Ultra-long wavelength Dirac plasmons in graphene capacitors

Graphene is a valuable 2D platform for plasmonics as illustrated in recent THz and mid-infrared optics experiments. These high-energy plasmons however, couple to the dielectric surface modes giving rise to hybrid plasmon-polariton excitations. Ultra-long-wavelengthes address the low energy end of the plasmon spectrum, in the GHz-THz electronic domain, where intrinsic graphene Dirac plasmons are essentially decoupled from their environment. However experiments are elusive due to the damping by ohmic losses at low frequencies. We demonstrate here a plasma resonance capacitor (PRC) using hexagonal boron-nitride (hBN) encapsulated graphene at cryogenic temperatures in the near ballistic regime. We report on a $100\;\mathrm{\mu m}$ quarter-wave plasmon mode, at $40\;\mathrm{GHz}$, with a quality factor $Q\simeq2$. The accuracy of the resonant technique yields a precise determination of the electronic compressibility and kinetic inductance, allowing to assess residual deviations from intrinsic Dirac plasmonics. Our capacitor GHz experiment constitutes a first step toward the demonstration of plasma resonance transistors for microwave detection in the sub-THz domain for wireless communications and sensing. It also paves the way to the realization of doping modulated superlattices where plasmon propagation is controlled by Klein tunneling.

Two dimensional electron systems (2DES) sustain both single-particle and collective lowenergy excitations, the latter being called plasmons. In graphene their interplay is controlled by the electron density n which rules the kinetic energy, interactions and damping. Free 2DES plasmons are dispersive with ω p ∝ √ q and a velocity v p diverging in the long wavelength limit [1]. However, plasmons are screened in the long wavelength limit , where d is the gate-2DES distance, and acquire an energy-independent velocity v p = v F (e 2 d/πǫ)(k F /hv F ), where v F and k F = √ πn are the Fermi velocity and wave vector, and ǫ ≃ 3ǫ 0 the hBN permittivity. Plasmons have been mostly investigated in the THz and mid-infrared (MIR) optics domains where the damping length α −1 > ∼ 10 µm was found to widely exceed the wavelength λ p < ∼ 1 µm [3][4][5][6]. In this high-energy range, plasmons couple to the dielectric surface modes forming plasmon-polaritons states [5]. Ultra-longwavelength plasmons (λ p ∼ 100 µm) belong to the GHz domain and do not suffer from this hybridization. They have been observed in high-mobility 2DESs in Ref.
[2], but remain elusive in graphene in spite of the interest of manipulating collective chiral Dirac fermion excitations. A constitutive element of plasmon propagation is the kinetic inductance which has been measured in Ref. [7]. High-mobility graphene [8] offers the opportunity to investigate GHz plasmonics with a transport approach. Motivations are manyfold; they include the demonstration of GHz plasma resonance devices [9], the investigation of plasmonic crystals in doping-modulated structures [10], that of interactions including viscous fluid effects at high temperature [11,12], or the coupling with hyperbolic phonon polaritons of the hBN dielectric [5,13,14] at high bias. In this letter we demonstrate a plasma resonance capacitor (PRC in Fig.1-a) where 100-µm Dirac plasmons propagate along a hBN-graphene-hBNmetal strip line of length L ≃ 24 µm and aspect ratio L/W = 3 ( Fig.1-c). We report on the fundamental mode, which is a quarter-wave resonance at f 0 = v p /4L ∼ 40 GHz, where we achieve a quality factor Q ≃ 2 at low temperature (T = 10 K) corresponding to a damping length α −1 = 2QL/π ∼ 30 µm.
Our graphene resonator is a T-shape hBN-encapsulated single-layer graphene sample covered by a top metallization serving both as radio-frequency (RF) port and DC gate (see methods, Fig.1-c and Figs.3). We have measured a series of eight devices, using both exfoliated and CVD graphene, but results presented below focus on sample PRC6 (L × W = 24×8 µm) which has the highest mobility and largest quality factors. The device is embedded in a coplanar waveguide and its RF gate-source admittance Y (f, n, T ) is measured in the range f = 0-40 GHz, n = 0-2×10 12 cm −2 and T = 10-300 K using standard vector-networkanalyzer (VNA) techniques (see methods and Ref. [15]). The strip line access, which has a broader width, constitutes the device source. It is equipped with low-resistance edge contacts ( Fig.1-c). The source end of the strip is an impedance short, securing a plasmon node, the open end of the strip being an antinode so that the PRC sustains odd harmonics of the We describe propagation by a distributed-line model ( Fig.1-b). The line capacitance C is the series addition of the insulator capacitance C ins = ǫW/d and the quantum capacitance . The large top-gate capacitance enhances the C Q contribution and gives access to a capacitance spectroscopy which is used below to determine the Dirac-point position. The line inductance is dominated by the kinetic contribution, which at low temperature writes [7] (1) From these expressions one recovers the plasmon velocity v p = 1/ √ L K C and the characteristic impedance Z ∞ = L K /C of the plasmonic strip line. The line resistance r = R/L accounts for plasmon losses which in principle include ohmic, viscous and dielectric contributions. The latter are negligible in our low-frequency range and viscous losses are minimized by using a plane wave geometry. Upon increasing the carrier density the plasmonic response, with L K ∝ 1/k F (single layer graphene), takes over single particle diffusive response, with r ∝ 1/k 2 F (constant mobility), allowing for low-damping plasmon propagation. The crossover occurs whenever the total strip resistance R < ∼ Z ∞ , or equivalently the plasmon damping length α −1 > ∼ L. Dealing with a resonant device, the PRC admittance, Y = jCω × tanh L jCω(r + jL K ω) / jCω(r + jL K ω) [16], is conveniently cast into the compact form : where x = ωL √ CL K is the reduced frequency. The capacitive character of the resonator is encoded in the low-frequency response Y /L = jCω. Eq.
(2) features a resonance behavior with a fundamental frequency at x 0 = π/2, an admittance peak amplitude ℜ(Y )(f 0 ) = Q/Z ∞ = 2/R and a width ∆f ≃ f 0 /Q. It includes higher harmonics, constituting the frequency comb f k = (2k + 1)f 0 with Q k ∼ (2k + 1)Q. The latter are discarded here due to the finite 40 GHz bandwidth of our cryogenic setup.
The rise of a plasma resonance in increasing electron density is illustrated in Figs.1-(dg). Admittance spectra, measured at T = 30 K, have been obtained after deembedding a contact resistance R c = 43 Ohms. Such a low value was obtained by back-gating the capacitor access region. Data are presented together with their fit with Eq.
At the lowest density the channel resistance takes over the kinetic inductance. The plasmon resonance f 0 ∼ 14 GHz is overdamped with Q ∼ 0.4 ( Fig.1-d). The admittance spectrum is reminiscent of the evanescent wave response reported in Refs. [16][17][18]. Fingerprints of a resonant behavior become perceptible for Q ≃ 0.5 with a shallow minimum of ℑ(Y ) at f ≃ f 0 ( Fig.1-e). A genuine resonance develops at n > ∼ 1 10 12 cm −2 ( Fig.1-f) and culminates at f 0 ≃ 38 GHz with Q = 1.69 for n = 2 10 12 cm −2 ( Fig.1-g). We deduce a plasmon velocity v p = 4Lf 0 = 3.6 10 6 m/s in agreement with the above estimate based on geometry, and a plasmon damping length α −1 = (2Q/π)L ≃ 26 µm. Plasmon losses correspond to a channel resistance R = 40 Ohms ( Fig.1-g) approaching the characteristic impedance Z ∞ = 35 Ohms, which is a suitable measuring condition. The same procedure has been reproduced by recording 400 complex spectra covering the density and temperature ranges, n = −0.1-2.25 10 12 cm −2 and T = 10-300 K, where we find a quality factor peaking to Q > ∼ 2 at 10 K (inset of Fig.2-a). Figs.2-(a,b) summarize the doping dependence of resonator parameters v p (n), Z ∞ (n) obtained using the same fitting procedure. From these values we calculate L K (n), C(n) (30 K-data in Fig.2-c) and find a good agreement with theoretical capacitance formula (black dashed line), including the dip of C Q (n) at neutrality. Still we observe a deviation of L K (n) with theory (blue dashed line) which exceeds our experimental uncertainties. This deviation is significant thanks to the increased accuracy of the resonant capacitor technique in the determination of C and L K when compared to non-resonant techniques [7]. It is too large ( > ∼ 100 pH) to be explained by geometrical inductance effects (µ 0 W ∼ 10 pH) or systematic errors in the deembedding procedure (±20 pH depending on frequency). From the channel resistance we estimate the conductivity and the mean free path l mf p (n, T ) which is plotted in (Fig.2-d) for typical temperatures. Mobility saturates near µ = 250000 cm 2 V −1 s −1 at low temperature (dashed black line in Fig.2-d). At high temperature we find a mean free path plateau l mf p (T ) ∼ 0.7 × 300/T µm, in agreement with theoretical estimates the acoustic phonon limited resistivity [19].
To summarize, we have shown that the plasma resonance capacitor principle works and provides an extensive characterization of equilibrium and transport graphene parameters including the compressibility C Q (n, T ), the kinetic inductance L K (n, T ) and mean-free-path l mf p (n, T ). The plasmon velocity matches expectations for the screened case. The weak doping dependencies of the plasmon velocity and characteristic impedance in Figs.2-a,b reflect theoretical expectations for massless graphene where v p , Z ∞ ∝ n − 1 4 as opposed to v p ∝ n − 1 2 for massive 2DESs such as bilayer graphene. The deviation from theory, observed in the inductance, gives rise to a saturation of the plasmon velocity v p < ∼ 4v F and characteristic impedance Z ∞ > ∼ 35 Ohms. A tentative explanation for this discrepancy is an additive mass contribution from the dilute 2DES in the back-gated silicon substrate, which loads graphene Dirac plasmons according to its capacitive coupling to graphene electrons, and eventually restricts electron mobility. Such a residual substrate coupling can easily be avoided by substituting the silicon back gate with a metallic bottom gate following Refs. [14,20].
In conclusion, we have demonstrated a graphene plasma resonance capacitor with a resonant frequency f 0 ≃ 40 GHz and a quality factor Q ≃ 2. Quality factors remain smaller than the Q ∼ 130 reported in the ultrashort wavelength (λ p = 0.1-0.2 µm) MIR domain of Ref. [5]. However the GHz damping length (28 µm) is comparable to the MIR value (10 µm in Ref. [5]), which is promising for applications. We have measured the doping dependence of plasmon velocity, line capacitance and kinetic inductance in good agreement with theory beside a small shift of the kinetic inductance. Our experiment paves the way to the realization of active plasma resonance transistors working in the 0.1 → 1 THz domain, above the natural cutoff ∼ 0.1 THz of conventional graphene field-effect transistors [21,22]. Building on this first demonstration performed at cryogenic temperature, a room temperature variant can be envisioned by scaling down the sample size and the plasmon wave length by a factor 10 to accommodate the phonon-limited mean free path of 0.7 µm at 300 K. Such an achievement would in particular allow realizing plasma resonance transistors in the 600 GHz frequency domain, highly desirable for high-resolution air-craft RADARs operating in the mm-range. The long plasmonic channels are compatible with the incorporation of bottom gate arrays to engineer doping modulation profiles and investigate the propagation of Dirac plasmons in bipolar superlattices.

Methods
Micromechanical exfoliation provides a pristine monolayer graphene, which is subsequently encapsulated between two layers of hexagonal boron nitride (hBN) in order to achieve a high electronic mobility (phonon limited at room temperature). The encapsulation is performed by means of a dry pick-up technique using in a polyvinyl alcohol (PVA) and polymethylmethacrylate (PMMA) stamp on a polydimethylsiloxane (PDMS) support [8]. A first top hBN flake (10 to 30 nm thick) is transferred from its substrate to the PMMA.
Then the graphene is picked up by the first hBN flake thanks to the strong van der Waals interactions and deposited on a second bottom hBN flake, which was previously exfoliated onto a high resistivity silicon substrate. The transfer process is carried out using a custom made alignment system with a heating plate operating between 30 and 130 • C. Finally, the PVA and PMMA polymers are dissolved in hot water (95 • C) and acetone, respectively.
The encapsulated graphene samples are generally hundreds of µm 2 in size and need to be patterned into a passive circuit. After characterization by Raman spectroscopy (fig. 3c) and atomic force microscopy (AFM, fig. 3a), we used e-beam nanolithography to define a T-shape capacitor (fig. 3b). The length of the channel, according to its mobility, defines the quarter wave plasma resonance and needs to be long enough to obtain a signal in the 0 − 40 GHz bandwidth. The etching of the hBN-graphene-hBN sandwich is performed using a mixture of CHF 3 /O 2 plasma etching through a temporary 40 nm thick aluminium mask. A chromium/gold (5/200 nm) edge contact is then deposited on the source edge of the T-shape heterostructures (using a comb design to enhance the contact length and reduce the contact resistance). In order to passivate the other edges of the heterostructures and to avoid any source-gate leakage current in the PRC, we oxidized 2 nm of aluminum followed by 10 nm of Al 2 O 3 by atomic layer deposition (ALD). Finally, a chromium/gold gate electrode is deposited on the top and the PRC is embedded into a coplanar waveguide (CPW). Similarly, thruline and dummy reference structures were defined on the same chip for de-embedding (see below).
High frequency admittance measurements were carried out in a Janis cryogenic probe station in the temperature range T = 10-300 K. The two-port scattering parameters S ij of the capacitor were measured using an Anritsu MS4644B vector network analyzer (VNA) in the 0 − 40 GHz range. Bias tees were used to decouple the DC gate voltages from the GHz probe signal. A short-open-load-reciprocal (SOLR) protocol was used to calibrate the wave propagation until the probe tips. We then measured S parameters of a symmetric thruline reference structure, calculated its ABCD (cascade) matrix A thru and took the inverse of the square root of this matrix A −1/2 thru . The wave propagation in the coplanar access of the PRC can now be de-embedded from its ABCD matrix: thru . The same procedure was carried out on a dummy reference structure which has the same contact and gate geometry as the PRC but does not contain encapsulated graphene. Finally, the ABCD matrices of the PRC and the dummy structure were converted to admittance (Y ) parameters and the Y matrix of the dummy structure was subtracted from that of the PRC in order to de-embed remaining stray capacitances. The Y i,j parameters should now all be the same (except for a minus sign in front of the off-diagonal elements), due to the symmetry of our two-port network. The admittance data shown in this article correspond to one of the off-diagonal elements −Y 12 , which was further de-embedded from the contact resistance:

Acknowledgments
The research leading to these results have received partial funding from the the European Union "Horizon 2020" research and innovation programme under grant agreement No. 785219 "Graphene Core", and from the ANR-14-CE08-018-05 "GoBN". * Electronic address: bernard.placais@lpa.ens.fr The dashed line is the theoretical expectation of Eq.(1) for the low temperature limit. Inset, quality factor : density dependence at 30 K (black), temperature dependence at 2 10 12 cm 2 /Vs (red). b) density dependence of the plasmonic line characteristic impedance (T = 30 K); dashed line is the theoretical prediction. c) density dependence of the capacitance (red dots) and kinetic inductivity (blue dots) L K W (T = 30 K). Black and blue dashed lines are the low-temperature theoretical expectations for the capacitance and inductivity L K W . d) density dependence of the electronic mean-free-path l mf p for a representative set of measuring temperatures. l mf p saturates at low temperature to a constant mobility regime l imp = µhk F /e with µ ≃ 250000 cm 2 V −1 s −1 (black dashed line). At high temperature, we retrieve the acoustic-phonon-limited scattering length plateau l ph ∼ 0.7 × (300/T ) µm.