Smooth periodic gauge satisfying crystal symmetry and periodicity to study high-harmonic generation in solids

Shicheng Jiang, Chao Yu, Jigen Chen, Yanwei Huang, Ruifeng Lu, and C. D. Lin

Phys. Rev. B 102, 155201 – Published 12 October 2020

Abstract

Intense lasers can easily drive nonadiabatic transitions of excited electron wave packets across the Brillouin zones, thus transition dipole moments (TDM) between energy bands of solids should be continuous, satisfying crystal symmetry, and periodic at zone boundaries. While current ab initio algorithms are powerful in calculating band structures of solids, they all introduced random phases into the eigenfunctions at each crystal momentum $k$ . Here we show how to choose a “smooth-periodic” gauge where TDMs can be smooth versus $k$ , preserving crystal symmetry, as well as maintaining periodic at boundaries. The symmetry properties of TDMs with respect to $k$ ensure the absence of even-order harmonics from MgO with inversion symmetry, while the TDM in the “smooth-periodic” gauge for broken-symmetry ZnO is responsible for even harmonics that were underestimated in previous simulations. These results reveal the importance of correctly treating the complex TDMs that satisfy crystal symmetry and continuous across zone boundaries in nonlinear laser-solid interactions, which has been elusive in most theories so far.

Received 13 March 2020
Accepted 24 September 2020

DOI:https://doi.org/10.1103/PhysRevB.102.155201

Physics Subject Headings (PhySH)

High-order harmonic generation Quantum description of light-matter interaction

Symmetries in condensed matter

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Shicheng Jiang ^1,2, Chao Yu¹, Jigen Chen ³, Yanwei Huang ², Ruifeng Lu ^1,*, and C. D. Lin^4,†

¹Institute of Ultrafast Optical Physics, Nanjing University of Science and Technology, Nanjing 210094, People's Republic of China
²State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, People's Republic of China
³Zhejiang Provincial Key Laboratory for Cutting Tools, Taizhou University, Taizhou 31800, People's Republic of China
⁴J. R. Macdonald Laboratory, Department of Physics, Kansas State University, Manhattan, Kansas 66506, USA

^*rflu@njust.edu.cn
^†cdlin@phys.ksu.edu

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Issue

Vol. 102, Iss. 15 — 15 October 2020

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Images

Figure 1
(a) Geometric structure of rocksalt MgO crystal. Red and green balls are O and Mg, respectively. (b) Band structure along the Γ-X axis. (c)–(h) Real (black line) and imaginary (red line) part of TDMs generated by the “smooth procedure” in the extended Brillouin zone.
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Figure 2
HHG spectrum from MgO calculated by 1D three-band SBEs. The equations are solved in the extended Brillouin zone and all the elements used in the SBE model are from Fig. 1. Laser parameter: 30 fs, 1300 nm, $1 \times 10^{13} W / c m^{2}$ .
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Figure 3
Comparison between experimental and calculated CEP dependent HHG from MgO. (a), (c), and (d) are reprinted with permission from Ref. [30] ®. The Optical Society. (b), (e), and (f) are the calculated spectrum along Γ-X. Laser parameter: 1700 nm, 10 fs, $4 \times 10^{13} W / c m^{2}$ .
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Figure 4
HHG spectra from MgO using different TDMs. Black line is the same as Fig. 3: harmonic spectrum calculated using correct dipole moments $[D_{c_{1} c_{2}} (k)$ is odd, $D_{v c_{2}} (k)]$ and $D_{v c_{1}} (k)$ are even functions of $k$ ). Red line: The same as the black line except that $D_{c_{1} c_{2}} (k)$ is changed to an even function of $k$ artificially. The subscripts $v, c_{1}, c_{2}$ represent the valence band, first conduction band and second conduction band, respectively. The intensities of the spectra in the right side of the vertical solid line were multiplied by a factor of 4.
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Figure 5
Illustration of transition paths of electrons for gases and solids with inversion symmetry. For the gas phase, a triangular system will never be formed because transition between states with same parity is forbidden. For the solid case where the energy levels are extended into bands, transitions between bands at these $k$ points away from Γ are not forbidden. The symmetry properties of the TDMs will ensure the absence of even order optical signal.
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Figure 6
Dipole phase of wurtzite ZnO crystal and simulated HHG. (a) Red line is the $k$ -dependent dipole phase generated by “smooth procedure”. Such a phase is not periodic because of the DC term in the additionally introduced phase. The black line is the $k$ -dependent dipole phase after the DC component is taken away as introduced in this article. (b) HHG spectra from ZnO. The green line is the experimental data, the blue line is calculated by two-band SBEs with elements obtained from ab initio software with the help of the present “smooth-periodic” procedure. The red line is from calculations where the dipole phases are calculated from the tight-binding model. The red line and experimental data are copied from our previous work [34]. To present clear comparison, the spectra are shifted vertically.
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