Superradiant terahertz lasing with graphene metamaterials

Terahertz (THz) solid-state lasers, which operate in the far-IR region of the electromagnetic spectrum, are used in spectroscopy and imaging. However, they are generally inefficient and do not function at room temperature. To address these issues, we have investigated the use of ultra-fast charge carrier relaxation and plasmon dynamics in graphene, to achieve superradiance and THz lasing. Our aim is to enable the fabrication of lasers that operate at room temperature and across the THz frequency range.

There is no commercially available microelectronic device that generates and detects electromagnetic waves over the entire THz frequency band,¹ despite developers' efforts in the past decade to achieve compact, tunable, and coherent sources. In one conventional electron device—the field-effect transistor—the application of an electric field controls the flow of carriers. Such devices increase their operating frequencies by scaling down the distance between the source and drain sides of the electron channel, but this reduces the output power. Photonic devices, such as semiconductor quantum-cascade lasers (QCLs), emit in the mid- and far-IR spectrum. These decrease their operating frequencies by scaling down the transition energy from the interband level (where electrons and holes move between the two adjacent energy state ‘bands’ of a semiconductor) to the inter-sub-band. However, QCLs are subject to significant thermal noise, which prevents them from operating at room temperature.

Figure 1. (Left) Energy band and carrier relaxation in graphene under IR laser pumping, where incident terahertz photons produce non-equilibrium charge carriers and stimulated THz photon emission. (Right) Exploiting this process to amplify THz waves in graphene.

To address these issues, we considered graphene, a one-atom-thick planar sheet of honeycomb-structured carbon crystal that has unique carrier-transport and optical properties.^2–4 The conduction and valence bands of graphene (the two continuous energy state bands) have a symmetrical conical shape around the edges of the Brillouin zone (a cell occupying the reciprocal space of a solid crystal lattice, and whose edges are generally at the minimum energy band gap in a semiconductor). In graphene this band gap is zero, and the energy of the electrons and holes is proportional to its momentum. The carriers lose their mass and behave as relativistic particles: in this case, relativistic Fermions (where one energy state can be occupied by only one electron or hole). The result is that they can be transported at ultrafast speeds, and are not subject to back-scattering.

Optical and electrical pumping of graphene results in negative dynamic conductivity, or gain, at THz frequencies.⁵ 2D plasmons in graphene dramatically enhance light-matter interaction, improving the quantum efficiency and leading to THz lasers with higher power output. When graphene is patterned into ordered structures of micrometer- to sub-micrometer size, these units become far smaller than the wavelength of THz waves, and the structures act as metamaterials,⁶ which are specially designed to have plasmonic responses that fall in the THz range. These can provide intense THz emission if the conditions at the plasmonic cavity boundary allow hydrodynamic instability,⁷ or the THz dynamic conductivity in the plasmonic cavity takes negative values through optical or electrical pumping.^7,8 Our work reviews recent advances in graphene plasmonics for stimulated emission of THz radiation.

We achieve interband population inversion (when there are more excited electrons and holes than carriers in a low-energy state) with optical pumping or carrier injection.⁵ Under sufficiently strong excitation, the interband stimulated emission of photons can prevail over the intraband (Drude) absorption. The dynamic conductivity of graphene, Re[σ(ω)], then becomes negative in the THz range because of its zero band gap. The stimulated emission of near-IR and THz photons from population-inverted graphene⁵ is limited by the quantum conductivity .^2,3 This is because the absorption of THz photons that can contribute to stimulated emission is made only by the interband transition process, whose absorbance is limited by .³ To overcome this, we considered introducing surface plasmon polaritons (SPPs), which have a shorter wavelength.^4,8 These waves have a speed three or four times slower than that of photons, enabling more effective interaction, and resulting in an increase in gain. Excitation of the SPPs in population-inverted graphene can exploit the resonant plasmon absorption in graphene metamaterials, such as micro ribbon arrays, micro disk arrays, and double- and multiple graphene structures, to achieve superradiant THz emission.

We studied theoretically the amplification of THz waves through stimulated generation of resonant plasmons in a graphene plasmonic metamaterial,⁸ see Figure 2(a). Graphene microcavities were confined between metal grating contacts on the flat surface of a dielectric substrate of silicon (silicon carbide can also be used). An external THz wave was incident upon the planar array of graphene microcavities normal to its plane, with polarization of the electric field across the metal grating contacts. The graphene is pumped either by optical illumination or by injection of electrons and holes from opposite metal contacts in each microcavity. In the latter case, the opposite ends of each cavity adjacent to the metal contacts must be p- and n-doped.^5,7 The array of microcavities amplifies the THz wave.⁸ We could numerically simulate the quasi Fermi energy (ε_F)—or the excited carrier population—and the carrier temperature to gauge the dynamic conductivity.

Figure 2(b) shows a contour map of the calculated absorbance as a function of the quasi-Fermi energy and the THz wave frequency for an array of the graphene microcavities. Here, the negative value of the absorbance yields the amplification coefficient. In Figure 2(b) the value of Re[σ_Gr(ω)] is negative above the black solid line, corresponding to Re[σ_Gr(ω)] = 0 (or transparent graphene). Above this boundary line, negative absorption (or amplification) takes place at all frequencies and pumping strengths.

Figure 2. (a) A graphene-metal micro ribbon array acting as a THz plasmonic metamaterial. The carrier population in graphene is inverted through optical or electrical pumping. Incident THz photons have an electric field intensity vector component E₀ perpendicular to the ribbon direction. Surface plasmon polaritons (SPPs) are excited and can produce significant gain at THz frequencies corresponding to the SPP modes. (b) Contour map of the absorbance as a function of both the quasi-Fermi energy and the frequency of incoming THz waves. This is for an array of graphene microcavities with period L =4μm and length a =2μm. The electron scattering time in graphene is assumed to be τ=10⁻¹²s. Blue and red arrows mark the quasi-Fermi energies for the maximum absorption and for the plasmonic lasing regime, respectively, at the fundamental plasmon resonance.

With increasing ε_F, the energy gain can balance the energy loss caused by the electron scattering in graphene, so that the net loss becomes zero. Above the transparency line Re[σ_Gr(ω)] = 0, the THz wave amplification at the plasmon resonance frequency is several orders of magnitude stronger than away from the resonances. The behavior of the amplification coefficient around the self-excitation regime is shown in Figure 3(a). Lasing occurs when the plasmon gain balances the electron scattering loss and the radiative loss, seen in Figure 3(b). Therefore, the plasmon oscillations are highly coherent. The quasi-Fermi energy corresponding to plasmonic lasing in the first plasmon resonance is marked by the red arrow in Figure 2(b).

Figure 3. (a) Variation of the power amplification coefficient along the fundamental plasmon-resonance amplification lobe, as shown in Figure 2(b). (b) Schematic illustration of the energy rate balance in the plasmon lasing regime. (c) Distribution of the normalized induced in-plane electric field at a moment in time corresponding to the maximum swing of plasma oscillations in the graphene microcavities at the fundamental plasmon amplification resonance.

The plasmons in different graphene microcavities oscillate in phase (even without the incoming electromagnetic wave) because the metal contacts act as synchronizing elements between adjacent microcavities. In Figure 3(c) we see the plasma oscillations in the array forming a single collective plasmon mode, distributed over the entire area of the array, and leading to enhanced superradiant THz emission.⁹

Numerical modeling reveals that the simulated gain coefficient exceeds 10⁴ cm⁻¹ in a wide THz range. Our work demonstrates the possibilities for stimulated emission of THz radiation in graphene using either photons or plasmons. In future, both channels may be used to develop graphene-based intense THz emitters and lasers operating at room temperature.

The authors wish to thank: D. V. Fateev at the Kotelnikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences (RAS), Saratov, Russia; A. A. Dubinov and Y. A. Aleshkin at the Institute for Physics of Microstructures, RAS; Nizhny Novgorod, Russia; V. Vyurkov, D. Svintsov, and I. Semenikin at the Institute of Physics and Technology, RAS, Moscow; S. Chan at the University of California, Santa Barbara, USA; V. Mitin at the University at Buffalo, State University of New York; M. S. Shur at Rensselaer Polytechnic Institute, New York; A. Satou, S. Boubanga Tombet, T. Watanabe, Y. Tanimoto, T. Fukushima, and T. Suemitsu at the Research Institute of Electrical Communication, Tohoku University, Japan; M. Ryzhii at the, University of Aizu, Japan; and E. Sano at Hokkaido University, Japan, for their extensive contributions. This work is financially supported in part by JST-CREST; JSPS-GA-SPR (#23000008); JSPS-Jpn-Russ; JSPS Core-to-Core, Japan; NSF-PIRE-TeraNano, USA; the Goverment of the Russian Federation (Cont. No. 11.G34.31.0030); and the Russian Academy of Sciences.