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Graphene’s unusual physical properties3,4 make it highly attractive for both fundamental research and new applications. Importantly, the conduction and valence bands in graphene show a relativistic massless dispersion and a chiral character of the electronic wavefunctions, resulting in non-equidistant Landau levels (LLs) (n=0,±1,±2…):

in a perpendicular magnetic field B, which includes an anomalous LL at zero energy E0=0 (e>0 is the elementary charge and vF≈106 m s−1 is the Fermi velocity). As a consequence, graphene shows the half-integer quantum Hall effect5,6, first observed in exfoliated graphene flakes7,8 and later in graphene epitaxially grown on SiC (refs 9, 10, 11).

The optical properties of graphene are equally unusual. The relativistic and gapless dispersion results in a universal and constant optical conductivity of σ0=e2/4ℏ (refs 6, 12, 13, 14, 15, 16, 17) for the photon energies ℏω above twice the Fermi energy (εF). In magnetic field, the non-equidistant spacing of the LLs gives rise to a spectacular series of absorption peaks, corresponding to transitions between various levels18,19,20,21. In contrast, only one peak, centred at the cyclotron frequency ωc=e B/mc, where mc is the cyclotron mass, is observed in conventional two-dimensional electron gases produced in semiconductor heterostructures.

Up to now, experimental magneto-optical studies of graphene18,19,21 have been focused on the diagonal conductivity component σx x(ω,B); the off-diagonal, or a.c. Hall22, conductivity σx y(ω,B) was addressed only theoretically20,23,24. The diagonal and Hall conductivities are directly related to the absorption 1−T and Faraday rotation angle θ (Fig. 1) respectively. Adopting the thin-film approximation and keeping only the terms linear in conductivity18,23, the following expressions are obtained:

where T is the substrate-normalized transmission (see the Methods section), Z0≈377 Ω is the impedance of vacuum and fs(ω) is a spectrally featureless function specific to the substrate (as described in Supplementary Information). Notably, the diagonal conductivity is formally independent of the sign of the charge carriers, whereas the d.c. Hall conductivity, which is sensitive to the carrier type, lacks spectral information. Therefore, studying the Faraday rotation is needed to complete the picture of the dynamics of electrons and holes in graphene.

Figure 1: Schematic diagram of the magneto-optical Faraday-rotation experiment.
figure 1

The polarization plane of the linearly polarized incoming beam is rotated by the Faraday angle θ after passing through graphene on a SiC substrate in a perpendicular magnetic field. Simultaneously, the polarization acquires a certain ellipticity. The positive direction of the magnetic field is along the z axis.

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Graphene, epitaxially grown on silicon carbide, is well adapted for magneto-optical studies because of its well-controlled morphology and essentially unlimited size25,26. Here, we used single- and multilayer graphene (SLG and MLG), grown respectively on the Si-terminated26 and C-terminated25 surfaces of 6H-SiC. The first sample underwent hydrogen passivation of the Si dangling bonds27,28, resulting in quasifreestanding SLG. The MLG consists of four to six rotated atomic layers (see the Methods section). These samples enabled us to explore the cases of both electron and hole doping as well as to access both the classical (high-doping) and quantum (low-doping) regimes as described below.

Figure 2a shows the Faraday angle θ measured on SLG at 5 K in magnetic fields up to 7 T in the far-infrared range as described in the Methods section. The spectra show a strongly field-dependent edge-like structure, giving rise to a positive rotation at low energies and negative θ at high energies. The maximum Faraday rotation exceeds 0.1 rad (∼6°), which is an exceptionally large effect given that it comes from a single layer. Measurements on the bare substrate did not reveal any Faraday effect, hence the observed rotation comes exclusively from the carbon monolayer. The inset shows the field dependence of θ at 10 and 27 meV. The curves follow an approximately linear dependence, with opposite slopes of +18.5 mrad T−1 and −4.5 mrad T−1 respectively.

Figure 2: Faraday rotation and magneto-optical transmission spectra of SLG.
figure 2

a, The Faraday angle θ at several fields up to 7 T at 5 K. The inset presents the magnetic-field dependence of θ(B) at ℏω=10 and 27 meV. The dashed lines are linear fits of the data points between 0 and 5 T. b, The transmission ratio T(B)/T(0) at the same fields. In the inset the absorption spectra 1−T(B) for B=0 T and 7 T are shown. The dashed lines in both panels and the inset of b are fits using the classical cyclotron resonance equations (4) and (5). c, The absorption of the same sample at room temperature in zero field (symbols), showing a clear step at ℏω=2|εF| due to Pauli blocking as shown in the inset. The blue line is a fitting curve using a phenomenological broadened step function a+btanh[(ℏω−2|εF|)/δ ε] giving |εF|=0.34±0.01 eV. d, The band dispersion near the K point measured by ARPES on another sample prepared in the same way. e, The cyclotron energy as a function of B. The dashed line is a theoretical dependence based on equation (6) using vF=1.02×106 and εF=−0.34 eV. f, The field dependence of the carrier mobility. The error bars in e and f represent the difference between the extreme parameter values obtained from a series of fits of the magneto-optical transmission and Faraday rotation spectra, where the input data were varied arbitrarily within the range of their systematic uncertainties.

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The zero-field-normalized transmission spectra T(B)/T(0) of SLG also reveal a strong magnetic-field dependence (Fig. 2b). The inset shows the absorption (1−T) at 0 and 7 T. We can clearly see a strong Drude peak, signalling a high doping level. The peak centre shifts from zero to a finite energy at 7 T, which is due to the cyclotron resonance as we discuss below. Although an estimate of the doping can be made from the integrated intensity of the Drude peak, the most direct measurement of the Fermi energy comes from the absorption spectra in the mid-infrared range (Fig. 2c), where an absorption edge is easily recognized. This is expected to be at 2|εF| owing to Pauli blocking6,12,13,16,17, giving an estimate for |εF|=0.34±0.01 eV.

Independently, we have carried out angle-resolved photoemission spectroscopy (ARPES) on a similar sample (Fig. 2d). The obtained dispersion curves show that the sample is p doped (the Fermi level is in the valence band). The Fermi energy inferred from optical measurements is in a reasonable agreement with an extrapolation of the occupied bands measured by ARPES. Below we show that the carrier type can also be directly extracted from the Faraday rotation.

The doping level and field strength used here puts the system in the classical regime, where the separation between the LLs at the Fermi energy is much smaller than εF (refs 17, 21). In this case, the Dirac quasiparticles are expected to exhibit the classical cyclotron resonance effect. Indeed, the data in Fig. 2a,b at each separate field are well fitted (dashed lines) using equations (2) and (3) and the classical Drude formulae:

where D is the Drude weight, ωc, the cyclotron frequency, is positive for electrons and negative for holes and τ is the scattering time. From here we see that the observed large Faraday rotation is associated with the cyclotron resonance. Furthermore, the curve θ(ω) reveals the value and sign of the cyclotron frequency, because |ωc| coincides with the position of the maximum absolute slope |dθ(ω)/dω|, and the sign of the slope matches the sign of ωc. In our case, the negative slope is an unmistakable signature of hole doping.

In the picture of non-interacting Dirac fermions, the Drude weight (at zero magnetic field) is equal to 2σ0|εF|/ℏ. Using this relation, we estimate |εF|=0.35±0.03 eV from the Drude weight extracted from the fit. This agrees with the estimate on the basis of the mid-infrared absorption threshold but is subject to relatively large error bars.

The cyclotron frequency (Fig. 2e) demonstrates an approximately linear growth with field, in agreement with the theoretical relation12,17

However, the theoretical curve using the value εF=−0.34 eV and vF=1.02×106 m s−1 extracted from the mid-infrared optical spectra and ARPES data respectively (dashed line in Fig. 2e) is somewhat lower than the experiment. Although the reason for this deviation is currently unknown, we can speculate about its possible relation to many-body effects. Interestingly, an electron–plasmon coupling was recently observed in similar samples29.

The carrier mobility μ can also be extracted using the relation |ωc|τ=μ|B| (Fig. 2f). The mobility varies weakly with field between 2,400 and 3,000 cm2 V−1 s−1. Even at 7 T, μ|B|∼1, which means that the cyclotron resonance is significantly damped owing to disorder. This can also be seen from the large width (1/τ∼10 meV) of the cyclotron peak in the inset of Fig. 2b.

The Faraday rotation and absorption observed in MLG are strikingly different (Fig. 3). Both the Faraday angle and absorption spectra show extra, strongly field-dependent, resonance structures, marked by arrows. As was found in previous studies18,19, they correspond to optical transitions between individual LLs. The series of transitions with energies E1−E0, E2−E1 and E3−E2 can be identified and fitted using equation (1), which gives vF=1.00±0.01×106 m s−1 (inset of Fig. 3b). Although inter-LL absorption peaks have been observed before18,19, the measurement of the Faraday rotation in the same conditions is the key novelty of the present work. Now we can distinguish transitions involving electrons from ones involving holes. In particular, the positive sign of dθ(ω)/dω at the transition energies, which coincide with the inflection points, unequivocally determines that the transitions are between electron bands. The low-frequency part of the spectra features a cyclotron-resonance-like structure, similar to the data on SLG. The sign of the Faraday angle now corresponds to electron doping.

Figure 3: Faraday rotation and magneto-optical transmission spectra of MLG.
figure 3

a,b, The Faraday angle θ (a) and the transmission ratio T(B)/T(0) (b) at several fields up to 7 T at 5 K. The curves in both panels are displaced for clarity; the offsets are given by the dashed lines of the same colours. The energies of the inter-LL transitions 01, 12 and 23 are indicated by the black, blue and dark red arrows respectively and shown as functions of magnetic field in the inset of b, where theoretical curves obtained using equation (1) with vF=1.00×106 m s−1 are also shown as dashed lines.

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As only transitions between occupied and empty levels are allowed, the simultaneous presence of 01, 12 and 23 transitions and the low-frequency cyclotron structure in MLG arises from a variation of the Fermi energy across different layers. The layer closest to the substrate is highly doped30; the doping in subsequent layers decreases exponentially with layer number. The most strongly doped innermost layer gives rise to the cyclotron resonance, as in SLG. Meanwhile, the individual inter-LL transitions originate from the weakly doped layers, which are in the quantum regime, as can be seen from the square-root field dependence of the transition energies. The maximum Faraday angle in this sample is smaller than in the monolayer sample, because the innermost layer in MLG is more weakly doped.

Summarizing the data on both samples, we point out that the Faraday rotation is a powerful contact-free tool to distinguish the carrier type and measure the mobility in graphene. In both the classical (high-doping) and quantum (low-doping) limits this technique reveals magneto-optical resonances owing to either the cyclotron effect or to inter-LL transitions. The technique has the potential to distinguish electronic contributions from separate graphene layers. However, our most striking observation is the giant value of the rotation (>0.1 rad for a single atomic layer). This is much larger than the fine-structure constant (∼10−2)—the predicted scale for the Faraday angle, associated with the quantized Hall conductance23,24,31,32. We have demonstrated that this enhancement is due to the proximity to the resonance frequency. Although in two-dimensional electron gases the cyclotron resonance gives rise to comparable absolute rotations2,32, the rotation comes from an effective layer that is about one order of magnitude thicker than graphene. Furthermore, although in two-dimensional electron gases the cyclotron frequency is doping independent, in graphene it can be tuned with doping (electrostatic or chemical) as seen from equation (6), and much higher frequencies can be achieved at the same fields.

The charge scattering determines the width of the cyclotron peak and inter-LL transitions. Improving the mobility will increase the rotations observed. As in the classical limit the Fermi–Dirac thermal filling factors play a minor role in the optical conductivity, the amount of rotation in this regime is not expected to be significantly reduced at elevated temperatures.

The Faraday effect and the associated magneto-optical Kerr effect are widely used in optical communication, data storage and computing. We suggest that the use of graphene for fast tunable ultrathin magneto-optical devices should be explored. Indeed, the possibility of easy and fast ambipolar doping together with a strong Faraday effect in a wide frequency range is a unique combination not present in other known materials.

Methods

Samples.

Single- and multilayer epitaxial graphene were made by graphitization of the surface of semi-insulating 6H-SiC substrates in an argon atmosphere26. The sample size was 10×10 mm2. SLG was obtained on the Si-terminated face of SiC by first preparing a () reconstructed surface (the so-called buffer layer) by annealing in 1 bar of Ar at a temperature of 1,450 °C, which was subsequently converted into quasifreestanding graphene by intercalation of hydrogen27,28. MLG was grown on the C-terminated face of SiC. In either case, the back side of the substrate was cleaned from undesirably grown graphene using Scotch tape and was checked to be graphene free using X-ray photo-emission. Likewise, the layer thicknesses of the SLG and MLG samples were determined using X-ray photo-emission measurements.

Magneto-optical experiment.

The sample was mounted in a split-coil superconducting magnet attached to a Fourier-transform spectrometer. A Globar light source and He-cooled bolometer detector were used. Two grid-wire gold polarizers (one fixed and one rotating) were mounted before and after the sample. The Faraday rotation was deduced from the angle of the rotating polarizer at which the intensity was at minimum. The substrate-normalized transmission T was measured with one polarizer only and is defined as the ratio between the intensity of the light transmitted through the graphene on SiC and through bare substrate.