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Structure gauges and laser gauges for the semiconductor Bloch equations in high-order harmonic generation in solids

Lun Yue and Mette B. Gaarde
Phys. Rev. A 101, 053411 – Published 7 May 2020
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  • INTRODUCTION
  • THEORETICAL METHODS
  • RESULTS FOR 1D ZNO
  • RESULTS FOR MONOLAYER hBN
  • SUMMARY AND OUTLOOK
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    The semiconductor Bloch equations (SBEs) are routinely used for simulations of strong-field laser-matter interactions in condensed matter. In systems without inversion or time-reversal symmetries, the Berry connections and transition dipole phases (TDPs) must be included in the SBEs, which in turn requires the construction of a smooth and periodic structure gauge for the Bloch states. Here, we illustrate a general approach for such a structure-gauge construction for topologically trivial systems. Furthermore, we investigate the SBEs in the length and velocity gauges and discuss their respective advantages and shortcomings for the high-harmonic generation (HHG) process. We find that in cases where we require dephasing or separation of the currents into interband and intraband contributions, the length-gauge SBEs are computationally more efficient. In calculations without dephasing and where only the total current is needed, the velocity-gauge SBEs are structure-gauge independent and are computationally more efficient. We employ two systems as numerical examples to highlight our findings: a one-dimensional model of ZnO and the two-dimensional monolayer hexagonal boron nitride (hBN). The omittance of Berry connections or TDPs in the SBEs for hBN results in nonphysical HHG spectra. The structure- and laser-gauge considerations in the current work are not restricted to the HHG process and are applicable to all strong-field matter simulations with SBEs.

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    • Received 12 February 2020
    • Accepted 13 April 2020

    DOI:https://doi.org/10.1103/PhysRevA.101.053411

    ©2020 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Geometric & topological phasesHigh-order harmonic generationMultiphoton or tunneling ionization & excitationStrong electromagnetic field effectsUltrafast phenomena
    1. Physical Systems
    SemiconductorsSingle crystal materials
    1. Techniques
    Nonperturbative methods
    Atomic, Molecular & Optical

    Authors & Affiliations

    Lun Yue* and Mette B. Gaarde†

    • Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803-4001, USA

    • *lun_yue@msn.com
    • †mgaarde1@lsu.edu

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    Vol. 101, Iss. 5 — May 2020

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    Images

    • Figure 1
      Figure 1

      Examples of computational complexities [Eqs. (20, 21, 22)], with nδt=5, NG=10. (a) Fixed Nb=2 and varying Nk. (b) Fixed Nk=2002 and varying Nb. Data points are shown for some parameters used for the 2D system in this work (calculations for VG SBE with dephasing are not performed for the 2D system, see text).

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    • Figure 2
      Figure 2

      (a) Band structure of the 1D ZnO model (two valence bands and three conduction bands are shown). (b) Three diagonal elements of the momentum operator. Two off-diagonal elements of the momentum operator, (c) before and (d) after application of the parallel transport gauge.

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    • Figure 3
      Figure 3

      High-harmonic spectrum for the laser-driven Mathieu potential calculated in (a) VG and (b) LG, without dephasing. The vertical dotted lines at around the 11th and 36th harmonics show, respectively, the band-gap energy and the largest energy between the highest valence band and the lowest conduction band. The VG SBEs in (a) are seen to be independent of the structure-gauge choice.

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    • Figure 4
      Figure 4

      High-harmonic spectrum for the laser-driven Mathieu potential calculated in LG (blue area) and VG (solid line). Upper panels show the (a) interband and (b) intraband contributions for the case of no dephasing. Lower panels (b) and (c) show the result for the case of 2-fs dephasing. The vertical dotted lines at around the 11th and 36th harmonics respectively show the band-gap energy and the largest energy between the highest valence band and the lowest conduction band.

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    • Figure 5
      Figure 5

      (a) Band structure of the 2D hBN model (one valence band and three conduction bands are shown). (b, c) Imaginary part of the x component of the momentum operator, Im(p12,xk), plotted in the first BZ as a function of the reduced coordinates along b1 and b2 [see Eq. (6)]. (b) No structure-gauge choice (random gauge); (c) parallel transport gauge; and (d) twisted parallel transport gauge.

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    • Figure 6
      Figure 6

      Berry connections for (a) the valence band and (b) the conduction band in the twisted parallel transport gauge. For better visualization, the arrow sizes in (b) are scaled by two with respect to (a). Berry curvatures for (c) the valence band and (d) the conduction band. The hexagon in the plots guides the eye and traces the first BZ.

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    • Figure 7
      Figure 7

      The intraband and interband contribution to the HHG spectrum, for a linearly polarized driver along Γ−K and parallely polarized harmonics, calculated with LG SBEs. The vertical dotted lines at around the 10th and 28th harmonics respectively show the smallest band gap at the K point and the maximal band gap at the Γ point. Dephasing time is set to ∞. The dominance of the interband over the intraband contribution is true for all driver and harmonic emission directions considered in this work.

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    • Figure 8
      Figure 8

      LG and VG convergence properties for HHG in hBN with respect to the number of bands, with (a, b) a linearly polarized driver along Γ−K, (c) along the Γ−M directions, and (d) a circular driver. The insets show the crystal structure, as well as the driver (large red arrows) and emission polarization directions (small blue arrows). The laser parameters are λ=1600 nm, A0=0.35, τ=29.4 fs. The vertical dotted lines at around the 10th and 28th harmonics respectively show the smallest band gap at the K point and the maximal band gap at the Γ point. Dephasing time is set to ∞. Note the fast convergence of the harmonics above the band-gap energy compared to that below the band-gap energy.

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    • Figure 9
      Figure 9

      HHG spectra with and without common approximations used in the literature (see text) for hBN driven by linearly polarized pulses. LPD is along (a, b) Γ−K and (c,d) Γ−M directions. Laser parameters and figure formats are the same as in Fig. 8.

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    • Figure 10
      Figure 10

      Same format as in Fig. 9 but for a circularly polarized pulse.

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