Anderson Localized Plasmon in Graphene with Random Tensile‐Strain Distribution

Abstract Anderson localization, the unusual phenomenon discovered in a disordered medium, describes the phase transition from the extended to localized state. Owing to the interference in multiple elastic scattering, this concept is firstly demonstrated in an electron system, then to photon and matter waves. However, Anderson localization has not been observed for polaritonic waves with its unique features of strong field confinement and tunability. Here, Anderson localization of plasmon polaritons is experimentally reported in a flat graphene sheet simultaneously with homogenous charge carrier and random tensile‐strain distributions. By selectively choosing different disordered levels, the transition from quasi‐expansion to weak localization, and finally Anderson localization are observed. Relying on the infrared nanoimaging technique, the spatial dependence of the localization is further studied, and finally the transition window from weak to Anderson localization of graphene plasmon polaritons is identified with the aid of the scaling theory. The experimental approach paves a new way to study Anderson localization in other polaritonic systems such as phonon, exciton, magnon polaritons, etc.


Optical and Raman characterization of graphene flakes
Our microcrystals of graphene are mechanically exfoliated from bulk graphite samples and then transferred to 285-nm thick SiO 2 /Si substrate. The fabricated single-layered graphene can be easily distinguished from multi-layered samples through optical microscopy due to its low optical contrast.
In Fig.S1, we show the optical images of graphene flakes with different disordered level. No matter how the strains distribute, the monolayer graphene shows similar optical contrast. The size of our graphene flakes is dozens of square microns.
In order to guarantee the high-quality of single-layered graphene, we conduct the Raman measurement on every fabricated samples. In Fig.S2, we show the representative Raman spectrum collected in central area of graphene flake. There are four distinctive Raman responses in our monolayer graphene: (1) The ratio between intensity of 2D and G band is about two; (2) The bandwidth of Raman band is narrow (FWHM ~28 cm -1 for 2D peak and FWHM ~9 cm -1 for G peak); (3) The defect-related D band at ~1350 cm -1 is absent; (4) The 2D band is great fitted by single Lorentz function. All these Raman properties show indisputable signs of our single-layered graphene without defects [1]. It is worth noting that graphene is easily contaminated under ambient atmosphere and the freshly fabricated samples are optimal candidate for the near-field measurements.

Determination of graphene edge through Raman Spectrum
The so-called D band is related to the breathing mode of sp 2 rings and requires a defect to be activated [2]. In our mechanically exfoliated graphene flakes, the D peak is usually unobservable due to few structure defects. However, the D band is seen at graphene edges (Fig.S3a), which act as onedimensional defects allowing elastic backscattering of electrons. Based on that phenomenon, we can determine the position of graphene edges through Raman line-mapping along the vertical direction of edge [2]. When the incident laser spot moves from substrate to graphene flake inside, the I (G) and I (2D) should monotonously increases the reach a saturation value in graphene center. Meanwhile, the ratio between I (2D) and I (G) remains two during the Raman mapping due to the single-layered feature. Differently, I (D) increases, reaches a maximum at edge area and then decreases. The maximum value is achieved when the incident laser spot crosses the edge. As shown in Fig.S3b, we fit the change of I (D) with Gauss method and I (G)/I (2D) with Boltzmann function. We define graphene edge where the maximum I (D) is, when I (G) and I (2D) are simultaneously half the saturation value.
The position of graphene edge is shown in Fig.1c-h as red dashed lines.

The separation of mechanical strains from charge doping in graphene
As reported [3], there are two factors which can induce the shift of graphene Raman peaks: mechanical strains (compressive or tensile) and charge doping from substrate (Hole-doping generally on the silicon dioxide substrate). We can separate the mechanical strain from charge doping through Raman Characterization. The details of measurement can be found in the Ref. [3]. Here, we just give a brief introduction. As shown in Fig.S4, the red and blue solid lines represent the Raman peak shift under different stains or different density of holes (electrons), respectively. The green dot (denoted O) is obtained from a freestanding graphene which is not affected by strain or charge doping. The coordinate is (1581.6±0.2cm -1 2676.9±0.7cm -1 ). We separate the strain of graphene from the charge doping using decomposition method as follows. The vector OP=ae T +be H , e T and e H are unit vectors for tensile strain and hole doping effects. Hence, the distribution of tensile strain in graphene sample can be extracted from the coefficient and the hole-doping effects can be extracted from the T G -2D space is divided into four quadrants including Q1, Q2, Q3 and Q4. Q4 is attributed to tensile strains and Q1 is attributed to compressive strains. The Q2 and Q3 are meaningless because that both hole and electron doping should induce the increase of G. Our mechanically exfoliated graphene samples on the silicon dioxide substrate are mostly dominated by tensile strains.

Extraction of characteristic length scale of the disorder
In order to characterize the spatial components of random disorder, we conduct the twodimensional fast Fourier transform (FFT) of strain maps, as shown in Fig. S5. When the Anderson transition occurs, the Fourier spectrum shows more and more widespread patterns, indicating that the disordered level of graphene system increases. The discrete distribution in the momentum space indicates that the disorder induced by strains is random, not periodic. In order to extract the characteristic length scale of disorders, we measure the size of every strain spot (green dashed circles in Fig.S6) and the distance between two neighboring spots (blue dashed lines in Fig. S6). For the weak and strong scattering condition, the average size of the strain spots is 263±101.4 nm and 304±170.3 nm, respectively. Given the fact that the plasmonic wavelength in our case is ~220 nm, the length scale of random disorders is in the same order of magnitude with mode wavelength.
Meanwhile, the average distance between two neighboring spots is 667±227.3 nm for weak scattering and 674±100 nm for strong scattering, which are both in the same order of magnitude with plasmonic wavelength. The characteristic length scale with the same order of magnitude compared with plasmonic wavelength is essential for the observation of Anderson localization.

Determination of surface roughness of graphene flakes
The rough surface induces the enhanced plasmonic amplitude due to the formation of H S ' [4,5]. This uneven distribution of plasmon cause an adverse impact when we study the Anderson localization. In conventional metal thin film, it is quite challenging to build an atomically flat surface with the limitation of preparation technologies. Hence, it is hardly distinguish the Anderson localization from the enhanced amplitude just caused by protuberances on surface [6].
The height of mechanically exfoliated graphene on substrate is less than 1.0 nm, which provides a relatively flat surface [7]. As shown in Fig the disorder level is, the roughness of graphene surface remains less than 0.2nm, which is very important for the study of Anderson localized plasmons.

Nano-infrared images of graphene plasmon with different localized state
We conduct optical nano-imaging of graphene plasmon with different localized status under different incident frequencies. The near-field amplitude is normalized by Si standard reference sample as following [8]: Here, and are the forth-order demodulated harmonics of the near-field amplitude probed for graphene flake and Si standard sample, respectively. In Fig.S8, we display the near-field amplitude of expanded plasmon. We observe longer plasmonic wavelength for lower incident frequency, according with well-known dispersion relation of graphene plasmon [8] (Fig.3d). The nanoimages of weak localized and Anderson localized plasmons are shown in Fig.S9 and Fig.S10, respectively. The near-field images contain intrinsic properties of graphene plasmon, including wavelength ( Fig.3 in main text), damping rate (Fig.S11), and dispersion (Fig.3g). All these intrinsic properties are independent of localized status of graphene plasmon, as mentioned in the main text.
All nano-infrared images are collected at ambient atmosphere.

Fitting method of the damping rate of graphene plasmon
Based on the reported fitting method [9], we extract the damping rate of graphene plasmon with different localized states. The fitting details can be found in the ref. [8] and we just show a brief description here. We conduct average line scans of the complex perpendicular to the graphene edge and subtract the background through two dimensional Fast Fourier Transform (2D-FFT). The could be fitted well with the formula: The first section is tip-launched plasmon and the second one is edge-launched. The edge-launched graphene plasmon is very weak and can be neglected in our condition [8,10]. So, the Re (q) and Im (q) can be extracted. The p can be calculated from Im(q)/Re(q) as a dimensionless figure of merit of propagation damping, as shown in Fig.S11. When fitting the plasmonic line-profiles, we exclude the first dominate fringe due to its complicated interference (edge-mode and sheet-mode) and tipsubstrate coupling effect [9]. The extracted damping rates increase when the Anderson transition occurs.

Calculation of mean free paths of graphene plasmons
We calculated the mean free paths of graphene plasmon (l*) based on scaling theory [11] as following: The extracted l* in the weak localization (Tab.S1) is larger than Anderson localization (Tab.S2).
Detailed explanation is provided in the main text. Localization length as a function of allows us to directly determine the transition window for Anderson localization of graphene plasmons (Fig.4b).

Theoretical calculation of dispersion of graphene plasmons
In order to get the dispersion, we calculate the complex reflectivity r p of graphene/SiO 2 structure. When we treat the system as a semi-infinite substrate, the reflectivity can be expressed as [12]: where 0 is the vacuum permittivity, 1 and 2 are relative permittivity of air and SiO 2 . is incident frequency and q is the plasmonic wavevector. The and represent the z-components of the wavevector of the incident and the transmitted plane-waves, respectively. The conductivity of graphene ( ) is derived with random-phase-approximation (RPA). Extracted maximum imaginary part of r p is shown as background color in Fig.3g.

The theory of Anderson localization in a 2D metal system
Theoretically, the distribution of polaritonic field in two-dimensional disordered metal system can be described by a set of Kirchhoff equations, whose detailed derivation can be found in Ref [6,[13][14][15][16]. Here, we represent the brief introduction as followings: In a semi-continuous film composed with metal grains randomly distributed on a dielectric substrate, the local field can be represented as [6]: where is the applied field and is the potential of the fluctuating field inside the film. The current density at the point r is given by Ohm s law: By the combination of current conservation law, the equation can be described as: In order to calculate the local electric field in the system, we need to discretize the above equation on disordered lattice [6]. The partial differential equation is reduced to a set of Kirchhoff equations.
The Kirchhoff equations are characterized by the Hamiltonian H [referred as the Kirchhoff for the Anderson transition problems [13], with both on-and off-diagonal correlated disorder as [14]: where and are the real-part and imaginary-part of the Kirchhoff Hamiltonian, associated with lattice potential. . E is the localized electric field, which is proportional to the applied field . The potential can be expressed in terms of the eigenfunctions ( ) of H as [6]:

Then
In the zeroth approximation, the local potential is solved as It strongly fluctuates and the averaged field intensity can be described as: where, is the dimensionless density of states for the H and a is structure parameter. Therefore, the field distribution in the two-dimensional disordered metal film can be described as a set of the KH eigenfunctions localized within , with the enhanced amplitude of [6].
Although the calculation is conducted in the 2D metal film, the main conclusions including Anderson localized field with one specific localization length and its enhanced amplitude are appropriate for graphene. Figure S1. Optical micrograph of our graphene flakes. (a-c) Flakes with low, medium and high disordered level, respectively. The single-layered graphene can be easily discerned through optical contrast. This optical contrast measurement cannot determine the different strains distribution in graphene S Figure S2. The representative Raman spectrum collected at central area of graphene flakes. All Raman responses including narrow spectral band of 2D and G peak, twofold ratio between 2D and G peak, great Lorentz fitting of 2D peak (shown in inset), and absence of D peak indicate that our mechanically exfoliated graphene is smooth and single-layered.         . We exclude the first fringes for fitting due to the complicated interference between edge mode and sheet mode and tip-substrate coupling effect.
---- Table S1. The scaling-theory extracted mean free paths (l*) of graphene plasmons and scattering strength (k p l*) in weak localization condition under different incident frequencies.
---- A   Table S2. The scaling-theory extracted mean free paths (l*) of graphene plasmons and scattering strength (k p l*) in Anderson localization condition under different incident frequencies.