Extremely efficient terahertz high-harmonic generation in graphene by hot Dirac fermions


Multiple optical harmonic generation—the multiplication of photon energy as a result of nonlinear interaction between light and matter—is a key technology in modern electronics and optoelectronics, because it allows the conversion of optical or electronic signals into signals with much higher frequency, and the generation of frequency combs. Owing to the unique electronic band structure of graphene, which features massless Dirac fermions1,2,3, it has been repeatedly predicted that optical harmonic generation in graphene should be particularly efficient at the technologically important terahertz frequencies4,5,6. However, these predictions have yet to be confirmed experimentally under technologically relevant operation conditions. Here we report the generation of terahertz harmonics up to the seventh order in single-layer graphene at room temperature and under ambient conditions, driven by terahertz fields of only tens of kilovolts per centimetre, and with field conversion efficiencies in excess of 10−3, 10−4 and 10−5 for the third, fifth and seventh terahertz harmonics, respectively. These conversion efficiencies are remarkably high, given that the electromagnetic interaction occurs in a single atomic layer. The key to such extremely efficient generation of terahertz high harmonics in graphene is the collective thermal response of its background Dirac electrons to the driving terahertz fields. The terahertz harmonics, generated via hot Dirac fermion dynamics, were observed directly in the time domain as electromagnetic field oscillations at these newly synthesized higher frequencies. The effective nonlinear optical coefficients of graphene for the third, fifth and seventh harmonics exceed the respective nonlinear coefficients of typical solids by 7–18 orders of magnitude7,8,9. Our results provide a direct pathway to highly efficient terahertz frequency synthesis using the present generation of graphene electronics, which operate at much lower fundamental frequencies of only a few hundreds of gigahertz.

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Fig. 1: Schematic and main results of the THz high-harmonic generation.
Fig. 2: Illustration of the mechanism of THz harmonic generation in graphene, based on calculations with the thermodynamic model of intraband nonlinear THz conductivity of graphene.
Fig. 3: Efficiency of THz high-harmonic generation.

Data and code availability

The datasets generated and analysed during this study, and the corresponding computer codes, are available from the corresponding authors on reasonable request.


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D.T. acknowledges financial support from the Deutsche Forschungsgemeinschaft (SFB 1242 ‘Non-Equilibrium Dynamics of Condensed Matter in the Time Domain’, TP B08), European Commission (EU Career Integration Grant EU CIG 334324 LIGHTER) and the Max Planck Society. M.G. and B.G. acknowledge support from the European Cluster of Advanced Laser Light Sources (EUCALL) project which has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement no 654220. K.J.T. acknowledges support through the Mineco Young Investigator Grant (FIS2014-59639-JIN). J.T. and U.L. acknowledge support from the EuCARD-2 project, which has received funding from the European Commission under grant agreement No 312453. We thank J. Lægsgaard, K. Krewer, E. Unger, W. Zhang, T. V. A. G. de Oliveira and M. Mittendorff for discussions. We thank the ELBE team for the operation of the TELBE facility.

Author information




D.T. and M.G. conceived and supervised the project. H.A.H., S.K., J.-C.D., Z.M., B.G., N.A., M.C., S.G., Z.W., D.T. and M.G. performed the nonlinear THz spectroscopy measurements and evaluated the experimental data. H.A.H. and D.T. performed the modelling, with contributions from Z.M. and K.-J.T. D.T., M.G. and M.B. interpreted the results, with contributions from all co-authors. Z.L., Z.C., A.N. and K.M. manufactured the samples. H.A.H. and D.T. characterized the linear THz properties of the graphene/fused silica samples. U.L. and J.T. provided for the special mode of high bunch charge operation of TELBE using the SRF photoinjector that enabled the observation of the seventh harmonic. M.B. initiated and supported the THz studies on graphene at MPI-P. D.T. and M.G. wrote the manuscript, with contributions from M.B., K.-J.T., S.K. and H.A.H. All co-authors discussed the results and commented on the manuscript.

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Correspondence to Michael Gensch or Dmitry Turchinovich.

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Extended data figures and tables

Extended Data Fig. 1 Experimental set-up.

Tunable multicycle THz pulses (red) from the undulator of the TELBE facility2 are used to irradiate the graphene sample. 100-fs pulses from a Ti:sapphire laser system (brown) are used to probe the transmitted and emitted THz pulses by free-space electro-optic sampling. PD, photodiode; BS, beamsplitter; Pol., polarizer.

Extended Data Fig. 2 Fundamental frequencies after bandpass filtering.

The bandwidths were determined from Gaussian fits to the spectra. MSA, mean square amplitude.

Extended Data Fig. 3 Scheme of the set-up for detection of multiple harmonics up to the seventh order.

Two 0.3-THz bandpass filters (BP) are used to suppress the undulator harmonic background. A single 2.1-THz bandpass filter after the sample attenuates the fundamental, third and fifth harmonics to an extent that they can still be detected by the EOS set-up.

Extended Data Fig. 4 HHG signal from graphene, reference signal from the SiO2 substrate and filter function of the 2.1-THz bandpass.

The red curve shows the as-measured HHG spectrum of the graphene sample. The black curve shows the reference spectrum taken from the bare SiO2 substrate. The measured transmission function of the 2.1-THz bandpass filter is also shown (grey line).

Extended Data Fig. 5 Schemes of the experimental configurations to determine the electric fields of the fundamental, THG and FHG pulses.

Measurements were performed with graphene/SiO2 and with the bare SiO2 substrate. a, Set-up for the THG experiment used to measure the fundamental and harmonic simultaneously. b, Set-up to measure the harmonic in the FHG experiment. Two filters were used before the sample and two after the sample, to optimize the signal-to-noise ratio. c, Set-up to determine the electric fields for the fundamental in the FHG experiment.

Extended Data Fig. 6 Filter function of the 1.93-THz bandpass and raw spectra from the THG experiment.

a, Amplitude transmission function of a single 1.93-THz bandpass filter. b, As-measured spectral amplitude in arbitrary units (a.u.), as determined from the bare SiO2 substrate (black) and from the graphene sample (red). The incident THz peak field of the fundamental at 0.68 THz was 61 kV cm−1.

Extended Data Fig. 7 Raw spectra from the FHG experiment.

These as-measured spectra show the spectral amplitude as determined from a measurement with the bare SiO2 substrate as a reference and a measurement of the graphene sample. The incident THz peak field in the fundamental at 0.37 THz was 40 kV cm−1 when using two filters in the incident beam. Insignificant transmission at the fundamental frequency and no spurious background at the FHG frequency band is observed in the reference field measurement.

Extended Data Fig. 8

Frequency-dependent phase difference induced by the 1.93-THz bandpass filter.

Extended Data Fig. 9 The frequency-dependent response function.

a, The bare substrate described by the amplitude transmission (black line) and the substrate-induced phase shift (blue line). b, A simulated acceptance function of the 1.9-mm-thick ZnTe detection crystal; amplitude (black curve) and phase shift (blue curve). Arrows indicate relevant axis.

Extended Data Fig. 10 Reconstruction of the harmonic fields from the measured FEOS signals.

This is an example of THG measurement with f = 0.68 THz → 3f = 2.04 THz. a, Measured FEOS signals (dimensionless). b, The corresponding fields transmitted through the incidence interface of the sample after deconvoluting the response functions of all the elements after the graphene film, including the 1.9-mm-thick ZnTe detection crystal, the 1.93-THz filter and the fused silica substrate from the FEOS signals in a. Black pulse is for the bare substrate, red for the graphene sample and blue for the difference. c, The pure THG field extracted from the blue field pulse in b.

Extended Data Fig. 11 Characterization of the graphene sample.

a, Raman spectrum of the graphene sample. b, Linear conductivity, real and imaginary, of the graphene film normalized to the universal conductivity σ0 = e2/(4ħ). The symbols represent the experimental data; the solid lines represent the Drude fit with a Fermi level energy EF = 170 meV (corresponding to a doping concentration Nc = 2.1 × 1012 cm−2) and a scattering time τ0 = 47 fs as fitting parameters. The error bars are the standard deviation in the measurements.

Extended Data Fig. 12 The nonlinear (THz-field-dependent) refractive index of the graphene film.

a, The THz refractive index of the graphene film as a function of frequency at various peak electric fields for the THz pump at 0.3 THz, showing reduction in the refractive index with both frequency and exciting field strength. b, The field dependence of the nonlinear THz refractive index at the harmonics 3f = 2.04 THz generated by 1f = 0.68 THz pump, 5f = 1.85 THz generated by 1f = 0.37 THz pump, and 7f = 2.1 THz generated by 1f = 0.3 THz pump.

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Hafez, H.A., Kovalev, S., Deinert, J. et al. Extremely efficient terahertz high-harmonic generation in graphene by hot Dirac fermions. Nature 561, 507–511 (2018). https://doi.org/10.1038/s41586-018-0508-1

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  • Terahertz
  • Seventh Harmonic
  • Single-layer Graphene
  • Free-space Electro-optic Sampling (FEOS)
  • Peak Electric Field Strength

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