An integrated DFT–FDTD design of plasmon-enhanced lead-free CsSn_xGe_1−xI₃ perovskite LEDs

Abstract

CsSn_xGe_1−xI₃ as lead-free perovskites are promising for next generation NIR emitting perovskite light emitting diodes (PeLEDs) due to their tunable bandgaps and stability. However, they suffer from poor light extraction efficiency (LEE), and accurate composition-specific optical data for these materials remain scarce. This study presents a density functional theory (DFT) informed finite-difference time-domain (FDTD) framework to optimize light extraction via compositional tuning and plasmonic enhancement. First, DFT calculations were performed to obtain composition-specific complex refractive index and extinction coefficient values for x = 0, 0.25, 0.5, 0.75, and 1. Results showed that the bandgap increased from 1.331 eV for CsSnI₃ to 1.927 eV for CsGeI₃ with increasing Ge content, while the refractive index ranged from 2.2 to 2.6 across compositions. These optical constants were then used as inputs for FDTD simulations of a PeLED structure with optimized Au/SiO₂ core–shell nanorods for plasmonic enhancement. A 12.1-fold Purcell enhancement was achieved for CsSn_0.25Ge_0.75I₃, while LEE reached 25% for CsSn_0.5Ge_0.5I₃. A LEE enhancement of 36% was obtained for CsSnI₃, and spectral overlap between emitter and plasmon resonance reached 96% for Sn-rich compositions. Among the studied compositions, CsSn_0.5Ge_0.5I₃ provides the best balance between emission enhancement, light extraction efficiency (25%), Purcell enhancement (5.3×), spectral matching (93%), and oxidation stability, while Ge-rich alloys exhibit stronger spontaneous emission rate enhancement. These results establish composition-aware design guidelines for lead-free perovskite emitters targeting flexible and wearable optoelectronic applications.

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1. Introduction

PeLEDs have rapidly emerged as promising candidates for next-generation light sources, including applications in displays,^1–3 wearable electronics, and flexible optoelectronics.^4,5 This rapid advancement has been made possible because of the exceptional optoelectronic properties of metal halide perovskites (MHPs), including high absorption coefficients, solution processability, and tunable emission characteristics.⁶ The external quantum efficiency (EQE) of these devices has surged from less than 1% in early demonstrations⁷ to over 28% for red,⁸ 30% for green,⁹ and 23% for blue emitters.¹⁰ This positions PeLEDs as serious competitors to already established organic LED (OLED) technology.¹

Apart from these efficiency gains, PeLEDs offer several other distinct advantages that make them attractive for commercial applications. These advantages, combined with narrow emission linewidths (FWHM typically <20 nm), bandgap tunability across visible to NIR wavelengths, and low-cost solution processing,^11–16 position PeLEDs as strong candidates for next-generation displays and flexible optoelectronics.

Although EQE has reached up to 32%,¹⁷ the inherent toxicity of Pb is a major barrier to commercialization of these optoelectronic devices.^9,18–20 Thus, developing lead-free alternatives is a research priority to ensure environmental safety and sustainability.^4,14,21 Also, for commercial applications, operational lifetimes should exceed 10 000 hours and manufacturing costs should remain below USD 100 m⁻².²² These targets are still unachieved for both lead-based and lead-free systems. This leaves significant room for innovation in both materials engineering and device architecture optimization.

The search for Pb substitutes has focused on tin (Sn²⁺) and germanium (Ge²⁺), which share comparable ionic radii and valence states with Pb²⁺.^18,21 Sn-based PeLEDs have demonstrated external quantum efficiencies (EQEs) of up to 20.29% with a narrow emission bandwidth of 24.9 nm,¹² while Ge–Pb mixed PeLEDs have exhibited photoluminescence quantum yields (PLQYs) of nearly 71% and EQEs around 13%.²³ Sn-based MHPs exhibit excellent optoelectronic properties with direct bandgaps ranging from 1.2 eV to 1.4 eV, emitting in the near-infrared (NIR) region.¹⁸ Recent experimental demonstrations have achieved CsSnI₃-based NIR PeLEDs with external quantum efficiencies of up to 7.6% and emission wavelengths of 930–950 nm.^24–26

However, both Sn- and Ge-based MHPs suffer from oxidation-induced instability, where Sn²⁺ and Ge²⁺ ions readily oxidize to Sn⁴⁺ and Ge⁴⁺, forming deep defects that degrade long-term performance.^12,18 This intrinsic chemical instability of Sn²⁺ arises from its relatively low Sn²⁺/Sn⁴⁺ redox potential, causing spontaneous oxidation in both precursor solutions and deposited thin films.²⁷

Partial substitution of Sn²⁺ with Ge²⁺ has proven particularly effective in mitigating this instability.^28,29 Ge alloying forms a native GeO_x layer on the perovskite surface, which acts as a protective barrier against environmental factors such as moisture and oxygen.³⁰ CsSn_0.5Ge_0.5I₃-based devices have demonstrated stable operation with less than 10% efficiency degradation over 500 hours of continuous operation,³⁰ representing a significant advancement in operational stability for lead-free systems. This concept, first demonstrated in perovskite photovoltaics, has led to simultaneous improvements in both power conversion efficiency and long-term stability.¹⁸ Compositionally tuned CsSn_xGe_1−xI₃ systems therefore represent a promising alternative to lead-based perovskites for next-generation optoelectronic devices.

However, the high refractive index (∼2.5) of these perovskites limits the outcoupling efficiency to approximately 8%.^1,31 This poor outcoupling happens because the perovskite's high refractive index traps most of the light. Studies show that nearly 80% of the generated photons never escape the device.³² The primary reason behind it is total internal reflection at material interfaces. This alone accounts for nearly 56% of total radiative power loss.^31,33 Thus, enhancing LEE is a critical step towards realizing highly efficient lead-free PeLEDs.

To model light extraction properly, we need accurate wavelength-dependent n and k values for the perovskite emitter.^34,35 Existing optical simulations often reuse generic or oversimplified n, k datasets resulting in substantial inaccuracies. Generic optical constants and purely geometric models cause inaccuracies in capturing the complex wave-optical behavior of light.³⁶ Such inaccuracies lead to spectral mismatch between the emitter's photoluminescence and the device's cavity modes. As a result, LEE predictions become unreliable and far-field radiation pattern characteristics become distorted.³⁴

First-principles DFT calculations are a robust approach to counter this problem. DFT calculations determine optimized geometrical structures⁵ along with accurate electronic structures and optical properties.³⁷ This methodology has been widely used to investigate the optical and intrinsic electronic properties of MHPs,^5,9,38 including Sn-based MHPs (CsSnX₃).² This approach eliminates reliance on extrapolated or approximate n, k values.

However, DFT cannot model the electromagnetic interactions that govern light propagation and extraction within PeLEDs.³¹ Therefore, to address this issue, DFT-driven electromagnetic modeling is required. The FDTD method is a powerful approach that numerically solves Maxwell's equations in the time domain.³⁹ This enables quantitative analysis of field distributions, resonant cavity effects, and photon outcoupling pathways.¹ Using DFT-derived n and k values as inputs, FDTD simulations can be used to calculate key performance metrics of PeLEDs. These include the Purcell factor, LEE, radiated power, spectral overlap, and far-field emission profiles.^31,39

Plasmonic coupling by embedding metallic nanoparticles is an effective strategy to improve light extraction in PeLEDs.³² This helps to overcome intrinsic optical limitations. Noble metal nanoparticles localize electromagnetic energy via surface plasmon resonance to enhance emission in PeLEDs.⁴⁰ Among different nanoparticle geometries, Au nanorods (NRs) offer superior chemical stability and oxidation resistance compared to Ag, along with enhanced local density of optical states (LDOS) that improves radiative recombination.⁴¹ The longitudinal plasmon resonance of Au NRs can be tuned via aspect ratio control to align with NIR emission wavelengths,⁴² making them particularly suitable for enhancing NIR-emitting CsSn_xGe_1−xI₃ through optimized spectral overlap and Purcell enhancement. Since Au is biocompatible and resistant to environmental degradation, it is well suited for wearable and flexible optoelectronic applications.⁴³

Despite recent progress in lead-free perovskites, the relationship between Ge substitution and the optical properties of CsSn_xGe_1−xI₃ PeLEDs has not been systematically investigated. Previous studies have focused primarily on stability improvements and bandgap tuning for photovoltaics, leaving the impact of Ge alloying on LED emission profiles, plasmonic interactions, and far-field characteristics largely unexplored.

We address this gap through an integrated DFT–FDTD framework that, for the first time, establishes quantitative links between Ge substitution, composition-specific optical properties, and device-level light extraction performance in CsSn_xGe_1−xI₃ PeLEDs. We compute wavelength-dependent n and k values directly from DFT for each composition, enabling physically accurate electromagnetic modeling of emission enhancement, plasmonic coupling with Au NRs, and far-field radiation characteristics. We also optimize NR geometries across x = 0 to 1 to achieve high LEE, Purcell enhancement, and spectral overlap. This makes NIR emission in device architectures compatible with flexible, wearable applications.

This paper is organized into four sections. Section 2 discusses computational methods for material and device analysis. Results and discussion are presented in Section 3, and Section 4 concludes the work.

2. Computational methods for material and device analysis

Our computational approach connects atomic-scale physics with device-scale optics by combining DFT and FDTD simulations. DFT provides the composition-dependent refractive index (n) and extinction coefficient (k) of CsSn_xGe_1−xI₃ alloys, capturing their underlying electronic structure. These optical constants are then used as inputs for FDTD simulations of the complete PeLED architecture, enabling quantitative analysis of light emission, plasmonic enhancement, and spectral overlap with compositional accuracy.

2.1. First-principles calculations of material properties

DFT calculations were performed using the CASTEP module in Materials Studio 2020. The study utilized generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional for geometry optimization. To mitigate bandgap underestimation by the PBE algorithm, we used the hybrid functional Heyd–Scuseria–Ernzerhof (HSE06) for band structure calculations. The properties were calculated by incorporating spin–orbit coupling (SOC). All calculations employed an ultrasoft pseudopotential with a plane-wave basis set and a cutoff energy of 500 eV. For the Brillouin zone sampling, a Monkhorst–Pack k-point grid of 6 × 6 × 6 was used. The geometry of CsSn_xGe_1−xI₃ was relaxed until the energy and force reached the criteria of convergence at 1.0 × 10⁻⁶ eV and 0.01 eV Å⁻¹ between two adjacent steps.

2.2. Device level optical modeling of the perovskites

3D FDTD simulations were performed using Lumerical FDTD (Ansys Inc.) to investigate the optical response of plasmonic CsSn_xGe_1−xI₃-based PeLEDs. The FDTD algorithm solves Maxwell's equations numerically in the time domain, allowing modeling of light propagation, near-field enhancement, and resonant-cavity effects in multilayer optoelectronic structures.⁴⁴ The computational workflow for FDTD simulations is shown in Fig. 1.

Fig. 1 Computational workflow for FDTD simulation of plasmonic CsSn_xGe_1−xI₃-based PeLEDs. The device setup incorporates DFT-derived optical constants for the perovskite layer alongside literature data for other materials. FDTD simulations employ refined meshing near critical interfaces and integrate Au/SiO₂ NRs at the ZnO/perovskite boundary. Performance analysis yields near-field and far-field distributions, Purcell factors, light extraction efficiency, and spectral overlap metrics, enabling composition-dependent device optimization.

The simulated PeLED adopted an architecture of ITO (100 nm)/Spiro-OMeTAD (35 nm)/CsSn_xGe_1−xI₃ (50 nm)/ZnO (40 nm)/Ag (100 nm), matching experimentally achievable stacks (Fig. 2a). Fig. 2b presents the energy-band diagram of the ITO/Spiro-OMeTAD/CsSn_xGe_1−xI₃/ZnO/Ag structure, showing hole injection from ITO into Spiro-OMeTAD, electron injection from Ag into ZnO, and carrier recombination within the perovskite layer. The present framework focuses exclusively on electromagnetic modeling of light emission and extraction. Electrical transport phenomena, including carrier injection barriers, mobility variations across compositions, and recombination zone distribution, are not explicitly simulated. Radiative recombination is assumed to occur uniformly within the emissive layer. Therefore, the model should be interpreted as an optical design study rather than a fully coupled electro-optical device simulation. Integration with drift-diffusion carrier transport modeling represents an important direction for future work.

Fig. 2 (a) Layered device architecture of the CsSn_xGe_1−xI₃-based PeLED, illustrating the anode/HTL/perovskite/ETL/cathode stack and the upward light-emission direction. (b) Energy-band diagram of the ITO/Spiro-OMeTAD/CsSn_xGe_1−xI₃/ZnO/Ag structure, showing hole injection from ITO into Spiro-OMeTAD, electron injection from Ag into ZnO, and carrier recombination within the perovskite layer.

Composition-dependent n, k spectra for the perovskite layer were derived from DFT calculations and imported into the FDTD material database. Optical constants for the remaining layers were taken from established datasets.^45–48 Spontaneous emission within the perovskite layer was represented by an electric dipole source positioned above the plasmonic nanostructure, radiating across the visible and NIR wavelengths corresponding to the emission window of CsSn_xGe_1−xI₃ compositions (Fig. 3). This source model treats electron–hole radiative recombination as an oscillating dipole and reproduces Purcell-enhanced emission in resonant photonic environments.⁴⁹ For plasmonic enhancements, a gold core–shell structured NR (Fig. 3) was positioned near the ZnO/perovskite interface, with SiO₂ acting as the shell to protect charge carriers from quenching.⁵⁰ If the emitter is placed too close to the metal surface, non-radiative energy transfer to the metal dominates and net quenching occurs. The SiO₂ shell surrounding the Au nanorod was therefore fixed at a thickness of 5 nm. This separation distance was selected on the basis of established distance-dependent fluorescence measurements,⁵¹ which show that the quenching regime dominates at separations below approximately 4 nm. By operating at 5 nm, the design places the emitter within the region of strong near-field enhancement while maintaining a margin above the quenching threshold. A full parametric sweep of shell thickness is identified as a clear direction for further work. The NR diameter and length were tuned so that the localized surface plasmon resonance (LSPR) spectrally overlapped with the composition-dependent emission peak. This produced strong near-field coupling and enhanced radiative decay.^52,53 The frequency-dependent dielectric function of gold was defined according to the Palik dataset.⁴⁶

Fig. 3 Schematic illustration of the CsSn_xGe_1−xI₃-based PeLED structure incorporating a plasmonic Au/SiO₂ nanorod. The device stack (ITO/Spiro-OMeTAD/CsSn_xGe_1−xI₃/ZnO/Ag) includes a dipole emitter positioned near the embedded Au/SiO₂ core–shell nanorod to enable plasmon–emitter coupling. The nanorod geometry is defined by the Au core length (l) and radius (r), surrounded by a SiO₂ shell.

Periodic boundary conditions were applied in the lateral (x–y) directions to emulate an infinite emitter array, while perfectly matched layer (PML) boundaries were implemented vertically (z) to suppress spurious reflections.⁵⁴ This configuration reproduces realistic outcoupling behavior and has been validated for photonic-band-structure and plasmonic-resonator analyses.⁵⁵ A non-uniform spatial mesh was adopted, with refinement to 0.5–1 nm around the NR and emitter interface, gradually coarsening elsewhere to maintain computational efficiency. Each simulation was executed for 1500 fs, ensuring full electromagnetic-field decay before Fourier transform. From the computed field data, several key optical metrics were extracted.

In this study, LEE is calculated from the power collected by a far-field transmission monitor positioned above the ITO layer and normalized to the emitted dipole power within the simulation domain. This approach quantifies the fraction of radiative power that escapes the device stack into free space. While this method accurately captures total far-field radiation, a detailed modal decomposition of individual loss channels such as surface plasmon polariton (SPP) modes, waveguide modes, and metal absorption pathways was not explicitly performed. Such separation would require additional eigenmode analysis or volumetric absorption monitors for each layer. Furthermore, the dipole source used in the FDTD simulations represents an idealized radiative recombination event. Non-radiative recombination mechanisms such as trap-assisted recombination, Auger recombination, polaronic effects, and defect-induced quenching are not included in the present electromagnetic model. Therefore, the calculated Purcell factors describe the enhancement of the radiative decay channel only. Consequently, the reported LEE assumes unity internal quantum efficiency (IQE). In realistic devices, the EQE is related to the LEE through EQE = IQE × LEE, so the reported LEE values represent an optical upper limit. Practical efficiencies will depend heavily on material defect density and carrier recombination dynamics. While Au nanorods enhance the local density of optical states, they also introduce intrinsic ohmic absorption losses governed by the imaginary part of the metal dielectric function. Because the optical constants of Au in our FDTD model are taken from the Palik experimental dataset, all frequency-dependent absorption is accounted for in the electromagnetic energy budget. Power absorbed within the nanorod contributes to the difference between the total Purcell enhancement and the far-field LEE. The present FDTD framework does not include thermal modeling. The time-averaged volumetric heat source density in the Au nanorod, , can be extracted from the computed field data and used as input to a heat diffusion solver in future work. For LED operation under spontaneous emission conditions at moderate power densities, the temperature rise from ohmic dissipation is expected to remain small. Nonetheless, a rigorous coupled electrothermal treatment is identified as an important next step, particularly for Sn-rich compositions where elevated local temperatures may accelerate Sn²⁺ oxidation.

The resulting far-field emission patterns and optical enhancement metrics are analyzed in Section 3 to establish composition–structure–performance relationships.

3. Results and discussion

3.1. Electrical and optical properties of CsSn_xGe_1−xI₃

The structural, electronic, and optical properties of CsSn_xGe_1−xI₃ (0 ≤ x ≤ 1) were systematically investigated using density functional theory (DFT) to establish the composition–property relationships governing light emission and plasmonic interactions in PeLED architectures. Five representative alloy configurations, pure CsSnI₃ (x = 1), CsSn_0.75Ge_0.25I₃, CsSn_0.5Ge_0.5I₃, CsSn_0.25Ge_0.75I₃, and pure CsGeI₃ (x = 0), were analyzed to understand the effect of doping gradually (Fig. 4). All structures converged to the orthorhombic perovskite phase, maintaining a corner-sharing octahedral structure. The lattice volume decreased monotonically with increasing Ge content, consistent with the smaller ionic radius of Ge²⁺ relative to Sn²⁺. This contraction leads to stronger metal–halide orbital overlap and influences both band dispersion and optical transition strength.

Fig. 4 Different doping arrangements of CsSn_xGe_1−xI₃. (a) x = 1, (b) x = 0.75, (c) x = 0.5, (d) x = 0.25 and (e) x = 0. Green, grey, brown, and violet indicate Ge, Sn, I, and Cs, respectively.

Fig. 5 shows the calculated electronic band structures and total density of states (DOS) of CsSn_xGe_1−xI₃ for varying Ge compositions. All compounds exhibit semiconducting behavior with a clear band gap at the Fermi level. The band gap increases systematically from 1.331 eV for CsSnI₃ to 1.927 eV for CsGeI₃, indicating effective band-gap tuning through Ge substitution. The DOS profiles further confirm the absence of states at the Fermi level and reveal gradual shifts in the valence and conduction band edges with increasing Ge content.

Fig. 5 Electronic band structures and the corresponding total density of states (DOS) of (a) CsSnI₃, (b) CsSn_0.75Ge_0.25I₃, (c) CsSn_0.5Ge_0.5I₃, (d) CsSn_0.25Ge_0.75I₃, and (e) CsGeI₃ along the high-symmetry k-path (M–Γ–X–M–R–X). The Fermi level is set to 0 eV (red dashed line). The calculated band gaps are 1.331 eV, 1.483 eV, 1.695 eV, 1.840 eV, and 1.927 eV with increasing Ge composition, respectively. A systematic widening of the band gap is observed across the CuSn_xGe_1−xI₃ series. The DOS plots further confirm the semiconducting nature of all compositions, showing negligible states at the Fermi level and gradual shifts of the valence and conduction band edges upon Ge substitution.

The absorption coefficient can be derived from the complex dielectric constant, ε = ε₁ + iε₂. The imaginary part (ε₂) of the dielectric constant (ε) can be calculated using the following formula from the electrical band structure⁵⁶:

where u is the vector defining the polarization of the incident light, is the matrix element, E^c_k is the conduction band energy, E^v_k is the valence band energy at wave number k, ω is the angular frequency of the electron, e is the charge, u·r is the momentum operator, and ħ is the reduced Planck's constant. The real part of the dielectric constant can be found using the Kramers–Kronig transformation:⁵⁷where P is the principal value of the integral. The optical absorption coefficient for the material is calculated as a function of the dielectric constant as follows:⁵⁸where ε₁ and ε₂ are the real and imaginary parts of the dielectric function, k is the extinction coefficient, and λ is the wavelength. The refractive index (n) and extinction coefficient (k) can be obtained from the dielectric constant using the following formulas:⁵⁹

The complex refractive index (n and k spectra) obtained from the frequency-dependent dielectric function reveals pronounced compositional dependence (Fig. 6). The optical constants were calculated within the independent-particle DFT framework; although excitonic effects were not explicitly included, their influence is expected to cause only minor quantitative shifts near the band edge without affecting the overall spectral trends used in the FDTD simulations. Increasing Ge substitution systematically blueshifts the n(λ) and k(λ) spectra, reflecting the widening electronic bandgap. Importantly, the magnitude of n remains high (∼2.2–2.6 in the visible/NIR region), corroborating the inherently strong light-confinement characteristics of Sn/Ge-based perovskites and explaining the limited intrinsic light extraction efficiency in planar PeLEDs.

Fig. 6 Optical properties of CsSn_xGe_1−xI₃: (a) refractive index and (b) extinction coefficient for various compositions of x.

The absorption coefficients in Fig. 7(b) further confirm the influence of Ge incorporation on optical transitions. Pure CsSnI₃ exhibits strong absorption extending into the NIR region, whereas Ge-rich compositions shift the absorption edge toward shorter wavelengths, consistent with increased bandgap energies. The absorption amplitude remains high (>10⁵ cm⁻¹ across the visible–NIR window), reaffirming the exceptional oscillator strength and suitability of these materials for high-radiance emitter applications. Alloy compositions (x = 0.25–0.75) show intermediate absorption edges, enabling fine-tuning of emission wavelengths for targeted plasmonic coupling with Au nanorods.

Fig. 7 (a) Bandgap of CsSn_xGe_1−xI₃ for various compositions of x. (b) Absorption coefficient of CsSn_xGe_1−xI₃ for various compositions of x.

The electronic bandgap analysis of the DFT optical spectra displays a nearly linear dependence on alloy composition (Fig. 7a). CsSnI₃ shows the smallest bandgap of 1.331 eV, while CsGeI₃ exhibits a significantly larger value of 1.927 eV, with mixed alloys occupying the intermediate range. Ge substitution reduces the antibonding interaction strength in the metal–iodide network, widening the bandgap and shifting emission toward shorter wavelengths. Such tunability is crucial because it enables precise matching of the emissive peak with the longitudinal plasmon resonance of Au/SiO₂ nanorods for maximized Purcell enhancement. While direct LED emission data for intermediate CsSn_xGe_1−xI₃ compositions remain limited, prior experimental and theoretical studies confirm monotonic bandgap widening with increasing Ge incorporation.^18,27,30 The calculated bandgap of CsGeI₃ (1.927 eV) agrees well with reported literature values near 1.9 eV.³⁰ Together with the excellent agreement between our predicted CsSnI₃ emission (931 nm) and experimentally reported values (930–950 nm),^24–26 this consistency for both end members supports the reliability of the predicted intermediate alloy behavior.

The DFT results establish a clear composition-dependent evolution of lattice structure, bandgap, and optical constants in CsSn_xGe_1−xI₃. These properties constitute the fundamental inputs for the subsequent FDTD modeling.

3.2. FDTD results

3.2.1. Purcell factor enhancement. The Purcell factor is a measure of the enhancement of an emitter's spontaneous decay rate when nanostructures are placed nearby.⁶² This enhancement occurs due to the modification of local photonic density of states (LDOS) at the emitter. The Purcell factor enhancement of the LED is shown in Fig. 8 for all values of x. Black markers indicate the peak values of Purcell across all x values. The highest Purcell factor of 12 was achieved for CsSn_0.25Ge_0.75I₃ for a NR length of 60 nm and a radius of 11 nm. This indicates strong coupling between the emitter and the plasmonic nanostructure. Table 1 shows the Purcell enhancement for all CsSn_xGe_1−xI₃ compositions and the corresponding NR length and radius values.

Fig. 8 Composition-dependent Purcell factor enhancement in CsSn_xGe_1−xI₃-based LEDs for (a) x = 1, (b) x = 0.75, (c) x = 0.5, (d) x = 0.25, and (e) x = 0.

Table 1 Comparison of the calculated band gaps (this work) with reported values (Ref.) and the corresponding FDTD simulation results of CsSn_xGe_1−xI₃ nanorod emitters

Composition	Bandgap (eV)		Emission	NR size (nm)		Purcell		LEE (%)		Sp. Ovlp.
	This work	Ref.	λ (nm)	Length	Radius	Peak	λ (nm)	Value	Enh.	Jcos
CsGeI3	1.927	1.90^30	643	55	11	4.4	643	24.9	17	0.89
CsSn0.25Ge0.75I3	1.840	1.84^60	674	60	11	12.1	695	19	34	0.80
CsSn0.5Ge0.5I3	1.695	1.69^60	731	70	17	5.3	721	25	33	0.93
CsSn0.75Ge0.25I3	1.483	1.48^60	836	120	17	4.9	837	23	28	0.96
CsSnI3	1.331	1.30^61	931	170	19	8.0	928	17.5	36	0.96

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Coincidence of the plasmon spectrum peak with the emitter peak creates resonance, increasing LDOS at that wavelength. This resonant condition in turn enhances the radiative decay rate of the emitter resulting in a large Purcell factor.⁶³ Under other conditions where the plasmonic response is weaker or not in coincidence with the emitter peak, LDOS may still be increased due to scattering.⁶⁴ This is responsible for Purcell enhancements that are not sharp. The measured Purcell behavior is a superposition of both resonance-driven enhancement, where spectral overlap is significant, and broadband off-resonant LDOS perturbation, where overlap is negligible.⁶³

3.2.2. LEE and LEE enhancement. The light extraction efficiency (LEE) quantifies the fraction of generated photons that escape the device structure.⁶⁵ Without plasmonic nanorods, the baseline LEE for these high-index perovskites is approximately 8%. Fig. 9 and 10 show the composition-dependent LEE in CsSn_xGe_1−xI₃-based LEDs. It can be seen that LEE consistently increases with Ge content, reaching a maximum of 25% for CsSn_0.5Ge_0.5I₃ using a NR length of 70 nm and a radius of 17 nm. The highest LEE enhancement of 36% was obtained for CsSnI₃ (x = 1) using a NR length of 170 nm and a radius of 19 nm.

Fig. 9 Composition-dependent LEE in CsSn_xGe_1−xI₃-based LEDs for (a) x = 1, (b) x = 0.75, (c) x = 0.5, (d) x = 0.25, and (e) x = 0.

Fig. 10 Composition-dependent LEE enhancement in CsSn_xGe_1−xI₃-based LEDs for (a) x = 1, (b) x = 0.75, (c) x = 0.5, (d) x = 0.25, and (e) x = 0.

Several mechanisms work in synergy when Au NRs are used for LEE and LEE enhancement. When NRs are placed near the perovskite emitter, they support localized surface plasmon resonance (LSPR), opening additional radiative channels and enhancing the local density of optical states.⁶⁶ The presence of NR breaks the symmetry of the device stack, scattering previously trapped waveguide modes into the far field and adding directionality to the emission.⁶⁷ This scattering contribution, combined with the Purcell-driven increase in radiative photon generation, increases the fraction of photons that escape the structure.⁶⁶ NR geometry also allows precise resonance tuning and large scattering cross-sections, further boosting both LEE and LEE enhancement across compositions.

Table 1 shows the achieved LEE and LEE enhancement values for all compositions along with the corresponding NR length and radius values. It is important to note that although plasmonic coupling enhances the spontaneous emission rate (Purcell factor), not all enhanced emission contributes to far-field radiation. The gap between Purcell enhancement and LEE indicates that a portion of the optical energy remains confined within the structure or is dissipated through metallic absorption. The present work focuses on radiative outcoupling trends across compositions, while a rigorous separation of individual loss channels (e.g., SPP modes versus waveguide modes) is identified as an important direction for future work.

3.2.3. Spectral overlap analysis. Spectral overlap presents the amount of alignment of the plasmon resonance spectrum with the emitter's photoluminescence spectrum.⁶⁸ Transverse mode LSPR is used, which can be effective for spectral overlap alignment in perovskites having refractive index n ≈ 2.0–2.6. Composition-dependent LEE and LEE enhancement values for all values of x in CsSn_xGe_1−xI₃ are shown in Fig. 11. The highest spectral overlap (J_cos) of about 96% has been achieved for both CsSn_0.75Ge_0.25I₃ and CsSnI₃ at the peak Purcell wavelength. J_cos is defined by⁶⁹

Fig. 11 Composition-dependent spectral overlap in CsSn_xGe_1−xI₃-based LEDs for (a) x = 1, (b) x = 0.75, (c) x = 0.5, (d) x = 0.25, and (e) x = 0.

The achieved J_cos values for all compositions of CsSn_xGe_1−xI₃ and the corresponding NR length and radius values are shown in Table 1.

The refractive index of perovskites influences the LSPR spectral position due to changes in the dielectric environment, enabling resonance tuning by the design of the nanostructure geometry.⁷⁰ This also helps in broadening the plasmon spectrum, thus covering a larger portion of the emitter spectrum and increasing overlap. Plasmons can boost emission where the perovskite spectrum falls off. This increases overlap even if the peaks are not perfectly aligned.⁷¹ The presence of vertical scattering channels also increases radiative contribution in the transverse mode.⁷²

3.2.4. Far-field emission profiles. Far-field plots depict the amount of light escaping the LED and radiating outward as freely propagating radiation.⁷³ Previously discussed mechanisms such as the enhancement of near-field to far-field conversion through plasmonic scattering, Purcell-enhanced radiative decay, symmetry breaking of waveguide modes, and transverse LSPR act together to provide strong far-field emission profiles.

Composition-dependent far-field plots for all values of x in CsSn_xGe_1−xI₃ are shown in Fig. 12. Sn-rich (x = 1) compositions emit in the deeper NIR region. Here the transverse LSPR scatters weakly, resulting in a dimmer far-field pattern. But as the Ge content increases, the emission shifts to shorter wavelengths. This blue shift is caused by the bandgap widening due to Ge alloying. These wavelengths couple more strongly to the Au NR resonance, producing stronger scattering and a brighter far-field map. So, the far-field intensity increases steadily with increasing Ge fraction in CsSn_xGe_1−xI₃ based LEDs.

Fig. 12 Far-field emission profiles of CsSn_xGe_1−xI₃-based LEDs for (a) x = 1, (b) x = 0.75, (c) x = 0.5, (d) x = 0.25, and (e) x = 0. The common color scale corresponding to the normalized far-field intensity used for all compositions.

Fig. 13a presents the angular line cuts extracted from the far-field emission maps for all CsSn_xGe_1−xI₃ compositions. The emission profiles exhibit strong composition dependence with Ge-rich alloys showing relatively narrow angular distributions. Sn-rich compositions display significantly broadened emission. This broadening is attributed to the higher refractive index of Sn-rich perovskites. This enhancement redistributes radiative power over wider emission angles.

Fig. 13 Far-field emission analysis of CsSn_xGe_1−xI₃-based LEDs. (a) Angular line cuts extracted from the far-field emission maps, illustrating composition-dependent emission directionality. (b) Beam width (FWHM) of the angular emission profiles, quantifying the transition from directional to diffuse emission with increasing Sn content. (c) Integrated far-field radiated power as a function of Sn composition, highlighting a non-monotonic extraction trend.

Fig. 13b summarizes the angular emission characteristics using the FWHM of the angular profiles. The FWHM increases with Sn content. This confirms the transition from directional emission in Ge-rich compositions to more diffuse radiation in Sn-rich alloys.

To quantify the total radiative output, the far-field intensity was integrated over all emission angles (Fig. 13c). The integrated power exhibits a non-monotonic dependence on composition, reaching a maximum near x = 0.5. This indicates that optimal far-field extraction does not coincide with the strongest Purcell enhancement but instead arises from a balance between emission rate enhancement and angular outcoupling.

3.3. Comparison with reported PeLED architectures

To place the performance of the CsSn_xGe_1−xI₃-based PeLEDs in context, Table 2 compares the Purcell factor and LEE reported for perovskite LED architectures published between 2021 and 2025. The table focuses mainly on lead-free and reduced-lead systems. Lead-based CsPbBr₃ devices are included only as reference benchmarks. Many earlier studies improve either the emission rate or the optical outcoupling, but not both at the same time. Clear and combined reporting of Purcell enhancement and LEE is still uncommon, especially for lead-free emitters.

Table 2 Comparison of the Purcell factor and light extraction efficiency (LEE) in perovskite LEDs reported between 2021 and 2025

Emitting material	Emission (nm)	Purcell factor	LEE/outcoupling	Optical structure	Methodology	Ref.
Lead-based CsPbBr3 devices are included only as benchmarks
CsSn0.25Ge0.75I3 (this work)	∼670	12× (sim.)	∼19%	Au/SiO2 nanorods	DFT n,k + FDTD	This work
CsSn0.5Ge0.5I3 (this work)	∼730	5.3× (sim.)	25%	Au/SiO2 nanorods	FDTD, near–far field	This work
FASnI3	∼880	PF inferred	Grating-assisted	DFB grating	Exp. + mode analysis	74
Cs2AgBiBr6	NIR	—	∼42% (sim.)	Metal microcavity	FDTD	39
Cs3Cu2I5	∼440	Implicit	Directional	VCSEL cavity	Experiment	75
CsPbBr3 (bench.)	∼530	∼2–3×	31–38%	Planar microcavity	TMM + dipole	76
CsPbBr3 (bench.)	∼520	∼2×	∼18×	Mie resonator	FDTD + TRPL	77

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As shown in Table 2, this work is among the few studies that quantitatively evaluate both Purcell enhancement and LEE within a single lead-free device platform. When compared with CsPbBr₃ microcavity and resonator benchmarks, the CsSn_0.5Ge_0.5I₃ composition shows competitive light extraction. At the same time, it maintains strong emission-rate enhancement. These results highlight the benefit of using composition-aware plasmonic design to balance light generation and light extraction.

3.4. Design considerations

The best Purcell factor does not necessarily lead to the best light extraction. This is due to a clear tradeoff between near-field enhancement and outcoupling. At x = 0.25, the emitter spectrum and the plasmon resonance are almost perfectly aligned. This strong match accelerates photon generation and drives the Purcell factor up to about 12×. The drawback is that this composition has a relatively high refractive index of roughly 2.5, which traps much of the emitted light inside the device.

At x = 0.5, the spectral match is slightly weaker, but the lower refractive index reduces total internal reflection, allowing more photons to escape. In practice, this means that x = 0.25 is more efficient at producing light, while x = 0.5 is more efficient at releasing it. This distinction shows that Purcell enhancement and far-field extraction should be treated as two separate but equally important design factors. Simply increasing the spontaneous emission rate does not automatically lead to the best overall device performance.

These trends also guide the choice of nanorod geometry. Sn-rich compositions emit farther into the NIR region and therefore couple best to longer nanorods of about 170 nm, which match the red-shifted plasmon resonance. Ge-rich compositions emit at shorter wavelengths and achieve stronger coupling with shorter nanorods in the range of 55 to 70 nm.

These competing effects are summarized in Fig. 14, which compares Purcell enhancement, spectral overlap, light extraction efficiency, and LEE enhancement across compositions. To provide contextual design guidance, compositional stability is represented qualitatively using the Ge fraction (1 − x) as a literature-supported indicator of improved oxidation resistance. Previous experimental studies have reported enhanced environmental stability in Sn–Ge mixed perovskites due to the formation of native GeO_x surface passivation layers.³⁰ However, it should be emphasized that no defect formation energy calculations or thermodynamic stability modeling were performed in the present work. The stability axis in Fig. 14 (labeled “Stability (Qual.)”) therefore serves only as a conceptual design parameter rather than a quantitatively computed metric.

Fig. 14 Spider plot comparing the normalized performance metrics of CsSn_xGe_1−xI₃ PeLEDs, including the Purcell factor, LEE, LEE enhancement, spectral overlap, and compositional stability. Compositional stability is represented using the Ge fraction (1 − x) as a qualitative proxy.

While Fig. 14 highlights discrete composition-dependent trends, the underlying tradeoffs can be more clearly visualized using continuous design maps. Fig. 15 presents two-dimensional landscapes that show how Purcell enhancement, light extraction, and spectral overlap evolve simultaneously with composition, emission wavelength, and plasmonic geometry. These maps complement the spider plot by revealing optimal design regions rather than isolated operating points.

Fig. 15 Design landscapes illustrating competing emission and extraction mechanisms in CsSn_xGe_1−xI₃ PeLEDs. (a) Purcell–LEE tradeoff map as a function of Sn composition, x, and emission wavelength, where the color scale denotes the Purcell factor and contour lines indicate regions of comparable light extraction efficiency. (b) Spectral overlap landscape (J_cos) between the emitter spectrum and plasmonic resonance as a function of nanorod length and emission wavelength, showing a systematic shift of optimal overlap toward longer nanorods for Sn-rich compositions.

3.5. Experimental validation and fabrication outlook

To validate our computational methodology, we compare our predictions for CsSnI₃ with recent experimental demonstrations. Our DFT calculations predict an emission wavelength of 931 nm for CsSnI₃, showing excellent agreement with experimentally measured values of 930–950 nm.^24–26 This confirms the accuracy of our DFT-derived optical constants.

Recent breakthroughs have demonstrated CsSnI₃ NIR PeLEDs with external quantum efficiencies of 7.6%,²⁴ a peak radiance of 226 W sr⁻¹ m⁻², and operational lifetimes exceeding 39 hours at 100 mA cm⁻².²⁵ These results confirm that Sn-based perovskites can function effectively as LED emitters.

Although mixed Sn–Ge compositions have not yet been extensively demonstrated in LED architectures, their successful synthesis and optical characterization in photovoltaic systems³⁰ confirm the structural feasibility and smooth compositional evolution of CsSn_xGe_1−xI₃ alloys. The present study therefore provides experimentally testable emission and plasmonic design targets for future realization of mixed Sn–Ge PeLEDs. Our framework specifically predicts optimal performance for CsSn_0.5Ge_0.5I₃ at an emission wavelength of 721 nm, utilizing Au/SiO₂ nanorods with a length of 70 nm and a radius of 17 nm.

Solution processability of both pure Sn²⁴ and mixed Sn–Ge³⁰ perovskites, along with established plasmonic nanoparticle integration methods,⁴¹ indicates experimental realization is feasible. Key considerations include anti-solvent treatment optimization, SnF₂ additives for oxidation suppression, and nanorod positioning control.

4. Conclusion

We have established a DFT–FDTD computational framework to design plasmon-enhanced lead-free CsSnI₃ perovskite LEDs. DFT calculations provided accurate optical constants for five compositions (x = 0, 0.25, 0.5, 0.75, and 1). Our predicted emission wavelength for CsSnI₃ is 931 nm. This shows excellent agreement with recent experimental demonstrations at 930–950 nm and validates our methodology.

Among all tested compositions, CsSn_0.5Ge_0.5I₃ delivers the most balanced performance. It achieves the highest light extraction efficiency (25%), strong Purcell enhancement (5.3×), and 93% spectral overlap. Improved oxidation resistance has been reported in prior experimental studies of Sn–Ge mixed perovskites.³⁰ It should be noted that the stability trends discussed here are literature-informed and were not directly calculated within the present framework. The results also suggest that longer nanorods suit Sn-rich NIR emitters, while shorter nanorods suit Ge-rich visible emitters.

The reported efficiencies are derived from far-field radiative power and do not explicitly decompose confined or absorptive loss channels. These simulations also do not account for non-radiative recombination or defect-induced quenching effects common in Sn-based perovskites. Thus, the calculated Purcell enhancement and LEE represent optical upper bounds assuming high internal quantum efficiency. Actual experimental device performance may be lower, depending heavily on defect passivation quality and recombination dynamics. Plasmonic enhancement inherently involves a trade-off between radiative rate enhancement and ohmic absorption losses in metallic nanostructures. Careful optimization of spacer thickness is necessary to suppress quenching in experimental realizations. Coupled electrothermal modeling of heat dissipation in the Au nanorod, beyond the scope of the present optical study, will be important for assessing device reliability in future work. Building on the successful experimental demonstration of CsSnI₃ LEDs, our validated computational framework provides systematic design guidelines for optimizing the full CsSn_xGe_1−xI₃ compositional range.

The combination of tunable emission wavelengths (643–931 nm), strong plasmonic response, and Ge-induced stability makes CsSn_xGe_1−xI₃ promising lead-free candidates for next-generation flexible and wearable optoelectronics. Experimental validation of these predictions through fabrication and characterization of mixed Sn–Ge LED devices is the logical next step.

Conflicts of interest

The authors declare no conflicts of interest.

Data availability

Representative finite-difference time-domain (FDTD) simulation files supporting this study are available at the following GitHub repository: https://github.com/debnath-shoumik/PeLED_FDTD_CsSnxGe1xI3.

Additional data may be obtained from the corresponding author upon reasonable request.

Acknowledgements

The authors acknowledge the financial support from the Basic Research Grant, Bangladesh University of Engineering and Technology (BUET), under Office Order No. Shongstha/R-60/Re-2413, dated 10 October 2023 (Professor Dr Md. Zahurul Islam). The authors also acknowledge the logistical support provided by the Department of Electrical and Electronic Engineering (EEE), BUET, throughout this work.

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