# Interband plasmon polaritons in magnetized charge-neutral graphene

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### The origin of interband GPPs in MCNG and their properties

Recall that typical GPPs in doped graphene come up from intraband digital transitions (ones between two states inside the identical band, both the conduction band or valence band, see Fig. 1a), and will be described inside the classical Drude mannequin of the conductivity24,25,26,27,28. From this viewpoint typical GPPs are much like the well-known floor plasmon polaritons in noble metals. In distinction, in MCNG intraband transitions are Pauli-blocked, e.g., being not doable, however interband transitions are allowed each between Landau ranges (LLs) of reverse signal and those involving the zero-th LL (see Fig. 1d). At that, ranges with damaging signal belong to the valence band, the degrees with constructive signal belong to the conduction band, and the zeroth degree belongs to each of them.

Even within the undoped system, electrons within the LL can couple to the time-periodic electromagnetic discipline and construct up an oscillating cost density attribute of a GPP (word that comparable collective oscillations happen in doped graphene in magnetic discipline10). Thus, interband GPPs in MCNG end result from Landau Degree quantization and, opposite to standard intraband GPPs, don’t have any classical counterpart.

The properties of GPPs in MCNG (their wavelength, lifetime, and so on.) are totally decided by the magneto-optical conductivity of MCNG, (hat sigma). Though usually (hat sigma) of a magnetized graphene is a tensor, in case of MCNG at zero temperature, (hat sigma) diagonalizes in order that, σxx = σyy = σ. This occurs because of the symmetry of the LL transitions −nn − 1 and −n + 1 → n (n is variety of LL) which contribute equally however with reverse signal to the non-diagonal conductivity11. Thus, equally to non-magnetized doped graphene the magneto-optical conductivity of MCNG at zero temperature is characterised by a scalar σ which will be calculated inside the linear-response approximation29.

All through the paper we take into account CNG and neglect any doable deviations from the cost neutrality level, (which could possibly be brought on, for instance, by the presence of the electron-hole puddles within the graphene pattern30). Nonetheless, as we present in Supplementary Notice 1 (Supplementary Fig. 1), interband GPPs can exist even in doped graphene, offered that the doping is low sufficient to keep away from Pauli blocking of the LL transitions. Particularly, the Fermi power, EF, needs to be smaller than one half of the LL transition power, (E_cleft( {sqrt {left| {n – 1} proper|} + sqrt } proper)), the place (E_cleft( B proper) = sqrt {2ehbar v_F^2B}) is cyclotron power, vF is Fermi velocity and B is the magnetic discipline. For instance, even assuming EF = Ec/2 = 24 meV (the focus of the cost carriers of three.1 × 1010 cm−2), Supplementary Fig. 1 exhibits that, at B = 1.3 T, the dispersion relation of TM GPP in doped graphene stays just about undistinguishable from that in MCNG. TE GPPs are extra delicate to doping. Nonetheless, rising the doping as much as 3.1 × 1010 cm−2 their dispersion relation at 11.3 THz shifts solely by Δok/okpl ≈ 0.007, the place okpl is wavevector of GPP in MCNG. For simplicity, we prohibit our evaluation of the GPPs dispersion relation to the case of a free standing MCNG. However, in a while, so as to mimic life like near-field and far-field experiments, we take into account MCNG encapsulated between skinny hBN slabs. Encapsulated graphene samples are generally used within the optical experiments, as they present record-high mobilities of charge-carriers, and thus the best GPPs lifetimes31. As well as, by the entire paper we are going to restrict ourselves to CNG at zero temperature, T = 0oOk. It’s price noticing that (as we present in Supplementary Fig. 2 in Supplementary Notice 1) the dispersion relation of TM GPPs in MCNG undergoes negligible modifications with a rise of temperature as much as 100 °Ok. Even at room temperature (T = 300°Ok) the dispersion relation of TM GPPs solely exhibits a slight frequency shift of the order of 0.5 THz, with an in any other case unaltered form of the dispersion curve. TE GPPs are extra delicate to temperature (at T = 300 °Ok, B = 1.3 T and frequency 11.24 THz the dispersion relation shifts by (frac{{{Delta}ok}}{{k_{pl}}} approx 0.01)), however they’re nonetheless observable at T = 300 °Ok. For definiteness and except explicitly said, in all calculations we are going to take into account a leisure time of cost carriers τ = 1 ps. Notice that this leisure time will depend on the joint density of states of LLs and is thus circuitously associated to the one for direct present transport. The worth of τ used within the paper supplies absorption spectra linewidths similar to these reported in measurements in hBN-encapsulated MCNG11. Different values used within the paper are: Fermi velocity υF = 1.15 × 106 ms−1 and exterior magnetic discipline B = 1.3 T (Ec = 48 meV) (reachable with neodymium magnets).

In Fig. 1b and 1e we present the true and imaginary elements of the dimensionless conductivity, α = 2πσ/c, as a operate of frequency,v, for non-magnetized doped graphene (with EF = 0.1 eV taken for reference) and MCNG, respectively. Each Re(α) and Im(α) of MCNG behave in a markedly completely different method in comparison with these of the non-magnetized doped graphene. Particularly, in MCNG Re(α) has a sequence of pronounced peaks. These peaks correspond to the interband LL transitions, |n − 1| → |n| and |n| → |n−1|, occurring on the discrete frequencies (nu _n = E_cleft( {sqrt {left| {n – 1} proper|} + sqrt } proper)/h) and mark the GPP bands.

The polarization of the GPP mode in every of the bands is set by the signal of lm(α). Certainly, when the graphene conductivity is a scalar, the dispersion relation of TM GPP reads26,32: 1/qz + α = 0, correspondingly, the place (q_z = sqrt {1 – q_{pl}^2})and (q_{pl} = k_{pl}/k_0)are the dimensionless out-of-plane and in-plane GPP wavevector parts, respectively, with ok0 = 2πv/c being the wavenumber in vacuum. For TE GPPs the dispersion relation satisfies qz + α = 0. The situation that the plasmon fields should decay away from the graphene layers indicate that TE plasmons solely exist for damaging Im(α), whereas the existence of TM plasmons require lm(α) > 0. In MCNG, Im(α) modifications its signal from damaging to constructive at frequencies vn. However, on the frequencies (tilde nu _n approx sqrt {left( {nu _n – nu _{n + 1}} proper)^2 – nu _{n + 1}nu _n}) (that are in-between of vn and vn+1) Im[α] modifications its signal from constructive to damaging. Due to this fact, TM GPPs exist within the frequency intervals (nu _n, <, nu, <, tilde nu _n) (proven by white areas in Fig. 1b,e). Oppositely, within the frequency intervals 0 < v < vn and (tilde nu _n, <, nu, <, nu _n) (proven by blue areas in Fig. 1b, e) Im[α] < 0, in order that TE GPPs will be supported.

In Fig. 1c, f we illustrate the dispersion curves for the GPPs in non-magnetized doped graphene (with EF = 0.1 eV) and MCNG, respectively. In every case, the dispersion of TM (TE) GPPs is proven by the blue (purple) curves. We see that GPPs in MCNG and in non-magnetized doped graphene behave very in another way. Particularly, whereas within the non-magnetized doped graphene each TM and TE intraband GPPs exist in two steady frequency intervals, in case of MCNG, the dispersion of the interband GPPs is cut up right into a set of slim frequency bands with the power band width (h{Delta}nu approx E_c) for the primary TM and TE bands. The band widths improve with the magnetic discipline and reduce with band index, n.

On the high-frequency border of the GPPs frequency bands, the dispersion curves of each TM and TE modes current a back-bending towards (q_{pl} = k_{pl}/k_0 = 0), happening because of the losses in graphene (given by Re(α)). Though the losses in MCNG inevitably restrict the propagation of the GPPs, their determine of benefit—outlined because the ratio between GPP propagation size, Lpl, and the GPP wavelength, λpl, ({mathrm{FOM}} = L_{pl}/lambda _{pl})—will be comparably massive.

Certainly, in case of TM GPP in MCNG, for small values of α, we are able to approximate ({lambda}_{pl} = lambda {mathrm{Im}}(alpha )) and (L_{pl} = frac{{lambda {mathrm{Im}}(alpha )^2}}{{2pi {mathrm{Re}}(alpha )}}) in order that ({mathrm{FOM}} = {mathrm{Im}}(alpha )/2pi {mathrm{Re}}(alpha ).) The situation (Im(alpha )gg Re(alpha )), offering excessive FOM, is fulfilled on the frequencies in the course of every TM band. Inside every band FOM grows proportionally to each the comfort time and the magnetic discipline as (propto left( {tilde nu _nleft( B proper) – nu _nleft( B proper)} proper)tau), the place each (tilde nu _nleft( B proper)) and vn(B) are proportional to (sqrt B). For the life like parameters thought-about in Fig. 1f, the FOM of TM GPP within the first frequency band reaches ~3.5 (see Supplementary Fig. 4a in Supplementary Notice 2). Thus, FOMs of TM GPPs in MCNG will be of the identical order of magnitude as FOM of phonon polariton in h-BN33. This end result shouldn’t be intuitive since GPPs current utterly completely different loss channels in comparison with phonon polaritons (electron-electron and electron-phonon scattering versus phonon phonon scattering, respectively), however it might be very helpful as, in opposite to the case of phonon polaritons, GPPs in MCNG are tunable.

To raised illustrate the robust confinement of TM GPPs in MCNG in Fig. 2a we present the spatial distribution of the vertical electrical discipline generated by a vertical electrical dipole positioned at z = 60 nm above the graphene, assuming the utilized magnetic discipline of 1.3 T, and v = 12 THz (λ0 = 25 μm). The fringes of the alternative polarities propagating away from the dipole area clearly point out the polaritonic wavelength (λpl = 4.3 μm) being a lot smaller than the free-space wavelength, λ0 = 25 μm, indicated by the horizontal black arrow (the corresponding frequency, v = 12 THz, is marked by the black level within the purple dispersion curve and by the horizontal dashed line in Fig. 2c).

The properties of TM GPP in MCNGs strongly rely upon the utilized magnetic discipline. In Fig. 2d we plot the true a part of the z-component of electrical discipline of TM GPP, excited by the dipole, as a operate of the gap alongside graphene, x, at two completely different values of the magnetic discipline: B = 1.3 T and B = 1 T. These discipline profiles illustrate that TM GPP wavelength will increase with B (λpl is sort of 5 occasions bigger for B = 1.3 T than for B = 1 T), the latter tendency of λpl(B) being additional confirmed by the purple curve in Fig. 2b. The specific dependence λpl(B) will be estimated from the dispersion relation, proven in Fig. 2c for a number of values of the magnetic discipline inside the first band. Away from the band-bending, the GPP wavelength will be written as (see Supplementary Notice 3)

$$lambda _{pl} = frac{{lambda _0}}{{{mathrm{q}}_{pl}}} approx frac{{2W_n}}{{pi left( {nu ^2 – nu _n^2} proper)}}$$

(1)

with (W_n = frac{{sigma _{uni}}}{pi }frac{{E_c^2}}{{h^2nu _n}}) being the spectral weight11 and (sigma _{uni} = frac{{e^2}}{{4hbar }}) the common optical conductivity, thus non-linearly scaling with B.

Other than the significance of FOM for purposes requiring propagating polaritons, it may be additionally vital that polaritons have massive lifetimes. The latter will be calculated as (tau _{life} = L_{pl}/v_{gr}), the place vgf is the group velocity, given by (v_{gr} = 2pi dnu /dk_{pl}). From Eq. (1) we are able to acquire (v_{gr} approx W_n/sqrt {W_nk_{pl} + pi ^2nu _n^2}). Thus, at a frequency 12 THz and for a magnetic discipline B = 1.3 T, for which Lpl ≈ 2 μm (see Fig. 2c), okpl ≈ 1.5 μm−1 (see Fig. 2d), and vgr = 106 m/s, we are able to estimate τlife ≈ 2 ps. This worth is comparable with that of long-lived phonon/exciton polaritons in vdW crystal slabs and 2D layered supplies19,20,33.

In robust distinction to the TM modes, the TE GPP in MCNG are far much less confined, as proven by the magnetic discipline snapshot in Fig. 3a (the place we use a vertical magnetic dipole supply positioned at z = 5 μm above the graphene on the frequency of v = 11.2 THz). The excitation of TE GPP in MCNG is confirmed by the diffraction shadow34 seen in Fig. 3a. It seems due to the damaging interference of the excited TE GPP in MCNG and the dipole radiation. That is because of the comparable wavelengths of free-scape radiation and TE GPPs. At a ample distance from the supply the TE plasmon is clearly distinguishable from the free-space fields. For a extra quantitative evaluation of the TE plasmons launched by the dipole, in Fig. 3d we plot the sector profile alongside graphene for 2 values of the magnetic discipline: B = 1.3 T (purple curve) and B = 1.4 T (blue curve). For comparability, the sector launched by the dipole in free-space with out graphene (with the wavelength λ0 = 26.77 μm) can also be proven by the black curve. In accordance with the sector profiles, the distinction between λ0 and the wavelength of TE GPP in MCNG doesn’t exceed 2%. Moreover, we see that, in distinction to TM modes, the wavelength of TE GPP in MCNG will depend on the utilized magnetic fields, see Fig. 3b (purple curve). However, because the plasmons propagate away from the supply, the section shift between the sector profiles for various magnetic fields turns into appreciable (see the shift between the purple and inexperienced discipline profiles in Fig. 3b). The propagation size of TE GPP in MCNG can also be rather more delicate to the magnetic discipline than their wavelength, as proven in Fig. 3b (in Fig. 3d the purple discipline profile clearly decays a lot faster than the inexperienced one). The latter will be defined by the vertical shift of the dispersion curves of the TE plasmons with the rise of the magnetic discipline, and thus the back-bending area (the place the absorption is elevated), see Fig. 3c. The sensitivity of each the section shift and the propagation size to the magnetic discipline will be doubtlessly used for the purposes of TE GPPs, notably in interferometers.

TE GPPs in MCNG are rather more confined to the graphene sheet than typical TE GPPs. The strongest confinement takes place near the back-bending of the dispersion curve with the worth ~1/qpl(max) the place qpl(max) is the utmost momentum (see Supplementary Notice 3), (q_{pl} = q_{pl}^{({mathrm{max}})} approx 1 + frac{1}{2}left[ {frac{{W_ntau }}{c}} right]^2), reached on the frequencies (nu ^{left( {{mathrm{max}}} proper)} approx nu _n – 2/left( {3pi tau } proper)). On the maximal momentum the confinement is 2 orders of magnitude smaller than in non-magnetized doped graphene16,17,18. At decrease frequencies TE GPP dispersion curve asymptotically tends to the sunshine line, qpl = 1 and thus turns into undistinguishable from the dispersion of the free-space waves.

GPPs in MCNGs will be doubtlessly noticed experimentally utilizing both near-field microscopy of non-structured MCNG or far-field transmission/reflection spectroscopy of periodically patterned MCNG. The near-field microscopy, nonetheless, is extra applicable for TM GPP in MCNG, because the tip of the near-field microscope usually presents a vertically polarized dipole supply, thus higher matching with the TM polarized excitations. Within the subsequent sections we mimic each near-field and far-field experiments.

### Prospects for potential spectral near-field experiments for the characterization of TM GPPs in MCNG

Scattering-type near-field optical microscopy (s-SNOM) makes use of a tip of an atomic drive microscope (AFM), which is illuminated with an exterior infrared laser35,36. The laser beam is concentrated on the apex of the tip, offering the mandatory momentum to launch GPPs in graphene, as illustrated in Fig. 4(a). GPPs emanate from the tip and upon reaching the pattern edge, they’re mirrored again. Because the tip is scanned towards the sting, the back-scattered sign is collected within the detector as a operate of the tip place2,3. Through the use of a broad-band mild supply, the near-field mild scans will be represented as a operate of frequency6. By Fourier remodeling the interferogram as a operate of frequency, the dispersion relation of GPPs will be retrieved.

In our numerical mannequin we characterize the illuminated AFM tip by a vertical dipole supply. As has been proven in37, absolutely the worth of the vertical part of electrical discipline taken at a long way under the dipole, |Ez|, can reproduce qualitatively the sign scattered by tip. Due to this fact, when modeling, we calculate |Ez| under the dipole, as a operate of the dipole place (x) and frequency (v), composing a near-field hyperspectral picture, |Ez(x,v)|.

An instance of the near-field hyperspectral picture of the one layer of MCNG encapsulated between two flakes of hBN (the thicknesses of the highest and backside hBN layers are 5 nm and 10 nm, respectively) is proven in Fig. 4b. For simplicity, in these simulations we take into account a free standing hBN-encapsulated CNG. However, in Supplementary Notice 4 we present (see Supplementary Fig. 5) {that a} 10 nm thick hBN movie is sufficient to make the affect of a substrate negligible. We carry out simulations on the frequencies of the primary frequency band of TM GPP in MCNG: 11.5–22.35 THz.

In Fig. 4b we clearly observe a number of vibrant fringes representing the alternating NF minima and maxima. They seem because of the interference between the GPP launched by the dipole and GPP mirrored by the sting. The perimeter spacing (the gap between neighboring NF minima or maxima) taken removed from the sting equals roughly to the half of the GPP wavelength, λpl/2. As v will increase the fringes develop into thinner and their inter-spacing decreases. The latter lower is in step with the dependence (lambda _{pl} propto left( {nu ^2 – nu _n^2} proper)^{ – 1}), see Eq. (1), the place for the primary band vn = v1 = 11.5THz. Moreover, the amplitude of the fringes decays with the rise of v, which will be attributed to the rise of MCNG losses (i.e., progress of Re(α)) notably, when approaching the limiting frequency, ({tilde {nu}} _1 = 22.35) THz.

In Fig. 4c we plot the Fourier rework of the near-field hyperspectral picture represented in Fig. 4b. The brilliant most seen within the coloration plot completely matches the GPP dispersion curve (the dashed inexperienced curve), assuming that the momenta of the GPPs are doubled, 2qpl(v). The latter is in step with the λpl/2 distance between the interference fringes in Fig. 4b. With our evaluation we conclude that the GPPs in hBN-encapsulated MCNG needs to be observable in s-SNOM experiments for life like parameters of graphene, even at reasonable exterior magnetic fields.

### Prospects for far-field spectroscopy experiments

For the coupling with GPPs from the far-field, the graphene will be structured into ribbons38,39,40,41. Such structuring permits one to keep away from the momentum mismatch between the GPPs and free-space waves. Relying upon the parameters of the grating, the excited GPPs current both “quantized” Fabry–Perot modes contained in the ribbons, or Bragg-resonances manifesting themselves a dip/peak within the transmission/reflection spectrum. In our simulations for the excitation of each TM and TE GPP in MCNGs we take into account a periodic array of micro-ribbons made in both free-standing or hBN-encapsulated graphene, as illustrated in Fig. 5a.

In Fig. 5b we present the absorption spectra, A(v), of the ribbon arrays (of various ribbon widths, W, and a hard and fast interval, L = 0.22 μm) illuminated by a normally-incident monochromatic aircraft wave. The latter is polarized alongside the x-course, thus matching with the polarization of TM GPPs. The stable and dashed curves in Fig. 5b correspond to the absorption by the hBN-encapsulated and free-standing MCNG ribbon arrays, respectively. In steady MCNG (stable black curve in Fig. 5b), the absorption spectra current just one most, matching with the interband transition frequency v1 = 11.5 THz. In distinction, the absorption spectrum of the MCNG ribbon array, (each for hBN-encapsulated and free-standing graphene), has a set of resonant maxima. For example, at W = 3L/4 the absorption spectra (stable blue curve) has one robust resonance at 12.5 THz and set of a lot weaker ones at greater frequencies. With reducing W, the resonances blueshift away from v1 dropping of their magnitudes. Equally to doped graphene ribbon arrays, the emergence of the absorption maxima will be defined by Fabry–Perot resonances in a single ribbon, forming whereas the GPPs replicate backwards and forwards from the ribbon edges. Within the resonance, the GPP refractive index, (q_{pl} = frac{{lambda _0}}{{lambda _{pl}}}), ought to fulfill42

$$q_{pl} = frac{{lambda _0left( {m + 3/4} proper)}}{{2W}},$$

(2)

the place m is an excellent quantity. Notice that the modes with the odd integers have antisymmetric distribution of the vertical electrical discipline with respect to the ribbon axis and don’t couple to normally-incident waves. For all W, the absorption resonances within the spectrum of hBN-encapsulated graphene ribbon array seem at decrease frequencies in comparison with those within the free-standing array. This may be defined by the stronger confinement of GPPs in encapsulated graphene than within the free-standing one. The robust confinement of TM GPPs makes them wonderful candidates for sensing the atmosphere43. In truth, modifications within the atmosphere will be detected through the ensuing frequency shifts of the plasmonic resonances within the far-field spectrum. The absorption resonances are corroborated by the dispersion curves within the backside panel of Fig. 5b (GPPs within the encapsulated graphene have bigger momenta). The vertical dotted traces connecting the absorption maxima (Fig. 5b, high panel) with the GPP dispersion curves (Fig. 5b, backside panel) mark the frequency positions of the peaks. The frequencies of the peaks match properly with those given by the straightforward Fabry–Perot 0th-order resonances in Eq.(2). Substituting Eq.(1) in Eq.(2) we are able to discover that the resonance frequencies are proportional to (sqrt {B/W}), which implies that they’re tunable by each magnetic discipline and the ribbon width (see Supplementary Notice 5).

To resonantly excite the TE GPPs, we now illuminate the hBN-encapsulated graphene ribbon arrays by a normally-incident monochromatic aircraft wave with electrical discipline pointing alongside the y course. Resulting from a lot smaller momenta of TE GPPs, the array interval have to be greater than two orders of magnitude bigger than the one thought-about for excitation of TM plasmons. In Fig. 5c we current A(v), as earlier than, for various W and for the interval L = 28 μm (see analogous simulations for smaller leisure occasions in Supplementary Notice 6). Every coloration curve demonstrates two robust resonant maxima. The broader resonances are associated to the absorption most on the interband transition frequency. Certainly, the place of the broader peaks coincides with the resonance within the steady MCNG (black curve). The looks of the second (slim) peak in every curve may be very completely different from the resonance within the case of the TM polarization. First, the slim peak seems at decrease frequency in comparison with the interband transition. That is clearly associated to the spectral area of the TE GPPs (positioned under the interband transition, as proven within the backside panel of Fig. 5c). Second, the frequencies of the slim resonance are virtually impartial upon the ribbon width. The inset in Fig. 5c displaying the zoomed-in frequency area of the slim resonances demonstrates the minor modifications within the magnitude of the height with change of W. The minor sensitivity of the frequency place of the slim resonance to the ribbon width is expounded to a distinct nature of the resonance. As a result of the momentum of the TE GPPs may be very near the sunshine line, they’ll simply leak out of the ribbon whereas reflecting from the sides, and subsequently can not construct the Fabry–Perot resonances in particular person ribbons. As a substitute, the TE GPPs within the grating represent the Bloch modes and result in the “collective” Bragg resonances. On this case the resonance situation for the TE GPP refractive index is linked to the grating’s interval, L, (relatively than to the ribbon width): (q_{pl} approx sin theta + mfrac{lambda }{L},) the place m = ±1,… is the diffraction order and θ is the angle of incidence (the dependence of the TE Bragg resonances upon each angle of incidence and frequency is illustrated in additional element within the Supplementary Notice 7).

Because of the weak confinement, TE GPPs are much less delicate to the dielectric atmosphere of graphene. Certainly, within the inset of Fig. 5c the absorption spectra by the free-standing array (dashed curves) improve of their amplitudes and barely shift from these by the hBN-encapsulated ribbon array (stable curves).

We thus exhibit that MCNG micro-ribbon arrays are a strong system for controlling the coupling between mild and each TM and TE GPP modes. This allows enhancement and tailoring of THz absorption by altering ribbon width and interval or utilized magnetic fields (in Supplementary Notice 5 we additionally illustrate the tunability of the absorption spectra of the MCNG ribbon arrays by a slight variation of the utilized magnetic discipline).