Terahertz spectroscopy detects and controls properties of matter with electromagnetic fields that are in the frequency range between a few hundred gigahertz 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. 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, electronic components and computers. At the same time, they constitute an interesting many-body system whose quantum properties can be modified, e.g., via a nanostructure 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, or optical rectification to produce well-defined THz sources. The resulting THz field can be characterized via its electric field ETHz(t). Present-day experiments can already output ETHz(t) that has a peak value in the range of MV/cm (megavolts per centimeter).[1] To estimate how strong such fields are, one can compute the level of energy change such fields induce to an electron over microscopic distance of one nanometer (nm), i.e., L = 1 nm. One simply multiplies the peak ETHz(t) with elementary charge e and L to obtain e ETHz(t) L = 100 meV. In other words, such fields have a major effect on electronic systems because the mere field strength of ETHz(t) can induce electronic transitions over microscopic scales. One possibility is to use such THz fields to study Bloch oscillations[2][3] where semiconductor electrons move through the Brillouin zone, just to return to where they started, giving rise to the Bloch oscillations.

The THz sources can be also extremely short,[4] 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 ETHz(t) provides vast new possibilities to systematic studies in semiconductors.

Besides the strength and duration of ETHz(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 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 excitons that are hydrogen-like states of matter. At the same time, the exciton binding energy is few to hundreds of meV that can be matched energetically with a THz photon. Therefore, the presence of excitons can be uniquely detected[5][6] based on the absorption spectrum of a weak THz field.[7][8] Also simple states, such as plasma and correlated electron–hole plasma[9] can be monitored or modified by THz fields.

Terahertz time-domain spectroscopy

Main article: 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 ETHz(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.

Using Terahertz for developing transmission images of packaged items.

By using short THz pulses,[4] a great variety of physical phenomena have already been studied. For unexcited, intrinsic semiconductors one can determine the complex permittivity or THz-absorption coefficient and refractive index, respectively.[10] The frequency of transversal-optical phonons, to which THz photons can couple, lies for most semiconductors at several THz.[11] Free carriers in doped semiconductors or optically excited semiconductors lead to a considerable absorption of THz photons.[12] 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.[13] 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 solids.

The Coulomb interaction can also pair electrons and holes into excitons, as discussed above. Due to their analog to the hydrogen atom, excitons have bound states that can be uniquely identified by the usual quantum numbers 1s, 2s, 2p, and so on. In particular, 1s-to-2p transition is dipole allowed and can be directly generated by ETHz(t) if the photon energy matches the transition energy. In gallium arsenide-type systems, this transition energy is roughly 4 meV that corresponds to 1 THz photons. At resonance, the dipole d1s,2p defines the Rabi energy ΩRabi = d1s,2p ETHz(t) that determines the time scale at which the 1s-to-2p 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.[4] Using this technique one can follow the formation dynamics of excitons[7][8] or observe THz gain arising from intraexcitonic transitions.[14][15]

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 optics[16] 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.[17][18]

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.[19] The Rabi splitting has also been measured with a short THz pulse[20] and also the onset to multi-THz-photon ionization has been detected,[21] 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.[22]

Theory of terahertz transitions

Terahertz transitions in solids can be systematically approached by generalizing the semiconductor Bloch equations[9] 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-like response[12] of plasma to extreme nonlinear effects of excitons.

See also

Terahertz nondestructive evaluation

Terahertz time-domain spectroscopy

Terahertz radiation

Ultrafast laser spectroscopy

Semiconductor Bloch equations

Cluster-expansion approach

Elliott formula

References

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

Feldmann, 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.46.7252

Ben 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

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

Timusk, 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)90216-8

Kira, 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

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/nature01676

Kira, 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

Kira, M.; Koch, S.W. (2006). "Many-body correlations and excitonic effects in semiconductor spectroscopy". Progress in Quantum Electronics. 30 (5): 155–296. Bibcode:2006PQE....30..155K. doi:10.1016/j.pquantelec.2006.12.002. ISSN 0079-6727.

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

Han, P. Y.; Zhang, X.-C. (1998). "Coherent, broadband midinfrared terahertz beam sensors". Applied Physics Letters 73 (21): 3049. doi:10.1063/1.122668

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

Huber, R.; Tauser, F.; Brodschelm, A.; Bichler, M.; Abstreiter, G.; Leitenstorfer, A. (2001). Nature 414 (6861): 286–289. doi:10.1038/35104522

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.93.076402

Huber, 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

Wegener, M. (2005). M. Extreme Nonlinear Optics: An Introduction. Springer. ISBN 978-3642060908

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.99.237401

Leinß, 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

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

Steiner, 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

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

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

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