A theoretical design of evanescent wave biosensors based on gate-controlled graphene surface plasmon resonance

A surface plasmon resonance (SPR) sensor based on gate-controlled periodic graphene ribbons array is reported. Different from the conventional methods by monitoring reflectivity variations with respect to incident angle or wavelength, this approach measures the change in SPR curve against the variation of graphene chemical potential (via dynamically tuning the gate voltage) at both fixed incident angle and wavelength without the need of rotating mirror, tunable filter or spectrometer for angular or wavelength interrogation. Theoretical calculations show that the sensitivities are 36,401.1 mV/RIU, 40,676.5 mV/RIU, 40,918.2 mV/RIU, and 41,160 mV/RIU for analyte refractive index (RI) equal to 1.33, 1.34, 1.35 and 1.36; their figure of merit (1/RIU) are 21.84, 24, 23.74 and 23.69, respectively. Significantly, the enhancement in the non-uniform local field due to the subwavelength graphene ribbon resonator can facilitate the detection in redistribution of protein monolayers modeled as dielectric bricks.

where ψ n (x) is the n th space harmonic given as where the Fourier basis functions ({ψ n }) form a complete set: Parameter δ mn is the Kronecker delta function, and k xn = k o √ ε inc sin θ + n2π/d x ; parameter ε inc is the relative dielectric constant in the incident region, for instance, Si-substrate in this research; parameters m and n are integers running from negative-to positive-infinities; symbol " † " stands for complex conjugate.
The tangential electric field component along the x-axis is written as: Where the Fourier amplitudes V n (x) and I n (z), satisfying transmission line equations, are the voltage and current amplitudes of the n th space harmonic, respectively. Their general solutions can be written as follows [12].
V n (z) = v + n exp(+ik zn z) + v − n exp(−ik zn z), with the propagation constant along the z-axis k zn = k 2 o ε s − k 2 xn and wave admittance Y n = ωε o ε s /k zn in a medium with relative dielectric constant ε s . Parameters v + n and v − n usually are termed as the amplitudes of forward-and backward-propagating waves of the n th diffraction order, respectively. Furthermore, the voltage and current amplitudes of each space harmonic can be collected and put into vectors given below.
where Y c is a diagonal matrix with the n th entry equal to Y n . The n th element in column vectors V (z), I(z), v + , and v − are V n (z), I n (z), v + n and v − n , respectively. The n th element in the diagonal matrix exp(±iK z z) is exp(±ik zn z).

Input-output relation of a uniform dielectric layer
Having the termination condition defined at the output of a uniform dielectric layer given below: where Y l , a full matrix in general, can be regarded as the load admittance matrix of the transmission lines between voltage (V ) and current (I) waves at z = t. Therefore, the relationship between the unknown amplitudes a n and b n (now are expressed in terms of vectors a and b) can be obtained through matrix operation, we obtain b = Γa. with With the reflection matrix Γ, the input admittance, which is defined by I(0) = Y in V (0), can be obtained as: where I is the identity matrix. Additionally, the voltage transfer matrix, which is defined as V (z = t) = T v (t; 0)V (0), is written below.

Input-output relation of a periodic graphene ribbons array
The discontinuity of tangential magnetic fields at the interface (periodic graphene ribbons array at z = 0) between the lower and upper dielectric layers produces the conduction current flowing on the graphene surface, yielding the equation given below.
where Y g is a full matrix with each entry related to the graphene conductivity and structure dimensions [12]; the n th element in column vectors I(0 ± ) and V (0) respectively are I n (0 ± ) and V n (0); the symbol ± represents the upper-and lower-mediums sandwiching the graphene ribbons array. The matrix Y g in Equation 14 establishes a relationship between voltage-and current-wave amplitudes across the periodic graphene ribbons layer; it is the so-called admittance matrix in microwave engineering [39]. Having the input-output relation of the graphene ribbons array in Equation 14 and input-output relation of a uniform dielectric layer described previously, the scattering characteristic of the overall structure in Fig.1(b) can be calculated through the building block approach by cascading each of the transfer matrix (input-output relation of each layer including graphene ribbons array, graphene sheet and uniform dielectric medium). The detail mathematical formulation concerning the Fourier Modal Method can be found in the literature [39][40][41][42]. As was reported [43,44], traditional approach of rigorous coupledwave analysis experienced a poor convergence in particular under the resonance of surface plasmonpolariton wave. In this research, the local basis functions inherently satisfying the electromagnetic edge conditions in both the graphene ribbons and slits were employed [12] to effectively improve the numerical convergence and eliminate the Gibbs phenomenon at the discontinuities. Notably, the mathematical formulation employed in this research is rigorous; nevertheless, a finite truncation in the Fourier series is inevitable due to numerical computation. The mathematical formulation and computational results regarding the scattering analysis of a periodic graphene ribbons array have been successfully confirmed in the report. More specifically, the sin-based local basis function vanishing at its both ends for any harmonic order is used to expand E x over the graphene strip, which is given as where 1 ; the graphene strip belongs to the region of [x 1 , x 2 ]; index n is ranging from 1 to N g . On the other hand, in the slit region, we have the singular basis functions with singularities at its two edges, which are commonly used to approximate the current parallel to the edges in a micro-strip line. They are expressed as follows: where x 2 ]. Parameter γ n = 2 for n = 0 and γ n = 1 for n = 0; index n runs from 0 to N s − 1. Notably, the denominator in s n (x) approximates √ w s √ ρ at the strip edge where x = w g + ρ (x (s) 1 = w g and x (s) 2 = d x ), which confirms the electric-field edge condition described previously. Now, the admittance matrix in Equation 14 can be rewritten as: Here, the sub-matrix [[G]] and [[S]] have the size N tot -by-N g and N tot -by-N s , respectively. Specifically, N tot , N g and N s satisfy the relationship: N tot = N g +N s . Moreover, the ratio between N g and N s equals to the ratio of w g to w s [12], therefore, we have N g = round[N tot · w g /(w g + w s )] and N s = N tot − N g ; the operator round[.] rounds a real number towards the nearest integer. In doing so, we have a square matrix where β ± mn = k xm ± nπ/w s and α ± mn = k xm ± nπ/w g ; function J o (·) is the zero order Bessel function of the first kind; function sinc(x) is the unnormalized sinc function defined as sinc(x) = sin(x)/x. Parameter n is the index of the local basis function.

Transmission-line network representation
The transmission-line network representation of the 2D model in Fig. 1B is shown in Fig. S1. Let's begin with the upper semi-infinite transmission line denoted as Air . Because of un-termination, the intrinsic admittance matrix Y a is regarded as the load admittance of the line denoted as Analyte ; Y A is the intrinsic admittance of the Analyte layer. Through the input-output relation given in the previous section, the input admittance looking into the analyte layer can be obtained as Y (A) in . Moreover, the admittance matrix (Y (RAs) g ) represents the input-output relation of the graphene ribbons array, shown in Equation (14). The input admittance looking into the graphene ribbons array equals to the shunt of Y in . This shunt admittance matrix is the load of the line termed as Silica . Again, the input-output relation in the silica layer can be applied to determine its input admittance matrix denoted as Y (Silica) in . Furthermore, the load admittance matrix of the line denoted as Si-Substrate equals to the shunt admittance of the graphene sheet and the input admittance looking into the silica layer equal to Y . The reflection of a incident wave is due to the mismatch between characteristic admittance matrix Y Si and the aforementioned load one. The reflection matrix is written below.
This completes the scattering analysis of the overall structure. Figure S1: Transmission-line network representation of the structure shown in Fig. 1(b) of the 2D model of biosensor structure; Y (RAs) g (a full matrix) is the admittance matrix of the graphene ribbons array and Y (S) g (a diagonal matrix) is that of the graphene-sheet electrode.

Convergence check for the scattering analysis
In the Fourier Modal Method, the electric and magnetic fields both are expressed in terms of the Fourier series. However, due to the finite computation resources, we have to truncate the infinite to a finite one. Consequently, the index of Fourier components under truncation ranges from -N to +N with total truncated order equal to 2N + 1. The convergence check for the absorption due to both graphene-sheet served as electrode (yellow curve) and periodic ribbons array (red curve), and reflectance (blue color) against the number of truncated order were carried out. Specifically, the convergence test is performed at the resonance condition of the SPPs. The excellent numerical convergence on the reflectivity and absorption, shown in Fig. S2, confirms the computation accuracy of the codes developed in this research. Figure S2: Scattering efficiency including the absorption efficiencies in both graphene sheet and graphene ribbons array and reflectance versus number of truncated order (2N + 1); the truncated space harmonic index is ranging from -N to +N : The structure dimensions of the biosensor are the same in Fig. 2. The analyte medium is modeled as a uniform dielectric layer with n A = 1.4393 + i0.00040824 and t A = 8nm; the operating wavelength is 4µm; µ c = 0.626eV and τ = 0.5ps.

S2: Contour of constant reflectivity against analyte RI and gate voltage
The analyte under test is modeled as a uniform dielectric layer having refractive index denoted as n A with thickness t A = 8nm. The incident angle and wavelength are 30 o and λ = 4µm, respectively. The structure parameters of the biosensor is the same as in Fig. 2. It is shown in Fig. S3 that the contour with smallest reflectivity (white color) is almost straight in the region where n A is ranging from 1.33 to 1.36. This can explain why the figure of merit are consistent for the four cases with n A = 1.33, n A = 1.34, n A = 1.35, and n A = 1.36. Figure S3: Contour of constant reflectivity against analyte RI (n A ) and applied gate voltage V dc