. The formalism in this subsection is similar to the one in Section 3.3, which treats the many-body case. In this subsection, all calculations are presented in detail so the reader can reproduce them, while in Section 3.3, several steps are omitted for the sake of clarity and brevity.
The time evolution of the system in the Heisenberg picture is not governed by the Schrödinger equation any more, but by the Heisenberg equation of motion
The Heisenberg equation allows us to calculate the equation of motion for arbitrary operators. Here, we obviously only consider processes that conserve the number of carriers in the system. We revert to the formalism of second quantization mentioned in the introduction and introduce operators describing the electron. In the following, ĉ_{1} (
) and ĉ_{2} ( ) describe annihilation (creation) operators for electrons in levels 1 and 2, respectively. The relevant particle-conserving two-operator quantities are thus , , , and . The first two quantities are the microscopic polarizations; it will be illustrated in Section 3.3 that the polarization is the relevant quantity when considering light–matter interaction. The latter two quantities and describe the carrier densities in levels 1 and 2, respectively. We observe that the equations of motion for these four quantities are coupled to each other, but form a closed system of equations.The Hamiltonian
of the two-level system may be written asInserting this Hamiltonian into the Heisenberg equation for the polarization
, we derive the equation of motion for this quantity:First, we remind ourselves of the Fermionic anticommutators to be used here
From Equation 3.24, we calculate
Let us now briefly discuss the electron–hole picture at this level. Starting from the ground state where the electrons are in level 1, often only a small fraction of the electrons is lifted to the upper level. It is then frequently advantageous to count the empty spaces in the ground state rather than the electrons. These “holes” carry positive charge because the missing electron implies lack of compensation of nuclear charge. We define
We now solve the polarization equation of motion in the relaxation time approach to illustrate that it leads to a Lorentzian lineshape of the polarization and absorption spectrum. Defining
, we calculateThe integral in Equation 3.34 may now be solved, yielding
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.
Figure 3.2 Schematic of a two-band direct-gap semiconductor.
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,
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,For the single particle contribution, we derive
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,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
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:
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 isThe complete Hamiltonian is thus
This Hamiltonian is now going to be used for the calculation of the optical polarization
,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
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
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: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,Neglecting the Coulomb interaction, we regain the optical Bloch equations:
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],
Equation 3.62 is thus obtained making, apart from common assumptions such as an infinite crystal lattice, the following approximations:
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.
Figure 3.3 Absorption of a single resonance for two cases with equal total width, but differing amounts of homogeneous and inhomogeneous broadening. The solid line corresponds to phonon broadening for a temperature of 30 K and disorder broadening of 0.287 meV. For the dashed line, the disorder strength was halved and an additional phenomenological homogeneous broadening of 0.066 meV was introduced. As expected, greater homogeneous broadening results in a more symmetric line shape and stronger tails.
Thus far, we have worked with semiclassical descriptions, i.e., including a classical light field. Yet it was shown in Section 3.2.1 that any system requires a spontaneous emission term even in a simple model, whereas in Section 3.2.2 it became apparent that the semiclassical model does not offer a term to that extent. Looking at the device aspect, we remark that the simplest optoelectronic device, the LED, emits light entirely due to spontaneous emission, and the laser diode depends on it for start-up. Quantum effects also play a role, e.g., in nanolasers and single quantum dots in resonators [17]. We thus briefly discuss quantization of the electromagnetic field. Due to the rich range of phenomena involved we leave the major discussion to the wealth of textbooks on the market, e.g., Refs. [18, 19] among many others.
Consider again a two-level atom. The Hamiltonian of the fully quantized system consists of three parts,
where
corresponds to the Hamiltonian of the undisturbed electron system (formerly Ĥ _{0}), is the Hamiltonian of the undisturbed light system, and describes the interaction between the two.Looking at a system in free space, we need to take into account a variety of photon modes characterized by their wave vectors q pointing in all directions. Spontaneous emission of a two-level atom can then be calculated. Starting from the excited two-level atom, the system wave function can be written as
We briefly discuss another case here, where there is only one photon mode at the energy
. Consider, e.g., a two-level atom in a cavity only allowing a specific photon mode, leading us to the Jaynes–Cummings model [21]. The undisturbed photon Hamiltonian isThe Jaynes–Cummings Hamiltonian may then be solved by writing it in a matrix representation and diagonalizing this matrix. Looking at Equation 3.65, the wave functions must be linear combinations of
(the atom is in the excited state, state 2, and there are n photons in the system) and state (the atom is in the ground state, state 1, and there are photons in the system). However, and are not the eigenstates of the system any more. They are often called the bare states while the true eigenstates, which result from diagonalization of Hamiltonian (3.64), are referred to as dressed states. On resonance, they are found to beAn interesting case also occurs if the two-level system interacts not with a single photon state
, but with a coherent superposition of photon states. In this case, the probability of finding the system in the excited state will dephase due to the different Rabi frequencies of the various photon states; however, at certain times there will be a revival of the excited state because of constructive interference. This “quantum revival” was predicted theoretically in 1980 [22] and experimentally observed in 1987 [23].Thus far, we have not considered the structure of the photon wave function at all. However, for the Jaynes–Cummings model (see Section 3.4), we consider the interaction of a two-level system with a single photon mode. For the preparation of this system, a high-quality microcavity is required. As we have seen, the strength of the light field at the location of the electron determines the interaction strength between light and matter. Therefore, resonators are widely used to engineer electron–photon interaction processes, and the last section deals with the theoretical handling of these structures.
In resonator structures, calculation of the appropriate photon modes is required to allow access to the strength of electron–photon coupling. This will be clarified using the example of a microcavity. Modes can be established by a transfer-matrix method, which will be briefly discussed. For simplicity, we only look at a planar structure and perpendicular light incidence in this book chapter, while for in-depth information, the reader is referred to Refs. [24, 25]. Information on the transfer matrix method may also be found in Ref. [26] and the references therein.
In free space or homogeneous matter, plane waves
are solutions to Maxwell's equations [4]. Any light field may be expanded in plane waves. The transfer matrix method uses this to express the light field as a plane wave in every layer of a microstructure and appropriately match the conditions at the interfaces.We consider a planar structure where the growth direction is taken as the z-axis. The index of refraction then only depends on z. The homogeneous wave equation reads
Consider first a single boundary between two layers L (left) and R (right). Generally, each mode
is a combination of plane waves traveling in forward and backward directions in each layer,Figure 3.4 Top: Central mode of a GaAs/AlAs microcavity with 10 (on air) and 20 (on substrate) λ/4 layer pairs of GaAs and AlAs with background refractive indices of n GaAs = 3.63 and n AlAs = 2.97, respectively. Bottom: Refractive index of the microcavity along the growth direction z.
Sir Isaac Newton, English natural philosopher (physicist and mathematician), 1643–1727.
Christiaan Huygens, Dutch physicist, astronomer, and mathematician, 1629–1695.
Paul Dirac, British physicist, 1902–1984.
Named after American physicist Isidor Isaac Rabi (1898–1988).
Named after Austrian-born Swiss and American physicist Wolfgang Pauli (1900–1958).
Named after Scottish physicist James Clerk Maxwell (1831–1879).
Named after Swiss physicist Felix Bloch (1905–1983).
See Ref. [7] for details.
Named after English mathematician and physicist Douglas Rayner Hartree (1897–1958) and Soviet physicist Vladimir Aleksandrovich Fock (1898–1974).
Named after German physicist and mathematician Max Born (1882–1970).
Victor Frederick Weisskopf, Austrian-born American physicist (1908–2002).
Eugene Paul Wigner, Hungarian-American physicist (1902–1995).