## Abstract

Control of propagating surface plasmon on a scale beyond the
diffraction limit is important for the development of integrated
nanophotonic circuits and optical information technology. In this
paper, a strain-based modulation mechanism for directional control of
propagating graphene plasmons was proposed. We demonstrated
numerically that the GPs can be directionally controlled by the
implementation of strain on graphene. The topologies of GPs excited by
a *z*-polarized optical emitter in unstrained and
strained graphene were illustrated both in real space and momentum
space. When imposing strain engineering to graphene in different
directions with a different modulus, multi-dimensional control of GPs
in any direction can be realized. The simulated propagation length
ratio *η* of the GPs can reach 3.5 when the
strain with a modulus of 0.20 is applied along or perpendicular to the
zigzag direction of graphene. Besides, the effect of PDMS on GPs was
investigated finally for the experiments to be carried out and we show
that the PDMS does not affect the generation of directional GPs under
strain engineering. Our proposed directional control of GPs not only
has the advantages of wide operating wavelength but does not require
additional coupling mechanisms, which is beneficial to the design of
integrated photonic devices.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

Information transmission and processing in nanoscale is an important subject in scientific research. As an excellent supporter for photon manipulation and integration in nanoscale, surface plasmons (SPs) have made outstanding achievements in nano-photonics circuits [1–3]. And the key to prompting its application is how to modulate the propagation direction of SPs more effectively. The directional control of SPs in noble metal has been studied extensively. By breaking the symmetry of the structure, the unidirectional excitation and modulation of SPs can be realized by using a decorated nanoslit [4–8] or asymmetric double slits [9]. Besides, by using a circularly or elliptically polarized beam [10,11], two interference beams with a particular phase difference [12], or oblique illumination [13,14], the directional SPs can also be excited and controlled by breaking the symmetry of the incident light beams. What’s more, it has been reported that photo-sensitive, temperature-sensitive, magneto-optical or nonlinear substrate can be developed as an external factor for the directional control SPs [15–20]. However, the Ohmic heat effect in metal results in a large loss of plasmons, which makes the design of the nano-photonic devices based on the metal SPs encounter a bottleneck [21,22]. The bottleneck has been broken by the successful preparation of graphene [23]. It has shown that doped graphene can also support the SPs, and the SPs in graphene are mainly in the mid-infrared to terahertz range [24], which broadens the operating wavelength of SPs-based photonic devices. Compared with the traditional metal SPs, graphene plasmons(GPs) have the advantages of high active controllability, extreme localization of electromagnetic field, longer plasmon life, etc. Therefore, the directional control of GPs has received extensive attention.

Similar to SPs in metal, the primary physical mechanism for directional control of propagating GPs is symmetry breaking. But the configurations in metal are not viable owing to the one atomic layer of graphene. The modulation of propagating GPs usually requires a design of composite structures or an additional coupling method. For graphene composite structures, the directional propagating GPs can be excited and controlled by linearly polarized beam [25], circularly or elliptically polarized beam [26], or the interference of two linearly polarized light beams [27] that are incident on the metal-graphene composited structure. What’s more, graphene-based meta-surface has also been designed to control the GPs by tuning the chemical potential of graphene [28–30]. Besides, GPs can also be controlled by changing the properties of the surrounding medium. According to transformation optics, Ashkan et al. [31]designed different microstructures on graphene substrates to modulate the propagation path of GPs through voltage tuning. Soljacic et al. [32]designed graphene-boron nitride heterostructure to realize full-angle negative refraction by tuning the carrier mobility of graphene or changing the thickness of boron nitride, thus realizing active and coordinated modulation of GPs. Shimura etal [33]also achieved unidirectional launching of propagating GPs only by designing asymmetric nano-ridges within the graphene itself. For additional coupling methods, the GPs can be successfully modulated by integrating graphene and waveguide [34] or Bragg reflector [35]together. Besides, magneto-optic [36] or nonlinear coupling [37] has also been found to be able to control the propagating direction of GPs. However, for composite structures, once the structure is given, the propagation direction of GPs is also determined, which makes it difficult to achieve the multi-dimensional or dynamic control of GPs. While for the method based on additional coupling, the complex coupling means lower efficiency in general. Moreover, these researches are mainly limited to light, electricity or magnetism modulation methods. Due to the good flexibility and other excellent mechanical properties of graphene, strain has become an effective way to tune graphene, the advantage of using strain to graphene is that it offers high repeatability [38–40]. In addition, the optical properties of graphene show anisotropy under strain engineering [41], which enables graphene to achieve more novel plasmon-based devices, such as the design of new flexible optoelectronic elements. However, in the mid-infrared to terahertz band, the modulation of GPs by strain engineering is rarely involved. Therefore, the in-depth study of a new modulation mechanism-strain engineering will provide a new platform for the development of GPs based devices.

In this paper, a strain-based modulation mechanism for directional control of propagating GPs was proposed. Considering graphene supported on a flexible substrate, we demonstrate that the propagating GPs in a particular direction can be excited by an emitter when strain engineering was imposed on the configuration. Without destroying the graphene lattice structure, we illustrate that multi-dimensional control of GPs can be realized by applying strain with different modulus in different directions of graphene. Furthermore, when the strain with a modulus of 0.20 is applied along or perpendicular to the zigzag direction of graphene, the direction ratio $\eta$ of the GPs propagation length can reach 3.5. Moreover, our results show that effective modulation can be achieved at any wavelength from mid-infrared to terahertz in support of GPs, and the modulation frequency can reach 300MHz, that is the proposed mechanism has the advantages of wide working wavelength and high modulation frequency. Besides, To provide more detailed guidance for experimental implementation, PDMS was taken as a flexible substrate of graphene, then how the PDMS affect the propagation properties of GPs were investigated extensively, and the results indicate that the generation of directional GPs was not influenced by the substrate. More importantly, our proposed configuration is simple in structure and does not require additional coupling mechanisms, which is beneficial to the design of integrated photonic devices. Our research results can provide detailed guidance for experimental implementation as well as theoretical support in micro-nano scale information transmission, biomedicine, wearable electronics, robot "skin" and other aspects, and further promote and broaden the practical application of the GPs based micro-nano photonics devices.

## 2. In-plane optical properties of graphene under strain engineering and details of FDTD simulation

Figure 1 displays our studied system and the axis orientation. A single-layer graphene was placed on a flexible substrate and the $x$-axis was chosen along the zigzag direction of the graphene. When the system is uniformly stretched along a prescribed direction in the range of elastic deformation, the strain tensor reads [42]

Here $\tau$ is the electron-phonon relaxation time estimated from $\tau =\mu E_F/ev_F^2$, $v_F\approx c/300$ is the Fermi velocity, $E_F$ is the Fermi level and $\mu$ is the DC mobility of graphene. In this study, the Fermi level is set as 0.4 eV, the DC mobility is 1000 cm$^2$/Vs and $T$ is 300 K.

To begin with, the in-plane conductivity $\sigma _{xx}$ and $\sigma _{yy}$ of graphene with or without strain engineering were calculated to investigate the effect of strain on graphene. Figure 2(a) shows the real and imaginary part of graphene conductivity with a strain modulus of $\kappa$=0.20 for the case of the tension T perpendicular to the zigzag direction (i.e. $\theta =90^{\circ }$). At the same time, the conductivity for the unstrained graphene was calculated for comparison, as the black line described in Fig. 2(a). One can find that the conductivity decreases along the stretch direction, and it increases in the vertical direction compared to the conductivity of unstrained graphene. Since the properties of GPs depend mainly on the imaginary part of the in-plane conductivity, then how the imaginary part of the conductivity changes with angle $\theta$ was depicted in Fig. 2(c), without loss of generality, the strain modulus was taken as $\kappa$ = 0.20 at a wavelength of 11.6 $\mu$m. The imaginary part of $\sigma _{xx}$ (red dashed line) and $\sigma _{yy}$ (blue dashed line) were compared with the case of $\kappa$ = 0 (black line). One can see that the in-plane conductivity is anisotropic under strain engineering, both the Im($\sigma _{xx}$) and Im($\sigma _{yy}$) of graphene periodically alters with the change of angle $\theta$. In addition, the Im($\sigma _{xx}$) and Im($\sigma _{yy}$) change in the opposite direction with the same amplitude, and the greatest increase or decrease in conductivity can be seen when the graphene was stretched along or perpendicular to the zigzag direction. By taking advantage of the anisotropy conductivity produced by strain engineering, anisotropy propagating GPs can be excited and controlled.

In this paper, the finite difference time domain (FDTD) method is used to simulate the properties of GPs, where the permittivity of graphene needs to be concerned. The permittivity of monolayer graphene can be expressed as $\epsilon _{g}(\omega )=1+i\widetilde {\sigma }(\omega )/(\omega \epsilon _0d)$, where d is the thickness of single-layer graphene sheet. Therefore, similar to Figs. 2(a) and 2(c), the in-plane permittivity of graphene with or without strain engineering was calculated in Figs. 2(b) and 2(d). By comparison, we find that the change of the dielectric constant is opposite to that of conductivity. The numerical calculations of GPs were performed by using the anisotropic permittivity from Figs. 2(b) and 2(d), the dimensions of the simulation region along $x$, $y$ and $z$ axis were set to 20 $\mu$m, 20 $\mu$m and 1 $\mu$m respectively. To avoid scattering caused by the boundary of the graphene sheet, the sheet extends out of the simulation region in the $x$-$y$ plane to mimic infinite graphene, and the thickness of graphene was taken as $d$ = 1 nm as a reasonable value. The point electric dipole was placed 10 nm above the center of the graphene layer, and the Perfect Matching Layers were chosen for all the sides of the simulation region. What’s more, the dimensions of the mesh covering the graphene layer and their near vicinity were set to 5 nm, 5 nm and 1 nm along the $x$, $y$ and $z$-axis respectively.

## 3. GPs on unstrained and strained graphene

Next, we turn to the excitation and modulation of propagating GPs supported by the unstrained and strained graphene. By using the FDTD method discussed above, the topologies of GPs excited by a $z$-polarized optical emitter in unstrained and strained graphene at a wavelength of 11.6 $\mu$m were presented in Fig. 3. Firstly, the electric field distribution $E_z$ and intensity $|E|$ on the unstrained ($\kappa$=0) graphene was shown in Figs. 3(a) and 3(c), we can see that the propagating GPs spread outward isotropically in $x$-$y$ plane with the same property. Then, considering the graphene was uniformly stretched along the $y$-axis (i.e.$\theta =90^{\circ }$), the electric field $E_z$ and $|E|$ on the strained graphene with a modulus $\kappa$=0.20 were simulated. As shown in Figs. 3(b) and 3(d), the GPs propagate anisotropically and most energy propagates along the $x$-axis on the surface. Applying tension to graphene can change the topology of propagating GPs from an isotropic circle to an anisotropic ellipse. The propagation distance of GPs will be significantly prolonged in the direction perpendicular to the tension. What’s more, It is obvious that by applying different strain modulus to graphene in different directions, the GPs propagating in the strained graphene can be controlled in multiple dimensions.

After that, to further understand the propagation behavior of GPs on the same graphene with or without strain engineering, it is necessary to calculate the dispersion relation of GPs analytically. Without considering the dissipation of energy in graphene, the dispersion relation of the supported GPs can be expressed as [47]

As a function of the conductivity, it can be concluded that different propagating GPs topologies can be generated with different $\sigma _{xx}$ and $\sigma _{yy}$ of graphene, that is, different $k$ vectors of GPs can be obtained. For unstrained graphene, taking $\text {Im}(\sigma _{xx})=\text {Im}(\sigma _{yy})=2.85\times 10^{-4}S$ at $\lambda _0 = 11.6 \mu$m, the calculated $k$-surface of GPs is plotted as the insert red curve shown in Fig. 4(a), where $k_{x} =k_{y}=18.7 k_{0}$, $\lambda _{x} = \lambda _{y}$ = 620 nm, the wavelength of GPs is compressed by 18.7 times, and there is no preferred direction for GPs. When the graphene is streched along the $y$-axis with $\kappa$ = 0.20, the isotropic conductivity of graphene changed to $\text {Im}(\sigma _{xx})=4.43\times 10^{-4}S$ and $\text {Im}(\sigma _{yy})=1.27\times 10^{-4}S$. The corresponding $k$-surface of GPs for the strained graphene turns into an ellipse, as the insert red curve shown in Fig. 4(b). Here the calculated wavevector $k$ and wavelength of GPs towards $x$-axis changed to $k_{x} = 11.8 k_{0}$, $\lambda _{x}$ = 983nm, while $k_{y}= 41.6 k_{0}$, $\lambda _{y}$ = 279 nm. Besides, performing Fourier transform on the electric field distribution Ez simulated in Fig. 3 through $\tilde {E}(k_x,k_y)=\int \int E(x,y)e^{-i(k_xx+k_yy)}dxdy$, the dispersion of GPs can be obtained, it can be found that the contour map coincides very well with the analytic results.

In the following, the normalized momenta and propagation length of the GPs modes in unstrained and strained graphene were calculated in Fig. 5 for comparison. The electric field of the electromagnetic eigenmode in graphene has the form $\textbf {E}(x,y)\text {exp}(i\beta k_0z)$, where $\textbf {E}(x,y)$ denotes the electric field distribution of GPs, $k_0 = \omega /c$ is the free-space wave vector. $\beta$ is the modal wave vector of GPs, here the real part of $\beta$ refer to the normalized momentum of the GPs mode, which can be written as $k/k_0=\text {Re}(\beta )$, while the imaginary part of $\beta$ provides the propagation length $L$ through $L/\lambda _0=1/[2\pi \cdot \text {Im}(\beta )]$, $L/\lambda _0$ is the called normalized propagation length of plasmons. Then eigensolutions of $\textbf {E}(x,y)$ and $\beta$ have been performed by using the MODE Solutions developed by Lumerical. Take the graphene was stretched along the $y$-axis (i.e. $\theta =90^{\circ }$) by a strain modulus of 0.20 as an example, the normalized momentum $k/k_0$ and normalized propagation length $L/\lambda _0$ in $x$ and $y$-axis as a function of wavelengths were obtained by solving the eigenmode. As shown in Figs. 5(a) and 5(b), the red and blue curves represent for GPs propagating along $x$ and $y$ direction respectively, while the black line represents the unstrained graphene. It can be seen that the normalized wave vector of GPs increases and the normalized propagation length decreases along the direction of tension, while in the direction perpendicularly direction, the normalized wave vector decreases, and the normalized propagation length increases. In addition, the change is consistent in trend for different wavelengths of the emitter. Specifically, the directional control of the propagating GPs in our proposed mechanism is mainly due to the anisotropy in-plane conductivity generated under strain engineering. Compared Fig. 5 with Fig. 2, we can intuitively see that an increase in conductivity leads to a decrease in the normalized wave vector and an increase in the propagation distance of GPs. For example, when the graphene sheet was stretched along the $y$-axis ( i.e. $\theta =90^{\circ }$ ), maximized Im$\sigma _{xx}$ and minimized Im$\sigma _{yy}$ obtained, resulting in the biggest increase of Lx and decrease of Ly. On the other hand, the zigzag direction of graphene can also be determined by studying the propagation property of GPs by stretching graphene in different directions.

Furthermore, by using nonretarded approximation and electrostatic limit, one can also obtain the analytical dispersion relation of GPs by the formula [24]

where the open circle lines in Figs. 5(a) and 5(b) represent the results of theoretical calculation, which are in good agreement with the eigensolution results. Next, how the propagating GPs tuned by different stretching directions $\theta$ or various strain modulus $\kappa$ were theoretically calculated in Figs. 5(c) and 5(d). Take the GPs propagating along the $x$-axis as an example, Fig. 5(c) describes the normalized momenta as a function of different angle $\theta$ at a wavelength of 11.6 $\mu$m, the blue and red line represent the graphene stretched along $y$-axis by a strain modulus of 0.10 and 0.20 respectively and $\kappa =0$ for the unstrained case. Besides, the normalized propagation distance as a function of angle $\theta$ at a wavelength of 11.6 $\mu$m was present in Fig. 5(d). One can find that the normalized momenta and propagation distance of GPs change periodically with the angle $\theta$. Therefore, Fig. 5 render, with the same excitation wavelength of emitter near the graphene, the momenta and propagation distance of GPs can be changed by different modulus or stretching directions, that is to say, the properties of propagated GPs can be effectively tuned by strain in multiple dimensions.To better describe the modulation of the directional GPs by strain stretching, the stretching direction $\theta$ dependent directional ratio $\eta$ of GPs in strained graphene was simulated in 2d polar coordinates, as shown in Fig. 6, the directional ratio was defined as $\eta =L_x/L_y$. Firstly, suppose a strain with a modulus of $\kappa$ = 0.20 was imposed to graphene, the ratio $\eta$ of GPs propagation distance in strained graphene as a function of $\theta$ was illustrated in Fig. 6(a), the red line and open circle represent the directional ratio $\eta$ at two different incident wavelengths of $\lambda$ = 11.6 $\mu$m and 4.6 $\mu$m respectively, while the black line represents the ratio $\eta$ of graphene in unstrained graphene. Two perfectly coincident curves can be observed when the graphene is stretched under the same condition. In other words, the propagating GPs modulated by strain engineering are independent of the excitation wavelengths of an emitter, namely the GPs can be controlled in a wide spectrum. Besides, maximized $\eta$ can obtain when the graphene was stretching along the $y$-axis. What’s more, according to the relation between $\sigma _{xx}$ and $\sigma _{yy}$ illustrated in Fig. 2(c), we can deduce that when the graphene was stretching along the zigzag direction (i.e. $\theta =0^{\circ }$), the propagation distance will increase in the $y$-axis and decreases in the $x$-axis with the same amplitude. Here the ratio $\eta$ should be defined as $\eta =L_x/L_y$, the max ratio $\eta$ will obtain at $\theta =0^{\circ }$, and the maximum value is still as high as 3.5. Then, chose an emitter with a wavelength of 11.6 $\mu$m and impose different modulus of strain on graphene, the propagation distance ratio $\eta$ of GPS as a function of $\theta$ for three different $\kappa$ was studied in Fig. 6(b). It can be found that the propagation ratio $\eta$ of GPs changes under different strain directions and modulus. Therefore, the propagation direction and propagation ratio of GPs can be effectively controlled by the direction and modulus of strain imposed on, so as to achieve multi-dimensional control of GPs.

It should be noted that all the above calculations are based on self-standing graphene, and it is well known that the GPs depend a lot on the dielectric environment due to their evanescent property. Take PDMS as a flexible substrate of graphene, where the optical properties of PDMS were taken from a Drude-Lorentz oscillator model with 15 oscillators, which is given by

Last but not least, the modulation rate of strain is also estimated, which defined as $d\kappa$ / dt = $v_{g} / L_{0}$. In the formula, $\kappa$ = $(L-L_{0}) / L_{0}$ represents the deformation ratio between the elongation $L-L_{0}$ and the original length $L_{0}$ of graphene, $v_{g}$ has the form of $v_{g}$=$\sqrt {\frac {dY}{3(1-2\rho )s}}$ in graphene, where $d$ = 0.35 nm, Y = 0.95 TPa , $\rho$ = 0.165 are the thickness, Young’s modulus and Poisson’s ratio of graphene, s =0.76 mg/$\mathrm {m^{2}}$ denotes the surface density of graphene sheet [50], so the estimated value of $v_{g}$ is approximately 15 km/s. Therefore, several hundred megahertz modulation rate can be obtained by controlling the GPs through strain engineering.(e.g. $L_{0}$ = 50 $\mathrm {\mu }$m, $d\kappa$ / dt = 300 MHz)

## 4. Conclusion

In conclusion, by taking advantage of the unique mechanical properties of graphene, we demonstrate numerically that the GPs can be directionally controlled by imposing strain in graphene. Different topologies of GPs excited by a $z$-polarized optical emitter were observed in unstrained and strained graphene due to the anisotropy conductivity generated in graphene under strain engineering. How the directivity of the GPs changes with different excitation wavelengths, strain modulus or strain directions were calculated theoretically to further reveal the parameter dependence of the proposed strain-based GPs modulation mechanism. Our simulation results indicated that not only effective modulation can be achieved at a wide spectrum in support of GPs, but multi-dimensional control of GPs in any direction can be realized. Moreover, when the strain with a modulus of 0.20 was implemented along or perpendicular to the zigzag direction of graphene, a maximized propagation ratio $\eta$ as high as 3.5 was obtained. Considering the specific experiments to be carried out, the PDMS-dependent momentum, propagation distance and direction ratio were investigated extensively, the results prove that the PDMS has no influence on the generation of directional GPs under strain engineering. Therefore, our theoretical results provide detailed guidance for experimental implementation and a new path for the application of integrated photonic devices.

## Funding

Research Project of Tianjin Municipal Education Commission (2020KJ088); The Science and Technology Development Fund of Tianjin Education Commission for Higher Education (2020ZD03); Tianjin Technical Expert Project under grant (20YDTPJC01770); Science and Technology Planning Project of Jin Nan District Tianjin under grant (20190111).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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