Self-biased Reconfigurable Graphene Stacks for Terahertz Plasmonics

The gate-controllable complex conductivity of graphene offers unprecedented opportunities for reconfigurable plasmonics at THz and mid-IR frequencies. However, the requirement of a gating electrode close to graphene and the single `control knob' that this approach offers for graphene conductivity limits the practical implementation and performance of graphene-controllable plasmonic devices. Herein, we report on graphene stacks composed of two or more graphene monolayers separated by electrically thin dielectrics and present a simple and rigorous theoretical framework for their characterization. In a first implementation, two graphene layers gate each other, thereby behaving as a controllable single equivalent layer but without any additional gating structure. Second, we show that adding an additional gate --a third graphene layer or an external gate-- allows independent control of the complex conductivity of each layer within the stack and hence provides enhanced control on the stack equivalent complex conductivity. The proposed concepts are first theoretically studied and then demonstrated experimentally via a detailed procedure allowing extraction of the parameters of each layer independently and for arbitrary pre-doping. These results are believed to be instrumental to the development of THz and mid-IR plasmonic devices with enhanced performance and reconfiguration capabilities.


Introduction
The strong graphene-light interaction has led to the rapid development of graphene plasmonics, 1,2 which benefit from the unique electrical properties of graphene in the terahertz and mid-infrared frequency bands. 3 The characterization of single-layer graphene structures has already been performed at microwaves, 4-7 terahertz 7-9 and optics, 3,10 and some promising applications such as modulators, [11][12][13][14][15][16] plasmonic waveguides 17,18 and Faraday rotators 19 have been developed. How-ever, the simple implementation and performance of these devices might be hindered by the presence of a gating electrode located close to graphene and the relatively weak control that this approach offers over the conductivity of graphene. 11,20 These limitations can be overcome using graphene stacks, structures composed of two or more isolated graphene layers separated by electrically thin dielectrics, which lead to increased conductivity and may provide novel reconfiguration strategies.
Optical plasmons and quantum transport in such structures have already been studied theoretically, [21][22][23] whereas some experimental studies have focused on the Anderson localization of Dirac electrons in one of the graphene layers at DC due to the screening effect. [24][25][26] Furthermore, the Coulomb drag of massless fermions has been experimentally measured, 27 while both intra-and inter-layer phenomena in structures surrounded by various dielectrics and their influence in the supported in-phase and out-of-phase plasmons have also been considered. 28,29 Potential applications of graphene stacks include modulators, 12 enhanced metasurfaces, 30 antennas, 31 or plasmonic parallel-plate waveguides, 18,32 among many others. Experimentally, graphene stacks have recently been applied to the development of vertical FET transistors. 33,34 In addition, the response of unbiased graphene stacks and devices at infrared frequencies has also been investigated. 20 In this context, the work herein demonstrates the concept of reconfigurable graphene stacks for THz plasmonics and presents a simple and rigorous theoretical framework for their characterization. Although the graphene monolayers within the stack are not close enough to couple through quantum effects, 26,34 their extremely small separation in terms of wavelength allows the stack to behave as a single equivalent layer of increased conductivity. The enhanced tunable capabilities of the proposed structure are experimentally demonstrated in different scenarios, including the mutual gating between the graphene layers and the independent control of each sheet through two different biasing gates. The measurement of the total stack conductivity σ S for various combinations of gate voltages permits not only the extraction of the different parameters that define each of the layers but also the determination of the effective gate capacitance of the surrounding dielectrics. The proposed formulation also allows the design of structures with the desired tunable conductivity be-havior. Our results show that reconfigurable graphene stacks boost the available range of complex conductivity values provided by single-layer structures, thus facilitating the easy implementation of THz and mid-infrared plasmonic devices with enhanced reconfiguration capabilities.

Results
Operation principle of reconfigurable graphene stacks.
The structure under analysis is shown in Figure 1a, where incident and transmitted beams required for THz time-domain measurements have been artistically rendered. The sample consists of two chemically vapor deposited (CVD) graphene monolayers separated by an electrically thin (d ∼ 80 nm) polymethylmethacrylate (PMMA) layer. Metal contacts, added using optical lithography followed by the evaporation of 50 nm of gold, have been included for biasing purposes. The sample is measured in the 0.5-2.5 THz frequency range using time-domain spectroscopy. The complex conductivity of the graphene stack is then retrieved using a dedicated formulation. 8,35,36 Details regarding the fabrication, measurement, and stack conductivity extraction process are provided in Methods. Because the dielectric separation layer between the graphene layers is extremely thin in terms of wavelengths (d/λ 0 ≪ 10 −3 ), 20 an incoming electromagnetic wave observes an stack where σ top and σ bot are the complex conductivity of the top and bottom graphene layers, respectively. Figure 1c plots the frequency-dependent real and imaginary parts of the extracted conductivity σ S for several DC biasing voltages applied between the graphene layers. In the low terahertz band, the real component of the sample conductivity does not vary with frequency, whereas the imaginary part, which facilitates the propagation of surface plasmons in this frequency band, 37 increases with frequency following a standard Drude model. Figure 2 shows the measured reconfiguration capabilities of the fabricated graphene stack at f = 1.5 THz in different scenarios. In the first case, depicted in Figure 2a, a gate voltage V DC is applied between the two graphene layers. The results clearly confirm the tunability of σ S and the ability of the stack to self-bias. The extracted chemical potentials corresponding to each graphene sheet, computed using the procedure detailed in Methods combined with the measured stack conductivity and further validated by Raman scattering measurements, 38 are depicted versus the applied gate voltage in Figure 3. Both graphene layers are p-doped, and they present slightly different Fermi levels. This difference can be due to the defects induced in the graphene layers during growth or transfer 39 and to the influence of the surrounding dielectrics. [40][41][42] In addition to the different morphology of the surrounding dielectrics, contamination during processing 43  . The examples illustrated in Figure 2 demonstrate the large potential of graphene stacks for THz plasmonics, as it is possible to control the behavior of the different layers within a unique stack to achieve the complex conductivity required for a desired application.
Static and dynamic characteristics.
The graphene stack is theoretically analyzed in two different but interdependent steps. First, the carrier density on each graphene layer is determined as a function of the applied gate voltages using an electrostatic approach. Second, this information is employed to compute the frequencydependent conductivity σ S of the stack. In a general case of two graphene sheets biased by different gate voltages V 1 and V 2 (see inset of Figure where −q is the electron charge, n p s is the total carrier density in the p graphene layer (with p={bottom,top}), n p s i corresponds to the pre-doping of the p sheet, and C p ox is the capacitance of the p dielectric layer. Once the carrier densities are known, the Fermi level of each graphene layer and the conductivity σ S , which determines the electromagnetic behavior of the whole stack, The simulated results, plotted in Figures 2a-3 together with measured data, confirm the accuracy of both the extraction procedure and the proposed model to characterize reconfigurable graphene stacks. The measured hysteresis behavior of the sample conductivity, which is mainly related to the charges trapped in the dielectrics surrounding the graphene layers, 45 is not considered in the model. In addition, the extracted values permits estimating a modulation speed of 6.2kHz for the stack (see Supplementary Note 1), similar to the one found in single-layer graphene structures. 12,46 This framework can be further employed to forecast the reconfiguration capabilities of a wide variety of graphene stacks, allowing the design of structures with desired plasmonic properties and tunable behavior.

Surface plasmons supported by graphene stacks.
The measured characteristics of the fabricated stack allows to simulate the frequency-dependent properties of the surface plasmons supported by the device. Specifically, the structure supports two different modes 32,47 (see Methods): an even TM and an odd quasi-TEM. The former can easily be seen as a usual TM plasmon propagating along a single-layer graphene sheet with a conductivity equal to the stack conductivity σ S . Figure 5a illustrates the characteristics of this mode, which presents lower field confinement and reduced tunability compared to plasmons in single-layer graphene. This behavior arises due to the increased imaginary component of the stack conductivity, which in turn reduces the kinetic inductance associated to this mode. The later is a perturbation of the TEM mode found in standard parallel-plate waveguides with two perfect electric conductors. Figure 5b confirms that this mode presents remarkable characteristics in terms of field confinement and tunability, clearly outperforming single-layer graphene structures. Note that the high losses associated to CVD graphene, 48 which prevent the propagation of the supported plasmons along many wavelengths, can be significantly mitigated employing high-quality graphene in the stack. 49 Supplementary Note 2 includes a comparison of the characteristics of plasmons supported by the stack and a single-layer graphene structure, and further discusses the influence of losses in both cases.

Discussion
This theoretical and experimental study of graphene stacks has demonstrated that the available range of complex conductivities in graphene stacks can be significantly boosted by two different approaches i) mutually biasing the graphene sheets without requiring the presence of any metallic bias, and ii) including a third gate source to control the conductivity of each layer independently.

Fabrication of single-layer and stack graphene structures.
The samples were fabricated using CVD graphene grown on Cu foil and transferred onto the substrate using the standard wet transfer method. 53 Supplementary Figure  The Raman spectra of the graphene employed in our devices is shown in Supplementary Figure 11. The G and 2D band points are located at 1589 and 2682 cm −1 with a full width at half maximum of 18 and 32 cm −1 , respectively. The intensity ratio of the 2D to the G band and of the D to the G band are 5.5 and 0.09. All of these numbers are typical hallmarks of monolayer graphene. 54 The use of spincoated PMMA as a separation layer between the graphene sheets allows the avoidance of problems associated with standard dielectric deposition techniques such as evaporation, sputtering 55 and ALD of oxides, which can induce defects in graphene. This approach is convenient for fabricating graphene stacks, allowing viable biasing schemes without the need of post-processing the graphene. Note that the DC isolation between the two graphene layers of the fabricated stack is not perfect, and some leakage current has been measured. However, it does not hinder the performance of the stack since i) the device does not operate at DC but in the THz band, and ii) graphene field's effect control is preserved as the DC biasing voltage source is able to provide the required bias voltage, hence the required electrical field, even when some leakage current occurs.
In addition, note that monolayer graphene devices have been annealed in a N 2 atmosphere at 200 • C during 4 hours. 56 The annealing aims removing possible graphene contamination by polymer residues and other impurities. 57 However, this process has not been applied to the graphene stack samples because it would remove the PMMA layer which isolates the two graphene sheets.

THz time-domain measurements.
The measurements at terahertz frequencies were performed using a commercial Time Domain The gating was applied using a 4-channel DC voltage source, Agilent N6700B. Only 2 channels were used for the measurements, and each channel was connected to a different gold contact corre-sponding to a graphene layer, whereas they both shared a common ground gold contact. For safety reasons and to prevent damaging the graphene stack, the maximum voltage (taking into account both sources) was limited to ±75 V. The sample was placed on an X-Y linear stage perpendicular to the THz beam, and everything was placed inside a sealed case purged with N 2 to keep a constant atmosphere during the duration of the measurements.

Stack conductivity extraction.
The stack conductivity σ S is extracted from the THz time-domain measurements using standard thin-film characterization techniques. 8,35,36,58 This approach is valid here thanks to the extreme fineness of the stack in terms of wavelength (d/λ 0 ≪ 10 −3 ). An example of the different set of measured pulses employed for the extraction procedure is shown in Supplementary Figure 13.
To keep the higher possible SNR, we have considered only the first transmitted pulse through the sample. Additional transmitted pulses that arise due to the internal reflections of the THz beam within the layers of the sample are clearly identified thanks to their temporal delay and subsequently removed.
The graphene stack is not free-standing but on top of a thick dielectric structure. Consequently, the influence of the dielectrics must be rigorously removed to extract the actual stack conductivity.
This procedure has been performed as follows: i) A pulse is transmitted without the presence of any sample to measure and store the response (including atmosphere and possible impurities) of the sealed cage. ii) A pulse is transmitted through an area of the sample free of graphene, which remains bare. The combination of this measured pulse with the pulse obtained in the previous step allows the extraction of the permittivity, loss tangent and thickness of the dielectrics using standard techniques. 36,58 iii) A pulse is transmitted through the graphene stack sample. Combining this measured pulse with the previous information, it is indeed possible to extract the conductivity of the graphene stack rigorously removing the influence of the dielectrics and surrounding atmosphere. 8,35,36 Graphene stack theory.
The frequency-dependent conductivity σ of a single graphene layer is modeled using the Kubo formalism 59 as where ω is the radian frequency, ε is energy, Γ = 1/(2τ) is a phenomenological electron scattering rate assumed independent of energy, τ is the electron relaxation time, T is temperature, −q is the charge of an electron,h is the reduced Planck's constant, and f d is the Fermi-Dirac distribution defined as being µ c the chemical potential and k B Boltzmann's constant. This model results from the long wavelength limit of the bosonic momentum (k || → 0) and takes into account both intraband and interband contributions of the graphene conductivity as well as a finite temperature.
In addition, the carrier density n s and chemical potential of the graphene layer are related through n s = n s e − n s h = 2sign(µ c ) where n s e and n s h are the electron and hole densities, respectively, ε is energy and v f is the Fermi velocity (∼ 10 8 cm s −1 in graphene).
Let us consider the case of two graphene layers closely located within a stack, as depicted in Supplementary Figure 1. As previously stated, the structure is analyzed first using an electrostatic approach, which determines the carrier density on the graphene layer, and then obtaining the electromagnetic behavior of the stack at THz. Following the superposition principle (see Supplementary Figure 1b), the carrier density on each layer are computed using Eq. (2)-(3). Note that this electrostatic approach approximates both graphene and polysilicon for infinite perfect conductors in order to compute the carrier density on each layer. Consequently, it cannot predict the presence of weak electrostatic fields that may arise due to i) the different DC conductivities that graphene and polysilicon present in practice, and ii) fringing effects at graphene borders. Combining these expressions with Eq. (??) permits the chemical potential on each graphene layer to be determined.
Once these potentials are known, the frequency-dependent complex conductivity of the individual graphene sheets is retrieved using Eq. (??), thus allowing the total graphene stack conductivity to be computed using Eq. (??). Note that in this approach we have neglected i) the influence of the separation layer located between the graphene sheets, which is electrically very small in the THz frequency range, and ii) the possible influence of the quantum capacitance, 60 which may be significant in the case of high permittivity or extremely thin (∼nm) dielectrics but is completely negligible here.
Note that the proposed approach approximates graphene's relaxation time as a constant quantity in each layer, and embeds all variations of graphene conductivity versus the applied bias in the chemical potential, 7 , 61 . 8 However, rigorous approaches indicate that the relaxation time not only depends on the defects in graphene (τ gr ), but also on the thermally excited surface polar phonons that may arise at the interface between graphene and the substrate (τ sb ), and on the frequencydependent electron-phonon coupling (τ e−ph ). 62 These values are related through the Matthiessen's rule 63 by τ −1 = τ −1 gr + τ −1 sb + τ −1 e−ph . In addition, graphene relaxation time and chemical potential are not totally independent. 62 In our particular experiments, the extracted relaxation times are very similar. Since the operation frequency is in the low THz range, well below the graphene optical phonon frequency, 62 we expect that electron-phonon phenomenon does not impact the τ decay mechanism. The similarities among the extracted relaxation times, which correspond to graphene layers surrounded by different substrates, suggests that the graphene/dielectric interface provides a high τ sb thus being graphene impurities and nonidealities (τ gr ) the main mechanism limiting the relaxation time, i.e. τ ≈ τ gr . Other possible effects such as carrier scattering by ionized impurities 64 and the electron-hole puddle effect 65 might also modify the measured relaxation time.

Extracting the characteristics of each graphene layer.
Let us consider a stack composed of two graphene layers, which are biased by two different gate sources as depicted in Supplementary Figure 1 where