The free electrons in the conduction band of doped narrow-gap semiconductors, for example, InSb, InAs and HgCdTe, behave as classic solid-state plasmas and have been examined through a number of infrared spectroscopy studies10,11. Owing 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 ωc, the plasma energy ωp, the Fermi energy EF, intra-donor transition energies and so on) can all lie within the same narrow energy range from 1 to 10 meV, or the terahertz frequency range (1 THz is equivalent to 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 new phenomena that can be exploited for future terahertz technology12,13,14.

Here, we have used a time-domain terahertz magneto-spectroscopy system15 (see the Methods section) with a linearly polarized, coherent terahertz beam to investigate magneto-plasmonic effects in a lightly n-doped InSb sample that shows a sharp plasma edge at 0.3 THz at zero magnetic field as well as sharp absorption and dispersion features around the cyclotron resonance (ωc/2π2 THz T−1). These spectral features can be sensitively controlled by changing the magnetic field and temperature, owing to the very small effective masses of electrons and low thermal excitation energy in this narrow-gap semiconductor. Furthermore, long decoherence times (up to 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, which can be revealed by polarization-sensitive measurements.

We found that the transmission of terahertz radiation through this plasma sensitively changes with the temperature, magnetic field and frequency. As an example, we show the temperature (Fig. 1a–c) and magnetic field (Fig. 1d–f) dependence of terahertz transmittance spectra. A striking feature in both Fig. 1a and Fig. 1d is a narrow range of temperature (Fig. 1a) and magnetic field (Fig. 1d) where the transmission of terahertz light is high, which we refer to as thermally induced transparency and magnetically induced transparency, respectively. Figure 1b shows a full contour map of the transmittance as a function of frequency (0.12–2.6 THz) and temperature (2–240 K) at a fixed magnetic field of 0.9 T. Figure 1c shows a calculated contour plot of the transmittance, based on a model to be discussed later. The spectra shown in these figures are characterized by two sharp ‘spikes’ in transmission (peaked at 1 and 2 THz) that are sandwiching a relatively flat transmission region (or a plateau) between them. The two spikes of transmission are stable with increasing temperature only up to 150 K, where they suddenly shift in frequency, one moving down and the other moving up. As a result, in a temperature region of 150–200 K there are two bands of transmission stretching towards both ends of our spectral range (see Fig. 1b). A horizontal cut of the contour at 0.25 THz is shown in Fig. 1a. Similarly, Fig. 1e and f show, respectively, measured and calculated contour plots of the transmittance as a function of frequency and magnetic field at a fixed temperature of 40 K. A horizontal cut of the contour at 0.25 THz is shown in Fig. 1d. In the following, we show how the induced transparency and opacity arise from coherent terahertz interference in a low-density magneto-plasma.

Figure 1: Temperature dependence of terahertz transmittance spectra for lightly doped InSb in a magnetic field.
figure 1

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,c Measured (b) and calculated (c) 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,f Measured (e) and calculated (f) 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.

At zero magnetic field, the only spectral feature appearing in our InSb samples is the plasma edge at the plasma frequency ωp/2π=[e2n/(m*κ ɛ0)]1/2/2π≈0.3 THz, where the free-electron density n=2.3×1014 cm−3, the electron effective mass m*=0.014 m0 at the band edge, m0=9.11×10−31 kg is the mass of free electrons in vacuum, the lattice dielectric constant κ=16 and ɛ0=8.85×10−12 F m−1 is the permittivity of vacuum. When a magnetic field is applied along the wave propagation direction (that is, the Faraday geometry), the incident linearly polarized terahertz 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 (see Fig. 2a). The CRA mode couples with the cyclotron motion of electrons. With increasing magnetic field, the plasma edge splits into the two magneto-plasmon frequencies given by

for the CRA (+) and CRI (−) modes, respectively, as plotted in Fig. 2b. These are the characteristic frequencies at which the refractive index becomes zero (neglecting dissipation). As the magnetic field increases, the frequency ω+ asymptotically approaches the electron cyclotron frequency ωc (the dotted line in Fig. 2b), whereas ω monotonically decreases and approaches zero.

Figure 2: Interaction of an incident linearly polarized terahertz wave with a plasma in a magnetic field in the Faraday geometry.
figure 2

a, Schematic of the CRA and CRI modes propagating in the magneto-plasma in n-InSb in the Faraday geometry. After transmitting through the sample, the x component of the terahertz field has acquired an extra interference term, cos[k0LRe(neno)], where ne and no are the extraordinary and ordinary indices of refraction, respectively, k0 is the wavenumber in vacuum and L is the sample thickness. b, Frequencies at which the real refractive indices of the CRA and CRI modes are equal to zero (red and blue curves, respectively), given by equation (1), and the electron cyclotron frequency ωc/2π (black dotted curve) as functions of magnetic field. ωp/2π=0.3 THz is the plasma frequency at zero magnetic field for an electron density of 2.3×1014 cm−3.

On the basis of these considerations, the measured terahertz magneto-transmittance spectra at low temperatures can be theoretically reproduced. We modelled the terahertz response of the InSb sample through a dielectric tensor for a classical magneto-plasma10,16,17 for both electrons and holes, including the effect of conduction-band non-parabolicity within the Kane model11. The phonon contribution to the dielectric permittivity was taken into account within the harmonic-oscillator approximation18. The contributions of holes and phonons turned out to be small, although non-negligible. The electron scattering rate was calculated, taking into account all relevant processes over the entire range of temperatures and magnetic fields (see Supplementary Information for details of our calculations of the carrier scattering rate and mobility as a function of temperature). Figure 3a and b show calculated 40 K transmittance spectra for only CRA (that is, |Ete|2 only) and only CRI (that is, |Eto|2 only), respectively, where Ete and Eto are the transmitted electric fields of the CRA and CRI modes, respectively (see Fig. 2a). The CRA wave experiences strong absorption and dispersion in the vicinity of ω+. On the other hand, the transmission of the CRI mode is nearly flat and featureless everywhere except at very low frequencies. Simple addition of the two (that is, |Ete|2+|Eto|2), however, does not produce any of the experimentally observed spectral features, except the clear plasma edge at zero field, as shown in Fig. 3c.

Figure 3: Experimental and theoretical terahertz transmittance spectra at various magnetic fields.
figure 3

ad, Calculated terahertz transmittance spectra for n-InSb at magnetic fields of 0, 0.25, 0.5, 1 and 1.5 T at a temperature of 40 K, including only the CRA wave (a), only the CRI wave (b), CRA and CRI without the interference term (c) and CRA, CRI including the interference term (d). The spectra were calculated using a magneto-plasma model described in the text. e, Experimental terahertz transmittance spectra for n-InSb at magnetic fields of 0, 0.25, 0.5, 1 and 1.5 T at a temperature of 40 K. All the peaks, dips and plateaux in e, arising from the interference between CRA and CRI modes are reproduced accurately in d.

The x component of the transmitted terahertz field (see Fig. 2), which is what we measure, is a superposition of the two fields Ete and Eto, that is, Etx=(Eto+Ete)/2. Hence, the signal intensity is given by

The last term on the right-hand side represents the interference between the CRA and CRI modes, as observed in one linear-polarization component (say, the x component) of the transmitted terahertz field through our polarization-selective detection scheme. This term depends on the index difference through cos[k0LRe(neno)], as shown in Fig. 2, where ne and no are the (complex) indices of the CRA and CRI modes, respectively, k0 is the wavenumber in vacuum and L (=0.8 mm) is the sample thickness. When ne experiences rapid changes owing to electron cyclotron resonance, whereas no stays almost constant (see Supplementary Information), the cosine oscillates between −1 and +1, creating sharp interference fringes. Adding the interference term (that is, 2Re(EteEto*)) in our simulation indeed totally modifies the spectra at finite magnetic fields, as shown in Fig. 3d. The agreement between theory (Fig. 3d) and experiment (Fig. 3e) is outstanding. The positions and shapes of all the transmission peaks, plateaux 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–magneto-plasmon excitations reaching tens of picoseconds.

Next, we consider the striking temperature (T) dependence of the measured terahertz transmittance at a fixed magnetic field, shown in Fig. 1a–c. As the temperature increases, electrical properties of the sample change because of, for example, temperature-dependent scattering processes (see Supplementary Information). However, the dominant process affecting the temperature dependence of the dielectric tensor at elevated temperatures is the thermal excitation of intrinsic carriers across the bandgap given by ni≈1.1×1014T1.5exp(−Eg/2kBT) cm−3, which leads to an exponentially growing plasma frequency. Here, Eg≈0.23 eV is the bandgap of InSb at zero temperature. The density of intrinsic carriers ni eventually exceeds the doping density of 2.3×1014 cm−3 at 180 K, as shown in Fig. 4a. Therefore, one would expect a weakly temperature-dependent transmittance below 180 K that would abruptly decrease above this temperature owing to the exponentially growing plasma frequency. The intensities of individually transmitted CRA and CRI modes, |Ete|2 and |Eto|2, respectively, indeed show this expected T dependence, as shown in Fig. 4b for a frequency of 0.25 THz.

Figure 4: Appearance of thermally induced terahertz transparency in a magneto-plasma as a result of coherence interference of CRA (extraordinary) and CRI (ordinary) waves.
figure 4

a, Density of intrinsic carriers, ni, versus temperature, excited across the bandgap. The dashed line represents the doping density for the sample (2.3×1014 cm−3). b, The three terms of the total transmittance, |Etx|2=|Ete|2+|Eto|2+2Re(EteEto*), as a function of temperature. The last term represents interference between the ordinary and extraordinary waves. c,d, Real (c) and imaginary (d) parts of the ordinary (no) and extraordinary (ne) indices of refraction. e,f, Measured (e) and calculated (f) transmittance of the sample as a function of temperature, corresponding to a horizontal slice of the experimental (Fig. 1b) and theoretical (Fig. 1c) contour maps of transmittance, respectively. All plots in bf are for a frequency of 0.25 THz and a magnetic field of 0.9 T.

However, as mentioned, the interference term in equation (2), 2Re(EteEto*), is proportional to cos[k0LRe(neno)]. With realistic parameters for our sample and experimental conditions, this interference term is negative and almost exactly cancels the other two terms in equation (2) below 160 K, as seen in Fig. 4b, leading to interference-induced opacity. Here, an incident linearly polarized electric field of unit amplitude (|Ei|=1) is assumed, and so the incident amplitudes of the CRA and CRI electric fields are equal to 1/2, that is, |Eie|=|Eio|=1/2. One can see from Fig. 4b that below 160 K the argument of the cosine function in the interference term, k0LRe(neno), is nearly constant and is close to π for our value of L (see equation (2)). When the temperature increases above 160 K, the difference between the refractive indices of the two normal waves, neno, starts growing exponentially, causing strong oscillations in the total transmittance owing to the interference term. These oscillations, however, are strongly damped above 200 K owing to the exponentially growing absorption coefficient for both normal modes, which is proportional to ωp2ν/ωc2, where ν is the carrier scattering rate. 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 in the side-by-side comparison between the observed and calculated temperature dependence of transmittance in Fig. 4e (experiment) and Fig. 4f (theory) as well as Fig. 1b,e (experiment) and Fig. 1c,f (theory). Thus, we conclude that, counter-intuitively, interference of normal modes causes high opacity of the low-density plasma and creates a transparency window when the plasma density exponentially increases.

These results demonstrate that free-carrier plasmas in lightly doped narrow-gap semiconductors are promising materials systems for terahertz physics, showing huge magnetic anisotropy effects and plasmon excitations in the terahertz range that are highly tunable with external fields, temperature and doping. In particular, we have shown that coherent interference phenomena, which are commonly observed and used in the visible and near-infrared range, can be extended into the terahertz regime. Conventional Fourier-transform infrared spectroscopy should in principle be able to detect these phenomena as long as one measures only one linear-polarization component of the transmitted beam, although multiple-reflection interference fringes might dominate the transmission spectra. Moreover, the observed 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 methods to manipulate terahertz waves13,14,19 as well as to probe more exotic phenomena in condensed-matter systems that occur owing to many-body interactions and disorder.


The time-domain terahertz magneto-spectroscopy system15 used in this study consisted of a chirped-pulse amplifier (CPA-2001, Clark-MXR) with a wavelength of 800 nm and a pulse width of 200 fs and a pair of 〈110〉 ZnTe crystals to generate and detect coherent radiation from 0.1 to 2.6 THz through surface rectification and electro-optic sampling, respectively. The electro-optic-sampling-based ZnTe detector is polarization-sensitive, and we placed it in such a way that we detect only the horizontal (or x) component of the transmitted terahertz field, to reveal the magnetic anisotropy-induced interference effects. A shaker, operating at 2 Hz, provided time delays up to 80 ps. We averaged over 1,000 scans for each spectrum. We used a magneto-optical cryostat to generate magnetic fields up to 10 T and varied the sample temperature from 1.6 to 300 K. The sample studied here was a large (0.8×20×30 mm3) crystal of Te-doped n-InSb with an electron density of 2.3×1014 cm−3 and a 2 K mobility of 7.7×104 cm2 V−1 s−1. For this particular sample, the Fermi energy at 0 T was 0.9 meV, or 0.21 THz, and the (angular) plasma frequency ωp=2π×0.28 THz. The Faraday geometry was used, where both the propagation direction of a linearly polarized terahertz wave and the magnetic field were perpendicular to the sample surface. We measured the transmitted terahertz waveforms through the sample and an empty hole (as a reference) at each magnetic field at a set of fixed temperatures. Then we Fourier-transformed the time-domain waveforms into the frequency domain and normalized the power spectra to obtain transmittance spectra. To eliminate interference fringes owing to multiple reflections within the sample, only the transmittance spectra of the first pulse that goes directly through the sample are shown in all figures. This windowing procedure effectively reduces the spectral resolution of the measurement to 50 GHz, which is more than adequate for the spectral features of interest.