Time-periodic light field has emerged as a control knob for manipulating quantum states in solid-state materials1,2,3, cold atoms4 and photonic systems5 through hybridization with photon-dressed Floquet states6 in the strong-coupling limit, dubbed Floquet engineering. Such interaction leads to tailored properties of quantum materials7,8,9,10,11, for example, modifications of the topological properties of Dirac materials12,13 and modulation of the optical response14,15,16. Despite extensive research interests over the past decade3,8,17,18,19,20, there is no experimental evidence of momentum-resolved Floquet band engineering of semiconductors, which is a crucial step to extend Floquet engineering to a wide range of solid-state materials. Here, on the basis of time and angle-resolved photoemission spectroscopy measurements, we report experimental signatures of Floquet band engineering in a model semiconductor, black phosphorus. On near-resonance pumping at a photon energy of 340–440 meV, a strong band renormalization is observed near the band edges. In particular, light-induced dynamical gap opening is resolved at the resonance points, which emerges simultaneously with the Floquet sidebands. Moreover, the band renormalization shows a strong selection rule favouring pump polarization along the armchair direction, suggesting pseudospin selectivity for the Floquetband engineering as enforced by the lattice symmetry. Our work demonstrates pseudospin-selective Floquet band engineering in black phosphorus and provides important guiding principles for Floquet engineering of semiconductors.
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The data that support the findings of this study are available from the corresponding author on request.
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We thank N.L. Wang, R.-B. Liu, D. Sun and Y.H. Wang for useful discussions. This work is supported by the National Key R&D Program of China (grant no. 2021YFA1400100), the National Natural Science Foundation of China (grant nos. 11725418, 12234011 and 11427903) and National Key R&D Program of China (grant nos. 2020YFA0308800 and 2016YFA0301004). P.Y. and W.D. acknowledge the support of the Basic Science Center Project of NSFC (grant no. 51788104). S.M. acknowledges supports from Ministry of Science and Technology (grant no. 2021YFA1400201), National Natural Science Foundation of China (grant no. 12025407) and Chinese Academy of Sciences (grant no. YSBR047).
The authors declare no competing interests.
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Extended data figures and tables
Extended Data Fig. 1 Experimental geometries of TrARPES measurements.
a–d, Schematics for four different combinations of pump polarizations and measurement directions along the AC or ZZ directions. The blue line indicates the ARPES measurement direction and the red arrow represents the pump polarization. The black ellipses represent the anisotropic electronic pocket of black phosphorus.
Extended Data Fig. 2 Pump-fluence-dependent hybridization gap and observation of Floquet sidebands.
a–f, TrARPES dispersions along the AC direction with different pump fluences (s-pol. pump, ħω = 420 meV, Δt = 0). The red and blue arrows point to the Floquet sideband of the VB and hybridization gap. g, The differential TrARPES dispersion with a pump fluence of 2.7 mJ cm−2 after subtracting the dispersion before pump. h–m, EDCs for data shown in a–f at the momentum point of the hybridization gap (dashed line in f) and fitting curves. n, The extracted hybridization gap as a function of pump fluence. The error bar of the hybridization gap is defined as the upper limit when the energy position is clearly offset from the peak in h–m and the error bar of the pump fluence is defined by the fluctuation of pump power (10%).
Extended Data Fig. 3 Undoped sample versus electron-doped sample (forbidden optical absorption) on near-resonance pumping.
a, Schematic illustration of the electronic structure of undoped black phosphorus with unoccupied CB. Direct optical transition from VB to CB is allowed on near-resonance pumping. b,c, TrARPES dispersion images measured in undoped sample along the AC direction at Δt = −1 ps (b) and Δt = 0 (c). The pump beam is polarized along the AC direction with photon energy of 380 meV and the pump fluence is 0.7 mJ cm−2. d,e, Second-derivative image and EDCs of TrARPES data shown in c. f, Extracted dispersions before pump (dashed black curve) and with pump (red curves). g, Schematic illustration of the electronic structure of electron-doped black phosphorus with occupied CB edge. Direct optical transition from VB to CB is forbidden on near-resonance pumping. h–l, Similar results as b–f but in electron-doped sample using pump pulses with photon energy of 400 meV and pump fluence of 3 mJ cm−2. The Fermi energy is obtained by the Fermi–Dirac fitting to the spectrum of electron-doped black phosphorus before pump (Δt = −1 ps).
Extended Data Fig. 4 Summary of Floquet band engineering for black phosphorus for s-pol. pump and p-pol. pump when the pump polarization is parallel to the AC direction.
The pump photon energy is 440 and 380 meV for s-pol. and p-pol. pump, respectively. The pump fluence is 0.7 mJ cm−2.
Extended Data Fig. 5 Theoretical calculation of Floquet band structure.
a–d, Floquet band structure with AC pump for dispersions along the AC and ZZ directions using the tight-binding model (a,b) and the k · p model (c,d). e–h, Similar results as a–d but with ZZ pump.
Extended Data Fig. 6 Comparison of experimental and theoretical dispersions along the AC direction for TrARPES with different pump photon energies.
a–j, TrARPES dispersions along the AC direction (a–e) and corresponding second-derivative images (f–j) with different pump photon energies at a pump fluence of 0.7 mJ cm−2. k–o, Calculated Floquet band structures obtained from the ab initio tight-binding calculations with the Floquet theory to compare with a–e. The dotted black curves show the dispersions for the VB and the CB in the equilibrium state for comparison.
Extended Data Fig. 7 Comparison of experimental and theoretical dispersions along the ZZ direction for TrARPES with different pump photon energies.
a–j, TrARPES dispersions along the ZZ direction (a–e) and corresponding second-derivative images (f–j) with different pump photon energies at a pump fluence of 0.7 mJ cm−2. k–o, Calculated Floquet band structures obtained from the ab initio tight-binding calculations with the Floquet theory to compare with a–e. The dotted black curves show the dispersions for the VB and the CB in the equilibrium state for comparison.
Extended Data Fig. 8 Atomic and electronic structure of black phosphorus.
a, Atomic structure of black phosphorus. The primitive cell of four atoms is on the right and the supercell of eight atoms is on the left. b, The BZ of the supercell. c, The BZ of the primitive cell. d, Comparison of the calculated band structures from DFT calculations and tight-binding calculations along high symmetry lines in the reduced BZ of the supercell. e,f, The calculated band structure around the Γ point along the AC direction (e) and along the ZZ direction (f). The Fermi level is set as zero.
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Zhou, S., Bao, C., Fan, B. et al. Pseudospin-selective Floquet band engineering in black phosphorus. Nature 614, 75–80 (2023). https://doi.org/10.1038/s41586-022-05610-3
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