The semiconductor
Bloch Bloch is a surname of German origin. Notable people with this surname include:
A–F
* (1859-1914), French rabbi
*Adele Bloch-Bauer (1881-1925), Austrian entrepreneur
*Albert Bloch (1882–1961), American painter
* (born 1972), German motor journal ...
equations
[
Lindberg, M.; Koch, S. W. (1988). "Effective Bloch equations for semiconductors". ''Physical Review B'' 38 (5): 3342–3350. do]
10.1103%2FPhysRevB.38.3342
/ref> (abbreviated as SBEs) describe the optical response of semiconductors
A semiconductor is a material which has an electrical resistivity and conductivity, electrical conductivity value falling between that of a electrical conductor, conductor, such as copper, and an insulator (electricity), insulator, such as glas ...
excited by coherent
Coherence, coherency, or coherent may refer to the following:
Physics
* Coherence (physics), an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference
* Coherence (units of measurement), a deri ...
classical light sources, such as lasers
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
. They are based on a full quantum theory, and form a closed set of integro-differential equation
In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function.
General first order linear equations
The general first-order, linear (only with respect to the term involving derivati ...
s for the quantum dynamics In physics, quantum dynamics is the quantum version of classical dynamics. Quantum dynamics deals with the motions, and energy and momentum exchanges of systems whose behavior is governed by the laws of quantum mechanics. Quantum dynamics is relevan ...
of microscopic polarization and charge carrier
In physics, a charge carrier is a particle or quasiparticle that is free to move, carrying an electric charge, especially the particles that carry electric charges in electrical conductors. Examples are electrons, ions and holes. The term is used ...
distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
* Probability distribution, the probability of a particular value or value range of a vari ...
.[
Schäfer, W.; Wegener, M. (2002). ''Semiconductor Optics and Transport Phenomena''. Springer. .
][
Haug, H.; Koch, S. W. (2009). ''Quantum Theory of the Optical and Electronic Properties of Semiconductors'' (5th ed.). World Scientific. p. 216. .
] The SBEs are named after the structural analogy to the optical Bloch equations
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
that describe the excitation dynamics in a two-level atom interacting with a classical electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
. As the major complication beyond the atomic approach, the SBEs must address the many-body
The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...
interactions resulting from Coulomb
The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI).
In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary char ...
force among charges and the coupling among lattice vibrations and electrons.
Background
The optical response of a semiconductor follows if one can determine its macroscopic polarization as a function of the electric field that excites it. The connection between and the microscopic polarization is given by
where the sum involves crystal-momenta of all relevant electronic states. In semiconductor optics, one typically excites transitions between a valence and a conduction band
In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies in w ...
. In this connection, is the dipole
In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways:
*An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system i ...
matrix element between the conduction and valence band and defines the corresponding transition amplitude.
The derivation of the SBEs starts from a system Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
that fully includes the free-particles, Coulomb interaction
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
, dipole interaction between classical light and electronic states, as well as the phonon
In physics, a phonon is a collective excitation in a periodic, Elasticity (physics), elastic arrangement of atoms or molecules in condensed matter physics, condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phon ...
contributions. Like almost always in many-body physics
The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...
, it is most convenient to apply the second-quantization formalism after the appropriate system Hamiltonian is identified. One can then derive the quantum dynamics of relevant observables
In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum physi ...
by using the Heisenberg equation of motion
Due to the many-body interactions within , the dynamics of the observable couples to new observables and the equation structure cannot be closed. This is the well-known BBGKY hierarchy
In statistical physics, the BBGKY hierarchy (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy, sometimes called Bogoliubov hierarchy) is a set of equations describing the dynamics of a system of a large number of interacting particles. The equ ...
problem that can be systematically truncated with different methods such as the cluster-expansion approach
The cluster-expansion approach is a technique in quantum mechanics that systematically truncates the BBGKY hierarchy problem that arises when quantum dynamics of interacting systems is solved. This method is well suited for producing a closed set ...
.[Kira, M.; Koch, S. W. (2011). ''Semiconductor Quantum Optics''. Cambridge University Press. .]
At operator level, the microscopic polarization is defined by an expectation value for a single electronic transition between a valence and a conduction band. In second quantization, conduction-band electrons are defined by fermionic
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
creation and annihilation operators
Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually ...
and , respectively. An analogous identification, i.e., and , is made for the valence band electrons. The corresponding electronic interband transition then becomes
that describe transition amplitudes for moving an electron from conduction to valence band ( term) or vice versa ( term). At the same time, an electron distribution follows from
It is also convenient to follow the distribution of electronic vacancies, i.e., the hole
A hole is an opening in or through a particular medium, usually a solid body. Holes occur through natural and artificial processes, and may be useful for various purposes, or may represent a problem needing to be addressed in many fields of en ...
s,
that are left to the valence band due to optical excitation processes.
Principal structure of SBEs
The quantum dynamics of optical excitations yields an integro-differential equation
In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function.
General first order linear equations
The general first-order, linear (only with respect to the term involving derivati ...
s that constitute the SBEs
These contain the renormalized Rabi energy
as well as the renormalized carrier energy
where corresponds to the energy of free electron–hole pairs and is the Coulomb matrix element, given here in terms of the carrier wave vector .
The symbolically denoted contributions stem from the hierarchical coupling due to many-body interactions. Conceptually, , , and are single-particle expectation values while the hierarchical coupling originates from two-particle correlations such as polarization-density correlations or polarization-phonon correlations. Physically, these two-particle correlations introduce several nontrivial effects such as screening
Screening may refer to:
* Screening cultures, a type a medical test that is done to find an infection
* Screening (economics), a strategy of combating adverse selection (includes sorting resumes to select employees)
* Screening (environmental), a ...
of Coulomb interaction, Boltzmann-type scattering of and toward Fermi–Dirac distribution Fermi–Dirac may refer to:
* Fermi–Dirac statistics
Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pa ...
, excitation-induced dephasing, and further renormalization
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering v ...
of energies due to correlations.
All these correlation effects can be systematically included by solving also the dynamics of two-particle correlations.[
Kira, M.; Koch, S.W. (2006). "Many-body correlations and excitonic effects in semiconductor spectroscopy". ''Progress in Quantum Electronics'' 30 (5): 155–296. do]
10.1016/j.pquantelec.2006.12.002
At this level of sophistication, one can use the SBEs to predict optical response of semiconductors without phenomenological parameters, which gives the SBEs a very high degree of predictability. Indeed, one can use the SBEs in order to predict suitable laser designs through the accurate knowledge they produce about the semiconductor's gain spectrum. One can even use the SBEs to deduce existence of correlations, such as bound excitons, from quantitative measurements.[
Smith, R. P.; Wahlstrand, J. K.; Funk, A. C.; Mirin, R. P.; Cundiff, S. T.; Steiner, J. T.; Schafer, M.; Kira, M. et al. (2010). "Extraction of Many-Body Configurations from Nonlinear Absorption in Semiconductor Quantum Wells". ''Physical Review Letters'' 104 (24). do]
10.1103/PhysRevLett.104.247401
The presented SBEs are formulated in the momentum space since carrier's crystal momentum follows from . An equivalent set of equations can also be formulated in position space.[
Stahl, A. (1984). "Electrodynamics of the band-edge in a direct gap semiconductor". ''Solid State Communications'' 49 (1): 91–93. do]
10.1016/0038-1098(84)90569-6
However, especially, the correlation computations are much simpler to be performed in the momentum space.
Interpretation and consequences
The dynamic shows a structure where an individual is coupled to ''all'' other microscopic polarizations due to the Coulomb interaction . Therefore, the transition amplitude is collectively modified by the presence of other transition amplitudes. Only if one sets to zero, one finds isolated transitions within each state that follow exactly the same dynamics as the optical Bloch equations
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
predict. Therefore, already the Coulomb interaction among produces a new solid-state
Solid state, or solid matter, is one of the four fundamental states of matter.
Solid state may also refer to:
Electronics
* Solid-state electronics, circuits built of solid materials
* Solid state ionics, study of ionic conductors and their use ...
effect compared with optical transitions in simple atoms.
Conceptually, is just a transition amplitude for exciting an electron from valence to conduction band. At the same time, the homogeneous part of dynamics yields an eigenvalue problem that can be expressed through the generalized Wannier equation. The eigenstates of the Wannier equation is analogous to bound solutions of the hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic, an ...
problem of quantum mechanics. These are often referred to as exciton
An exciton is a bound state of an electron and an electron hole which are attracted to each other by the electrostatic Coulomb force. It is an electrically neutral quasiparticle that exists in insulators, semiconductors and some liquids. The ...
solutions and they formally describe Coulombic binding by oppositely charged electrons and holes.
However, a real exciton is a true two-particle correlation because one must then have a correlation between one electron to another hole. Therefore, the appearance of exciton resonances in the polarization does not signify the presence of excitons because is a single-particle transition amplitude. The excitonic resonances are a direct consequence of Coulomb coupling among all transitions possible in the system. In other words, the single-particle transitions themselves are influenced by Coulomb interaction making it possible to detect exciton resonance in optical response even when true excitons are not present.[
Koch, S. W.; Kira, M.; Khitrova, G.; Gibbs, H. M. (2006). "Semiconductor excitons in new light". ''Nature Materials'' 5 (7): 523–531. do]
10.1038/nmat1658
Therefore, it is often customary to specify optical resonances as exciton''ic'' instead of exciton resonances. The actual role of excitons on optical response can only be deduced by quantitative changes to induce to the linewidth
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to iden ...
and energy shift of excitonic resonances.
The solutions of the Wannier equation produce valuable insight to the basic properties of a semiconductor's optical response. In particular, one can solve the steady-state solutions of the SBEs to predict optical absorption spectrum analytically with the so-called Elliott formula
The Elliott formula describes analytically, or with few adjustable parameters such as the dephasing constant, the light absorption or emission spectra of solids. It was originally derived by Roger James Elliott to describe linear absorption base ...
. In this form, one can verify that an unexcited semiconductor shows several excitonic absorption resonances well below the fundamental bandgap energy. Obviously, this situation cannot be probing excitons because the initial many-body system does not contain electrons and holes to begin with. Furthermore, the probing can, in principle, be performed so gently that one essentially does not excite electron–hole pairs. This gedanken experiment illustrates nicely why one can detect excitonic resonances without having excitons in the system, all due to virtue of Coulomb coupling among transition amplitudes.
Extensions
The SBEs are particularly useful when solving the light propagation through a semiconductor structure. In this case, one needs to solve the SBEs together with the Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
driven by the optical polarization. This self-consistent
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
set is called the Maxwell–SBEs and is frequently applied to analyze present-day experiments and to simulate device designs.
At this level, the SBEs provide an extremely versatile method that describes linear as well as nonlinear phenomena such as exciton
An exciton is a bound state of an electron and an electron hole which are attracted to each other by the electrostatic Coulomb force. It is an electrically neutral quasiparticle that exists in insulators, semiconductors and some liquids. The ...
ic effects, propagation effects, semiconductor microcavity effects, four-wave-mixing, polaritons
In physics, polaritons are quasiparticles resulting from strong coupling of electromagnetic waves with an electric or magnetic dipole-carrying excitation. They are an expression of the common quantum phenomenon known as level repulsion, also ...
in semiconductor microcavities, gain spectroscopy, and so on.[
Klingshirn, C. F. (2006). ''Semiconductor Optics''. Springer. .
] One can also generalize the SBEs by including excitation with terahertz (THz) fields that are typically resonant with intraband transitions. One can also quantize the light field and investigate quantum-optical effects that result. In this situation, the SBEs become coupled to the semiconductor luminescence equations
The semiconductor luminescence equations (SLEs)Kira, M.; Jahnke, F.; Koch, S.; Berger, J.; Wick, D.; Nelson, T.; Khitrova, G.; Gibbs, H. (1997). "Quantum Theory of Nonlinear Semiconductor Microcavity Luminescence Explaining "Boser" Experiments". ...
.
See also
* Absorption
Absorption may refer to:
Chemistry and biology
* Absorption (biology), digestion
**Absorption (small intestine)
*Absorption (chemistry), diffusion of particles of gas or liquid into liquid or solid materials
*Absorption (skin), a route by which ...
* Semiconductor-luminescence equations
* Elliott formula
The Elliott formula describes analytically, or with few adjustable parameters such as the dephasing constant, the light absorption or emission spectra of solids. It was originally derived by Roger James Elliott to describe linear absorption base ...
* Quantum-optical spectroscopy
Quantum-optical spectroscopyKira, M.; Koch, S. (2006).
"Quantum-optical spectroscopy of semiconductors". ''Physical Review A'' 73 (1).
doibr>10.1103/PhysRevA.73.013813 .Koch, S. W.; Kira, M.; Khitrova, G.; Gibbs, H. M. (2006). "Semiconductor exc ...
* Optical Bloch equations
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
* Wannier equation
* Gain spectroscopy of semiconductors
* Semiconductor laser theory
Semiconductor lasers or laser diodes play an important part in our everyday lives by providing cheap and compact-size lasers. They consist of complex multi-layer structures requiring nanometer scale accuracy and an elaborate design. Their theore ...
* Nonlinear theory of semiconductor lasers
Laser theory of Fabry-Perot (FP) semiconductor lasers proves to be nonlinear, since the gain,Noppe M G On Nonlinear Theory for Semiconductor Lasers. 2016 Laser Phys. 26055004 (doi:10.1088/1054-660X/26/5/055004) the refractive indexPartovi and E ...
Further reading
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* {{cite book, last1=Kira, first1=M., last2=Koch, first2=S. W., title=Semiconductor Quantum Optics, year=2011, publisher=Cambridge University Press, isbn=978-0521875097
References
Theoretical physics
Semiconductor analysis
Quantum mechanics
Equations of physics