Coherent effects in semiconductor optics

Macroscopically, Maxwell's equations show that in the absence of free charges and currents an electromagnetic field interacts with matter via the optical polarization ${\displaystyle {\mathbf {P} }}$. The wave equation for the electric field ${\displaystyle {\mathbf {E} }}$ reads ${\displaystyle (\nabla \cdot \nabla -{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}){\mathbf {E} }({\mathbf {r} },t)=\mu _{0}{\frac {\partial ^{2}}{\partial t^{2}}}{\mathbf {P} }({\mathbf {r} },t)}$ and shows that the second derivative with respect to time of ${\displaystyle {\mathbf {P} }}$, i.e., ${\displaystyle {\frac {\partial ^{2}}{\partial t^{2}}}{\mathbf {P} }}$, appears as a source term in the wave equation for the electric field ${\displaystyle {\mathbf {E} }}$. Thus, for optically thin samples and measurements performed in the far-field, i.e., at distances significantly exceeding the optical wavelength ${\displaystyle \lambda }$, the emitted electric field resulting from the polarization is proportional to its second time derivative, i.e., ${\displaystyle {\mathbf {E} }\propto {\frac {\partial ^{2}}{\partial t^{2}}}{\mathbf {P} }}$. Therefore, measuring the dynamics of the emitted field ${\displaystyle {\mathbf {E} }(t)}$ provides direct information on the temporal evolution of the optical material polarization ${\displaystyle {\mathbf {P} }(t)}$.

Microscopically, the optical polarization arises from quantum mechanical transitions between different states of the material system. For the case of semiconductors, electromagnetic radiation with optical frequencies is able to move electrons from the valence (${\displaystyle v}$) to the conduction (${\displaystyle c}$) band. The macroscopic polarization ${\displaystyle {\mathbf {P} }}$ is computed by summing over all microscopic transition dipoles ${\displaystyle p_{cv}}$ via ${\displaystyle {\mathbf {P} }={\frac {1}{V}}\sum _{c,v}({\mathbf {d} }_{cv}p_{cv}+\mathrm {c.c.} )}$,[2] where ${\displaystyle {\mathbf {d} }_{cv}}$ is the dipole matrix element which determines the strength of individual transitions between the states ${\displaystyle v}$ and ${\displaystyle c}$, ${\displaystyle \mathrm {c.c.} }$ denotes the complex conjugate, and ${\displaystyle V}$ is the appropriately chosen system's volume. If ${\displaystyle \epsilon _{c}}$ and ${\displaystyle \epsilon _{v}}$ are the energies of the conduction and valence band states, their dynamic quantum mechanical evolution is according to the Schrödinger equation given by phase factors ${\displaystyle \mathrm {e} ^{-\mathrm {i} \epsilon _{c}\,t/\hbar }}$ and ${\displaystyle \mathrm {e} ^{-\mathrm {i} \epsilon _{v}\,t/\hbar }}$, respectively. The superposition state described by ${\displaystyle p_{cv}}$ is evolving in time according to ${\displaystyle \mathrm {e} ^{-\mathrm {i} (\epsilon _{c}-\epsilon _{v})t/\hbar }}$. Assuming that we start at ${\displaystyle t=0}$ with ${\displaystyle p_{cv}(t=0)=p_{cv,0}}$, we have for the optical polarization

${\displaystyle {\mathbf {P} }(t)=\sum _{c,v}({\mathbf {d} }_{cv}p_{cv,0}\,\mathrm {e} ^{-\mathrm {i} (\epsilon _{c}-\epsilon _{v})t/\hbar }+\mathrm {c.c.} )}$.

Thus, ${\displaystyle {\mathbf {P} }(t)}$ is given by a summation over the microscopic transition dipoles which all oscillate with frequencies corresponding to the energy differences between the involved quantum states. Clearly, the optical polarization ${\displaystyle {\mathbf {P} }(t)}$ is a coherent quantity which is characterized by an amplitude and a phase. Depending on the phase relationships of the microscopic transition dipoles, one may obtain constructive or destructive interference, in which the microscopic dipoles are in or out of phase, respectively, and temporal interference phenomena like quantum beats, in which the modulus of ${\displaystyle {\mathbf {P} }(t)}$ varies as function of time.

Ignoring many-body effects and the coupling to other quasi particles and to reservoirs, the dynamics of photoexcited two-level systems can be described by a set of two equations, the so-called optical Bloch equations.[6] These equations are named after Felix Bloch who formulated them in order to analyze the dynamics of spin systems in nuclear magnetic resonance. The two-level Bloch equations read

${\displaystyle \mathrm {i} \hbar {\frac {\partial }{\partial t}}p_{cv}=\Delta \epsilon \,p_{cv}+{\mathbf {E} }\cdot {\mathbf {d} }I}$

and

${\displaystyle \mathrm {i} \hbar {\frac {\partial }{\partial t}}I=2{\mathbf {E} }\cdot {\mathbf {d} }(p_{cv}-p_{cv}^{\star }).}$

Here, ${\displaystyle \Delta \epsilon =(\epsilon _{c}-\epsilon _{v})}$ denotes the energy difference between the two states and ${\displaystyle I}$ is the inversion, i.e., the difference in the occupations of the upper and the lower states. The electric field ${\displaystyle {\mathbf {E} }}$ couples the microscopic polarization ${\displaystyle p}$ to the product of the Rabi energy ${\displaystyle {\mathbf {E} }\cdot {\mathbf {d} }}$ and the inversion ${\displaystyle I}$. In the absence of the driving electric field, i.e., for ${\displaystyle {\mathbf {E} }=\mathbf {0} }$, the Bloch equation for ${\displaystyle p}$ describes an oscillation, i.e., ${\displaystyle p_{cv}(t)\propto \mathrm {e} ^{-\mathrm {i} \Delta \epsilon \,t/\hbar }}$.

The optical Bloch equations enable a transparent analysis of several nonlinear optical experiments. They are, however, only well suited for systems with optical transitions between isolated levels in which many-body interactions are of minor importance as is sometimes the case in atoms or small molecules. In solid state systems, such as semiconductors and semiconductor nanostructures, an adequate description of the many-body Coulomb interaction and the coupling to additional degrees of freedom is essential and thus the optical Bloch equations are not applicable.

For a realistic description of optical processes in solid materials, it is essential to go beyond the simple picture of the optical Bloch equations and to treat many-body interactions that describe the coupling among the elementary material excitations by, e.g., the see article Coulomb interaction between the electrons and the coupling to other degrees of freedom, such as lattice vibrations, i.e., the electron-phonon coupling. Within a semiclassical approach, where the light field is treated as a classical electromagnetic field and the material excitations are described quantum mechanically, all above mentioned effects can be treated microscopically on the basis of a many-body quantum theory. For semiconductors the resulting system of equations are known as the semiconductor Bloch equations. For the simplest case of a two-band model of a semiconductor, the SBEs can be written schematically as[2]

${\displaystyle \mathrm {i} \hbar {\frac {\partial }{\partial t}}p_{\mathbf {k} }=\Delta \varepsilon _{\mathbf {k} }\,p_{\mathbf {k} }+\Omega _{\mathbf {k} }\,(n_{\mathbf {k} }^{c}-n_{\mathbf {k} }^{v})+\mathrm {i} \hbar {\frac {\partial }{\partial t}}p_{\mathbf {k} }|_{\text{corr}}\,,}$

${\displaystyle \mathrm {i} \hbar {\frac {\partial }{\partial t}}n_{\mathbf {k} }^{c}=(\Omega _{\mathbf {k} }^{\star }\,p_{\mathbf {k} }-\Omega _{\mathbf {k} }\,p_{\mathbf {k} }^{\star })+\mathrm {i} \hbar {\frac {\partial }{\partial t}}n_{\mathbf {k} }^{c}|_{\text{corr}}\,,}$

${\displaystyle \mathrm {i} \hbar {\frac {\partial }{\partial t}}n_{\mathbf {k} }^{v}=-(\Omega _{\mathbf {k} }^{\star }\,p_{\mathbf {k} }-\Omega _{\mathbf {k} }\,p_{\mathbf {k} }^{\star })+\mathrm {i} \hbar {\frac {\partial }{\partial t}}n_{\mathbf {k} }^{v}|_{\text{corr}}\,.}$

Here ${\displaystyle p_{\mathbf {k} }}$ is the microscopic polarization and ${\displaystyle n_{\mathbf {k} }^{c}}$ and ${\displaystyle n_{\mathbf {k} }^{v}}$ are the electron occupations in the conduction and valence bands (${\displaystyle c}$ and ${\displaystyle v}$), respectively, and ${\displaystyle \hbar {\mathbf {k} }}$ denotes the crystal momentum. As a result of the many-body Coulomb interaction and possibly further interaction processes, the transition energy ${\displaystyle \Delta \varepsilon _{\mathbf {k} }}$ and the Rabi energy ${\displaystyle \Omega _{\mathbf {k} }}$ both depend on the state of the excited system, i.e., they are functions of the time-dependent polarizations ${\displaystyle p_{\mathbf {k} '}}$ and occupations ${\displaystyle n_{\mathbf {k} '}^{c}}$ and ${\displaystyle n_{\mathbf {k} '}^{v}}$, respectively, at all crystal momenta ${\displaystyle \hbar {\mathbf {k} '}}$.

Due to this coupling among the excitations for all values of the crystal momentum ${\displaystyle \hbar {\mathbf {k} }}$, the optical excitations in semiconductor cannot be described on the level of isolated optical transitions but have to be treated as an interacting many-body quantum system.

A prominent and important result of the Coulomb interaction among the photoexcitations is the appearance of strongly absorbing discrete excitonic resonances which show up in the absorption spectra of semiconductors spectrally below the fundamental band gap frequency. Since an exciton consists of a negatively charged conduction band electron and a positively charged valence band hole (i.e., an electron missing in the valence band) which attract each other via the Coulomb interaction, excitons have a hydrogenic series of discrete absorption lines. Due to the optical selection rules of typical III-V semiconductors such as Galliumarsenide (GaAs) only the s-states, i.e., 1s, 2s, etc., can be optically excited and detected, see article on Wannier equation.

The many-body Coulomb interaction leads to significant complications since it results in an infinite hierarchy of dynamic equations for the microscopic correlation functions that describe the nonlinear optical response. The terms given explicitly in the SBEs above arise from a treatment of the Coulomb interaction in the time-dependent Hartree–Fock approximation. Whereas this level is sufficient to describe excitonic resonances, there are several further effects, e.g., excitation-induced dephasing, contributions from higher-order correlations like excitonic populations and biexcitonic resonances, which require one to treat so-called many-body correlation effects that are by definition beyond the Hartree–Fock level. These contributions are formally included in the SBEs given above in the terms denoted by ${\displaystyle |_{\text{corr}}}$.

The systematic truncation of the many-body hierarchy and the development and the analysis of controlled approximations schemes is an important topic in the microscopic theory of the optical processes in condensed matter systems. Depending on the particular system and the excitation conditions several approximations schemes have been developed and applied. For highly excited systems, it is often sufficient to describe many-body Coulomb correlations using the second order Born approximation.[7] Such calculations were, in particular, able to successfully describe the spectra of semiconductor lasers, see article on semiconductor laser theory. In the limit of weak light intensities, signature of exciton complexes, in particular, biexcitons, in the coherent nonlinear response have been analyzed using the dynamics controlled truncation scheme.[8][9] These two approaches and several other approximation schemes can be viewed as special cases of the so-called cluster expansion[10] in which the nonlinear optical response is classified by correlation functions which explicitly take into account interactions between a certain maximum number of particles and factorize larger correlation functions into products of lower order ones.

By nonlinear optical spectroscopy using ultrafast laser pulses with durations on the order of ten to hundreds of femtoseconds, several coherent effects have been observed and interpreted. Such studies and their proper theoretical analysis have revealed a wealth of information on the nature of the photoexcited quantum states, the coupling among them, and their dynamical evolution on ultrashort time scales. In the following, a few important effects are briefly described.

The few examples given above represent only a small subset of several further phenomena which demonstrate that the coherent optical response of semiconductors and semiconductor nanostructures is strongly influenced by many-body effects. Other interesting research directions which similarly require an adequate theoretical analysis including many-body interactions are, e.g., phototransport phenomena where optical fields generate and/or probe electronic currents, the combined spectroscopy with optical and terahertz fields, see article terahertz spectroscopy and technology, and the rapidly developing area of semiconductor quantum optics, see article semiconductor quantum optics with dots.

• Semiconductor luminescence equations
• Semiconductor Bloch equations
• Cluster-expansion approach
• Semiconductor laser theory
• Quantum beats
• Spin echo
• Stark effect
• Superradiance

• Allen, L.; Eberly, J. H. (1987). Optical Resonance and Two-Level Atoms. Dover Publications. ISBN 978-0486655338.
• Mandel, L.; Wolf, E. (1995). Optical Coherence and Quantum Optics. Cambridge University Press. ISBN 978-0521417112.
• Schäfer, W.; Wegener, M. (2002). Semiconductor Optics and Transport Phenomena. Springer. ISBN 978-3540616146.
• Meier, T.; Thomas, P.; Koch, S. W. (2007). Coherent Semiconductor Optics: From Basic Concepts to Nanostructure Applications (1st ed.). Springer. ISBN 978-3642068966.
• Haug, H.; Koch, S. W. (2009). Quantum Theory of the Optical and Electronic Properties of Semiconductors (5th ed.). World Scientific. ISBN 978-9812838841.
• Kira, M.; Koch, S. W. (2011). Semiconductor Quantum Optics. Cambridge University Press. ISBN 978-0521875097.
• Ridley, B. (2000). Quantum Processes in Semiconductors. Oxford University Press. ISBN 978-0198505792.