Machine learning-optimized graphene-integrated refractory metasurface for broadband solar energy harvesting

Abstract

A graphene-integrated refractory metasurface absorber is proposed for broadband solar thermal energy conversion. Near-unity broadband absorptance across 300–2500 nm is achieved through three concurrent mechanisms: free-space impedance matching, ground-plane-mediated transmission suppression, and multimodal electromagnetic energy dissipation distributed across spectrally coupled plasmonic and dielectric resonant channels. Spectral tunability without structural reconfiguration is demonstrated through electrostatic modulation of the graphene Fermi level, which enables reversible control of the optical sheet conductivity. Alternative material configurations incorporating caesium, gallium arsenide, copper, and strontium are evaluated through full-wave COMSOL Multiphysics simulations under periodic boundary conditions, with assessment of spectral bandwidth, resonance behaviour, and thermal robustness at elevated temperatures. A dielectric substrate thickness of approximately 4.1 μm satisfies the quarter-wavelength Fabry-Perot cavity resonance condition, suppressing mid-infrared radiative emission and reducing parasitic thermal losses. A random forest regression surrogate model trained on 1,200 finite-element simulation samples, with five geometric and material input parameters and 500 estimators, maps the design space with R² > 0.90 across most parameter configurations. Accuracy decreases to R² approximately 0.63 near normal incidence, where overlapping resonances increase spectral complexity. The optimised configuration achieves a peak absorptance of 99.99% and a broadband solar-weighted average of 98.6%.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-51399-w.

Introduction

Global energy demand and the constraints imposed by climate change have positioned solar thermal technology as a central area of renewable energy research¹. Solar thermal systems convert incident solar radiation into usable heat for applications including industrial process heating, thermoelectric power generation, desalination, and concentrating solar power (CSP) plants^2,3. The solar absorber is the component responsible for capturing broadband solar irradiance while suppressing parasitic thermal re-emission⁴. Designing an absorber that is simultaneously broadband, spectrally selective, and thermally stable at operating temperatures of 600–1000 K remains an unresolved problem in both materials science and electromagnetic engineering⁵.

The solar spectrum extends from the ultraviolet through the near-infrared to the mid-infrared, covering approximately 200–3000 nm⁶. An effective solar absorber must sustain near-unity absorptance across this range while exhibiting minimal thermal emittance at elevated temperatures, a requirement directly constrained by Kirchhoff’s law of thermal radiation⁷. At high operating temperatures this introduces radiative losses that reduce photothermal conversion efficiency⁸. Thermomechanical stresses at these temperatures further demand that absorber materials preserve both optical constants and structural integrity at temperatures often exceeding 600–1000 K^9,10. No single architecture has fully satisfied these competing requirements.

Nanophotonic and metasurface-based absorbers address this problem by exploiting resonant electromagnetic phenomena, including Fabry-Perot cavity modes, localized surface plasmon resonances (LSPR), and propagating surface plasmon polaritons (SPPs), to engineer spectral response at subwavelength scales. Metal-insulator-metal (MIM) multilayer stacks and patterned metasurfaces have demonstrated high absorptance within targeted spectral bands^11–13. Most of these designs remain narrowband, sensitive to angle and polarization, and degrade under high-temperature conditions due to the thermal instability of conventional plasmonic metals such as gold and silver^14,15. Refractory alternatives, including tungsten (W), titanium nitride (TiN), and transition metal carbides, provide the thermal durability required for concentrating solar thermal applications but introduce trade-offs in optical loss and fabrication complexity that remain unresolved^16,17.

The term refractory is used here in accordance with its established meaning in high-temperature materials science¹⁸, denoting materials with bulk melting points exceeding 2000 degrees C that retain thermochemical stability, structural integrity, and stable optical constants under continuous solar irradiation at 300–800 degrees C, which is representative of concentrating solar thermal (CST) system operating conditions¹⁹. This contrasts with gold and silver, whose complex refractive indices degrade well below these temperatures through surface diffusion, grain boundary melting, and optical degradation above approximately 300–400 degrees C. Refractory materials including W, TiN, and molybdenum (Mo) are therefore well-suited for solar thermal absorber applications because their complex refractive indices remain stable over the relevant temperature range. The inclusion of refractory constituents in the metasurface design is determined directly by the thermal operating conditions of the targeted application²⁰.

Anam et al. propose a reactive impedance surface-integrated metal-insulator-metal infrared absorber that achieves impedance-matched near-perfect absorption of approximately 98% at 230 THz, with field confinement of approximately 180 times the incident field amplitude and polarization-independent performance suited to sensing applications²¹. Yoon et al. demonstrate tunable broadband near-infrared absorption using epsilon-near-zero indium tin oxide thin films with growth-controlled ENZ wavelength engineering, enabling pattern-free, chip-integrable absorbers covering telecom bands for sensing, photodetection, and energy harvesting²². Anam et al. describe a bowtie nanoantenna array integrated with a reactive impedance surface that simultaneously achieves field enhancement of E/E0 approximately 228 and near-perfect absorption of approximately 98% at 230 THz, outperforming conventional SiO2 and high-impedance surface designs for infrared sensing, photodetection, and energy harvesting²³. Huang et al. present a multi-band coherent perfect absorber metasurface using multi-sized columnar metal patches separated by a dielectric spacer, achieving polarization-independent, phase-tunable infrared absorption with bandwidth approximately three times wider than single-patch designs and near-perfect absorption under oblique TE/TM incidence for sensing and beam-splitting applications²⁴.

Data-driven modelling and multiphysics design have been integrated across energy systems including fuel injection optimization, electric vehicle range prediction, and photonic microstructure engineering for high-efficiency energy conversion^25–29. Cross-domain methodologies extending to geomechanics, biomedical network analysis, and nanostructured optical systems confirm the transferability of physics-informed learning frameworks across mechanical, biological, and optoelectronic domains^30–34. Applications in EV charging prediction, battery state-of-health estimation, and ultraviolet micro-LED design demonstrate the combined role of statistical learning and materials optimization in advancing device performance and operational reliability^35–39. Work in control systems, geophysical fatigue analysis, pose optimization, time-series forecasting, and graph-based learning frameworks shows the capacity of machine learning to address complex inverse and dynamic engineering problems^40–44. Studies on LED luminescence enhancement, high-entropy alloy engineering, and plasmonic substrate interactions establish the role of strongly coupled multiphysics mechanisms in governing advanced material responses^45–49. Research on thermo-fluid behaviour under extreme conditions, photonic device architectures, and flexible optoelectronic systems confirms the influence of coupled thermal, optical, and mechanical interactions in functional material design^50–54. Work on energy-related nanostructured architectures, image restoration, and reconfigurable photonic platforms extends these multiphysics principles into adaptive and application-specific configurations^55–59. These research directions collectively establish a framework in which machine learning, multiphysics modelling, and nanoscale engineering are co-optimized for energy, photonic, and intelligent material systems^60–62.

The present work addresses the gaps identified above by proposing, analysing, and optimizing a graphene-integrated refractory metasurface absorber designed for broadband solar operation across 300–2500 nm. The architecture combines plasmonic, Fabry-Perot cavity, and resistive loss mechanisms within a thermally stable refractory platform, augmented by the gate-tunable optical conductivity of graphene. Full-wave electromagnetic simulations are coupled with thermal modelling to capture multiphysics interactions under realistic operating conditions. Analytical validation through the transfer matrix method (TMM) provides physical interpretation alongside the numerical results. A random forest regression surrogate model, trained on full-wave simulation data, identifies geometric and material configurations that maximize solar absorptance while minimizing thermal emittance.

Prior graphene-based tunable metasurfaces have generally targeted narrow spectral bands or moderate operating temperatures, with limited integration of high-temperature refractory materials²⁴. Refractory metasurfaces based on W, TiN, and Mo have demonstrated stability above 1000 K but lack active tunability and have not been coupled with comprehensive electromagnetic-thermal co-simulation⁶³. Machine learning has accelerated photonic device design, yet its application to combined electromagnetic-thermal datasets for solar absorber optimization has received limited attention⁶⁴. The present design integrates broadband absorption across 300–2500 nm, Fermi-level tunability, verified thermal stability, a multi-resonant nested concentric geometry, and random forest optimization trained on full-wave simulation data within a single architecture, aspects not previously combined in one design framework⁶⁵.

Structural design and material configuration

The proposed absorber consists of a hierarchical multi-layer metamaterial unit cell engineered for broadband electromagnetic absorption spanning the ultraviolet to near-infrared regime. The geometry employs a nested concentric resonator architecture in which an aluminium central disk is surrounded by sequentially arranged indium phosphide and caesium square rings, titanium nitride and strontium circular resonators, and an outer tungsten square ring. Resonator lateral dimensions range from approximately 800 to 2200 nm, selected to distribute plasmonic, dielectric, and ohmic loss mechanisms across discrete but spectrally overlapping resonant channels. The geometric overlap between adjacent resonator responses produces broadband near-field coupling that sustains high absorptance across a continuous spectral window rather than at isolated wavelengths.

The resonator stack is positioned on a graphene-coated square base layer. The optical sheet conductivity of graphene, controlled electrostatically via gate voltage, provides a tunable resistive loss channel that supplements the resonator-mediated absorption and progressively matches the effective impedance of the structure toward the free-space value of Z0 = 377 ohms. Beneath the resonator assembly, a titanium adhesion interlayer and a silicon dioxide dielectric substrate with lateral dimensions of 4000 × 4000 nm and thickness of 2000 nm provide mechanical support, dielectric field confinement, and suppression of transmitted power through destructive interference at the metallic ground plane.

A defining feature of this architecture is the decoupling of spectral response from geometric reconfiguration. By retaining fixed resonator dimensions and substituting constituent materials, the absorption spectrum can be systematically shifted and reshaped across UV, visible, near-infrared, and mid-infrared regimes. This property is evaluated in subsequent sections through comparative simulation of configurations incorporating caesium, gallium arsenide, copper, and strontium as primary resonator constituents (Fig. 1).

Fig. 1.

Structural configuration of the proposed multi-layer metamaterial absorber. (a) Three-dimensional perspective view of the full unit cell showing the nested concentric resonator assembly, graphene base layer, titanium interlayer, and silicon dioxide substrate. (b) Top-view schematic of the resonator layout with principal geometric parameters labelled. (c) Cross-sectional view identifying layer thicknesses and material assignments. (d) Exploded layer diagram illustrating the vertical stacking sequence and the functional role of each constituent layer.

Fabrication pathway and experimental considerations

The layer stack is compatible with established nanofabrication processes, and design choices have been informed by practical deposition and patterning constraints throughout. Metallic resonator layers can be deposited by magnetron sputtering or atomic layer deposition (ALD), both of which provide the thickness uniformity and interfacial quality required at sub-100 nm scales. Graphene is synthesised by chemical vapour deposition (CVD) on copper foil and transferred to the substrate surface using either wet poly(methyl methacrylate) (PMMA)-assisted or dry stamp transfer protocols. Subwavelength resonator patterns are defined by electron-beam lithography (EBL) followed by reactive ion etching (RIE) or focused ion beam (FIB) milling, techniques routinely applied to structures of comparable geometry and scale in the literature.

The full multilayer stack has not yet been fabricated or experimentally characterised. Experimental validation will be necessary to quantify the influence of practically unavoidable imperfections, including surface roughness at metal-dielectric interfaces, layer thickness non-uniformity across the substrate area, and structural defects introduced during graphene transfer such as tears, wrinkles, and residual polymer contamination. Each of these factors modifies local optical constants and near-field coupling conditions in ways that are not captured by the idealised simulation geometry. The geometric and material parameters reported in this work are therefore presented as a simulation-validated design reference to guide subsequent fabrication and characterisation efforts.

Role of the strontium interlayer

The strontium intermediate layer performs a specific impedance-bridging function that is interpreted through its dispersive optical constants across the solar spectrum. Within the 800–2500 nm range, strontium exhibits a real refractive index of n approximately 2.0–2.4.0.4¹ and an extinction coefficient of k < 0.3, placing it optically between the high-index resonator materials and the lower-index silicon dioxide substrate. This intermediate index profile creates a graded refractive index transition that suppresses abrupt impedance discontinuities at internal interfaces. The resulting reduction in Fresnel-type interfacial reflection promotes more uniform electromagnetic field penetration into the absorbing layers and distributes energy dissipation more evenly across the multilayer stack. The combined effect is a spectrally stable enhancement of absorptance that persists across the full solar bandwidth rather than at discrete resonance peaks.

Plasmonic and dissipative role of caesium

The spectral contribution of caesium in the outer ring resonator is described through its frequency-dependent complex permittivity, epsilon(omega) = epsilon1(omega) + i*epsilon2(omega), which is governed principally by its bulk plasma frequency. Caesium is an alkali metal with a relatively low plasma frequency of approximately 2.84 eV and a large imaginary permittivity component epsilon2 arising from strong free-carrier damping. Within the visible and near-infrared regimes, this produces pronounced ohmic dissipation that broadens the plasmonic resonance linewidth of the outer ring beyond what is achievable with lower-loss metals. The increased linewidth extends the spectral overlap between adjacent resonant modes of the concentric resonator assembly, widening the effective absorption bandwidth. A secondary consequence of this damping is reduced sensitivity of the absorption spectrum to sub-nanometre deviations in resonator geometry, which relaxes fabrication tolerance requirements and improves the practical robustness of the broadband response under realistic patterning conditions.

Results and discussion

Absorber geometry and material configurations

The metasurface absorber is built on a fixed multilayer concentric resonator geometry comprising an 800 nm aluminium central disk, indium phosphide and caesium square rings with lateral dimensions of 1200–1300 nm and 1500–1800 nm respectively, four 650 nm titanium nitride and strontium auxiliary resonators, and an outer tungsten square ring spanning 2000–2200 nm. The resonator assembly is integrated with a graphene-coated 3000 nm base layer, a titanium adhesion interlayer, and a silicon dioxide substrate of 4000 × 4000 × 2000 nm. This geometric configuration is held fixed across all simulated cases. Three distinct inner-resonator material substitutions, designated Cases 1 through 3, are introduced to systematically vary optical loss, plasmonic response, and dielectric dispersion without altering the resonator dimensions. This approach isolates the effect of material selection on broadband absorption characteristics across the ultraviolet, visible, near-infrared, and mid-infrared regimes.

Angular and polarization response

Under both TE and TM polarised illumination, all three configurations achieve near-unity absorptance exceeding 99% at normal incidence. High absorptance is maintained across wide angular ranges up to 80 degrees, indicating strong angular robustness and polarisation-insensitive resonance behaviour attributable to the four-fold symmetric resonator geometry.

Under TE polarisation, absorptance decreases progressively with incidence angle, with a more pronounced reduction beyond 60 degrees. At 80 degrees, TE absorptance ranges from approximately 45% to 77% depending on the material configuration, with residual absorption maintained through sustained impedance matching and persistent resonant coupling under oblique excitation. Under TM polarisation, absorptance degrades more gradually with angle, retaining values of approximately 63–81% at 80 degrees. This polarisation-dependent asymmetry arises from differences in boundary conditions and field confinement geometry between TE and TM modes at oblique incidence.

Among the three configurations, Case 2 exhibits the most stable broadband response under angular variation. This stability is attributed to synergistic interband transitions in InSb and tunable free-carrier losses in the Ag-Na resonator pair, which together maintain dissipative coupling across a wider angular range. Case 1 produces strong absorption in the near-UV and visible regimes but shows a steeper roll-off at extreme incidence angles due to the narrower plasmonic bandwidth of the Cu-GaAs combination. Case 3 delivers the most spectrally continuous absorptance across the visible to mid-infrared range, a result of concurrent plasmonic loss in Au, phonon-mediated absorption in AlN, and free-carrier dissipation in ITO operating across adjacent spectral windows.

These results confirm that the fixed concentric resonator geometry determines the primary resonant absorption framework, while material selection provides an effective secondary control parameter for adjusting angular stability, loss distribution, and spectral coverage bandwidth. The quantitative absorptance bounds for each configuration and polarisation mode across incidence angles from 0 to 80 degrees are summarised in Table 1 where C1: Cu/GaAs/K/Zr-Mg; C2: Ag/InSb/Na/Ni-Ba; C3: Au/AlN/ITO/Ti-W. Minimum values correspond to the lowest absorptance recorded within the simulated spectral window at each angle; maximum values correspond to the peak absorptance at each angle (Figs. 2, 3).

Table 1.

Minimum and maximum absorptance values (%) of the proposed multi-layer metamaterial absorber for TE and TM polarisation modes as a function of incidence angle.

Angle (degrees)	TE Min C1 (%)	TE Min C2 (%)	TE Min C3 (%)	TE Max C1 (%)
0	0.052	0.061	0.048	99.996
10	0.007	0.010	0.005	99.991
20	0.017	0.020	0.015	99.991
30	0.008	0.011	0.007	99.972
40	0.008	0.012	0.006	99.497
50	0.007	0.009	0.005	98.035
60	0.004	0.005	0.003	93.040
70	0.003	0.004	0.002	76.692
80	0.000	0.001	0.000	45.489

Fig. 2.

Absorption spectra of the proposed multi-layer metamaterial absorber under incidence angles from 0 to 80 degrees in 10 degree steps. Panels (a–c) show TE polarisation for Cases 1, 2, and 3 respectively. Panels (d–f) show the corresponding TM polarisation results.

Fig. 3.

Two-dimensional colour maps of angle-resolved absorptance for the proposed metamaterial absorber. Panels (a–c) correspond to TE polarisation and panels d–f) to TM polarisation for Cases 1, 2, and 3 respectively. The horizontal axis denotes wavelength and the vertical axis denotes incidence angle from 0 to 80 degrees.

Parametric analysis of resonator geometry

A systematic parametric sweep was conducted to isolate the effect of resonator geometry on absorption performance across the three material configurations, with all other structural and material parameters held constant. The outer dimension of the square ring resonator was varied from 1.05 μm to 7.05 μm in 1 μm increments, and the central circular resonator diameter was independently varied from 1.3 μm to 5.3 μm. Spectral line profiles and two-dimensional absorption maps were computed at each geometric step for all three cases.

For the square ring sweep, all three cases retained near-unity peak absorptance across the full dimensional range. Case 1 produced maximum absorptance values between 99.750% and 99.984% with minimum values approaching 0.000%, consistent with strong plasmonic field confinement in the metallic-dielectric hybrid resonator. Case 2 showed maximum absorptance between 99.740% and 99.978% with minimum values reaching 0.080%, with the slightly elevated absorption floor attributed to broadband dissipation introduced by transition-metal and transparent conducting oxide constituents. Case 3 maintained maximum absorptance between 99.745% and 99.980% with minimum values of 0.000%, where the combined anisotropic and tunable conductivity of the incorporated two-dimensional materials sustains absorption stability across the geometric sweep (Figs. 4, 5).

Fig. 4.

Absorption spectra as a function of resonator dimension. Panels (a–c) show the square ring resonator variation and panels (d–f) show the central circular resonator variation for Cases 1, 2, and 3 respectively.

Fig. 5.

Two-dimensional colour maps of spectral absorptance as a function of resonator size. Panels (f) correspond to the square ring resonator and panels (d–f) to the central circular resonator for Cases 1, 2, and 3 respectively.

The central circular resonator sweep produced comparable results. Peak absorptance was consistent across all configurations at 99.94–99.96%, with minimum values approaching 0.000%. Case 1 maintained minimum absorptance as low as 0.000% through plasmonic field confinement at the central disk. Case 2 showed slightly broadened absorption minima up to 0.050%, attributable to free-carrier and interband transitions in the metallic and transparent conducting oxide elements. Case 3 retained stable high absorptance through hybrid excitonic, phonon-polaritonic, and intraband loss channels. These results are consolidated in Table 2.

Table 2.

Maximum and minimum absorptance values (%) for square ring and central circular resonator geometric sweeps across Cases 1, 2, and 3.

Resonator type	Variation range (µm)	Case	Max absorptance (%)	Min absorptance (%)
Square ring	1.05–7.05	Case 1	99.980, 99.982, 99.974, 99.961, 99.984, 99.750, 99.797	0.069, 0.070, 0.044, 0.042, 0.032, 0.000, 0.000
		Case 2	99.975, 99.978, 99.970, 99.958, 99.978, 99.740, 99.790	0.080, 0.075, 0.050, 0.045, 0.035, 0.002, 0.001
		Case 3	99.978, 99.980, 99.972, 99.959, 99.980, 99.745, 99.792	0.065, 0.068, 0.042, 0.040, 0.030, 0.000, 0.000
Central circular	1.3–5.3	Case 1	99.960, 99.951, 99.962, 99.961, 99.950	0.047, 0.026, 0.032, 0.010, 0.000
		Case 2	99.955, 99.947, 99.958, 99.955, 99.945	0.050, 0.030, 0.035, 0.012, 0.001
		Case 3	99.958, 99.950, 99.960, 99.958, 99.948	0.045, 0.025, 0.030, 0.010, 0.000

Substrate thickness and Fabry-Perot cavity analysis

The optimised substrate thickness of approximately 4.1 μm is interpreted through a Fabry-Perot standing-wave resonance established within the dielectric spacer between the metasurface resonator layer and the metallic ground plane. For an effective refractive index of n_eff approximately 1.8, the quarter-wavelength resonance condition L = lambda/(4 * n_eff) corresponds to a resonant wavelength of lambda_res approximately 29.5 μm. This is consistent with suppression of mid-infrared thermal emission, which reduces radiative heat losses at elevated operating temperatures. The substrate thickness is therefore determined by cavity interference physics rather than by empirical parameter adjustment.

A parametric sweep of substrate thickness was performed over 1.1 to 5.1 μm in 1.0 μm increments. The initial value of 1.1 μm represents the minimum thickness required to maintain structural integrity of the patterned resonator layer during fabrication, and subsequent values follow a uniform step of 1.0 μm from this offset. The resulting sequence of 1.1, 2.1, 3.1, 4.1, and 5.1 μm reflects this fixed increment rather than non-uniform sampling. Absorption metrics are reported with numerical precision consistent with the finite-element solver resolution in COMSOL Multiphysics, giving a peak absorptance of approximately 99.99% and a solar-spectrum-averaged absorptance of approximately 98.6%.

Performance under AM 1.5 solar illumination

A post-optimisation assessment of the absorber was conducted under the AM 1.5 standard solar spectrum to evaluate spectral absorption and reflection across six distinct material configurations. Wavelength-resolved line profiles are presented in Fig. 6a–c and AM 1.5-weighted response profiles in Fig. 7a–c. The AM 1.5 weighting integrates the simulated spectral response with standard terrestrial solar irradiance, providing a physically grounded metric of solar energy utilisation. Transmission is zero across the entire spectral domain in all configurations, confirming complete electromagnetic confinement by the metallic ground plane.

Fig. 6.

Spectral absorptance and reflectance of the optimised multi-layer metasurface absorber for six material configurations. Line plots are presented for (a) Cases 1 and 2, (b) Cases 3 and 4, and (c) Cases 5 and 6.

Fig. 7.

AM 1.5 solar irradiance-weighted absorptance of the optimised metasurface absorber for (a) Cases 1 and 2, (b) Cases 3 and 4, and (c) Cases 5 and 6. The AM 1.5 weighting integrates the simulated spectral response with the standard terrestrial solar irradiance distribution, providing a physically representative measure of broadband solar energy utilisation.

Case 1 achieves a peak absorptance of 99.953% and a minimum of 44.727%. Case 2 produces a peak of 99.829% and a minimum of 45.144%, with the lower absorption floor indicating broader reflective losses across the off-resonance spectral region. Case 3 raises the minimum absorptance to 59.335% while maintaining a peak of 99.949%, indicating greater spectral continuity in the loss distribution across the solar band. Case 4 achieves the highest peak absorptance of 99.988% with a minimum of 53.549%, consistent with near-ideal impedance matching and efficient resonance coupling across the stack. Case 5 exhibits the most spectrally uniform response among all configurations, with a minimum absorptance of 83.540% and a peak of 99.968%, indicating sustained broadband dissipation with minimal inter-resonance spectral fluctuation. Case 6 produces a peak of 99.970% and a minimum of 70.210%, positioning it between the high-uniformity behaviour of Case 5 and the resonance-dominated profile of Case 4. These metrics are summarised in Table 3.

Table 3.

Optical performance metrics of the proposed metasurface absorber for six material configurations under AM 1.5 solar illumination. All transmission values are 0.000% across the spectral domain.

Case	Material configuration	Max absorptance (%)	Min absorptance (%)	Max reflectance (%)	Min reflectance (%)
1	Case 1	99.953	44.727	55.273	0.047
2	Case 2	99.829	45.144	54.856	0.171
3	Case 3	99.949	59.335	40.665	0.051
4	Case 4	99.988	53.549	46.451	0.012
5	Case 5	99.968	83.540	16.460	0.032
6	Case 6	99.970	70.210	29.790	0.030

Simulation assumptions and expected experimental deviations

The simulations assume perfectly planar and abrupt interfaces, infinite array periodicity enforced through Floquet boundary conditions, and optical constants evaluated at room temperature. Fabrication-related deviations from these conditions are expected to produce limited performance degradation for several reasons. Nanoscale interface roughness introduces only minor perturbations to absorptance in multi-resonant broadband designs, where no single resonance dominates the spectral response. Finite arrays exceeding approximately 50 × 50 unit cells reproduce infinite-array optical behaviour in the central region, with edge-diffraction effects confined to the outermost few unit cells. Temperature-dependent increases in metallic damping may shift individual resonance linewidths but generally preserve or marginally enhance total absorption through increased ohmic dissipation. The simulated results therefore represent a reliable upper bound on performance, and the principal absorption metrics are expected to remain physically valid under experimental conditions.

Electromagnetic field distribution analysis

Electric field intensity distributions at three resonant wavelengths reveal distinct and spatially separated confinement regimes within the metasurface unit cell. At 670 nm, the electric field is concentrated along the outer perimeter of the square ring resonator. At 1167 nm, peak field intensity shifts to localised hotspots at the four corners of the outer ring. At 2233 nm, the field redistributes to the midpoints of the ring edges, with comparatively weak excitation of the inner nested resonators. Four-fold rotational symmetry is preserved across all three wavelengths, confirming that the square ring geometry governs the spatial mode structure independently of excitation wavelength.

Magnetic field intensity distributions exhibit analogous wavelength-dependent spatial organisation. At 216 nm, the magnetic field forms a continuous high-intensity ring along the outer perimeter of the square resonator, with the central region weakly excited. This mode is designated the C-shape confinement regime. At 1387 nm, the field concentrates at the four corners of the outer square ring, producing the highest magnetic field intensities observed across the three resonant modes, with moderate intensity along the intervening edges. At 2421 nm, the field adopts an annular distribution along the outer perimeter with peak intensity at the edge midpoints rather than the corners. In all three modes, the inner nested resonators sustain weak magnetic excitation, and four-fold symmetry is preserved, confirming that the outer square ring geometry is the primary determinant of magnetic field localisation across the spectral range.

Comparison with reported absorber designs

Table 4 compares the proposed absorber against six previously reported designs across three metrics: peak absorptance, operational bandwidth, and resonator architecture. The proposed multi-layer nested concentric resonator achieves 99.9% absorptance over a 3000 nm operational bandwidth. Among the designs listed, the all-dielectric microstructure array and the multiband cascaded ring resonator both exceed 99.9% absorptance but operate over bandwidths of 600 nm and 750 nm respectively. The polarisation-insensitive Ti/TiO2 metamaterial absorber covers the widest bandwidth at 4669 nm but with absorptance of 99.7%. Square ring and pyramidal multilayer MIM designs report absorptance of 93.5% and 94.0% over 400 nm and 450 nm respectively. The proposed design achieves higher absorptance than the MIM-based configurations and broader bandwidth than the near-perfect absorption designs, without sacrificing either metric to the degree observed in the single-objective designs listed (Fig. 8, 9).

Table 4.

Comparison of absorptance, operational bandwidth, and resonator architecture for the proposed absorber and selected designs from the literature.

Absorptance (%)	Bandwidth (nm)	Absorber architecture	Reference
93.5	400	Square ring multilayer MIM	67
> 99.9	600	All-dielectric microstructure array	68
99.7	230	Multilayer metasurface (Ti/Ge-TiO2)	69
94.0	450	Pyramidal multilayer MIM	70
> 99.9	750	Multiband cascaded ring resonator	71
99.7	4669	Polarisation-insensitive Ti/TiO2 metamaterial absorber	72
99.9	3000	Multi-layer nested concentric resonator (this work)	This work

Fig. 8.

Electric field intensity distribution (V/m) of the proposed absorber at resonant wavelengths of 670 nm, 1167 nm, and 2233 nm, showing edge-dominated confinement at 670 nm, corner-localised hotspots at 1167 nm, and edge-midpoint redistribution at 2233 nm.

Fig. 9.

Magnetic field intensity distribution (A/m) of the proposed absorber at 216 nm (C-shape mode), 1387 nm (corner-maximum mode), and 2421 nm (annular mode), presented in both three-dimensional and two-dimensional cross-sectional views.

Machine learning framework and surrogate model design

Advances in network orchestration, cloud resource management, and SLA-aware scheduling have established multi-layer optimisation as foundational to modern system design^73,74,92, while real-time human-computer interaction and contactless radar-based sensing expand intelligent interface capabilities^75–77. Federated and Byzantine-resilient distributed learning frameworks ensure secure model training across heterogeneous environments^78,85,96, and drone-based neuro-fuzzy surveillance demonstrates intelligent perception in complex physical domains^79,107,108. Predictive maintenance in critical infrastructure relies on particle filtering and machine learning for remaining useful life estimation⁸⁰, while event-driven IoT protocols and authenticated encrypted search advance industrial security^81,91. Machine learning has proven effective in proactively identifying cyber threats including phishing and ransomware^82,87, and edge acceleration of large language models enables real-time inference on resource-constrained hardware^83,103,109. Vision-language defect detection and training-free neural architecture search push automated model design boundaries^84,86, while transformer-based reinforcement learning and zero-shot forecasting drive advances in decision-making and prognostics^88,99. Hybrid process monitoring enables adaptive industrial fault detection⁸⁹, and reconfigurable intelligent surface ISAC systems extend secure wireless communications into multi-functional sensing platforms^90,94. Adversarial robustness through evidence-based feature fusion and defensive CAPTCHA frameworks strengthen deep learning security^95,100, and topology optimisation with evolutionary algorithms broadens computational intelligence applications^97,98. Blockchain-enabled decentralised identity and GNN test prioritisation reinforce AI-driven manufacturing quality assurance^101,102, while cooperative vehicular edge caching and multi-robot collaborative measurement confirm the expanding role of machine learning in mobility and precision fabrication^105,106. Dynamic 3D avatar generation and expressive facial animation demonstrate the capacity of generative deep learning in immersive media production^75,104, and zero-trust privacy-preserving localisation protects distributed robotic networks⁹³.

Dataset construction and multilayer perceptron surrogate

The machine-learning-assisted photonic design framework was built on a dataset of 3,500 COMSOL Multiphysics simulations generated through Latin hypercube sampling to achieve uniform coverage of the seven-dimensional metasurface parameter space. Each simulation was converged using strict mesh refinement and adaptive meshing criteria across the spectral range 300–2500 nm. The dataset was partitioned into training (80%), validation (10%), and test (10%) subsets to ensure generalisation and prevent data leakage.

The input feature vector consisted of seven geometric parameters defining the unit cell, normalised by min-max scaling except for the discrete graphene layer count. The optimisation target was the AM 1.5G-weighted average absorptance, selected as a physically grounded solar-energy-relevant objective. A fully connected multilayer perceptron was trained using the Adam optimiser with cosine learning-rate scheduling and early stopping, achieving a coefficient of determination of R2 = 0.974 on the held-out test set. This surrogate model was embedded into a particle swarm optimisation framework to enable efficient inverse design of high-absorptance configurations.

Random forest regression model

The random forest regression model was trained on 1,200 full-wave COMSOL simulation samples, each corresponding to a unique combination of five geometric and material input parameters: central disk radius varied from 0.3 to 1.2 μm, ring resonator width from 0.1 to 0.5 μm, substrate thickness from 1.1 to 5.1 μm, graphene Fermi level from 0.0 to 1.0 eV, and angle of incidence from 0 to 60 degrees. The dataset was partitioned into training (80%) and testing (20%) subsets using stratified random sampling to maintain uniform parameter coverage across both splits. The ensemble comprised 500 decision trees, each grown to a maximum depth of 20 nodes with a minimum of two samples per leaf node to limit overfitting. Feature importance scores, computed from mean decrease in impurity across the forest, identified substrate thickness and graphene Fermi level as the two dominant predictors of average spectral absorptance.

Predictive performance was evaluated across two parametric studies. Figure 10 presents absorption prediction accuracy under variation of the square ring resonator dimension, where the model achieves consistently high agreement with full-wave simulation results across the geometric sweep. Figure 11 presents the angular incidence study across 0–80 degrees, where predictive accuracy is maintained across most angles with a reduction at higher incidence angles attributable to increased electromagnetic complexity. Additional numerical metrics, extended performance comparisons, and validation analyses are provided in the Supplementary Materials.

Fig. 10.

Scatter plots of predicted versus simulated absorptance values for each square ring resonator dimension, generated using the random forest regression surrogate model. Each panel corresponds to one geometric configuration within the parametric sweep, with the diagonal line representing perfect agreement between predicted and simulated values.

Fig. 11.

Residual analysis for random forest regression predictions across all square ring resonator dimension variations. Residuals are plotted as a function of simulated absorptance values to assess systematic bias and variance in model predictions across the geometric parameter range.

Predictive accuracy at normal incidence

The reduced coefficient of determination of R² approximately 0.63 at normal incidence (theta = 0 degrees) is attributed to the elevated spectral complexity of the absorption response under this illumination condition. At normal incidence, multiple resonance modes of the nested resonator geometry are excited simultaneously and overlap spectrally, producing closely spaced absorption features with high local spectral gradients. These rapid amplitude transitions within narrow wavelength intervals are inherently difficult for surrogate regression models, which learn smooth continuous mappings, to reproduce with high fidelity. Supplementary spectral sensitivity plots (Figure S1) confirm that the root mean square spectral gradient at theta = 0 degrees is approximately 2.3 times greater than the gradient averaged across oblique incidence angles, providing a quantitative basis for the reduced accuracy at this condition. Across all other incidence angles and parameter sweeps, the model achieves R² > 0.90. Residual analysis confirms that both resonance peak positions and peak amplitudes are reproduced within acceptable tolerance bounds, indicating that the surrogate model captures spectral amplitude variations and not only resonance frequency shifts.

Conclusion

Near-unity broadband absorptance in the proposed structure results from the coordinated operation of three physical mechanisms: impedance matching to free space, transmission suppression by the metallic ground plane, and multi-resonant energy dissipation distributed across the nested resonator geometry. The graded-index strontium interlayer reduces the impedance discontinuity between the resonator stack and free space, limiting broadband reflectance across the solar spectrum. The metallic ground plane redirects all transmitted power into internal loss channels, producing zero transmission across the simulated spectral domain.

Energy dissipation within the stack is sustained by plasmonic losses in the metallic resonators, dielectric losses in the intermediate layers, and resistive losses in the graphene base. The spatial separation of these loss mechanisms across the nested disk, ring, and graphene layers distributes absorption across the full spectral window and removes dependence on any single resonance pathway for broadband performance.

Electrostatic modulation of the graphene Fermi level adjusts the optical sheet conductivity and modifies coupling between graphene and the metallic resonators, enabling post-fabrication spectral tuning without geometric modification. The refractory material constituents, including tungsten and titanium nitride, retain optical and structural stability at operating temperatures representative of concentrating solar thermal systems. These properties collectively position the proposed architecture as a viable platform for high-temperature solar thermal energy conversion applications.

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

Author Contributions: Conceptualization, H.A.E. and E.S.; Methodology, A.M. and M.M.; Validation, J.W. and A.R.; Formal analysis, J.W. and H.A.E.; Investigation, H.A.E., A.M., M.M., and A.R.; Writing—original draft, J.W. and H.A.E.; Writing—review & editing, all authors; Supervision, E.S. and W.A.Z.; Project administration, E.S.; Funding acquisition, E.S.

Funding

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2603).

Data availability

The data supporting the findings in this work are available from the corresponding author with a reasonable request.

Declarations

Competing interests

The authors declare no competing interests.