Continuous graphene

We first discuss the intrinsic MO terahertz effects in continuous (unpatterned) graphene. A basic device used in our experiments is a large-area CVD (chemical vapour deposition)-graphene field-effect transistor (g-FET) sketched in Fig. 1a. It allowed us to combine ambipolar control of doping n (positive for electrons and negative for holes) with magneto-transport and MO measurements in a perpendicular magnetic field B (positive towards the light source). The THz transmission T(ω) and the FR spectra θ F (ω) were obtained while graphene was illuminated by a linearly polarized light with the angular frequency ω. The Fabry–Perot interference in the substrate was suppressed intentionally to simplify the spectra analysis. From these measurements, we extracted the transmission spectra for the right-hand (RH)/left-hand (LH) circular polarizations, T ± (ω)28, which fully characterizes the MO response of the device.

Figure 1: Doping-dependent magneto-resistance and magneto-optical extinction spectra of continuous graphene. (a) Schematic representation of a terahertz g-FET device and the optical experiment. (b) Two-terminal source-drain resistance at B=0, 3 and 7 T as a function of the gate-induced doping. (c) Extinction spectra at B=0 T for various doping levels and polarities. The open circles represent the best Drude fit for n=−7.9 × 1012 cm−2. (d) Doping dependence of the Drude weight relative to the CNP (black symbols) and the scattering rate (red symbols). The black line is the theoretical prediction for Drude weight of Dirac fermions for v F =106 m s−1. The red line is the guide to the eye. (e) The extinction (for the LH circular polarization) at n=−7.9 × 1012 cm−2 for different values of magnetic field from 0 to 7 T with the step of 1 T. The open circles represent the best Drude fits at 4 and 7 T. The inset presents the field dependence of the CR frequency (circles) and the linear fit (dashed line). (f) The extinction spectra at 7 T as a function of doping. The colour legend is the same as in c. The spectra are shown for the LH/RH circular polarizations for the p-/n-doped regimes respectively, that is, for polarizations exhibiting the cyclotron resonance. The inset presents the doping dependence of the experimental cyclotron frequency (circles) and a fit using a Dirac-fermion model as described in the text (dashed lines). All measurements are done at T=250 K. Full size image

The source-drain resistance as a function of the gate voltage V g in all our devices has a characteristic peaked shape (Fig. 1b) with the charge neutrality point (CNP) at a positive voltage, indicating a residual p-type doping. The curves show a strong positive magnetoresistance and are electron–hole symmetric, demonstrating the high quality of our samples. The charge-carrier mobility μ≈3,500 cm2 V−1 s−1 and the doping inhomogeneity δn≈5 × 1011 cm−2 are deduced using a standard analysis of the transport curves29. Figure 1c shows the zero-field extinction spectra 1−T(ω)/T CNP (ω), which are representative of the optical absorption in graphene (as specified in Supplementary Note 2 and shown in Supplementary Fig. 2). The strong exctinction increase at low frequencies is due to the Drude absorption by the doped charges3,4,5,6,9. The fact that almost half of terahertz photons are stopped by gate-injected carriers in an atomic monolayer demonstrates the remarkable efficiency of the Drude response. Importantly, the extinction curves at the matching p- and n-type doping levels are similar, which is consistent with the symmetric shape of the transport curves.

Optical conductivity is described almost perfectly (as shown by the open circles in Fig. 1c) using a semiclassical Drude model:

where D is the Drude weight, ω c is the cyclotron frequency (positive/negative for n-/p-type doping), τ is the scattering time and ± refer to RH/LH circular polarizations. A relation between σ ± and T ± that takes the substrate into account is given in Supplementary Note 1 and illustrated in Supplementary Fig. 1.

Before we analyse the dependence of the Drude parameters on the doping and the magnetic field, we stress that the Dirac-fermion theory (in the semiclassical limit) predicts26 that

and

where v F is the Fermi velocity, ħ the reduced Planck’s constant and e the (positive) elementary charge. This behaviour is different from the case of conventional 2DEGs25, where the Drude weight is proportional to n, while the cyclotron frequency is n-independent. Notably, equation (3) was not tested so far since in all terahertz spectroscopic measurements the electrostatic gating was done separately from applying a magnetic field.

The Drude weight and the scattering rate 1/τ extracted by fitting the zero-field extinction curves (where ω c =0) are strongly doping dependent (Fig. 1d). A theoretical dependence (dashed line) for a typical value of v F =106 m s−1 agrees reasonably well with our experiment, with the experimental values being about 20% lower, in a qualitative agreement with a previous report3. We did not observe any significant change of the Drude weight with the magnetic field. As observed in a recent report9, the scattering rate decreases as the doping increases for both charge polarities. This effect, which is beneficial for MO applications, is consistent with either a doping-induced screening of charged impurities30,31,32 or with the resonant scattering33.

In a magnetic field, the terahertz spectra are dominated by the cyclotron resonance (CR) (Fig. 1e,f). The extinction curves for a strongly p-type-doped sample (n=−7.9 × 1012 cm−2) for the LH circular polarization, where the CR is observed in this case, are shown in Fig. 1e for different magnetic field values up to 7 T. The Drude fit works equally well in magnetic field, as shown by open circles for 4 and 7 T. The cyclotron frequency shifts linearly with B (inset of Fig. 1e) and a record extinction of 43% is reached at 5 T at a frequency as high as 2.5 THz. Furthermore, Fig. 1f reveals that the cyclotron frequency strongly increases (up to 10 THz) with the reduction of the charge concentration for both doping regimes. This is a spectacular fingerprint of Dirac fermions, as theory indeed predicts ω c to be doping dependent, according to equation (3). This relation fits the experimental data very well (inset of Fig. 1f), where similar Fermi velocity values for p- and n-type doping regimes are found (1.01±0.02 and 0.92±0.05 × 106 m s−1, respectively).

Figure 2 shows the MCD (defined as the difference between the extinction coefficients for the RH and LH circular polarizations) and the FR spectra at 7 T for different doping values. Both quantities can be non-zero only if the time reversal symmetry is broken and are the key parameters for the non-reciprocal light manipulation. We note that the high values of MCD and FR (in the present case, 35% and 0.11 rad, respectively) demonstrate the potential of graphene as a MO material in this frequency range. Furthermore, Fig. 2 reveals that in addition to tuning the frequency and modulating the intensity of the magneto-extinction, the electrostatic doping also allows the inversion of the magneto-circular dichroism and the FR at a fixed magnetic field. Indeed, in both cases the spectra corresponding to equal absolute carrier concentrations but opposite doping types show a high degree of symmetry with respect to zero. This fully agrees with the Dirac-fermion theory predicting the inversion of the cyclotron frequency (equation (3)) at a constant Drude weight (equation (2)) as n changes sign. The inversion of the MCD is thus due to the fact that the polarization where the CR is observed is defined by the doping type. This new effect that enables novel applications (as discussed below) is thus directly related to the unique possibility of ambipolar doping control in graphene10.

Figure 2: Electrostatic sign inversion of magnetic circular dichroism and Faraday rotation in continuous graphene. (a,b) Spectra of MCD=(T − −T + )/T CNP and FR θ F at different doping levels at 7 T. The inset compares the polarization ellipses of the transmitted radiation and illustrates the opposite signs of the MCD and the FR for the p- and n-type doping. All measurements are done at T=250 K. Full size image

Patterned graphene

As demonstrated above, the non-reciprocal MO effects in unpatterned graphene are especially strong at low frequencies (below 5–7 THz). However, in the higher THz region the Drude response is less efficient. In order to broaden further the tuning range of the magneto-teraherz response, one can use the new physical properties arising from the excitation of magneto-plasmons (that is, collective charge oscillations in external magnetic field) in graphene nanostructures23,34,35,36,37. To this end, we patterned graphene with a periodic antidot array (Fig. 3a) that acts as a two-dimensional plasmonic crystal24. The transport curves (presented in Supplementary Fig. 3) are similar to the ones in continuous graphene (Fig. 1b), indicating that patterning does not strongly degrade the charge mobility. In order to better understand the magnetic field dependence, we first show in Fig. 3b the zero-field extinction spectra of patterned graphene with a period of 3 μm and a hole diameter of 2 μm that were studied previously by us24. They feature a Drude-like peak (A) as in continuous graphene and two additional peaks (B and C) due to Bragg scattering of graphene plasmons on the periodic structure. Our previous study revealed that peak B is mostly due to the (0, 1)-order Bragg reflection, while peak C has a mixed (1, 1)–(0, 2) character. Importantly, they both demonstrate a dependence typical for Dirac fermions (Fig. 3e)4,14,15 that allows controlling the resonance frequency by the gate voltage, in addition to tuning it by the antidot dimensions and spacing24.

Figure 3: Electro-magneto-optical terahertz experiment and simulations in antidot-patterned graphene. (a) Schematic illustration of the patterned antidot array. Electrical contacts and terahertz light are the same as in Fig. 1a and are not shown. The superimposed colour image exemplifies a distribution of the perpendicular AC electric field, when a magneto-plasmon mode (B − ) is excited. (b) Experimental extinction spectra at B=0 T for various carrier concentrations (p-doping). Symbols A, B and C indicate the resonant peaks. (c) Experimental MO extinction spectra for RH and LH polarizations at n=−9.3 × 1012 cm−2 at the selected magnetic fields. The black dashed lines are guides to the eye to follow the evolution of peaks B and C. (d) Finite-element simulation of the experimental data at the same doping level and the magnetic field values as in c. As in c, the black dashed lines follow peaks B and C. (e) Doping dependence of the Bragg peaks B and C extracted from b. (f) Distribution of AC electric field E z corresponding to the modes A − , B − and C − at 7 T as shown in d. (g) Spectra of MCD and FR corresponding to c. (h) Magnetic field dependence of experimental (solid triangles) and simulated (open circles) magneto-plasmon frequencies corresponding to c,d. The black dashed line indicates the bulk cyclotron resonance ω c (B). All measurements are done at T=250 K. Full size image

A comparison of Figs 1c and 3b reveals that the extinction in patterned graphene is in fact lower than or comparable to the extinction in continuous graphene at the same frequencies, even close to plasmonic resonances. A certain reduction of the extinction is because of the removal of graphene by patterning and the corresponding reduction of the filling factor (about 40% in our case). Second, this is a consequence of a relatively low mobility, which broadens and reduces both the plasmon and the Drude peaks. Therefore, in patterned samples with a high enough mobility, where the peaks are expected to be sharper and higher, the extinction should be higher than in continuous graphene.

The magnetic field affects strongly the extinction curves (Fig. 3c) and shifts the peak frequencies (solid symbols in Fig. 3h). Peak A shifts upwards in the LH polarization while keeping the intensity constant. Although this resembles the behaviour of the CR peak in continuous graphene (Fig. 1e), the frequency of peak A is much lower than the bulk CR for the same values of n and B and shows a sublinear field dependence. A similar excitation was observed in 2DEGs38 and identified as a cyclotron mode at low magnetic fields crossing over to a so-called edge magneto-plasmon circulating around the holes as the field is increased. Bragg magneto-plasmon peaks B and C exhibit a dramatic field dependence and a totally different behaviour for the RH and LH circular polarizations (where we denote them B + , C + and B − , C − respectively). Most notable is a strong increase/decrease of their intensity for the LH/RH polarization, accompanied by field-induced shifts as indicated by the dashed lines in Fig. 3c. The polarization frequency splitting is most prominent for peak B, where opposite shifts are observed in the two polarizations. Associated with these spectacular magneto-plasmonic effects are the resonant MCD and FR structures (Fig. 3g), which are present at much higher frequencies than resonances in continuous graphene (Fig. 2), confirming that patterning indeed extends significantly the spectral range of the strong MO activity.

These results agree qualitatively with low-temperature experiments38,39,40 and theory41,42 in conventional 2DEG antidot arrays. However, it is imperative to quantitatively understand the present spectra in order to predict the performance of graphene THz devices. To this end, we performed a finite-element electromagnetic simulation of our system, where the Drude weight D and the cyclotron frequency ω c were determined using equations (2) and (3), while the scattering time τ was set to 0.1 ps in order to match the observed width of the magneto-plasmon peaks. The simulated extinction spectra for various magnetic field values are shown in Fig. 3d and the corresponding mode frequencies are shown as open circles in Fig. 3h One can notice an excellent agreement with the experimental curves (Fig. 3c), in terms of the spectral shapes and the evolution of the peak positions and intensities as a function of B. In particular, the simulation reproduces perfectly the sublinear field dependence of peak A and the opposite field-induced shifts of the B − and B + peaks.