Interference-Induced Terahertz Transparency in a Semiconductor Magneto-plasma

Junichiro Kono

[This is an invited article based on recently published work by the author and his collaborators from Rice University, Texas A&M University, and Los Alamos National Laboratory -- 2Physics.com]

Author: Junichiro Kono
Affiliation: Dept of Electrical and Computer Engineering, Rice University, Houston, Texas 77005, USA.

Maximum modulation of light transmission occurs when an initially opaque medium is suddenly made transparent. This dramatic phenomenon, induced transparency, indeed occurs in atomic and molecular gases through different mechanisms [1,2], while there remains much room for further studies in solids. A plasma is an illustrative system exhibiting opacity, where light is completely reflected if its frequency is smaller than the plasma frequency. Light-plasma interaction theory provides a universal framework to describe such diverse phenomena and systems as radiation in space plasmas, diagnostics of laboratory plasmas, and collective excitations in condensed matter. However, quite surprisingly, induced transparency in plasmas is a rather uncharted area of research.

In a paper published online in Nature Physics on December 6, 2009, researchers at Rice University, Texas A&M University, and Los Alamos National Laboratory reported a novel type of thermally- and magnetically-induced transparency in a semiconductor plasma, revealed by coherent terahertz (THz) magneto-spectroscopy [3]. They observed a sudden appearance and disappearance of transmission through a slab of electron-doped InSb over narrow temperature and magnetic field ranges.

To explain these striking observations, the researchers developed a theoretical model based on coherent interference between the left- and right-circularly polarized eigenmodes of the low-density magneto-plasma in InSb. Detailed simulations demonstrated how the observed THz modulation and interference effects depend sensitively on the magnetic field, as well as on the temperature through the intrinsic carrier density of narrow-gap semiconductors. Excellent agreement between experiment and theory demonstrated surprisingly long-lived coherence of magnetoplasmon excitations.

The free electrons in the conduction band of doped narrow-gap semiconductors, e.g., InSb, InAs, and HgCdTe, behave as classic solid-state plasmas and have been examined through a number of infrared spectroscopy studies [4,5]. Due to the low electron densities achievable in these materials and to the electrons’ small effective mass and high mobility, most of the important energy scales (the cyclotron energy, the plasma energy, the Fermi energy, intra-donor transition energies, etc.) can all lie within the same narrow energy range from ~1 to 10 meV, or the THz frequency range (1 THz = 4.1 meV). The interplay between these material properties, which are tunable with magnetic field, doping density, and/or temperature, make doped narrow-gap semiconductors a useful material system in which to probe and explore novel phenomena that can be exploited for future THz technology.

The Rice researchers used a time-domain THz magneto-spectroscopy system [6] with a linearly-polarized, coherent THz beam to investigate magneto-plasmonic effects in a lightly n-doped InSb sample that exhibits a sharp plasma edge at ~0.3 THz at zero magnetic field as well as sharp absorption and dispersion features around the cyclotron resonance. These spectral features can be sensitively controlled by changing the magnetic field and temperature due to the very small effective masses of electrons and low thermal excitation energy in this narrow-gap semiconductor. Furthermore, long decoherence times (< 40 ps) of electron cyclotron oscillations give rise to sharp interference fringes and coherent beating between different normal modes (coupled photon-magneto-plasmon excitations) of the semiconductor plasma.

Figure 1 (click on the image to see hi resolution version) Temperature dependence of THz transmittance spectra for lightly-doped InSb in a magnetic field. a, Transmittance versus temperature at 0.25 THz at a magnetic field of 0.9 T (corresponding to a horizontal cut in the contour map of b), showing thermally induced transparency. b, Measured and c, calculated transmittance contour as a function of temperature (2-240 K) and frequency (0.12-2.6 THz) at a fixed magnetic field of 0.9 T. d, Transmittance versus magnetic field at 0.25 THz at a temperature of 40 K (corresponding to a horizontal cut in the contour map of e), showing magnetically induced transparency. e, Measured and f, calculated transmittance contour as a function of magnetic field (0-2 T) and frequency (0.12-2.6 THz) at a fixed temperature of 40 K.

As an example, the temperature (Figs. 1a, 1b, and 1c) and magnetic field (Figs. 1d, 1e, and 1f) dependence of THz transmittance spectra are shown. A striking feature in both Figs. 1a and 1d is a narrow range of temperature (1a) and magnetic field (1d) where the transmission of THz light is high. Figure 1b shows a full contour map of the transmittance as a function of frequency and temperature at a fixed magnetic field. Figure 1c shows a calculated contour plot of the transmittance, based on their model. A horizontal cut of the contour at 0.25 THz is shown in Fig. 1a. Similarly, Figs. 1e and 1f show, respectively, measured and calculated contour plots of the transmittance as a function of frequency and magnetic field. A horizontal cut of the contour at 0.25 THz is shown in Fig. 1d.

At zero magnetic field, the only spectral feature appearing in our InSb samples is the plasma edge at the plasma frequency (0.3 THz). When a magnetic field is applied along the wave propagation direction, the incident linearly-polarized THz wave propagates in the sample as a superposition of the two transverse normal modes of the magneto-plasma: the left-circularly-polarized mode, called the ‘extraordinary’ or cyclotron resonance active (CRA) wave, and the right-circularly-polarized mode, called the ‘ordinary’ or cyclotron resonance inactive (CRI) wave. The CRA mode couples with the cyclotron motion of electrons. With increasing magnetic field, the plasma edge splits into the two magnetoplasmon frequencies for the CRA and CRI modes.

The THz response of the InSb sample was modeled through a dielectric tensor for a classical magneto-plasma for both electrons and holes, including the effect of conduction band non-parabolicity. The CRA wave experiences strong absorption and dispersion, while the transmission of the CRI mode is nearly flat and featureless everywhere except at very low frequencies. Simple addition of the two, however, does not produce any of the experimentally observed spectral features. What is measured experimentally is a superposition of the two fields, which contains the interference between the CRA and CRI modes. The interference term depends on the index difference between the two modes, and its inclusion in the simulation indeed totally modified the spectra at finite magnetic fields. The agreement between theory and experiment is outstanding. The positions and shapes of all the transmission peaks, plateaus, and dips in the spectra are accurately reproduced in great detail, confirming the accuracy of our interpretation and theoretical model and indicating the long coherence times of coupled photon-magnetoplasmon excitations reaching tens of ps.

The dominant process affecting the temperature dependence of the dielectric tensor at elevated temperatures is the thermal excitation of intrinsic carriers across the band gap given, which leads to an exponentially growing plasma frequency. The density of intrinsic carriers eventually exceeds the doping density at ~180 K. Therefore, one would expect a weakly temperature-dependent transmittance below ~180 K that would abruptly decrease above this temperature due to the exponentially growing plasma frequency. The intensities of individually-transmitted CRA and CRI modes indeed exhibit this expected temperature dependence.

However, again, one has to include the interference term in calculating the transmission. With realistic parameters for the sample and experimental conditions, this interference term is negative and almost exactly cancels the other two terms below 160 K, leading to interference-induced opacity. As the temperature increases above 160 K, the difference between the refractive indices of the two modes starts growing exponentially, causing strong oscillations in the total transmittance due to interference. These oscillations, however, are strongly damped above 200 K due to the exponentially growing absorption coefficient for both normal modes. As a result, only one strong peak remains prominent, followed by a few progressively smaller peaks, explaining the existence of the observed transparency bands. This is further illustrated by the excellent agreement between the observed and calculated temperature dependence of transmittance in Figs. 1b and 1e (experiment) and 1c and 1f (theory).

These results demonstrate that free carrier plasmas in lightly-doped narrow-gap semiconductors are promising materials systems for THz physics, exhibiting huge magnetic anisotropy effects and plasmon excitations in the THz range that are highly tunable with external fields, temperature, and doping. In particular, coherent interference phenomena, which are commonly observed and used in the visible and near-infrared range, can be extended into the THz regime. Moreover, the observed novel interference phenomena depend sensitively on plasma properties and carrier interactions, and thus, can be used to study solid-state plasmas over a vast range of external fields and temperatures from the classical limit to the ultra-quantum limit. This experimental finding may open up further new opportunities for using coherent THz methods to probe more exotic phenomena in condensed matter systems that occur due to many-body interactions and disorder.

References
[1] Harris, S. E. "Electromagnetically induced transparency". Phys. Today 50, 36-42 (1997). Abstract.
[2] McCall, S. L. & Hahn, E. L. "Self-induced transparency by pulsed coherent light", Phys. Rev. Lett. 18, 908-911 (1967). Abstract.
[3] X. Wang, A. A. Belyanin, S. A. Crooker, D. M. Mittleman, and J. Kono, “Interference-Induced Terahertz Transparency in a Semiconductor Magneto-plasma,” Nature Physics, published online on December 6, 2009. Abstract. Rice University Press Release.
[4] Palik, E. D. & Furdyna, J. K. "Infrared and microwave magnetoplasma effects in semiconductors". Rep. Prog. Phys. 33, 1193-1322 (1970). Abstract.
[5] McCombe, B. D & Wagner, R. J. in "Advances in Electronics and Electron Physics", Vol 37 (eds Marton, L.) 1-79 (Academic Press, 1975).
[6] Wang, X., Hilton, D. J., Ren, L., Mittleman, D. M., Kono, J. & Reno, J. L. "Terahertz time-domain magnetospectroscopy of a high-mobility two-dimensional electron gas". Optics Lett. 32, 1845-1847 (2007). Abstract.

Filming Photons of Nanoscale Structures with Electrons

Ahmed Zewail [Image courtesy: Caltech]

In two recent papers published in the December 17 issue of Nature [1] and the October 30 issue of Science [2], a team of researchers from the California Institute of Technology (Caltech) reported the invention of techniques that allow the real-time, real-space visualization of fleeting changes in the structure of nanoscale matter. These novel techniques have been used to image the evanescent electrical fields produced by the interaction of electrons and photons, and to track changes in atomic-scale structures.

Four-dimensional (4D) microscopy-the methodology upon which the new techniques were based-was developed at Caltech's Physical Biology Center for Ultrafast Science and Technology. The center is directed by Ahmed Zewail, the Linus Pauling Professor of Chemistry and professor of physics at Caltech, and winner of the 1999 Nobel Prize in Chemistry.

Zewail was awarded the Nobel Prize for pioneering the science of femtochemistry, the use of ultrashort laser flashes to observe fundamental chemical reactions occurring at the timescale of the femtosecond (one-millionth of a billionth of a second). The work "captured atoms and molecules in motion," Zewail says, but while snapshots of such molecules provide the "time dimension" of chemical reactions, they don't give the dimensions of space of those reactions-that is, their structure or architecture.

Zewail and his colleagues were able to visualize the missing architecture through 4D microscopy, which employs single electrons to introduce the dimension of time into traditional high-resolution electron microscopy, thus providing a way to see the changing structure of complex systems at the atomic scale.

Image 1: The diffraction obtained for silicon with 4D electron microscopy. From the patterns the structure can be determined on the nanoscale. [Credit: AAAS / Science / Zewail / Caltech]

In the research detailed in the Science paper [2], Zewail and postdoctoral scholar Aycan Yurtsever were able to focus an electron beam onto a specific nanoscale-sized site in a specimen, making it possible to observe structures within that localized area at the atomic level.

In electron diffraction, an object is illuminated with a beam of electrons. The electrons bounce off the atoms in the object, then scatter and strike a detector. The patterns produced on the detector provide information about the arrangement of the atoms in the material. However, if the atoms are in motion, the patterns will be blurred, obscuring details about small-scale variations in the material.

The new technique devised by Zewail and Yurtsever addresses the blurring problem by using electron pulses instead of a steady electron beam. The sample under study - in the case of the Science paper [2], a wafer of crystalline silicon-is first heated by being struck with a short pulse of laser light. The sample is then hit with a femtosecond pulse of electrons, which bounce off the atoms, producing a diffraction pattern on a detector.

Since the electron pulses are so incredibly brief, the heated atoms don't have time to move much; this shorter "exposure time" produces a sharp image. By adjusting the delay between when the sample is heated and when the image is taken, the scientists can build up a library of still images that can be strung together into a movie.

"Essentially all of the specimens we deal with are heterogeneous," Zewail explains, with varying compositions over very small areas. "This technique provides the means for examining local sites in materials and biological structures, with a spatial resolution of a nanometer or less, and time resolution of femtoseconds."

The new diffraction method allows the structures of materials to be mapped out at an atomic scale. With the second technique-introduced in the Nature paper [1], which was coauthored by postdoctoral scholars Brett Barwick and David Flannigan - the light produced by such nanostructures can be imaged and mapped.

The concept behind this technique involves the interaction between electrons and photons. Photons generate an evanescent field in nanostructures, and electrons can gain energy from such fields, which makes them visible in the 4D microscope.

Image 2: Photons imaged in nanoscale structures (carbon nanotubes) using pulsed electrons at very high speed. Shown are the evanescent fields for two time frames and for two polarizations. [Credit: Zewail/Caltech]

In what is known as the photon-induced near-field electron microscopy (PINEM) effect, certain materials-after being hit with laser pulses-continue to "glow" for a short but measurable amount of time (on the order of tens to hundreds of femtoseconds).

In their experiment, the researchers illuminated carbon nanotubes and silver nanowires with short pulses of laser light as electrons were being shot past. The evanescent field persisted for femtoseconds, and the electrons picked up energy during this time in discrete amounts (or quanta) corresponding to the wavelength of the laser light. The energy of an electron at 200 kilo-electron volts (keV) increased by 2.4 electron volts (eV), or by 4.8 eV, or by 7.2 eV, etc.; alternatively, an electron might not change in energy at all. The number of electrons showing a change is more striking if the timing is just right, i.e., if the electrons are passing the material when the field is at its strongest.

The power of this technique is that it provides a way to visualize the evanescent field when the electrons that have gained energy are selectively identified, and to image the nanostructures themselves when electrons that have not gained energy are selected.

"As noted by the reviewers of this paper, this technique of visualization opens new vistas of imaging with the potential to impact fields such as plasmonics, photonics, and related disciplines," Zewail says. "What is interesting from a fundamental physics point of view is that we are able to image photons using electrons. Traditionally, because of the mismatch between the energy and momentum of electrons and photons, we did not expect the strength of the PINEM effect, or the ability to visualize it in space and time."

References
[1] "Photon-induced near-field electron microscopy",
Brett Barwick, David J. Flannigan & Ahmed H. Zewail,
Nature, Vol. 462, pp 902-906 (17 December 2009). Abstract.
[2] "4D Nanoscale Diffraction Observed by Convergent-Beam Ultrafast Electron Microscopy",
Aycan Yurtsever and Ahmed H. Zewail,
Science, Vol. 326. no. 5953, pp 708 - 712 (October 30, 2009). Abstract.

[We thank Media relations, Caltech for materials used in this posting]

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Creation of ‘Synthetic Magnetic Fields’ for Neutral Atoms

Ian Spielman [photo courtesy: Joint Quantum Institute, University of Maryland]

The current (December 3) issue of the journal 'Nature' carries an article describing the creation of the so-called “synthetic” magnetic fields for ultracold gas atoms, in effect “tricking” neutral atoms into acting as if they are electrically charged particles subjected to a real magnetic field.

This important new capability in ultracold atomic gases is achieved by a team of researchers at the Joint Quantum Institute (JQI), a collaboration of the University of Maryland and the National Institute of Standards and Technology (NIST). The demonstration of this capability not only paves the way for exploring the complex natural phenomena involving charged particles in magnetic fields, but may also contribute to an exotic new form of quantum computing.

As researchers have become increasingly proficient at creating and manipulating gaseous collections of atoms near absolute zero, these ultracold gases have become ideal laboratories for studying the complex behavior of material systems. Unlike usual crystalline materials, they are free of obfuscating properties, such as impurity atoms, that exist in normal solids and liquids.

However, studying the effects of magnetic fields is problematic because the gases are made of neutral atoms and so do not respond to magnetic fields in the same way as charged particles do. So how would you simulate, for example, such important exotic phenomena as the quantum Hall effect, in which electrons can “divide” into quasiparticles carrying only a fraction of the electron’s electric charge?

The answer Ian Spielman and his colleagues came up with is a clever physical trick to make the neutral atoms behave in a way that is mathematically identical to how charged particles move in a magnetic field. A pair of laser beams illuminates an ultracold gas of rubidium atoms already in a collective state known as a Bose-Einstein condensate. The laser light ties the atoms' internal energy to their external (kinetic) energy, modifying the relationship between their energy and momentum. Simultaneously, the researchers expose the atoms to a real magnetic field that varies along a single direction, so that the alteration also varies along that direction.

A pair of laser beams (red arrows) impinges upon an ultracold gas cloud of rubidum atoms (green oval) to create synthetic magnetic fields (labeled Beff). (Inset) The beams, combined with an external magnetic field (not shown) cause the atoms to "feel" a rotational force; the swirling atoms create vortices in the gas [Image courtesy: JQI]

In a strange inversion, the laser-illuminated neutral atoms react to the varying magnetic field in a way that is mathematically equivalent to the way a charged particle responds to a uniform magnetic field. The neutral atoms experience a force in a direction perpendicular to both their direction of motion and the direction of the magnetic field gradient in the trap. By fooling the atoms in this fashion, the researchers created vortices in which the atoms swirl in whirlpool-like motions in the gas clouds. The vortices are the “smoking gun,” Spielman says, for the presence of synthetic magnetic fields.

A harbinger of the synthetic magnetic fields is the formation of vortices (spots). These spots, the number of which increases with increasing synthetic field, mark the points about which atoms swirled with a whirlpool-like motion. The measurement units in each panel indicate the size of the external magnetic field gradient applied to the gas of atoms, with larger external fields producing more vortices. [Image courtesy: JQI]

Previously, other researchers had physically spun gases of ultracold atoms to simulate the effects of magnetic fields, but rotating gases are unstable and tend to lose atoms at the highest rotation rates.

In their next step, the JQI researchers plan to partition a nearly spherical system of 20,000 rubidium atoms into a stack of about 100 two-dimensional “pancakes” and increase their currently observed 12 vortices to about 200 per-pancake. At a one-vortex-per-atom ratio, they could observe the quantum Hall effect and control it in unprecedented ways. In turn, they hope to coax atoms to behave like a class of quasiparticles known as “non-abelian anyons,” a required component of “topological quantum computing,” in which anyons dancing in the gas would perform logical operations based on the laws of quantum mechanics.

Reference
"Synthetic magnetic fields for ultracold neutral atoms"
Y.-J. Lin, R. L. Compton, K. Jiménez-García, J. V. Porto & I. B. Spielman.
Nature, 462, 628-632 (3 December, 2009). Abstract.

[We thank National Institute of Standards and Technology for materials used in this report]