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High-harmonic spectroscopy of strongly bound excitons in solids

Simon Vendelbo Bylling Jensen, Lars Bojer Madsen, Angel Rubio, and Nicolas Tancogne-Dejean
Phys. Rev. A 109, 063104 – Published 4 June 2024
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  • INTRODUCTION
  • THEORETICAL MODEL AND METHODS
  • RESULTS AND DISCUSSION
  • SUMMARY AND CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We explore the nonlinear response of ultrafast strong-field-driven excitons in a one-dimensional solid with ab initio simulations. We demonstrate from our simulations and analytical model that a finite population of excitons imprints unique signatures to the high-harmonic spectra of materials. We show the exciton population can be retrieved from the spectra. We further demonstrate signatures of exciton recombination and that a shift of the exciton level is imprinted into the harmonic signal. The results open the door to high-harmonic spectroscopy of excitons in condensed-matter systems.

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    • Received 7 July 2023
    • Revised 22 March 2024
    • Accepted 8 May 2024

    DOI:https://doi.org/10.1103/PhysRevA.109.063104

    ©2024 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    ExcitonsHigh-order harmonic generation
    1. Techniques
    Density functional theory
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Simon Vendelbo Bylling Jensen1,*, Lars Bojer Madsen1, Angel Rubio2,3,†, and Nicolas Tancogne-Dejean2,‡

    • *simon.jensen@mpsd.mpg.de
    • †angel.rubio@mpsd.mpg.de
    • ‡nicolas.tancogne-dejean@mpsd.mpg.de

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    Vol. 109, Iss. 6 — June 2024

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    • Figure 1
      Figure 1

      Part of the band structure with the valence band and lowest-energy conduction bands. The band gap is found to be 9.45 eV.

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    • Figure 2
      Figure 2

      Linear absorption spectrum calculated through TDHF with exponential dampening of the electronic current, corresponding to a Lorentzian broadening of the absorption spectrum. The vertical lines denote the two applied pump frequencies, the exciton-resonant one, at ωex=3.86 eV, and the band-gap-resonant one, at ωbg=9.45 eV. We associate the peak just below the band-gap energy to be a signature of excited excitonic states.

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    • Figure 3
      Figure 3

      (a) First moment, m of Eq. (3), of the approximated exciton wave function for different pump frequencies and intensities. The areas denote regimes where the ωex pump generates dominantly bound excitons (blue) or free carriers (red). (b) Excitation pathways from the valence band (VB) to conduction band (CB) or to exciton (Ex). The corresponding energies are ωbg=9.45 eV and ωex=3.86 eV. [(c)–(e)] Exciton density for a hole at x′=0, after excitation by a 25-fs pump pulse with intensities of 106, 109, and 1011Wcm−2, respectively. (f) Number of pumped free carriers Nfc and excitons Nex in the system as of Eqs. (A1) and (A3). The dashed lines show the linear perturbative scaling behavior for resonant excitation. The highest intensity value for the ωex pump is omitted due to the excitation exceeding the damage threshold predicted under the electron-hole plasma model of ≈10% of excited electrons [88, 89, 90]. The quantities of [(a)–(f)] are evaluated after the pump preparation, just before the system is driven to produce HHG.

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    • Figure 4
      Figure 4

      Degree of ionization of the exciton induced by a pump laser versus different intensities utilizing identical laser parameters as for Fig. 3 in the main text. See main text for details.

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    • Figure 5
      Figure 5

      Time-frequency analysis of the harmonic radiation of the region above the exciton binding energy for, respectively, an unpumped (a) and an exciton-seeded sample (b), obtained by an exciton-resonant pump with an intensity of 107Wcm−2. The 2000-nm driving electric field is sketched in green and the time-frequency analysis is performed with a Gabor transform window of σ=0.40 fs.

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    • Figure 6
      Figure 6

      Norm-squared exciton wave function as a function of distance and time, during the HHG process. Comparing an (a) unpumped system with (b) an exciton-seeded system with parameters of Fig. 5. The exciton wave function is considered within the temporal region of the probe pulse, which electric field is inserted in green.

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    • Figure 7
      Figure 7

      Magnified time-frequency analysis of Figs. 5 and 5 for, respectively, an unpumped and exciton-seeded sample. Obtained with parameters of Fig. 5. Trajectories from our exciton-extended semiclassical model are depicted with dotted lines. Black color depicts trajectories recombining to the valence band and the blue color corresponds to recombination in form of a bound exciton.

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    • Figure 8
      Figure 8

      Time-frequency analysis of the harmonic radiation in the spectral region below the band gap for, respectively, an (a) unpumped and an (b) exciton-seeded sample, obtained by an exciton-resonant pump similarly to Fig. 5. A window of σ=10 fs is applied for the Gabor transform. Dashed lines denote the locations of the exciton peak, as well as the first exciton sidebands.

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    • Figure 9
      Figure 9

      Time-frequency analysis of the below-band-gap harmonic radiation for an exciton-seeded sample, obtained by an exciton-resonant pump with an intensity of 107Wcm−2. The probe-pulse intensity is scanned in the regime of 1011 to 2×1012Wcm−1 and given in the caption. We used a window of σ=10 fs for the Gabor transform. The positions of the exciton peak and the first exciton sidebands are denoted with dashed lines.

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    • Figure 10
      Figure 10

      [(a)–(c)] HHG spectra for various wavelengths for the unpumped system versus the system prepared by an exciton-resonant pump. The HHG driving probe wavelength is scanned across 1600, 2000, and 2600 nm, respectively, for [(a)–(c)]. The colored areas denote the exciton resonance (blue) and the first excitonic associated sidebands, at ωex±2ω (green). (d) Harmonic yield enhancement of the exciton resonance and sidebands for λ=2000 nm as a function of bound exciton population, utilizing a 105–108Wcm−2 exciton-resonant pump. Dashed lines are explained in the main text.

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    • Figure 11
      Figure 11

      High-harmonic generation spectra for the systems prepared with a large population of bound excitons with ωex or free carriers with ωbg. The excited systems are prepared with a 108−Wcm−2ωex pump or a 1012−Wcm−2ωbg pump to generate a strong excitation of, respectively, bound excitons or free carriers. The total excitation prepared by the ωbg pump is 2.3 times larger than the excitation generated by the ωex pump, as given in Fig. 1 of the main text. The harmonics is obtained when driven with a 2000-nm probe of intensity 1012Wcm−2. The exciton resonance and the band-gap energy are marked with a blue dashed and red dashed-dotted line, respectively. For illustrative purposes, the spectra have been smoothed. See text in Sec. 3 for pulse durations and delay.

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    • Figure 12
      Figure 12

      Number of excitons Nex and free carriers Nfc as a function of time, in blue for pump and probe, and in black for the probe-only case. Parameters of Fig. 11 are used, alongside a probe-only calculation.

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    • Figure 13
      Figure 13

      Ratio of the number of excitons Nex before and after the probe pulse (circles), and ratio of the number of free carriers Nfc before and after the probe pulse (crosses), as a function of pump intensity.

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