1. Introduction

In 1968, Veselago numerically predicted that negative refraction can be obtained using material with simultaneously negative effective permittivity and permeability, so-called negative refractive index (NRI) material [1]. Because of the unavailable NRI material in nature, metamaterial (MM), an artificial microstructure made of periodic resonant sub-wavelength electromagnetic units have awakened great interests due to its unusual electromagnetic properties [2–5], especially the possibility of attaining double negative (DNG) index material by a well-known fishnet MM structure composed of nanohole array embedded through metal-dielectric-metal (MDM) multilayers [6–13]. The unique electromagnetic properties of this MDM fishnet MM strongly depend on the design of the resonance elements rather than the composition [14], and ascribed to the plasmonic waveguide modes stemming from surface plasmon polaritons (SPPs). The negative permeability of the MDM fishnet MM is due to the excitation of displacement current (J_D) loops [13], where J_D is constructed by an internal SPP mode flowing through the inner metal-dielectric interfaces of the structure [15,16]. Notably, it has also been shown that impinging light can couple to different orders of SPP modes through the holes to excite multiple magnetic resonances that could be potentially used to obtain multi-band DNG MMs [12,13].

Multi-band MMs have attracted significant attentions due to their multi-frequency applications. Up to this point, people have demonstrated dual-band [15–20] or multi-band MMs [21–23] to broaden the frequency bandwidth of the negative index. Despite these progressive advancements, the negative index frequency regions in MMs are difficult to tune, which is a considerable challenge to practical applications. To this end, Garcίa-Meca et al. present a tunable double-negative index fishnet MM in the visible spectrum by adjusting geometry parameters of the resonance units [13], however this response is usually fixed at the time of fabrication yielding materials that are essentially passive and operate over a limited bandwidth. Therefore, the active modulations of the negative index MMs by integrating dynamic components into the unit cell design have been widely studied to solve the problem. Such actively tuned negative index has, in particular been studied using Liquid Crystal (LC) based MMs [24–26]. However, the tunability is still somewhat limited and the integration of the required layer structures as well as the electrodes for the LC tuning can be difficult in MDM multilayer structure. Very recently, a ferrite-based MM has been applied to obtain tunable dual-band negative permeability [27], whereas this MM works only in the gigahertz (GHz) regime and its structural geometry is complicated to attain experimentally in the optical region. Dani et al. altered the transmission at two negative index resonances in the near infrared (N-IR) region by photoexcitation of carriers in the amorphous silicon dielectric layer of the MM [28], however its pump intensity is very high, usually of the order of several gigawatts per square centimetre. Cao et al. proposed a tunable MM based on phase change material (PCM) [29], whereas it is a single-band NRI MM; moreover, the photoexcitation and temporal evolution of the transient states in the structure were not explored. To date, it is still a formidable challenging to obtain an ultrafast, low-power all optical tunable multi-band DNG MM under low pump intensity.

In this work, we propose a structure possessing a configuration of elliptical nanohole array (ENA) embedded in MDM tri-layers where the dielectric core layer is a prototypical phase change material, Ge₂Sb₂Te₅ [30,31]. We consider the case when the chirp is introduced by displacing the neighbouring elliptical holes towards each other from their central positions. We show that this chirped MM gives rise to a DNG index in the two different optical regions. It is because two internal SPP modes at the inner metal-dielectric interfaces can be enhanced to couple to impinging light, hence providing a dual-band DNG index. Moreover, by introducing Ge₂Sb₂Te₅ into a MDM-ENA as a dielectric layer, it is hypothesized that one will observe a large wavelength shift of dual-band DNG index resonances in the mid-infrared (M-IR) region after switching the Ge₂Sb₂Te₅ between its amorphous and crystalline states. Importantly, a heat model is constructed to investigate the temporal variation of the temperature of Ge₂Sb₂Te₅ layer in the structure. The model shows that the temperature of the amorphous Ge₂Sb₂Te₅ layer can be raised from room temperature to > 433 K (phase transition point of Ge₂Sb₂Te₅) [32,33] in just 0.4 ns under excitation of a low pump light intensity of 7.3 μW/μm². Therefore, our proposed structure can fast and pronouncedly tune the dual-band DNG index with low power in the M-IR region. This phase change metamaterial (PCMM) possesses a simple geometry which remains compatible with standard photolithography patterning and can be easily realized in the optical region. Finally, it should be noted that PCMs do not require any energy to maintain the structural state of the material. Thus, once the MM has been switched it will retain its dual-band NRI until it is switched again. This clearly makes tunable dual-band NRI in PCMM interesting from a 'green technology' perspective.

2. Tunable dual-band DNG index chirped metamaterial and simulation method

The normal fishnet MM consists of a two gold layers (40nm thick Au) spaced by a dielectric interlayer (80nm thick Ge₂Sb₂Te₅) with an inter-penetrating two dimensional square array of elliptical holes shown in Figs. 1(a)-1(b). In Figs. 1(c)-1(d), we simultaneously displace rows 1 and 2, and rows 3 and 4 towards each other from their centers by a distance “δ” to create a chirped fishnet MM. The unit cell is shown in Figs. 1(b) and 1(d) for both normal and chirped MM respectively, where the lattice constant along the long axis of the elliptical hole is L_y = 400nm, L_x1 and L_x2 are the chirped lattice constants along the short axis of the elliptical aperture, where L_x1 = L_x - 2δ and L_x2 = L_x + 2δ, the diameters of the elliptical holes are d₁ = 360nm and d₂ = 200nm, β is a cross-section plane of the structure. The z-axis is normal to the structure surface and the x-y plane is parallel to the structure surface. This simulated structure is suspended in the vacuum, deep etching of a silicon support substrate can be used to achieve this. The structure is periodically extended along the x and y axes. The Au bottom layer interacts with the upper Au layer to give rise to close loops of J_D to excite strong magnetic resonances. Au is selected as the metal due to its stability and low ohmic loss. The geometry of the unit cell and the thicknesses of the sandwich layers have been chosen to allow for the impedance matching between the MM and impinging plane wave [34]. A commercial software Comsol based on the Finite Element Method (FEM) is used to calculate the S-parameters corresponding to the frequency dependent reflection r(ω) and transmission t(ω) coefficients of the structure. These S-parameters are then used to retrieve the effective parameters for the structure. The dielectric properties of Au as given by Johnson & Christy are used [35]. The structures are excited by a source with a wavelength range from 1800nm to 6800 nm, propagating along the z direction with the E field polarized along the small axis of the elliptical hole as shown in Figs. 1(a) and 1(b). In this work, two kinds of excitation sources are used to study the different cases: A plane wave source is simulated at normal incidence to the PCMM for retrieving its effective parameters; a Gaussian pulse is used as the excitation source to evaluate the required time to switch from the amorphous to the crystalline state of the Ge₂Sb₂Te₅. In particular, the Gaussian source has a repetition rate, f_r = 25 kHz and pulse duration of 2.6ns. The light fluence shining on the sample from a single pulse is written as [36]

 

Fig. 1 (a) Schematic of the normal MM consisting of a 80nm thick Ge₂Sb₂Te₅ dielectric layer between two 40nm thick Au films perforated with a square array of elliptical holes suspended in a vacuum. The lattice constant is L_x = L_y = 400nm and hole diameters are d₁ = 360nm, d₂ = 200nm. (b) Illustration of ENA lattice in the normal MM. (c) Schematic of the chirped MM consisting of a 80nm thick Ge₂Sb₂Te₅ dielectric layer between two 40nm thick Au films perforated with a rectangular array of elliptical holes suspended in a vacuum. The lattice constant along the long axis of the elliptical hole is L_y = 400nm, L_x1 and L_x2 are the chirped lattice constants along the short axis of the elliptical aperture varying with the different values of δ, and hole diameters are d₁ = 360nm, d₂ = 200nm. (d) Illustration of ENA lattice in the chirped MM.

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The phase-change material Ge₂Sb₂Te₅ was selected due to its significantly different optical properties between the amorphous and crystalline phases. The real, ɛ₁(ω) and imaginary,ɛ₂(ω) parts of the dielectric function of Ge₂Sb₂Te₅ at different phases are obtained from the experimental data in [30] and for the M-IR spectral range the dielectric function is shown in Fig. 2. A very large change in the real part of the dielectric function across the M-IR is obtained after switching the thin film’s structure from the amorphous to the crystalline phase. The change in the imaginary part of the dielectric constant is negligible and remains at a low value, as one would expect for below band-gap irradiation, thus implying low optical losses. It should be mentioned that the reversible phase transition in Ge₂Sb₂Te₅ is highly repeatable and more than a billion cycles have been experimentally demonstrated in data storage devices [31].

 

Fig. 2 Dielectric constant (a) ɛ₁(ω) vs wavelength,(b) ɛ₂(ω) vs wavelength for both amorphous and crystalline phases of Ge₂Sb₂Te₅.

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This proposed PCMM can be represented as an effective medium since their dimensions are much smaller than the operating wavelength [12]. The effective refractive index, n_eff, and impedance, $\eta$, of the PCMM can be derived from the complex coefficients of transmission $t=T_a{e}^{i{\phi }_a}$ and reflection $r=R_a{e}^{i{\phi }_{ra}}$ using the Fresnel formula [37], where T_a is the amplitude and φ_a is the phase of the transmission coefficient and, R_a is the amplitude and φ_ra is the phase of the reflection coefficient. For an equivalent isotropic homogenous slab of thickness h surrounded by semi-infinite media with refractive index n₁ and n₃ under normal incidence, we have

The so-called material parameters ε_eff and µ_eff of double negative index fishnet metamaterial are extracted using the well-known Nicholson-Ross-Weir (NRW) method [38,39]. Therefore, once n_eff and η are evaluated, the effective permittivity and permeability are calculated using

3. Simulation results and discussion

Given the elliptical holes in the MM, the response of the structure is different for different polarizations. As shown in Fig. 1(a), s polarization means the incident electric field vector is parallel to the long axis of the ENA and the incident electric field vector perpendicular to the long axis of the ENA is then denoted by p polarization. Here, the structure has a lower transmission for s polarized light due to the electric field’s orientation with respect to the metallic stripe width [42] and thus the polarization of the incident wave was set to be p polarized.

Figure 3 shows both the transmission (t) and Real(µ_eff) of the chirped PCMM with amorphous Ge₂Sb₂Te₅ for p polarization at different values of δ. In Fig. 3(a), we find that two regions of extraordinary optical transmission (EOT) can be observed if we modify the x direction periodicity (L_x1 and L_x2) of ENA by moving the neighbouring elliptical holes towards each other from their centers (i.e., increase the value of δ). These EOTs are caused by the double magnetic resonances, which can in turn contribute to the dual negative permeability bands shown in Fig. 3(b). The occurrences of the double magnetic resonances are due to the excitations of the two internal SPP modes with the different diffraction orders [13,16]. The coupling of the impinging light to the different inner modes running between metals is based on the existence of the vertical lattice constants (L_x1 and L_x2) providing the parallel momentums equal to the different order SPP modes. With increasing δ, the mode in the first resonance region shifts to the shorter wavelength and the second resonance mode shifts to the longer wavelength. This chirped PCMM possesses a low transmission (high loss) associated with electron scattering in the thin metal films [6]. It could be improved using lower-loss metallic films or compensated for by adding a gain material. As shown in Fig. 3(b), the larger the δ is, the more negative the Real(µ_eff) in the second resonance, whereas the opposite happens to the first resonance.

 

Fig. 3 3D FEM simulation of (a) transmission;(b) the real part of permeability of the amorphous Ge₂Sb₂Te₅ for the different δ with p polarization at normal incidence.

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Taking into account the thickness of the PCMM, we retrieve the effective parameters from transmission and reflection coefficients shown in Figs. 4(a) and 4(b) for the different states of Ge₂Sb₂Te₅ with δ = 40nm. It can be seen that the resonance frequency of the MM possesses a 46% tunable range when the structural phase of Ge₂Sb₂Te₅ switches from amorphous to crystalline. The two resonance wavelengths of 2100 nm and 3318nm for the amorphous phase, and 3078 nm and 4776nm for the crystalline phase are attributed to the magnetic responses of the structure. With the magnetic resonance, the structure is impedance matched, hence possessing a low reflection corresponding to the dips in reflection of Fig. 4(b). Furthermore, the calculated phase of the transmission (t) and reflection (r) coefficients are shown in Figs. 4(c) and 4(d). Importantly, changing the structural phase of the Ge₂Sb₂Te₅ offers dual-band transmission and reflection phase tunability which can be very useful in a number of applications.

 

Fig. 4 3D FEM simulation of (a) transmission; (b) reflection; (c)transmission phase; (d)reflection phase for different states of Ge₂Sb₂Te₅ with δ = 40nm for p polarization at normal incidence.

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In Fig. 5, we have simulated the permeability μ_eff and permittivity ɛ_eff for the structure. For the amorphous and crystalline phases of Ge₂Sb₂Te₅, we observe that the EOT windows overlap with the frequency regions where negative Real(ɛ_eff) and Real(μ_eff) [see Figs. 5(a) and 5(c)] coincide, allowing the structure to behave as a tunable dual-band DNG index MM shown in Fig. 6(a). However as the phase of Ge₂Sb₂Te₅ changes to crystalline, the absolute values of Real(μ_eff) decrease correspondingly due to the weaker magnetic resonance in the crystalline Ge₂Sb₂Te₅, shown in Fig. 5(a).

 

Fig. 5 3D FEM simulation of (a) real part of permeability; (b) imaginary part of permeability; (c) real part of permittivity; (d) imaginary part of permittivity for different states of Ge₂Sb₂Te₅ with δ = 40nm for p polarization at normal incidence.

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Fig. 6 3D FEM simulation of (a) real part of n_eff; (b)imaginary part of n_eff; (c) figure-of -merit; (d) absorbance for different states of Ge₂Sb₂Te₅ with δ = 40nm for p polarization at normal incidence.

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A large tuning range for the dual-band DNG index of approximately 978 nm is shown to be possible. Furthermore, the bandwidth of Real(n_eff) becomes wider for the crystalline phase in Fig. 6(a), but at the cost of an accompanied higher value of Imag(n_eff) in Fig. 6(b). Regarding losses, the figure-of-merit (FOM) for the chirped MM defined as FOM = Real(n_eff)/ Imag(n_eff) is taken to show the overall performance of the structure. As shown in Fig. 6(c), the FOM for the amorphous phase is higher than that of the crystalline phase. As mentioned above, the DNG refractive index of the structure is associated with the excitation of internal SPP mode between the two Au layers, and the plasmon modes can be seen as absorbance peaks. Figure 6(d) presents the trapped plasmon mode absorbance lines of the structure for different forms of Ge₂Sb₂Te₅, in which reflectance $R={\left|r\right|}^2$, transmittance $T={\left|t\right|}^2$, and absorbance, A = 1-R-T. The Ge₂Sb₂Te₅ variable dielectric function gives rise to a concomitant tunability in the MM absorbance. It has been shown that the absorbance peak of the structure is broader in the crystalline phase due to increased damping of the plasmon resonance [43].

In the field maps of Fig. 7, the arrows show current whereas the colour shows the magnitude of the magnetic field. For the amorphous PCMM with δ = 40nm, the displacement current J_D and total magnetic field intensity distributions of all the components H_x, H_y and H_z ($H=\sqrt{{\left|H_x\right|}^2+{\left|H_y\right|}^2+{\left|H_z\right|}^2}$) for the two resonance modes (𝛌 = 2100nm, 3318nm) are plotted at β plane in Figs. 7(a) and 7(b). The total magnetic field intensities at all of the resonance wavelengths are clearly concentrated on the Ge₂Sb₂Te₅ dielectric interlayer, as expected for the internal SPP resonances [13]. At these two resonance modes, it can be observed that anti-parallel currents are excited at top and bottom internal metallic interfaces, closed by J_D. Current loops between the metallic layers are formed to excite the magnetic resonance response of negative permeability. Therefore, we can further confirm that these resonance modes are internal SPP modes since the external SPP modes do not exhibit magnetic response owing to the non-looped current [16]. As transiting the amorphous Ge₂Sb₂Te₅ to the crystalline, at the resonance modes of 3078nm and 4776nm, Figs. 7(c) and 7(d) show that J_D also forms virtual current loops with the electric current, which implies that the internal SPP modes can be excited to create magnetic responses in the crystalline state. However, the total magnetic field intensities for the crystalline phase shown in Figs. 7(c) and 7(d) are weaker than the amorphous phase shown in Figs. 7(a) and 7(b). It indicates that the crystalline structure has smaller magnetic dipolar moments than the amorphous phase and thus leading to the smaller FOMs.

 

Fig. 7 A map of the normalized total magnetic field intensity distribution H (colour bar) and displacement current J_D (red arrows) along β plane (a) at 2100nm resonance wavelength for the amorphous Ge₂Sb₂Te₅, (b) at 3318nm resonance wavelength for the amorphous Ge₂Sb₂Te₅, (c) at 3078nm resonance wavelength for the crystalline Ge₂Sb₂Te₅, (d) at 4776nm resonance wavelength for the crystalline Ge₂Sb₂Te₅.

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For inducing the crystallization process (amorphous to crystalline phase transition) of Ge₂Sb₂Te₅, a temperature above the crystallization temperature is necessary. A heat transfer model developed from our previous work [44], is used here to obtain the temporal variation of temperature of Ge₂Sb₂Te₅ layer using the Finite Element Method (FEM) solver within COMSOL. The parameters for thermal crystallization can be found in [44]. Figure 8 presents the heat source power Q_s(r, t) and the temperature of the amorphous Ge₂Sb₂Te₅ layer, where the structure is located at the center of light source. As can be seen, the temperature within the amorphous Ge₂Sb₂Te₅ dielectric layer is a function of the incident radiation flux and can exceed the amorphous to crystalline phase transition temperature of 433K (the phase transition point of Ge₂Sb₂Te₅) after 0.4 ns and has a maximum temperature of 463K after 0.62ns under a threshold incident flux of 7.3μW/μm². Due to heat dissipation to the surroundings, the temperature starts dropping after 0.62ns before the next pulse arrives.

 

Fig. 8 3D- FEM simulation of heat power irradiating on an amorphous MDM-ENA (δ = 40nm) located at the beam center, where the solid red line presents the heat power irradiating on the structures under normal incident intensity of 7.3 μW/μm², the dash red line is the temperature of the amorphous Ge₂Sb₂Te₅ layer during one pulse.

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The temperature distributions of the structure at 0.4ns and 0.62ns along the plane β are shown in Figs. 9(a) and 9(b) respectively. One can observe that the temperature within the amorphous Ge₂Sb₂Te₅ layer is uniform, and the dominant temperature gradient is towards the top and bottom Au films.

 

Fig. 9 The temperature distribution of the unit cell of an amorphous rectangular periodic MDM-ENA along the β plane at (a) 0.4ns and (b) 0.62ns, where the color image indicates the temperature distribution and the arrows indicate the heat flux.

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As the amorphous Ge₂Sb₂Te₅ has a smaller value of ɛ₂(ω) than that of the crystalline Ge₂Sb₂Te₅, the switch state is “ON (transparent)” for the amorphous Ge₂Sb₂Te₅ and “OFF (opaque)” for the crystalline Ge₂Sb₂Te₅ [45–49]. A large difference in the dielectric function between the amorphous and crystalline structural phase can be used to enhance the “ON/OFF” ratio, so called as extinction ratio [45]. In Fig. 10(a), we show the spectra of transmission losses($\left|20\times {\log }_{10}(t)\right|$)of the chirped PCMM for the different structural phases, where the spectra of the transmission losses for the amorphous (ON-state) and crystalline (OFF-state) is expressed by solid red line and dashed red line, respectively. The maximum transmission loss of the “On-state” is 36.5dB at the wavelength of 3458 nm. For the “OFF-state”, the maximum transmission loss of 37.8 dB is obtained at the wavelength 5048 nm. Figure 10(b) shows the spectra of the extinction ratio (∆t_ex) of the structure, where the ∆t_ex in dB is the difference between the transmission losses for the amorphous state ($\left|20\times {\log }_{10}(t_{ON})\right|$) and the transmission losses for the crystalline state ($\left|20\times {\log }_{10}(t_{OFF})\right|$). As can be seen, the ∆t_ex can achieve the minimum and maximum values of −15.8 dB and13.6 dB at 3458 nm and 5048 nm, respectively. The ripples of the spectra is owing to the interference between the incident light and its reflection. In the conventional Ge₂Sb₂Te₅ based optical switch, the rise and fall times can be observed for amorphization and crystallization, respectively [46]. Here, we believe that the rise and fall times (switching time) may be further improved or shortened using the MDM-PCMM. It is because the MDM-PCMM exhibits strong surface-plasmon polariton (SPP) resonances hence concentrating the electromagnetic field more efficiently on the Ge₂Sb₂Te₅ dielectric interlayer, compared to the Ge₂Sb₂Te₅ material without plasmonic resonances. This will enable the incident light to heat up the Ge₂Sb₂Te₅ more sufficiently, thus shortening the switching time compared with the conventional Ge₂Sb₂Te₅ based optical switch. Therefore, the MDM-PCMM could be a promising candidate for ultrafast optical switch.

 

Fig. 10 (a) Spectra of the transmission losses for both ON-state (amorphous) and OFF-state (crystalline) of the structure. (b) Spectra of the extinction ratio of the structure.

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Another noteworthy feature is this thin chirped PCMM slab can be considered as a basic building block to construct a much thicker multilayer MM so called as 3D PCMM. The multilayer 3D PCMM can excite a strong magneto-inductive coupling between neighboring functional layers under a normally incident light (pulsed laser irradiation) [10]. As predicted by the LC circuit model, the tight coupling between adjacent LC resonators through mutual inductance results in a low loss [50,51]. Moreover, the loss can be further reduced owing to the destructive interference of the anti-symmetric currents across the metal film, effectively cancelling out the current flow in the centre of the film [10]. Thanks to the low loss in the multilayer 3D PCMM, the incident light could be sufficiently absorbed to heat up each Ge₂Sb₂Te₅ dielectric layer in the 3D PCMM, thus satisfying the temperature of the phase transition. This may present a new route to come up with a tunable 3D MM using the Ge₂Sb₂Te₅ .

In this study, a MDM-PCMM perforated with a periodic array of elliptical nanoholes is adopted as the base configuration because it is able to provide a degree of flexibility to adjust the values of its effective permittivity and permeability hence obtaining the impedance match, by simultaneously varying the three free parameters: short axis of the ellipse, long axis of the ellipse, and lattice constant. It will have an improved impedance matching in moving from circular/square to elliptical apertures, since circular holes can only adjust two parameters (namely the hole radius and the lattice constant) to achieve the impedance match [52,53]. Nevertheless, recent study shows that one can acquire additional freedom by varying one further parameter, such as the dielectric spacer thickness, to again have three free parameters and it appears possible to achieve comparable performance with circular/square holes than with elliptical/rectangular holes [51]. Thereby, we anticipate that the Ge₂Sb₂Te₅ may also offer new possibilities for achieving tunable 3D PCMM with polarization-independence in the visible region.

4. Conclusion

In conclusion, a chirped MM is numerically proposed to achieve an ultrafast tunable dual-band DNG index in the optical region. The dual-band DNG index is due to the excitation of two internal SPP modes in the inner metal-dielectric interfaces of the MM. An amorphous-to-crystalline switching of the Ge₂Sb₂Te₅ state is obtained under excitation of a low pump light intensity of 7.3 μW/μm². The fast response time of 0.4 ns was achieved. A large wavelength shift of 46% for the dual-band DNG index can be obtained in the M-IR spectral region by use of the Ge₂Sb₂Te₅ PCM. This work may offer an innovative and practical paradigm for the development of fast tunable multi-band metamaterial cloaks, antennas and absorbers with a low power. Moreover leveraging off multi-bit phase-change random access memory (PCRAM) research, it should be possible to achieve continuous negative refractive index tuning across the 978 nm bandwidth by partially crystallizing the phase change layer.