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Efficient generation of extreme terahertz harmonics in three-dimensional Dirac semimetals

Jeremy Lim, Yee Sin Ang, F. Javier García de Abajo, Ido Kaminer, Lay Kee Ang, and Liang Jie Wong
Phys. Rev. Research 2, 043252 – Published 18 November 2020
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  • INTRODUCTION
  • MODELING NONPERTURBATIVE ELECTRON…
  • SUPERCRITICAL REGIME OF NONLINEAR OPTICS
  • SUBCRITICAL REGIME OF NONLINEAR OPTICS
  • COMPARING VARIOUS 3D AND 2D DSMs
  • DISCUSSION
  • SUMMARY
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We show that three-dimensional (3D) Dirac semimetals (DSMs) can achieve highly efficient terahertz high-order harmonic generation (HHG) up to the 31st harmonic with input intensities ≈10MW/cm2—over 105 times lower than required in conventional terahertz HHG systems. Our theory reveals that this extreme nonlinearity is made possible by the existence of an operation regime that differs from previous demonstrations of lower order harmonic generation. We also reveal an unexpected regime in which emitted harmonics abruptly become negligible beyond the third order. This unprecedented vanishing of higher order nonlinearity has a geometrical origin related to the combination of conical dispersion and extra dimensionality in 3D DSMs, breaking the common notion that 3D DSMs share the essential physics of two-dimensional DSMs. Our findings pave the way to unlocking the full potential of 3D DSMs as efficient platforms for terahertz light sources and optoelectronics at moderate intensities.

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    • Received 15 June 2020
    • Revised 19 August 2020
    • Accepted 14 October 2020

    DOI:https://doi.org/10.1103/PhysRevResearch.2.043252

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    High-order harmonic generationNonlinear opticsUltrafast optics
    1. Physical Systems
    Dirac semimetal
    Condensed Matter, Materials & Applied PhysicsNonlinear DynamicsAtomic, Molecular & Optical

    Authors & Affiliations

    Jeremy Lim1, Yee Sin Ang1,*, F. Javier García de Abajo2,3, Ido Kaminer4, Lay Kee Ang1,†, and Liang Jie Wong5,‡

    • 1Science, Math, and Technology, Singapore University of Technology and Design, 8 Somapah Road, Singapore, Singapore 487372
    • 2ICFO-Institut de Ciencies Fotoniques, Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
    • 3ICREA-Institució Catalana de Recerca i Estudis Avançats, Passeig Lluís Companys 23, 08010 Barcelona, Spain
    • 4Department of Electrical Engineering, Technion, Haifa 3200003, Israel
    • 5School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore, Singapore 639798

    • *yeesin_ang@sutd.edu.sg
    • †ricky_ang@sutd.edu.sg
    • ‡liangjie.wong@ntu.edu.sg

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    Vol. 2, Iss. 4 — November - December 2020

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    • Figure 1
      Figure 1

      Highly efficient generation of extreme harmonics (up to 31st order) in the 3D DSM material Cd3As2 at modest driving field strengths. (a) HHG in a 3D DSM occurs when a driving laser pulse (incoming red wave) induces carrier oscillations (current density depicted at one instant in time on the Dirac cone) and transitions that lead to the emission of high-harmonic light (outgoing multicolored waves). Driving a Cd3As2 thin film with a linearly polarized pulse of central frequency 1 THz and peak field strength of 10 MV/m [(b) inset] produces the emitted spectrum shown in panel (b), where we see that harmonics up to the 31st order and beyond can be generated at energy conversion efficiencies well beyond 10−5. Panels [(c)–(g)] show the change in the output energy spectrum and conversion efficiency as a function of field strength of the external driving laser (markers, with connecting lines as visual guides). We see that extreme harmonic generation continues to remain relatively efficient even at much lower field strengths. We consider a 250-nm-thick Cd3As2 thin film of radius 1 mm and Fermi energy EF=60 meV, uniformly illuminated by a 2-ps long pulse of 1-THz peak frequency.

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    • Figure 2
      Figure 2

      Vanishing of the high-order intraband emission from 3D DSMs in the subcritical regime (Φmax<Φcrit). An incident laser field induces oscillating carriers in 3D Dirac cones (a) and 2D Dirac cones (b). However, the emission profile resulting from these oscillations differs drastically between the 3D case and the 2D case, breaking the notion that 3D DSMs are merely bulk versions of 2D DSMs. While optical nonlinearities abruptly disappear beyond the third order in 3D DSMs (c), they persist at every order in 2D DSMs (d). This phenomenon holds across a broad range of field strengths, as shown in panels (e) and (f). The contribution of the interband current is very weak and has been verified to fall below the intensity range displayed. In panels (e) and (f), we also see that the results of our fully closed-form, nonperturbative expressions (curves) are in excellent agreement with rigorous numerical simulations (solid circles). The reason why higher order nonlinearities vanish in 3D DSMs and not in 2D DSMs lies in the extra dimension that 3D DSMs possess compared to 2D DSMs. Despite sharing the same expression for the current [Eq. (1)], the required integration over 3D momentum space (g) for 3D DSMs, in contrast to integration over 2D momentum space (h) for 2D DSMs, leads to very different behavior in these materials. In panels (g) and (h), the colored solid areas represent the regions of integration, which are shifted by the applied laser field. In this comparison, we consider Fermi velocities vx=vy=vz=106 m/s, Fermi energy EF=250 meV at temperature T=0 K, and no carrier scattering.

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    • Figure 3
      Figure 3

      High-order harmonic generation in the 3D DSM materials Cd3As2 (a) and Na3Bi (b), and in 2D DSM graphene (c). In the subcritical regime (field corresponding to critical potential marked by vertical black dashes), emitted harmonics beyond the third order are greatly suppressed in 3D DSMs. No such suppression occurs in 2D DSMs. In the supercritical regime, however, the intensity of the emitted harmonics in 3D DSMs rapidly increase with increasing field strength. In each panel, markers and lines denote numerical (intraband and interband emission) and analytical (intraband emission only) results, respectively. The good agreement between them indicates the dominance of intraband emission in our regime of interest. For the sake of clarity, we plot only up to the ninth harmonic.

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    • Figure 4
      Figure 4

      Existence of a unique operation regime (upper right quadrant) in which highly efficient generation of extreme THz harmonics (≥29th orders) is possible. Our theory reveals that this distinct regime is complementary to existing demonstrations of harmonic generation of up to the third [31] (cyan cross) and seventh [32] (cyan circle) order: By combining a potential amplitude Φmax/Φcrit similar to that used in Ref. [32], and a scattering time ω0τ similar to that of Ref. [31], the extreme nonlinearity that becomes possible only within this regime can be leveraged for highly efficient THz HHG up to the 31st order and beyond, as we predict in Fig. 1 (red diamond). The color map is computed using Eqs. (2) and (3) at T=0 K. Only normalized harmonic intensities ≥10−6 are considered.

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    • Figure 5
      Figure 5

      Representation of the discretization method used in the SBC FDTD scheme for graphene. At spatial grid point j+1/2, where the graphene sheet is located, Hy is evaluated on either side of the sheet.

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    • Figure 6
      Figure 6

      HHG intensity spectrum in the supercritical regime at different temperatures. While the harmonic intensity peaks decrease as the temperature increases from T=0 K to T=300 K, the changes are not substantial in the supercritical regime. We consider a Cd3As2 thin film doped to EF=250 meV (at T=0 K) driven by a 2-ps-long, 1-THz-centered pulse of peak field strength Ex,0=10 MV/m inside the material. We assume that the induced current is spatially uniform throughout the 250-nm-thick thin film of radius 1 mm. We neglect the effects of scattering.

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    • Figure 7
      Figure 7

      Dependence of the HHG peak intensity on scattering time and Fermi level EF. For all harmonics (frequency labeled in the bottom right corner of each panel), we observe that the peak intensities saturate as the scattering time increases. We find that in the supercritical regime, saturation occurs at a smaller scattering time for lower doping levels. As temperature has little effect on the HHG performance of Cd3As2 in the supercritical regime, we work in the T→0 K limit. The values presented here are computed using Eqs. (B14) and (B15), with the vector potential modified to include intraband scattering. Unless otherwise stated, we consider the same parameters as in Fig. 6.

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    • Figure 8
      Figure 8

      High-order harmonic spectrum of Cd3As2 for various scattering times. The red curves and squares represent the situation in which scattering is neglected, which we plot in Fig. 1 in the main text. For a scattering time of 150 fs (black curves and filled circles), which is similar to the experimentally determined value of 145 fs obtained in Ref. [31], we see that the energy spectral density (left vertical axis) remains within the same order of magnitude as in the absence of scattering. The energy conversion efficiency (right vertical axis) of the 31st harmonic is about 8.7×10−6—within one order of magnitude of the case with no scattering. The decrease in both the energy spectral density and energy conversion efficiency is more dramatic for scattering times of the order of tens of fs, but the generation of harmonics up to the 31st order and beyond remains possible. Unless otherwise stated, the same parameters as in Fig. 1 are considered.

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    • Figure 9
      Figure 9

      Impact of polarization and Fermi velocity anisotropy on HHG in DSMs. The laser polarization is controlled by varying the relative phase between a superposed x-polarized and y-polarized laser pulse of the same amplitude E0/2 (labeled on the left). We choose the driving field amplitudes E0=1.41 and E0=4.24 MV/m, which correspond to the subcritical and supercritical regimes, respectively. In the subcritical regime of 3D DSMs, the strong suppression of HHG beyond the third harmonic persists for all polarizations (a)–(c), in contrast to 2D DSMs, where the fifth and ninth harmonics are relatively strong (d). In the supercritical regime (e)–(h), both 2D and 3D DSMs produce relatively high intensities of higher harmonics. Efficient HHG is most favored by linearly polarized (as opposed to circularly polarized) driving pulses in both 2D and 3D DSMs, a result that agrees with a previous study for graphene [67]. For 3D DSMs, we further observe that a larger Fermi velocity anisotropy generally leads to more effective HHG when driven by circularly polarized pulses.

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    • Figure 10
      Figure 10

      Strong suppression of harmonics beyond third order from Cd3As2 in the subcritical regime at T=4 K (a), T=77 K (b), and T=300 K (c) for various scattering times (different curves within each panel). While the suppression of HHG above third order is most obvious at T=4 K and T=77 K, we see that this suppression still persists even up to room temperature (T=300 K). We also see that, at all temperatures, the inclusion of finite scattering only emphasizes the suppression. The spectra presented here are computed using Eq. (1), which accounts for both interband and intraband contributions. The driving field is a 2-ps-long laser pulse linearly polarized along x with 1-THz central frequency and peak amplitude Ex,0=1.1 MV/m. The Fermi level is EF=250 meV. The temperature dependence of the chemical potential μ(T) is modeled using the Sommerfeld expansion, which is valid for highly doped samples [68].

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    • Figure 11
      Figure 11

      Comparison of high-order harmonic generation in Cd3As2 (maroon curves, squares) and graphene (blue curves, circles). By virtue of a finite interaction volume, the intensity (left vertical axes) of the HHG in Cd3As2 can be more than two orders of magnitude greater than the HHG intensity output of a single atomic layer of graphene under the same conditions. For both materials, we assume a Fermi energy EF=60 meV at T=0. We consider a cylindrical sample of radius R=1 mm and thickness D=250 nm uniformly illuminated by a 1-THz-centered, 2-ps-long (intensity FWHM) linearly polarized (along x) laser pulse of peak amplitude Ex,0=10 MV/m. These parameters and methodology are the same as those used in Fig. 1, with the FDTD algorithm used for graphene being replaced by a modified version detailed in Appendix pp1-s5.

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    • Figure 12
      Figure 12

      3D DSM HHG intensity in the supercritical regime as a function of the Fermi velocities in each direction, as computed using Eqs. (2) and (3). We assume vx=vy for simplicity. We consider a Fermi energy EF=250 meV, a temperature T=0 K, and no carrier scattering. The incident field is a 2-ps-long, 1-THz-centered pulse linearly polarized along x with a peak field strength Ex,0=10 MV/m. The color maps indicate that the performance of HHG varies greatly depending on Fermi velocity values and anisotropy.

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    • Figure 13
      Figure 13

      Harmonic spectra for different combinations of parameters. For all panels, the spectrum represented by the thin red curve corresponds to the following parameters: EF=118 meV, f0=ω0/2π=0.3 THz, and vx=vy=vz=0.78×106 m/s. The spectrum denoted by the thick blue line corresponds to the following parameters: EF=60 meV, f0=ω0/2π=1 THz, and (vx,vy,vz)=(1.28,1.3,0.327)×106 m/s. For both cases, a two-cycle-long (intensity FWHM) driving pulse is considered. In each panel, the scattering times τ and potential amplitudes Φmax are chosen such that the quanitites ω0τ and Φmax/Φcrit are the same for both sets of parameters. We see that the normalized harmonic spectra for both sets of parameters overlap exactly in all panels. All results shown here are computed at T=0 K using our closed-form analytical expressions given by Eqs. (2) and (3).

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    • Figure 14
      Figure 14

      Demonstration that the deviation of the predicted number of harmonics in Fig. 4 from experimentally reported results in Ref. [32] is due to temperature difference. No intraband harmonics above the third order are predicted in the low-temperature limit T=0 K (solid blue curve). At the reported experimental temperature T=300 K, intraband-only simulations using the same set of parameters indicate that higher order intraband harmonics are generated (red dash-dotted curve)—consistent with the result reported in Ref. [32]. We consider the following parameters: f0=ω0/2π=0.3 THz, EF=118 meV, vx=vy=vz=0.78×106 m/s, E0,x=6 MV/m, and τ=10 fs. We use a 14.7-ps-long (intensity FWHM) driving THz pulse.

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