Abstract
Studying the collective excitations in charge neutral graphene (CNG) has recently attracted a great interest because of unusual mechanisms of the charge carrier dynamics. The latter can play a crucial role for formation of recently observed in twisted bilayer CNG graphene plasmon polaritons (GPPs) associated with the interband transitions between the flat electronic bands. Besides, GPPs in CNG can be a tool providing insights into various quantum phenomena in CNG via optical experiments. However, the properties of interband GPPs in CNG are not known, even in the simplest configurations. Here, we show that magneticallybiased singlelayer CNG can support interband GPPs of both transverse magnetic and transverse electric polarizations (particularly, at zero temperature). GPPs exist inside the absorption bands originating from the electronic transitions between Landau levels and are tunable by the magnetic field. We place our study into the context of potential nearfield and farfield optical experiments.
Introduction
Graphene plasmon polaritons (GPPs)—electromagnetic fields coupled to oscillating Dirac charge carriers in graphene—exhibit nanoscale field confinement allowing for an active control of their wavelength via gating, and are thus highly interesting for nanophotonics and optoelectronics^{1}. To support GPPs, graphene needs to host a sufficient amount of free charge carriers, which can be achieved by doping (chemical or via gating)^{2,3}, thermal fluctuations^{4,5} or optical pumping^{6}. In doped graphene, currently considered as a standard platform for exploring GPPs, the energies of GPPs (and thus frequencies at which they can be excited) are limited by the Fermi energy, E_{F}, which can reach a few tenths of an electronvolt. There is a strong desire to extend GPP spectral range exploiting additional ways to tune GPPs at higher energies^{7}. One way to tune GPPs in doped graphene is to apply a static magnetic field, resulting in so called graphene magnetoplasmons^{8,9,10}. However, the performance of the magnetoplasmons in doped graphene is still limited by the high electron scattering rates and the need of high magnetic fields (typically, several Tesla).
Recently, it has been realized that in certain conditions charge neutral graphene (CNG) could become an alternative to doped graphene for GPP applications, since CNG possesses specific optical properties. For instance, being exposed to reasonably low magnetic fields, CNG manifests the colossal magnetooptical activity in the midinfrared and terahertz frequency ranges^{11}. Besides, as reported in^{12}, CNG shows the giant intrinsic photoresponse due to the small electronelectron scattering rates. Interestingly, collective effects in twisted bilayer CNG can enable the formation of GPP associated with the interband transitions^{13,14} between the flat electronic bands that appear at the magic angle. Intriguingly, these interband GPP can be closely related with the superconducting states appearing in periodically strained singlelayer CNG^{15} and magic angle twisted bilayer graphene^{16,17,18}. Up to now, although having a high interest for superconducting states formation and increasing of operational frequencies for graphenebased plasmonic devices, GPPs in CNG have been barely explored.
Here, motivated by the recent experimental data^{11}, we show that a singlelayer CNG biased by external static magnetic field, or, simply, magnetized CNG (MCNG), can support interband GPPs of both transverse magnetic (TM) and transverse electric (TE) polarizations in both infrared and terahertz (THz) frequency range. To the best of our knowledge, GPPs in MCNG are the unique case of lowfrequency interband plasmons. Importantly, their wavelength and energy can be controlled by an external magnetic field, while their figure of merit (the ratio between GPP propagation length and GPP wavelength) and lifetime are comparable with those of the lowloss phonon/exciton polaritons recently discovered in van der Waals (vdW) crystal slabs^{19,20}. Besides, the interband TE GPPs have two orders of magnitude larger confinement of the electromagnetic fields than TE intraband GPPs^{21,22,23}. A much better confinement of the fields in the interband TE GPPs favors their excitation via Bragg resonances in a graphene grating, as we demonstrate in our calculations. On the other hand, we show that TM GPPs can be very efficiently excited either via Fabry–Perot resonances in an array of graphene ribbons (in the farfield experiments) or by means of nearfield microscopy (in the nearfiled experiments).
Results and discussion
The origin of interband GPPs in MCNG and their properties
Recall that conventional GPPs in doped graphene arise from intraband electronic transitions (ones between two states within the same band, either the conduction band or valence band, see Fig. 1a), and can be described within the classical Drude model of the conductivity^{24,25,26,27,28}. From this point of view conventional GPPs are similar to the wellknown surface plasmon polaritons in noble metals. In contrast, in MCNG intraband transitions are Pauliblocked, e.g., being not possible, but interband transitions are allowed both between Landau levels (LLs) of opposite sign and the ones involving the zeroth LL (see Fig. 1d). At that, levels with negative sign belong to the valence band, the levels with positive sign belong to the conduction band, and the zeroth level belongs to both of them.
Even in the undoped system, electrons in the LL can couple to the timeperiodic electromagnetic field and build up an oscillating charge density characteristic of a GPP (note that similar collective oscillations occur in doped graphene in magnetic field^{10}). Thus, interband GPPs in MCNG result from Landau Level quantization and, contrary to conventional intraband GPPs, have no classical counterpart.
The properties of GPPs in MCNG (their wavelength, lifetime, etc.) are fully determined by the magnetooptical conductivity of MCNG, \(\hat \sigma\). Although in general \(\hat \sigma\) of a magnetized graphene is a tensor, in case of MCNG at zero temperature, \(\hat \sigma\) diagonalizes so that, σ_{xx} = σ_{yy} = σ. This happens due to the symmetry of the LL transitions −n → n − 1 and −n + 1 → n (n is number of LL) which contribute equally but with opposite sign to the nondiagonal conductivity^{11}. Thus, similarly to nonmagnetized doped graphene the magnetooptical conductivity of MCNG at zero temperature is characterized by a scalar σ which can be calculated within the linearresponse approximation^{29}.
Throughout the paper we consider CNG and neglect any possible deviations from the charge neutrality point, (which could be caused, for example, by the presence of the electronhole puddles in the graphene sample^{30}). However, as we show in Supplementary Note 1 (Supplementary Fig. 1), interband GPPs can exist even in doped graphene, provided that the doping is low enough to avoid Pauli blocking of the LL transitions. Specifically, the Fermi energy, E_{F}, has to be smaller than one half of the LL transition energy, \(E_c\left( {\sqrt {\left {n  1} \right} + \sqrt {\left n \right} } \right)\), where \(E_c\left( B \right) = \sqrt {2e\hbar v_F^2B}\) is cyclotron energy, v_{F} is Fermi velocity and B is the magnetic field. For example, even assuming E_{F} = E_{c}/2 = 24 meV (the concentration of the charge carriers of 3.1 × 10^{10} cm^{−2}), Supplementary Fig. 1 shows that, at B = 1.3 T, the dispersion relation of TM GPP in doped graphene remains virtually undistinguishable from that in MCNG. TE GPPs are more sensitive to doping. However, rising the doping up to 3.1 × 10^{10} cm^{−2} their dispersion relation at 11.3 THz shifts only by Δk/k_{pl} ≈ 0.007, where k_{pl} is wavevector of GPP in MCNG. For simplicity, we restrict our analysis of the GPPs dispersion relation to the case of a free standing MCNG. Nevertheless, later on, in order to mimic realistic nearfield and farfield experiments, we consider MCNG encapsulated between thin hBN slabs. Encapsulated graphene samples are commonly used in the optical experiments, as they show recordhigh mobilities of chargecarriers, and thus the highest GPPs lifetimes^{31}. In addition, through the whole paper we will limit ourselves to CNG at zero temperature, T = 0^{o}K. It is worth noticing that (as we show in Supplementary Fig. 2 in Supplementary Note 1) the dispersion relation of TM GPPs in MCNG undergoes negligible changes with an increase of temperature up to 100 °K. Even at room temperature (T = 300°K) the dispersion relation of TM GPPs only shows a slight frequency shift of the order of 0.5 THz, with an otherwise unaltered shape of the dispersion curve. TE GPPs are more sensitive to temperature (at T = 300 °K, B = 1.3 T and frequency 11.24 THz the dispersion relation shifts by \(\frac{{{\Delta}k}}{{k_{pl}}} \approx 0.01\)), but they are still observable at T = 300 °K. For definiteness and unless explicitly stated, in all calculations we will consider a relaxation time of charge carriers τ = 1 ps. Note that this relaxation time depends on the joint density of states of LLs and is thus not directly related to the one for direct current transport. The value of τ used in the paper provides absorption spectra linewidths comparable to those reported in measurements in hBNencapsulated MCNG^{11}. Other values used in the paper are: Fermi velocity υ_{F} = 1.15 × 10^{6} ms^{−1} and external magnetic field B = 1.3 T (E_{c} = 48 meV) (reachable with neodymium magnets).
In Fig. 1b and 1e we show the real and imaginary parts of the dimensionless conductivity, α = 2πσ/c, as a function of frequency,v, for nonmagnetized doped graphene (with E_{F} = 0.1 eV taken for reference) and MCNG, respectively. Both Re(α) and Im(α) of MCNG behave in a markedly different manner compared to those of the nonmagnetized doped graphene. Namely, in MCNG Re(α) has a series of pronounced peaks. These peaks correspond to the interband LL transitions, n − 1 → n and n → n−1, occurring at the discrete frequencies \(\nu _n = E_c\left( {\sqrt {\left {n  1} \right} + \sqrt {\left n \right} } \right)/h\) and mark the GPP bands.
The polarization of the GPP mode in each of the bands is determined by the sign of lm(α). Indeed, when the graphene conductivity is a scalar, the dispersion relation of TM GPP reads^{26,32}: 1/q_{z} + α = 0, correspondingly, where \(q_z = \sqrt {1  q_{pl}^2}\)and \(q_{pl} = k_{pl}/k_0\)are the dimensionless outofplane and inplane GPP wavevector components, respectively, with k_{0} = 2πv/c being the wavenumber in vacuum. For TE GPPs the dispersion relation satisfies q_{z} + α = 0. The condition that the plasmon fields must decay away from the graphene layers imply that TE plasmons only exist for negative Im(α), while the existence of TM plasmons require lm(α) > 0. In MCNG, Im(α) changes its sign from negative to positive at frequencies v_{n}. On the other hand, at the frequencies \(\tilde \nu _n \approx \sqrt {\left( {\nu _n  \nu _{n + 1}} \right)^2  \nu _{n + 1}\nu _n}\) (which are inbetween of v_{n} and v_{n+1}) Im[α] changes its sign from positive to negative. Therefore, TM GPPs exist in the frequency intervals \(\nu _n\, <\, \nu\, <\, \tilde \nu _n\) (shown by white regions in Fig. 1b,e). Oppositely, in the frequency intervals 0 < v < v_{n} and \(\tilde \nu _n\, <\, \nu\, <\, \nu _n\) (shown by blue areas in Fig. 1b, e) Im[α] < 0, so that TE GPPs can be supported.
In Fig. 1c, f we illustrate the dispersion curves for the GPPs in nonmagnetized doped graphene (with E_{F} = 0.1 eV) and MCNG, respectively. In each case, the dispersion of TM (TE) GPPs is shown by the blue (red) curves. We see that GPPs in MCNG and in nonmagnetized doped graphene behave very differently. Namely, while in the nonmagnetized doped graphene both TM and TE intraband GPPs exist in two continuous frequency intervals, in case of MCNG, the dispersion of the interband GPPs is split into a set of narrow frequency bands with the energy band width \(h{\Delta}\nu \approx E_c\) for the first TM and TE bands. The band widths increase with the magnetic field and decrease with band index, n.
At the highfrequency border of the GPPs frequency bands, the dispersion curves of both TM and TE modes present a backbending toward \(q_{pl} = k_{pl}/k_0 = 0\), taking place due to the losses in graphene (given by Re(α)). Although the losses in MCNG inevitably limit the propagation of the GPPs, their figure of merit—defined as the ratio between GPP propagation length, L_{pl}, and the GPP wavelength, λ_{pl}, \({\mathrm{FOM}} = L_{pl}/\lambda _{pl}\)—can be comparably large.
Indeed, in case of TM GPP in MCNG, for small values of α, we can approximate \({\lambda}_{pl} = \lambda {\mathrm{Im}}(\alpha )\) and \(L_{pl} = \frac{{\lambda {\mathrm{Im}}(\alpha )^2}}{{2\pi {\mathrm{Re}}(\alpha )}}\) so that \({\mathrm{FOM}} = {\mathrm{Im}}(\alpha )/2\pi {\mathrm{Re}}(\alpha ).\) The condition \(Im(\alpha )\gg Re(\alpha )\), providing high FOM, is fulfilled at the frequencies in the middle of each TM band. Inside each band FOM grows proportionally to both the relaxation time and the magnetic field as \(\propto \left( {\tilde \nu _n\left( B \right)  \nu _n\left( B \right)} \right)\tau\), where both \(\tilde \nu _n\left( B \right)\) and v_{n}(B) are proportional to \(\sqrt B\). For the realistic parameters considered in Fig. 1f, the FOM of TM GPP in the first frequency band reaches ~3.5 (see Supplementary Fig. 4a in Supplementary Note 2). Thus, FOMs of TM GPPs in MCNG can be of the same order of magnitude as FOM of phonon polariton in hBN^{33}. This result is not intuitive since GPPs present completely different loss channels compared to phonon polaritons (electronelectron and electronphonon scattering versus phonon phonon scattering, respectively), but it may be very useful as, in contrary to the case of phonon polaritons, GPPs in MCNG are tunable.
To better illustrate the strong confinement of TM GPPs in MCNG in Fig. 2a we show the spatial distribution of the vertical electric field generated by a vertical electrical dipole positioned at z = 60 nm above the graphene, assuming the applied magnetic field of 1.3 T, and v = 12 THz (λ_{0} = 25 μm). The fringes of the opposite polarities propagating away from the dipole region clearly indicate the polaritonic wavelength (λ_{pl} = 4.3 μm) being much smaller than the freespace wavelength, λ_{0} = 25 μm, indicated by the horizontal black arrow (the corresponding frequency, v = 12 THz, is marked by the black point in the red dispersion curve and by the horizontal dashed line in Fig. 2c).
The properties of TM GPP in MCNGs strongly depend on the applied magnetic field. In Fig. 2d we plot the real part of the zcomponent of electric field of TM GPP, excited by the dipole, as a function of the distance along graphene, x, at two different values of the magnetic field: B = 1.3 T and B = 1 T. These field profiles illustrate that TM GPP wavelength increases with B (λ_{pl} is almost five times larger for B = 1.3 T than for B = 1 T), the latter tendency of λ_{pl}(B) being further confirmed by the red curve in Fig. 2b. The explicit dependence λ_{pl}(B) can be estimated from the dispersion relation, shown in Fig. 2c for several values of the magnetic field within the first band. Away from the bandbending, the GPP wavelength can be written as (see Supplementary Note 3)
with \(W_n = \frac{{\sigma _{uni}}}{\pi }\frac{{E_c^2}}{{h^2\nu _n}}\) being the spectral weight^{11} and \(\sigma _{uni} = \frac{{e^2}}{{4\hbar }}\) the universal optical conductivity, thus nonlinearly scaling with B.
Apart from the importance of FOM for applications requiring propagating polaritons, it can be also important that polaritons have large lifetimes. The latter can be calculated as \(\tau _{life} = L_{pl}/v_{gr}\), where v_{gf} is the group velocity, given by \(v_{gr} = 2\pi d\nu /dk_{pl}\). From Eq. (1) we can obtain \(v_{gr} \approx W_n/\sqrt {W_nk_{pl} + \pi ^2\nu _n^2}\). Thus, at a frequency 12 THz and for a magnetic field B = 1.3 T, for which L_{pl} ≈ 2 μm (see Fig. 2c), k_{pl} ≈ 1.5 μm^{−1} (see Fig. 2d), and v_{gr} = 10^{6} m/s, we can estimate τ_{life} ≈ 2 ps. This value is comparable with that of longlived phonon/exciton polaritons in vdW crystal slabs and 2D layered materials^{19,20,33}.
In strong contrast to the TM modes, the TE GPP in MCNG are far less confined, as shown by the magnetic field snapshot in Fig. 3a (where we use a vertical magnetic dipole source positioned at z = 5 μm above the graphene at the frequency of v = 11.2 THz). The excitation of TE GPP in MCNG is confirmed by the diffraction shadow^{34} seen in Fig. 3a. It appears because of the destructive interference of the excited TE GPP in MCNG and the dipole radiation. This is due to the comparable wavelengths of freescape radiation and TE GPPs. At a sufficient distance from the source the TE plasmon is clearly distinguishable from the freespace fields. For a more quantitative analysis of the TE plasmons launched by the dipole, in Fig. 3d we plot the field profile along graphene for two values of the magnetic field: B = 1.3 T (red curve) and B = 1.4 T (blue curve). For comparison, the field launched by the dipole in freespace without graphene (with the wavelength λ_{0} = 26.77 μm) is also shown by the black curve. According to the field profiles, the difference between λ_{0} and the wavelength of TE GPP in MCNG does not exceed 2%. Besides, we see that, in contrast to TM modes, the wavelength of TE GPP in MCNG depends on the applied magnetic fields, see Fig. 3b (red curve). Nevertheless, as the plasmons propagate away from the source, the phase shift between the field profiles for different magnetic fields becomes considerable (see the shift between the red and green field profiles in Fig. 3b). The propagation length of TE GPP in MCNG is also much more sensitive to the magnetic field than their wavelength, as shown in Fig. 3b (in Fig. 3d the red field profile clearly decays much quicker than the green one). The latter can be explained by the vertical shift of the dispersion curves of the TE plasmons with the increase of the magnetic field, and thus the backbending region (where the absorption is increased), see Fig. 3c. The sensitivity of both the phase shift and the propagation length to the magnetic field can be potentially used for the applications of TE GPPs, particularly in interferometers.
TE GPPs in MCNG are much more confined to the graphene sheet than conventional TE GPPs. The strongest confinement takes place close to the backbending of the dispersion curve with the value ~1/q_{pl}^{(max)} where q_{pl}^{(max)} is the maximum momentum (see Supplementary Note 3), \(q_{pl} = q_{pl}^{({\mathrm{max}})} \approx 1 + \frac{1}{2}\left[ {\frac{{W_n\tau }}{c}} \right]^2\), reached at the frequencies \(\nu ^{\left( {{\mathrm{max}}} \right)} \approx \nu _n  2/\left( {3\pi \tau } \right)\). At the maximal momentum the confinement is two orders of magnitude smaller than in nonmagnetized doped graphene^{16,17,18}. At lower frequencies TE GPP dispersion curve asymptotically tends to the light line, q_{pl} = 1 and thus becomes undistinguishable from the dispersion of the freespace waves.
GPPs in MCNGs can be potentially observed experimentally using either nearfield microscopy of nonstructured MCNG or farfield transmission/reflection spectroscopy of periodically patterned MCNG. The nearfield microscopy, however, is more appropriate for TM GPP in MCNG, since the tip of the nearfield microscope typically presents a vertically polarized dipole source, thus better matching with the TM polarized excitations. In the next sections we mimic both nearfield and farfield experiments.
Prospects for potential spectral nearfield experiments for the characterization of TM GPPs in MCNG
Scatteringtype nearfield optical microscopy (sSNOM) utilizes a tip of an atomic force microscope (AFM), which is illuminated with an external infrared laser^{35,36}. The laser beam is focused at the apex of the tip, providing the necessary momentum to launch GPPs in graphene, as illustrated in Fig. 4(a). GPPs emanate from the tip and upon reaching the sample edge, they are reflected back. As the tip is scanned toward the edge, the backscattered signal is collected in the detector as a function of the tip position^{2,3}. By using a broadband light source, the nearfield light scans can be represented as a function of frequency^{6}. By Fourier transforming the interferogram as a function of frequency, the dispersion relation of GPPs can be retrieved.
In our numerical model we represent the illuminated AFM tip by a vertical dipole source. As has been shown in^{37}, the absolute value of the vertical component of electric field taken at some distance below the dipole, E_{z}, can reproduce qualitatively the signal scattered by tip. Therefore, when modeling, we calculate E_{z} below the dipole, as a function of the dipole position (x) and frequency (v), composing a nearfield hyperspectral image, E_{z}(x,v).
An example of the nearfield hyperspectral image of the single layer of MCNG encapsulated between two flakes of hBN (the thicknesses of the top and bottom hBN layers are 5 nm and 10 nm, respectively) is shown in Fig. 4b. For simplicity, in these simulations we consider a free standing hBNencapsulated CNG. Nevertheless, in Supplementary Note 4 we show (see Supplementary Fig. 5) that a 10 nm thick hBN film is enough to make the influence of a substrate negligible. We perform simulations at the frequencies of the first frequency band of TM GPP in MCNG: 11.5–22.35 THz.
In Fig. 4b we clearly observe several bright fringes representing the alternating NF minima and maxima. They appear due to the interference between the GPP launched by the dipole and GPP reflected by the edge. The fringe spacing (the distance between neighboring NF minima or maxima) taken far from the edge equals approximately to the half of the GPP wavelength, λ_{pl}/2. As v increases the fringes become thinner and their interspacing decreases. The latter decrease is consistent with the dependence \(\lambda _{pl} \propto \left( {\nu ^2  \nu _n^2} \right)^{  1}\), see Eq. (1), where for the first band v_{n} = v_{1} = 11.5THz. Besides, the amplitude of the fringes decays with the increase of v, which can be attributed to the increase of MCNG losses (i.e., growth of Re(α)) particularly, when approaching the limiting frequency, \({\tilde {\nu}} _1 = 22.35\) THz.
In Fig. 4c we plot the Fourier transform of the nearfield hyperspectral image represented in Fig. 4b. The bright maximum seen in the color plot perfectly matches the GPP dispersion curve (the dashed green curve), assuming that the momenta of the GPPs are doubled, 2q_{pl}(v). The latter is consistent with the λ_{pl}/2 distance between the interference fringes in Fig. 4b. With our analysis we conclude that the GPPs in hBNencapsulated MCNG should be observable in sSNOM experiments for realistic parameters of graphene, even at moderate external magnetic fields.
Prospects for farfield spectroscopy experiments
For the coupling with GPPs from the farfield, the graphene can be structured into ribbons^{38,39,40,41}. Such structuring allows one to avoid the momentum mismatch between the GPPs and freespace waves. Depending upon the parameters of the grating, the excited GPPs present either “quantized” Fabry–Perot modes inside the ribbons, or Braggresonances manifesting themselves a dip/peak in the transmission/reflection spectrum. In our simulations for the excitation of both TM and TE GPP in MCNGs we consider a periodic array of microribbons made in either freestanding or hBNencapsulated graphene, as illustrated in Fig. 5a.
In Fig. 5b we show the absorption spectra, A(v), of the ribbon arrays (of different ribbon widths, W, and a fixed period, L = 0.22 μm) illuminated by a normallyincident monochromatic plane wave. The latter is polarized along the xdirection, thus matching with the polarization of TM GPPs. The solid and dashed curves in Fig. 5b correspond to the absorption by the hBNencapsulated and freestanding MCNG ribbon arrays, respectively. In continuous MCNG (solid black curve in Fig. 5b), the absorption spectra present only one maximum, matching with the interband transition frequency v_{1} = 11.5 THz. In contrast, the absorption spectrum of the MCNG ribbon array, (both for hBNencapsulated and freestanding graphene), has a set of resonant maxima. For instance, at W = 3L/4 the absorption spectra (solid blue curve) has one strong resonance at 12.5 THz and set of much weaker ones at higher frequencies. With decreasing W, the resonances blueshift away from v_{1} dropping in their magnitudes. Similarly to doped graphene ribbon arrays, the emergence of the absorption maxima can be explained by Fabry–Perot resonances in a single ribbon, forming while the GPPs reflect back and forth from the ribbon edges. In the resonance, the GPP refractive index, \(q_{pl} = \frac{{\lambda _0}}{{\lambda _{pl}}}\), should satisfy^{42}
where m is an even number. Note that the modes with the odd integers have antisymmetric distribution of the vertical electric field with respect to the ribbon axis and do not couple to normallyincident waves. For all W, the absorption resonances in the spectrum of hBNencapsulated graphene ribbon array appear at lower frequencies compared to the ones in the freestanding array. This can be explained by the stronger confinement of GPPs in encapsulated graphene than in the freestanding one. The strong confinement of TM GPPs makes them excellent candidates for sensing the environment^{43}. In fact, changes in the environment can be detected via the resulting frequency shifts of the plasmonic resonances in the farfield spectrum. The absorption resonances are corroborated by the dispersion curves in the bottom panel of Fig. 5b (GPPs in the encapsulated graphene have larger momenta). The vertical dotted lines connecting the absorption maxima (Fig. 5b, top panel) with the GPP dispersion curves (Fig. 5b, bottom panel) mark the frequency positions of the peaks. The frequencies of the peaks match well with the ones given by the simple Fabry–Perot 0thorder resonances in Eq.(2). Substituting Eq.(1) in Eq.(2) we can find that the resonance frequencies are proportional to \(\sqrt {B/W}\), which means that they are tunable by both magnetic field and the ribbon width (see Supplementary Note 5).
To resonantly excite the TE GPPs, we now illuminate the hBNencapsulated graphene ribbon arrays by a normallyincident monochromatic plane wave with electric field pointing along the y direction. Due to much smaller momenta of TE GPPs, the array period must be more than two orders of magnitude larger than the one considered for excitation of TM plasmons. In Fig. 5c we present A(v), as before, for different W and for the period L = 28 μm (see analogous simulations for smaller relaxation times in Supplementary Note 6). Each color curve demonstrates two strong resonant maxima. The broader resonances are related to the absorption maximum at the interband transition frequency. Indeed, the position of the broader peaks coincides with the resonance in the continuous MCNG (black curve). The appearance of the second (narrow) peak in each curve is very different from the resonance in the case of the TM polarization. First, the narrow peak appears at lower frequency compared to the interband transition. This is clearly related to the spectral region of the TE GPPs (located below the interband transition, as shown in the bottom panel of Fig. 5c). Second, the frequencies of the narrow resonance are almost independent upon the ribbon width. The inset in Fig. 5c showing the zoomedin frequency region of the narrow resonances demonstrates the minor changes in the magnitude of the peak with change of W. The minor sensitivity of the frequency position of the narrow resonance to the ribbon width is related to a different nature of the resonance. Because the momentum of the TE GPPs is very close to the light line, they can easily leak out of the ribbon while reflecting from the edges, and therefore cannot build the Fabry–Perot resonances in individual ribbons. Instead, the TE GPPs in the grating constitute the Bloch modes and lead to the “collective” Bragg resonances. In this case the resonance condition for the TE GPP refractive index is linked to the grating’s period, L, (rather than to the ribbon width): \(q_{pl} \approx \sin \theta + m\frac{\lambda }{L},\) where m = ±1,… is the diffraction order and θ is the angle of incidence (the dependence of the TE Bragg resonances upon both angle of incidence and frequency is illustrated in more detail in the Supplementary Note 7).
Due to the weak confinement, TE GPPs are less sensitive to the dielectric environment of graphene. Indeed, in the inset of Fig. 5c the absorption spectra by the freestanding array (dashed curves) increase in their amplitudes and slightly shift from those by the hBNencapsulated ribbon array (solid curves).
We thus demonstrate that MCNG microribbon arrays are a powerful system for controlling the coupling between light and both TM and TE GPP modes. This enables enhancement and tailoring of THz absorption by changing ribbon width and period or applied magnetic fields (in Supplementary Note 5 we also illustrate the tunability of the absorption spectra of the MCNG ribbon arrays by a slight variation of the applied magnetic field).
Conclusion
Summarizing, our work demonstrates that a singlelayer CNG biased by an external magnetic field supports magneticallytunable interband GPPs of both TM and TE polarizations. Some differences between interband GPPs in MCNG and conventional plasmon in doped graphene are worth stressing. From the fundamental side, GPPs in MCNG are rooted in LL quantization and have no classical counterpart, while conventional plasmons can be described by a classical Drudelike response. Figures of merit and lifetimes of GPPs in MCNG are comparable to those of phonon/exciton polaritons in vdW materials. For example, for the applied magnetic field B = 1.3 T the maximal figure of merit and lifetime of TM GPP in the first frequency band reaches ~3.5 and 23 ps, respectively, thus being similar to the corresponding values for phonon polaritons in hBN^{33}. However, in difference to phonon polaritons, which are supported by polar slabs within very narrow Reststrahlen bands (determined by the lattice parameters and thus being hardly tunable), the equally narrow frequency bands of interband GPP in MCNG are easily tunable by the applied magnetic field. Notably, the interband TE GPPs in MCNG are two orders of magnitude more confined than TE GPPs in doped graphene, favoring their excitation and potential use in applications.
We have simulated realistic experimental setups (both nearfield optical microscopy and farfield spectroscopy), which demonstrate that GPP in MCN are detectable with currently available techniques. Importantly, our calculations show that rather low values of magnetic fields (B < 2 T) are already suitable for the observations.
From a different perspective, we believe that our analysis of plasmons in a singlelayer magnetized CN graphene can be a tool providing insights into various intriguing quantum phenomena in CNG via optical experiments^{12,13,15}. We foresee that the intriguing phenomena recently discovered in the singlelayer CN graphene, as for instance the giant intrinsic photoresponse^{12}, colossal magnetooptical activity^{11} and markedly reduced electronic noise^{44}, combined with the ability to support GPPs, make magnetized CN graphene promising for a graphenebased magnetically controllable nanoplasmonic and optoelectronic devices, such as gas sensors or biosensors^{44,45,46} or magnetically tunable plasmonassistant photodetectors^{47}, among other.
Methods
First principle numerical simulation
Full wave electromagnetic simulations were performed using the COMSOL software based on finiteelement methods in frequency domain. The graphene layer was modeled as a surface current, implemented in the boundary conditions. In order to achieve convergence, the mesh element size in the vicinity of graphene was much smaller than the plasmon wavelength.
Simulation of NF experiment
In the simulation, the tip was modeled by a vertical point dipole source (polarized along zaxis). We assume that the vertical component of the field below the dipole, E_{z}, approximates the scattered signal in a sSNOM experiment^{28}. We thus simulated the nearfield profiles by recording the calculated E_{z} as a function of the dipole position, x (due to the translational symmetry of the problem along the yaxis, E_{z} does not depend upon y), and dipole frequency. The NF value of E_{z} is taken 5 nm above the top hBN interface. The dispersion diagram of GPP in MCNG is obtained from the realspace Fourier transform of these NF values taken in space–frequency domain (x, w).
Data availability
The data supporting the findings of this study are included in the main text and in the Supplementary Information files, and are also available from the corresponding authors upon reasonable request.
References
Gonçalves, P. A. D. and Peres, N. M. R. An introduction to graphene plasmonics, (World Scientific Publishing Co. Pte. Ltd., 2016).
Fei, Z. et al. Gatetuning of graphene plasmons revealed by infrared nanoimaging. Nature 487, 82–85 (2012).
Chen, J. et al. Optical nanoimaging of gatetunable graphene plasmons. Nature 487, 77–81 (2012).
Vafek, O. Thermoplasma polariton within scaling theory of singlelayer graphene. Phys. Rev. Lett. 97, 266406 (2006).
AlonsoGonzález, P. et al. Acoustic terahertz graphene plasmons revealed by photocurrent nanoscopy. Nat. Nanotech. 12, 31–35 (2017).
Ni, G. et al. Ultrafast optical switching of infrared plasmon polaritons in highmobility graphene. Nat. Photon. 10, 244–247 (2016).
Bezares, F. J. et al. Intrinsic Plasmon–Phonon Interactions in Highly Doped Graphene: A NearField Imaging Study. Nano Lett. 17, 5908–5913 (2017).
Yan, H. et al. Infrared spectroscopy of tunable Dirac terahertz magnetoplasmons in graphene. Nano Lett. 12, 3766–3771 (2012).
Crassee et al. Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene. Nano Lett. 12, 2470–2474 (2012).
Ferreira, A., Peres, N. M. R. & Castro Neto, A. H. Confined magnetooptical waves in graphene. Phys. Rev. B 85, 205426 (2012).
Nedoliuk, I. O., Hu, S., Geim, A. K. & Kuzmenko, A. B. Colossal infrared and terahertz magnetooptical activity in a twodimensional Dirac material. Nat. Nanotechnol. 14, 756–761 (2019).
Ma, Q. et al. Giant intrinsic photoresponse in pristine graphene. Nat. Nanotech. 14, 145–150 (2019).
Sharma, G. et al. Superconductivity from collective excitations in magicangle twisted bilayer graphene. Physical Review Research 2, 022040 (2020).
Hesp, N. et al. Collective excitations in twisted bilayer graphene close to the magic angle. Preprint at https://arxiv.org/abs/1910.07893 (2019).
Peltonen, T. & Heikkilä, T. Flatband superconductivity in periodically strained graphene: meanfield and Berezinskii–Kosterlitz–Thouless transition. Journal of Physics: Condensed Matter 32, 365603 (2020).
Cao, Y. et al. Unconventional superconductivity in magicangle graphene superlattices. Nature 556, 43–50 (2018).
Yankowitz, M. et al. Tuning superconductivity in twisted bilayer graphene. Science 363, 1059 (2019).
Lu, X. et al. Superconductors, orbital magnets and correlated states in magicangle bilayer graphene. Nature 574, 653–657 (2019).
Dai, S. et al. Tunable Phonon Polaritons in Atomically Thin van der Waals Crystals of Boron Nitride. Science 343, 1125–1129, https://doi.org/10.1126/science.1246833 (2014).
Ma, W. et al. Inplane anisotropic and ultralowloss polaritons in a natural van der Waals crystal. Nature 562, 557–562 (2018).
Mikhailov, S. A. & Ziegler, K. New electromagnetic mode in graphene. Phys. Rev. Lett. 99, 016803 (2007).
Jablan, M., Buljan, H. & Soljacic, M. Transverse electric plasmons in bilayer graphene. Opt. Express 19, 11236–41 (2011).
Pellegrino, F. M. D., Angilella, G. G. N. & Pucci, R. Linear response correlation functions in strained graphene. Phys. Rev. B 84, 195407 (2011).
Hanson, G. W. J. Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene. Appl. Phys. 103, 064302 (2008).
Jablan, M., Buljan, H. & Soljacic, M. Plasmonics in graphene at infrared frequencies. Phys. Rev. B 80, 245435 (2009).
Falkovsky, L. A. Optical properties of graphene and IV–VI semiconductors. Phys. Uspekhi 51, 887–897 (2008).
Hwang, E. H. & Das Sarma, S. Dielectric function, screening, and plasmons in twodimensional graphene. Phys. Rev. B 75, 205418 (2007).
Wunsch, B., Stauber, T., Sols, F. & Guinea, F. Dynamical polarization of graphene at finite doping. N. J. Phys. 8, 318–318 (2006).
Gusynin, V. P., Sharapov, S. G. & Carbotte, J. P. Magnetooptical conductivity in graphene. J. Phys. Condens. Matter 19, 026222 (2007).
Katsnelson, M. I., Novoselov, K. S. & Geim, A. K. Chiral tunnelling and the Klein paradox in graphene. Nat. Phys. 2, 620–625 (2006).
Woessner, A. et al. Highly confined lowloss plasmons in graphene–boron nitride heterostructures. Nat. Mater. 14, 421–425 (2015).
Nikitin, A. Y. World Scientific Handbook of Metamaterials and Plasmonics. World Sci. Ser. Nanosci. Nanotechnol. 4, 307–338 (2017).
Yoxall, E. et al. Direct observation of ultraslow hyperbolic polariton propagation with negative phase velocity. Nat. Photon 9, 674–678 (2015).
Nikitin, A. Y., Rodrigo, S. G., GarcíaVidal, F. J. & MartínMoreno, L. In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region. N. J. Phys. 11, 123020 (2009).
Ocelic, N., Huber, A. & Hillenbrand, R. Pseudoheterodyne detection for backgroundfree nearfield spectroscopy. Appl. Phys. Lett. 89, 101124 (2006).
Yang, H. U. et al. A cryogenic scatteringtype scanning nearfield optical microscope. Rev. Sci. Instrum. 84, 023701 (2013).
Nikitin, A. Y. et al. Realspace mapping of tailored sheet and edge plasmons in graphene nanoresonators. Nat. Photon 10, 239–243 (2016).
Ju, L. et al. Graphene plasmonics for tunable terahertz metamaterials. Nat. Nanotechnol. 6, 630–634 (2011).
Yan, H. et al. Damping pathways of midinfrared plasmons in graphene nanostructures. Nat. Photonics 7, 394–399 (2013).
Nikitin, A. Y., Guinea, F., GarciaVidal, F. J. & MartinMoreno, L. Surface plasmon enhanced absorption and suppressed transmission in periodic arrays of graphene ribbons. Phys. Rev. B 85, 081405 (2012).
García de Abajo, F. J. Graphene plasmonics: challenges and opportunities. ACS Photonics 1, 135–152 (2014).
Nikitin, A. Y. Low, Tony, MartinMoreno. Luis Phys. Rev. B 90, 041407 (2014).
Vasić, B., Isić, G. & Gajić, R. Localized surface plasmon resonances in graphene ribbon arrays for sensing of dielectric environment at infrared frequencies. J. Appl. Phys. 113, 013110 (2013).
Fu, W. et al. Biosensing near the neutrality point of graphene. Sci. Adv. 3, e1701247 (2017).
Kanti, P. R., Badhulika, S., Saucedo, N. M. & Mulchandani, A. Graphene nanomesh as highly sensitive chemiresistor gas sensor. Anal. Chem. 84, 8171–8178 (2012).
SalehiKhojin, A. et al. Polycrystalline Graphene Ribbons as Chemiresistors. Adv. Mater. 24, 53–57 (2012).
Koppens, F. et al. Photodetectors based on graphene, other twodimensional materials and hybrid systems. Nat. Nanotech. 9, 780–793 (2014).
Acknowledgements
We acknowledge funding from Spain’s MINECO under Grant No. MAT201788358C3 and funding from the European Union Seventh Framework Programme under grant agreement no. 881603 Graphene Flagship for Core 3 . L.M.M. acknowledge Aragon Government through project QMAD. A.Y.N. acknowledges the Basque Department of Education (grant numbers PIBA202010014). The research for A.B.K. was supported by the Swiss National Science Foundation.
Author information
Authors and Affiliations
Contributions
A.B.K., A.Y.N. and L.M.M. conceived the work. T.M.S., A.Y.N. and L.M.M. performed numerical and analytical calculations. T.M.S. and A.Y.N. wrote the paper with the input from all coauthors. J.M.P. contributed to the interpretation of the results. All the authors contributed to the discussion of the results.
Corresponding authors
Ethics declarations
Competing interests
The author(s) declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Slipchenko, T.M., Poumirol, JM., Kuzmenko, A.B. et al. Interband plasmon polaritons in magnetized chargeneutral graphene. Commun Phys 4, 110 (2021). https://doi.org/10.1038/s42005021006072
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s42005021006072
This article is cited by

Manipulating polaritons at the extreme scale in van der Waals materials
Nature Reviews Physics (2022)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.