Threshold of Terahertz Population Inversion and Negative Dynamic Conductivity in Graphene Under Pulse Photoexcitation
We present a theoretical study of population inversion and negative dynamic conductivity in intrinsic graphene in the terahertz (THz) frequency range upon pulse photoexcitation at near-/mid-infrared wavelengths. The threshold pulse fluence required for population inversion and negative dynamic conductivity can be orders of magnitude lower when the pulse photon energy is lower, because of *Reptinted with permission from A. Satou, V. Ryzhii, Y. Kurita, and T. Otsuji (2013). Threshold of terahertz population inversion and negative dynamic conductivity in graphene under pulse photoexcitationJ.Appl. Phys., 113, 143108. Copyright © 2013 AIP Publishing LLC.
Graphene-Based Terahertz Electronics and Plasmonics: Detector and Emitter Concepts Edited by Vladimir Mitin, Taiichi Otsuji, and Victor Ryzhii Copyright © 2021 Jenny Stanford Publishing Pte. Ltd.
ISBN 978-981-4800-75-4 (Hardcover), 978-0-429-32839-8 (eBook) www.jennystanford.com the inverse proportionality of the photoexcited carrier concentration to the pulse photon energy and because of the weaker carrier heating. We also investigate the dependence of dynamic conductivity on momentum relaxation time. Negative dynamic conductivity takes place either in high- or low-quality graphene, where Drude absorption by carriers in the THz frequency is weak.
Graphene has attracted much attention for a wide variety of devices, owning to its exceptional electronic and optical properties [1-3]. In particular, THz devices such as lasers [4-9] and photodetectors [10-12] which take advantage of the high carrier mobility and gapless dispersion of graphene have been investigated . We have demonstrated that population inversion can occur in optically pumped graphene in the THz/far-infrared range of frequencies; hence lasing in this range is possible, utilizing the gapless linear energy spectrum and relatively high optical phonon (OP) energy in graphene [4-8]. Due to the energy spectrum e = tVpfik, where t% ^ 10s cm/s is the Fermi velocity and к is the wavenumber, the Fermi energy %at equilibrium in intrinsic graphene is equal to zero. Hence, the electron and hole distribution functions at the bottom of the conduction band and the top of the valence band have values of/e(0) = /h(0) = 1/2. This implies that upon photoexcitation, the values of the distribution functions at low energies can be greater than one half,/e(f) =/h(f) > 1/2, corresponding to the population inversion. This type of population inversion means that the total dynamic conductivity in graphene at THz/far-infrared frequencies is negative, and that lasing in graphene at these frequencies is possible.
Recently, we measured the carrier relaxation and recombination dynamics in optically pumped epitaxial graphene on silicon  and in exfoliated graphene  by THz time-domain spectroscopy based on an optical pump/THz & optical probe technique, and we observed the amplification of THz radiation by stimulated emission from graphene under pulse excitation. To our knowledge, those are the first observation of THz amplification using optically pumped graphene, and they demonstrate the possibility of realizing THz lasers based on graphene.
The carrier dynamics in optically pumped graphene strongly depend on the initial temperature of carriers and the intensity of the optical pumping. For sufficiently low carrier concentrations, that is, at low temperatures under weak pumping, photoexcited carriers accumulate effectively near the Dirac point via the cascade emission of OPs. It is predicted that population inversion can be achieved efficiently under these conditions [4, 5, 8]. In contrast, at room temperature or under stronger pumping, where the carrier concentration is high (~1012 cm"2), carrier-carrier (CC) scattering plays a crucial role in the dynamics after pulse excitation, because of the fast quasi-equilibration of the carriers. Ultrafast optical pump-probe spectroscopy on graphene has indicated that the quasiequilibration by CC scattering occurs on a time scale of 10-100 fs [16-19], which is much faster than a single OP emission. In this case, the pulse excitation makes carriers initially very hot initially, and the energy relaxation and recombination via OP emission follow. The short time scale for the quasi-equilibration is partly caused by the ineffective dielectric screening of the Coulomb potential by relatively low dielectric layers surrounding graphene, which is typically supported by a SiC or Si02 substrate and exposed to air.
We have previously shown that population inversion and negative dynamic conductivity in intrinsic graphene at room temperature under pulse excitation with a photon energy of 0.8 eV, which corresponds to a wavelength of 1.55 pm, can be achieved with a pulse energy fluence above a certain threshold [7, 20].
In the present chapter, we investigate the dependence on the pulse photon energy of the threshold pulse fluence for THz negative dynamic conductivity, by extending our previous model to take into account the effect of the Pauli blocking [7, 20]. We also examine the dependence of dynamic conductivity on momentum relaxation time, r. We consider the two limiting cases where cor> 1, which correspond to high-quality graphene, and where сот «1, which corresponds to low-quality graphene.