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Terahertz spectroscopy detects and controls properties of matter with
electromagnetic fields In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
that are in the frequency range between a few hundred
gigahertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), often described as being equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose formal expression in terms of SI base un ...
and several terahertz (abbreviated as THz). In many-body systems, several of the relevant states have an energy difference that matches with the energy of a THz
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
. Therefore, THz spectroscopy provides a particularly powerful method in resolving and controlling individual transitions between different many-body states. By doing this, one gains new insights about many-body quantum kinetics and how that can be utilized in developing new technologies that are optimized up to the elementary quantum level. Different electronic excitations within semiconductors are already widely used in
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'' originated as an acronym for light amplification by stimulated emission of radi ...
, electronic components and
computer A computer is a machine that can be Computer programming, programmed to automatically Execution (computing), carry out sequences of arithmetic or logical operations (''computation''). Modern digital electronic computers can perform generic set ...
s. At the same time, they constitute an interesting many-body system whose quantum properties can be modified, e.g., via a
nanostructure A nanostructure is a structure of intermediate size between microscopic and molecular structures. Nanostructural detail is microstructure at nanoscale. In describing nanostructures, it is necessary to differentiate between the number of dimen ...
design. Consequently, THz spectroscopy on semiconductors is relevant in revealing both new technological potentials of nanostructures as well as in exploring the fundamental properties of many-body systems in a controlled fashion.


Background

There are a great variety of techniques to generate THz radiation and to detect THz fields. One can, e.g., use an antenna, a quantum-cascade laser, a
free-electron laser A free-electron laser (FEL) is a fourth generation light source producing extremely brilliant and short pulses of radiation. An FEL functions much as a laser but employs relativistic electrons as a active laser medium, gain medium instead of using ...
, or
optical rectification Electro-optic rectification (EOR), also referred to as optical rectification, is a non-linear optical process that consists of the generation of a quasi-DC polarization in a non-linear medium at the passage of an intense optical beam. For typica ...
to produce well-defined THz sources. The resulting THz field can be characterized via its electric field ''E''THz(''t''). Present-day experiments can already output ''E''THz(''t'') that has a peak value in the range of MV/cm (megavolts per centimeter).Junginger, F.; Sell, A.; Schubert, O.; Mayer, B.; Brida, D.; Marangoni, M.; Cerullo, G.; Leitenstorfer, A. et al. (2010). "Single-cycle multiterahertz transients with peak fields above 10 MV/cm". ''Optics Letters'' 35 (15): 2645. doi:10.1364/OL.35.002645 To estimate how strong such fields are, one can compute the level of energy change such fields induce to an
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
over microscopic distance of one nanometer (nm), i.e., ''L'' = 1 nm. One simply multiplies the peak ''E''THz(''t'') with
elementary charge The elementary charge, usually denoted by , is a fundamental physical constant, defined as the electric charge carried by a single proton (+1 ''e'') or, equivalently, the magnitude of the negative electric charge carried by a single electron, ...
''e'' and ''L'' to obtain ''e'' ''E''THz(''t'') ''L'' = 100 meV. In other words, such fields have a major effect on electronic systems because the mere field strength of ''E''THz(''t'') can induce electronic transitions over
microscopic scale The microscopic scale () is the scale of objects and events smaller than those that can easily be seen by the naked eye, requiring a lens (optics), lens or microscope to see them clearly. In physics, the microscopic scale is sometimes regarded as ...
s. One possibility is to use such THz fields to study Bloch oscillationsFeldmann, J.; Leo, K.; Shah, J.; Miller, D.; Cunningham, J.; Meier, T.; von Plessen, G.; Schulze, A.; Thomas, P.; Schmitt-Rink, S. (1992). "Optical investigation of Bloch oscillations in a semiconductor superlattice". ''Physical Review B'' 46 (11): 7252–7255. doi:10.1103/PhysRevB.46.7252Ben Dahan, Maxime; Peik, Ekkehard; Reichel, Jakob; Castin, Yvan; Salomon, Christophe (1996). "Bloch Oscillations of Atoms in an Optical Potential". ''Physical Review Letters'' 76 (24): 4508–4511. doi:10.1103/PhysRevLett.76.4508 where semiconductor electrons move through the
Brillouin zone In mathematics and solid state physics, the first Brillouin zone (named after Léon Brillouin) is a uniquely defined primitive cell in reciprocal space Reciprocal lattice is a concept associated with solids with translational symmetry whic ...
, just to return to where they started, giving rise to the Bloch oscillations. The THz sources can be also extremely short,Jepsen, P.U.; Cooke, D.G.; Koch, M. (2011). "Terahertz spectroscopy and imaging - Modern techniques and applications". ''Laser & Photonics Reviews'' 5 (1): 124–166. doi:10.1002/lpor.201000011 down to single cycle of THz field's oscillation. For one THz, that means duration in the range of one picosecond (ps). Consequently, one can use THz fields to monitor and control ultrafast processes in semiconductors or to produce ultrafast switching in semiconductor components. Obviously, the combination of ultrafast duration and strong peak ''E''THz(''t'') provides vast new possibilities to systematic studies in semiconductors. Besides the strength and duration of ''E''THz(''t''), the THz field's photon energy plays a vital role in semiconductor investigations because it can be made resonant with several intriguing many-body transitions. For example, electrons in
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 ...
and holes, i.e., electronic vacancies, in valence band attract each other via the Coulomb interaction. Under suitable conditions, electrons and holes can be bound to
exciton An exciton is a bound state of an electron and an electron hole which are attracted to each other by the electrostatic Coulomb's law, Coulomb force resulting from their opposite charges. It is an electrically neutral quasiparticle regarded as ...
s that are hydrogen-like states of matter. At the same time, the exciton
binding energy In physics and chemistry, binding energy is the smallest amount of energy required to remove a particle from a system of particles or to disassemble a system of particles into individual parts. In the former meaning the term is predominantly use ...
is few to hundreds of meV that can be matched energetically with a THz photon. Therefore, the presence of excitons can be uniquely detectedTimusk, T.; Navarro, H.; Lipari, N.O.; Altarelli, M. (1978). "Far-infrared absorption by excitons in silicon". ''Solid State Communications'' 25 (4): 217–219. doi:10.1016/0038-1098(78)90216-8Kira, M.; Hoyer, W.; Stroucken, T.; Koch, S. (2001). "Exciton Formation in Semiconductors and the Influence of a Photonic Environment". ''Physical Review Letters'' 87 (17). doi:10.1103/PhysRevLett.87.176401 based on the absorption spectrum of a weak THz field.Kaindl, R. A.; Carnahan, M. A.; Hägele, D.; Lövenich, R.; Chemla, D. S. (2003). "Ultrafast terahertz probes of transient conducting and insulating phases in an electron–hole gas". ''Nature'' 423 (6941): 734–738. doi:10.1038/nature01676Kira, M.; Hoyer, W.; Koch, S.W. (2004). "Terahertz signatures of the exciton formation dynamics in non-resonantly excited semiconductors". ''Solid State Communications'' 129 (11): 733–736. doi:10.1016/j.ssc.2003.12.015 Also simple states, such as plasma and correlated electron–hole plasma can be monitored or modified by THz fields.


Terahertz time-domain spectroscopy

In optical spectroscopy, the detectors typically measure the intensity of the light field rather than the electric field because there are no detectors that can directly measure electromagnetic fields in the optical range. However, there are multiple techniques, such as antennas and electro-optical sampling, that can be applied to measure the time evolution of ''E''THz(''t'') directly. For example, one can propagate a THz pulse through a semiconductor sample and measure the transmitted and reflected fields as function of time. Therefore, one collects information of semiconductor excitation dynamics completely in time domain, which is the general principle of the terahertz time-domain spectroscopy. By using short THz pulses, a great variety of physical phenomena have already been studied. For unexcited,
intrinsic semiconductor An intrinsic semiconductor, also called a pure semiconductor, undoped semiconductor or i-type semiconductor, is a semiconductor without any significant dopant species present. The number of charge carriers is therefore determined by the properties ...
s one can determine the
complex permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ( epsilon), is a measure of the electric polarizability of a dielectric material. A material with high permittivity polarizes mo ...
or THz-absorption coefficient and refractive index, respectively.Grischkowsky, D.; Keiding, Søren; Exter, Martin van; Fattinger, Ch. (1990). "Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors". ''Journal of the Optical Society of America B'' 7 (10): 2006. doi:10.1364/JOSAB.7.002006 The frequency of transversal-optical
phonon A phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. In the context of optically trapped objects, the quantized vibration mode can be defined a ...
s, to which THz photons can couple, lies for most semiconductors at several THz.Han, P. Y.; Zhang, X.-C. (1998). "Coherent, broadband midinfrared terahertz beam sensors". ''Applied Physics Letters'' 73 (21): 3049. doi:10.1063/1.122668 Free carriers in doped semiconductors or optically excited semiconductors lead to a considerable absorption of THz photons.Zhang, W.; Azad, Abul K.; Grischkowsky, D. (2003). "Terahertz studies of carrier dynamics and dielectric response of n-type, freestanding epitaxial GaN". ''Applied Physics Letters'' 82 (17): 2841. doi:10.1063/1.1569988 Since THz pulses passes through non-metallic materials, they can be used for inspection and transmission of packaged items.


Terahertz-induced plasma and exciton transitions

The THz fields can be applied to accelerate electrons out of their equilibrium. If this is done fast enough, one can measure the elementary processes, such as how fast the screening of the Coulomb interaction is built up. This was experimentally explored in Ref.Huber, R.; Tauser, F.; Brodschelm, A.; Bichler, M.; Abstreiter, G.; Leitenstorfer, A. (2001). ''Nature'' 414 (6861): 286–289. doi:10.1038/35104522 where it was shown that screening is complete within tens of femtoseconds in semiconductors. These insights are very important to understand how electronic plasma behaves in
solid Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...
s. The Coulomb interaction can also pair electrons and holes into excitons, as discussed above. Due to their analog to the
hydrogen atom A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral hydrogen atom contains a single positively charged proton in the nucleus, and a single negatively charged electron bound to the nucleus by the Coulomb for ...
, excitons have
bound state A bound state is a composite of two or more fundamental building blocks, such as particles, atoms, or bodies, that behaves as a single object and in which energy is required to split them. In quantum physics, a bound state is a quantum state of a ...
s that can be uniquely identified by the usual
quantum number In quantum physics and chemistry, quantum numbers are quantities that characterize the possible states of the system. To fully specify the state of the electron in a hydrogen atom, four quantum numbers are needed. The traditional set of quantu ...
s 1''s'', 2''s'', 2''p'', and so on. In particular, 1''s''-to-2''p'' transition is dipole allowed and can be directly generated by ''E''THz(''t'') if the photon energy matches the transition energy. In
gallium arsenide Gallium arsenide (GaAs) is a III-V direct band gap semiconductor with a Zincblende (crystal structure), zinc blende crystal structure. Gallium arsenide is used in the manufacture of devices such as microwave frequency integrated circuits, monoli ...
-type systems, this transition energy is roughly 4 meV that corresponds to 1 THz photons. At resonance, the dipole ''d''1''s'',2''p'' defines the Rabi energy ΩRabi = ''d''1''s'',2''p'' ''E''THz(''t'') that determines the time scale at which the 1''s''-to-2''p'' transition proceeds. For example, one can excite the excitonic transition with an additional optical pulse which is synchronized with the THz pulse. This technique is called transient THz spectroscopy. Using this technique one can follow the formation dynamics of excitons or observe THz gain arising from intraexcitonic transitions.Kira, M.; Koch, S. (2004). "Exciton-Population Inversion and Terahertz Gain in Semiconductors Excited to Resonance". ''Physical Review Letters'' 93 (7). doi:10.1103/PhysRevLett.93.076402Huber, Rupert; Schmid, Ben; Shen, Y.; Chemla, Daniel; Kaindl, Robert (2006). "Stimulated Terahertz Emission from Intraexcitonic Transitions in Cu2O". ''Physical Review Letters'' 96 (1). doi:10.1103/PhysRevLett.96.017402 Since a THz pulse can be intense and short, e.g., single-cycle, it is experimentally possible to realize situations where duration of the pulse, time scale related to Rabi- as well as the THz photon energy ħω are degenerate. In this situation, one enters the realm of extreme nonlinear opticsWegener, M. (2005). M. ''Extreme Nonlinear Optics: An Introduction''. Springer. where the usual approximations, such as the rotating-wave approximation (abbreviated as RWA) or the conditions for complete state transfer, break down. As a result, the Rabi oscillations become strongly distorted by the non-RWA contributions, the multiphoton absorption or emission processes, and the dynamic Franz–Keldysh effect, as measured in Refs.Danielson, J.; Lee, Yun-Shik; Prineas, J.; Steiner, J.; Kira, M.; Koch, S. (2007). "Interaction of Strong Single-Cycle Terahertz Pulses with Semiconductor Quantum Wells". ''Physical Review Letters'' 99 (23). doi:10.1103/PhysRevLett.99.237401Leinß, S.; Kampfrath, T.; v.Volkmann, K.; Wolf, M.; Steiner, J.; Kira, M.; Koch, S.; Leitenstorfer, A. et al. (2008). "Terahertz Coherent Control of Optically Dark Paraexcitons in Cu2O". ''Physical Review Letters'' 101 (24). doi:10.1103/PhysRevLett.101.246401 By using a free-electron laser, one can generate longer THz pulses that are more suitable for detecting the Rabi oscillations directly. This technique could indeed demonstrate the Rabi oscillations, or actually the related Autler–Townes splitting, in experiments.Wagner, Martin; Schneider, Harald; Stehr, Dominik; Winnerl, Stephan; Andrews, Aaron M.; Schartner, Stephan; Strasser, Gottfried; Helm, Manfred (2010). "Observation of the Intra-exciton Autler-Townes Effect in GaAs/AlGaAs Semiconductor Quantum Wells". ''Physical Review Letters'' 105 (16). doi:10.1103/PhysRevLett.105.167401 The Rabi splitting has also been measured with a short THz pulseSteiner, J.; Kira, M.; Koch, S. (2008). "Optical nonlinearities and Rabi flopping of an exciton population in a semiconductor interacting with strong terahertz fields". ''Physical Review B'' 77 (16). doi:10.1103/PhysRevB.77.165308 and also the onset to multi-THz-photon ionization has been detected,Ewers, B.; Köster, N. S.; Woscholski, R.; Koch, M.; Chatterjee, S.; Khitrova, G.; Gibbs, H. M.; Klettke, A. C.; Kira, M.; Koch, S. W. (2012). "Ionization of coherent excitons by strong terahertz fields". ''Physical Review B'' 85 (7). doi:10.1103/PhysRevB.85.075307 as the THz fields are made stronger. Recently, it has also been shown that the Coulomb interaction causes nominally dipole-forbidden intra-excitonic transitions to become partially allowed.Rice, W. D.; Kono, J.; Zybell, S.; Winnerl, S.; Bhattacharyya, J.; Schneider, H.; Helm, M.; Ewers, B.; Chernikov, A.; Koch, M.; Chatterjee, S.; Khitrova, G.; Gibbs, H. M.; Schneebeli, L.; Breddermann, B.; Kira, M.; Koch, S. W. (2013). "Observation of Forbidden Exciton Transitions Mediated by Coulomb Interactions in Photoexcited Semiconductor Quantum Wells". ''Physical Review Letters'' 110 (13). doi:10.1103/PhysRevLett.110.137404


Theory of terahertz transitions

Terahertz transitions in solids can be systematically approached by generalizing the semiconductor Bloch equations and the related many-body correlation dynamics. At this level, one realizes the THz field are directly absorbed by two-particle correlations that modify the quantum kinetics of electron and hole distributions. Therefore, a systematic THz analysis must include the quantum kinetics of many-body correlations, that can be treated systematically, e.g., with the cluster-expansion approach. At this level, one can explain and predict a wide range of effects with the same theory, ranging from
Drude In German folklore, a drude (, , pl. ''Druden'') is a kind of malevolent nocturnal spirit (an alp, kobold or hag) associated with nightmares, prevalent especially in Southern Germany. Druden were said to participate in the Wild Hunt and we ...
-like response of plasma to extreme nonlinear effects of excitons.


See also

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References

{{Branches of Spectroscopy Spectroscopy Terahertz technology