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Fizeau drag in graphene plasmonics

Abstract

Dragging of light by moving media was predicted by Fresnel1 and verified by Fizeau’s celebrated experiments2 with flowing water. This momentous discovery is among the experimental cornerstones of Einstein’s special relativity theory and is well understood3,4 in the context of relativistic kinematics. By contrast, experiments on dragging photons by an electron flow in solids are riddled with inconsistencies and have so far eluded agreement with the theory5,6,7. Here we report on the electron flow dragging surface plasmon polaritons8,9 (SPPs): hybrid quasiparticles of infrared photons and electrons in graphene. The drag is visualized directly through infrared nano-imaging of propagating plasmonic waves in the presence of a high-density current. The polaritons in graphene shorten their wavelength when propagating against the drifting carriers. Unlike the Fizeau effect for light, the SPP drag by electrical currents defies explanation by simple kinematics and is linked to the nonlinear electrodynamics of Dirac electrons in graphene. The observed plasmonic Fizeau drag enables breaking of time-reversal symmetry and reciprocity10 at infrared frequencies without resorting to magnetic fields11,12 or chiral optical pumping13,14. The Fizeau drag also provides a tool with which to study interactions and nonequilibrium effects in electron liquids.

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Fig. 1: Plasmonic Fizeau drag in graphene: theory and modelling.
Fig. 2: Experimental demonstration of the plasmonic Fizeau drag.
Fig. 3: Quantitative analysis of the plasmonic Fizeau drag.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability

The code used to analyse data are available from the corresponding author upon reasonable request.

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Acknowledgements

Research on the physics and imaging of the plasmonic Fizeau effect at Columbia was supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under award no. DE-SC0018426. D.A.B. acknowledges the support from MIT Pappalardo Fellowship. M.M.F. is supported by the Office of Naval Research under grant ONR-N000014-18-1-2722. Work in the P.J.-H. group was supported by AFOSR grant FA9550-16-1-0382, the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF9643, and Fundacion Ramon Areces. The development of new nanofabrication and characterization techniques enabling this work has been supported by the US DOE Office of Science, BES, under award DE-SC0019300. The development of the universal cryogenic platform used for scanning probe measurements is supported as part of the Energy Frontier Research Center on Programmable Quantum Materials funded by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under award no. DE-SC0019443. The development of infrared nano-optics is supported by Vannevar Bush Faculty Fellowship to D.N.B., ONR-VB: N00014-19-1-2630. D.N.B. is Moore Investigator in Quantum Materials EPIQS #9455. This work also made use of the Materials Research Science and Engineering Center Shared Experimental Facilities supported by the National Science Foundation (NSF) (grant no. DMR-0819762). Support from the Materials Engineering and Processing programme of the National Science Foundation, award number CMMI 1538127 for hBN crystal growth is also greatly appreciated.

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Authors

Contributions

D.A.B. and D.N.B. conceived and supervised the project. Y.D., L.X., S.Z., D.A.B. and D.N.B. designed the experiments. I.Y.P. and D.A.B. fabricated the devices. S.L. and J.H.E. provided the isotopic hBN crystals. Y.D., L.X., A.S.M. and R.P. performed the experimental measurements. Y.D., L.X., R.J., F.L.R., D.A.B. and D.N.B. analysed the experimental data. Z.S., M.M.F., A.J.M. and L.S.L. developed the theoretical analysis of the experimental data with input from P.J.-H., H.G. and Z.D. Y.D., L.X., Z.S., M.M.F., D.A.B. and D.N.B. co-wrote the manuscript with input from all co-authors.

Corresponding authors

Correspondence to D. A. Bandurin or D. N. Basov.

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The authors declare no competing interests.

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Peer review information Nature thanks Joel Cox, Jiahua Duan and Hugen Yan for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Fig. 1 Gate-dependent transport of a typical device.

Two-terminal resistance R2pt of a typical device as a function of the back-gate voltage Vg at T = 170 K. Inset shows the transport measurement configuration, where a source meter (Keithley 2450) was used to source gate voltage, and a lock-in amplifier (SR 830) was used to measure the resistance of the entire device.

Extended Data Fig. 2 Experimental configuration for measuring current-gating effects.

Voltage is applied across the source/drain electrode and SPP imaging is performed close to the drain. The black streamlines symbolize the d.c. current. The voltmeter measures the electrostatic potential of the SPP launcher as a function of the biasing current.

Extended Data Fig. 3 Current–voltage characteristics of a typical device.

The black symbols represent the source–drain voltage while sourcing current through the device. The blue symbols represent the simultaneously measured voltage on the SPP launcher, representing the current-gating effect induced by the biasing current. Inset shows a magnified view of the low-voltage region.

Extended Data Fig. 4 Characteristic real-space nano-infrared and topography images.

a, Near-field image taken in the vicinity of a gold launcher at T = 170 K and Vg = 50 V. Gold-launched λp fringes and tip-launched λp/2 fringes near the graphene edge are clearly visible. Dashed rectangles mark the regions magnified in c and d. b, AFM topography image taken simultaneously with the near-field image in a. The graphene region is uniform with minimal topographic variations. c, d, Magnified near-field images near graphene edges, showing the tip-launched and edge-reflected SPP fringes with λp/2 periodicity.

Extended Data Fig. 5 Real-space nano-infrared and topography line profiles under current.

ad, Representative results for simultaneously taken topography (a, b) and near-field (c, d) data as a function of current density at T = 170 K and Vg = −50 V. a, AFM topography collected in the vicinity of a gold launcher on the left of the field of view. The 2D plot is assembled from AFM line profiles measured at different current densities while scanning along the same line in real space. Red and black arrows and dashed lines indicate positions where the averaged line profiles in b are acquired. b, Averaged line profiles of AFM topography for current densities of ±0.75 mA μm−1, whose topography signals are essentially the same. One of the line profiles is shifted vertically for clarity. c, Near-field data, taken simultaneously with a. A standard one-dimensional Fourier filter was applied here to reduce noise. d, Averaged line profiles of the near-field signal in c for the same current densities as the topography data in b. A Fizeau shift is clearly visible.

Extended Data Fig. 6 Uncertainty analysis for fitted SPP wavelength.

a, Distribution of the least-squares estimate of the SPP wavelength in equation (7) generated by Monte Carlo simulation. b, Examples of typical simulated SPP line profiles used for analysis in a. c, I: Dependence of variance (bright and dark red lines) and bias (blue line) of wavelength estimate on spatial resolution (SR). Bright red line corresponds to zero signal noise (σy/A = 0). Dark red and blue lines correspond to σy/A = 10%; II: Dependence of bias and variance in wavelength estimate on pixel size in units of wavelength. The pixel size has minimal effect on σλ as long as one samples above the Nyquist rate, as indicated by the vertical green dashed line. d, I: Strong dependence of error in wavelength estimate on signal noise σy. The variance of the wavelength estimate (bright red, dark red and green lines) will increase roughly linearly with σy until about 25%, and the bias (bright and dark blue lines) is less than 1 nm for SR = 20 nm; II: Dependence of error in wavelength estimate on SPP propagation length 1/q2. Both the variance (bright and dark red lines) and the bias (blue line) of the wavelength estimate improve with 1/q2, even more so when there is positioning noise σx (dark red line). e, Assessing the statistical significance of the Fizeau shift using an F test. Solid red and purple lines represent the dependence of F statistics on the sample wavelength-shift standard deviation sΔλ. Purple line assumes α = 0 and red line assumes α is finite (see text). Cyan shaded region corresponds to F statistics that reject the null hypothesis of no Fizeau shift (F > Fcrit = 2.65). Vertical dashed line corresponds to wavelength-shift standard deviation sΔλ estimated from data in Fig. 3a, b.

Extended Data Fig. 7 Additional datasets revealing plasmonic Fizeau drag in graphene.

Other representative data when scanning along the same line at different d.c. currents (averaged ±25 μA μm−1 for each profile) for different gate voltages, temperatures and devices. Within a set of polariton line profiles, the first polariton fringes are aligned to enable better visual inspection of Fizeau shifts. Line profiles are shifted vertically for clarity. Within each panel, the fitted line profiles of the smallest and largest current densities are shown in the lower panel for visual comparison. The images of the devices are near-field scattering amplitude measured at 170 K without gating. a, Device 1, T = 170 K, Vg = −47 V; b, Device 1, T = 170 K, Vg = +47 V; c, Device 1, T = 170 K, Vg = −60 V; d, Device 1, T = 60 K, Vg = +60 V; e, Device 2, T = 170 K, Vg = +50 V; f, Device 3, T = 60 K, Vg = +60 V.

Supplementary information

Supplementary Information

This file contains a list of the notations, details regarding the theory of Fizeau plasmon drag in graphene (this section includes Supplementary Figures 1-4), Supplementary Table 1 and the Supplementary References.

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Dong, Y., Xiong, L., Phinney, I.Y. et al. Fizeau drag in graphene plasmonics. Nature 594, 513–516 (2021). https://doi.org/10.1038/s41586-021-03640-x

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