Signatures of nonthermal carriers in nonlinear photoluminescence

Abstract

Ultrafast light emission from plasmonic nanostructures provides a sensitive probe of the energetic electrons generated by intense optical excitation. The way this emission scales with excitation intensity has long been read through the lens of heated electron populations at an elevated temperature, yet this picture cannot reconcile the variety of behaviors seen across different materials and experiments. We argue that transient nonthermal electrons play a far larger role than has been appreciated. We present a unified description that reproduces emission behavior across diverse systems and outline experiments needed to resolve their signatures.

Nonlinear light emission from plasmonic nanostructures is conventionally interpreted through heated electron populations at elevated temperature. Here, authors show that transient nonthermal electrons dominate more often than recognized and present a framework to identify them.

Introduction

Ultrafast optical excitation of plasmonic nanostructures triggers a cascade of nonequilibrium carrier dynamics that culminate in broadband nonlinear photoluminescence (NPL). The nonlinear response of this emission provides a sensitive probe of the transient electron distribution. Under intense optical fields, collective plasmonic oscillations drive strong near-field confinement and absorption, rapidly populating high-energy electron states and producing transient distributions that can deviate markedly from thermal equilibrium.

In this context, NPL, which is characterized by its power-law dependence on excitation intensity, serves as a powerful probe of the elevated thermal Fermi–Dirac populations and offers insight into the carrier energy flow in plasmonic metals¹. A key advantage of nonlinear analysis is its ability to average out system-specific complexities, such as photonic density of states, local field enhancement, and resonance lineshapes, which often complicate raw spectral fitting. Moreover, modulation and characterization of carrier distributions are increasingly relevant for emerging applications, including enhanced hot carrier photochemistry², plasmonic tunnel junctions^3–5, enhanced light-matter interactions^6–8, and nanoscale thermometry⁹.

Since seminal studies on the nonlinear character of broadband continuum emission from rough plasmonic metal films¹⁰, extensive experimental and theoretical efforts have sought to rigorously test and extend the scope of these initial observations. Early work illuminated the role of intraband photoluminescence (IBPL) in down-converted emission, while attributing the up-converted region to two-photon absorption (2PA) and fluorescence. However, NPL from rough metal films was revisited by Haug et al.¹ and showed that earlier analysis obscured key nonlinear optical signals due to coarse binning of the up- and down-converted spectral regions. By instead analyzing the emission power-scaling wavelength by wavelength, they revealed a spectrally varying power-law exponent (PLE) with a characteristic lineshape. Crucially, fitting this lineshape provides direct access to internal state parameters of the carrier distribution, specifically the hot-electron temperature.

The strength of PLE analysis lies in its ability to convert power-dependent spectra into a diagnostic lineshape that is less sensitive to static spectral weighting from the photonic environment. This point has been developed from complementary perspectives across the field. Haug, Lupton, and co-workers established wavelength-resolved PLE analysis as a way to reveal hot-electron intraband luminescence that can be hidden by spectral binning^1,11. Over time, multiple schools of thought have emerged and evolved through studies that extend beyond films to isolated particles and well-controlled nanoscale geometries. Building on PLE analysis, recent studies on isolated nanostructures rather than bulk films have revealed highly variable PLE lineshapes^11,12. Even subtle variations provide valuable insight into the mechanisms of energy deposition and redistribution among electrons in metallic nanostructures. Many observed PLEs point to alternative processes, such as multiphoton absorption¹³ (MPA) and Raman scattering¹⁴ (RS) as the dominant contributions. The excitation scheme is also critical: while pulsed excitation is conventionally employed for nonlinear optics, continuous wave (CW) excitation has been shown to produce broadband emission with distinct PLE variations that appear incompatible with solely hot-electron PL alone^12,15.

Traditionally, NPL has been interpreted through hot-electron thermal models invoking elevated effective Fermi–Dirac distributions. While this framework has been useful, growing evidence indicates that it cannot fully account for the diverse PLE lineshapes observed across materials and excitation regimes. Sivan and co-workers further emphasized the crossover between thermal and nonthermal photoluminescence and showed how nonthermal carrier populations can produce distinct nonlinear scaling behavior^9,16. García de Abajo and co-workers have likewise advanced microscopic descriptions of plasmon-driven carrier generation and nonlinear emission, clarifying how nonequilibrium carrier distributions emerge from optical excitation in metallic nanostructures¹⁷. Recent results, including our prior work on metal-dielectric-metal nanogaps¹⁸, demonstrate that thermal models alone may be insufficient. We showed that precise sample geometries with tunable confinement enable engineering of carrier distributions beyond quasi-equilibrium, providing controlled access to high-k optical transitions and momentum-forbidden channels that generate significant nonthermal contributions to emission, as evident by the observed PLEs. Together, these contributions establish PLE lineshapes as more than empirical fitting curves: they provide experimentally accessible fingerprints of the carrier distribution and the excitation pathways that generate it.

The existence of such nonthermal carriers is further supported by the now well-established three-temperature model (3TM)^{14,16,19–21}, which extends the conventional two-temperature model (2TM) by introducing an explicit nonthermal carrier population within the excitation pulse alongside the hot-electron and lattice temperatures (Box 1). More rigorous approaches based on the Boltzmann transport equation (BTE) likewise predict the transient formation of nonthermal carrier distributions under strong optical excitation and indicate that these carriers can contribute measurably to radiative processes^19,22,23.

In this perspective, we propose a straightforward analytical framework for interpreting PLEs that sidesteps the need for complex simulations, such as 3TMs. Rather than serving as a replacement for detailed modeling, this approach provides an experimentally accessible means to fit PLE lineshapes and to disentangle the relative roles of thermal and nonthermal carriers. What emerges is a diagnostic signature that can be compared across materials, geometries, and excitation conditions, offering a common reference point for interpreting nonlinear light emission. To illustrate its utility, we apply the model to readily available data drawn from a broad set of recent studies, enabling direct comparison across diverse experimental contexts.

Our aim is to encourage the community to examine NPL through the prism of nonthermal carrier generation and to recognize the regimes where these signatures may appear and even dominate. Careful attention to the full PLE lineshape, including both up- and down-converted regions, is critical for distinguishing between thermal and nonthermal contributions. Framed this way, PLE analysis becomes a tool not only for clarifying past ambiguities but also for unifying observations across disparate systems. By broadening the experimental perspective in this direction, we hope to stimulate new ways of probing nonlinear emission and to deepen understanding of the role of nonthermal carriers in plasmonics and beyond.

In the remainder of this perspective, we first develop a general analytical framework for interpreting PLEs of NPL and their spectral dependence. We then derive the expected PLE lineshapes for purely thermal, purely nonthermal, and mixed carrier distributions, highlighting the distinct signatures associated with each case. Building on this foundation, we apply the framework to experimentally relevant systems, including precisely engineered plasmonic nanogaps and a broad survey of published NPL measurements spanning metals, nanoparticles, conducting oxides, and semiconductors. We further examine how experimental practices, such as spectral binning and limited spectral access, can obscure or misidentify nonlinear mechanisms. Finally, we discuss extensions to electrically driven plasmonic systems and outline practical guidelines for designing experiments and analyses that enable robust identification of nonthermal carrier contributions, concluding with an outlook on open questions and future directions.

Box 1: three-temperature model.

The 3TM¹⁹ provides a general description of energy flow in free-carrier systems following ultrafast excitation. It extends the familiar 2TM, which accounts only for coupling between the electronic and lattice temperatures, T_e and T_l, respectively²³. The 3TM introduces an additional nonthermal carrier population, N, allowing energy stored in highly non-equilibrium carriers to be treated explicitly. In this framework, N, T_e, and T_l all depend on space and time, (r, t).

A common minimal form is^14,16,20,21

$$\frac{\partial N}{\partial t}=-{\gamma }_{\mathrm{e}-\mathrm{e}}N-{\gamma }_{\text{e}-\text{ph}}N+P_{\mathrm{abs}}\left(\mathbf{r},t\right),C_e\frac{\partial T_e}{\partial t}=-\nabla \left(-{\kappa }_{\mathrm{e}}\nabla T_{\mathrm{e}}\right)-G\left(T_{\mathrm{e}}-T_l\right)+{\gamma }_{\text{e}-\mathrm{e}}N,C_l\frac{\partial T_l}{\partial t}={\kappa }_l{\nabla }^2{T}_l+G\left(T_{\mathrm{e}}-T_l\right)+{\gamma }_{\mathrm{e}-\mathrm{ph}}N,$$

where γ_e-e and γ_e-ph are the characteristic relaxation rates for nonthermal carriers via electron-electron and electron-phonon scattering, respectively. The remaining symbols denote heat capacities (C_e and C_l), thermal conductivities (κ_e and κ_l), electron-phonon coupling (G), and absorbed laser power (P_abs).

The 3TM treats nonthermal carriers as an energy reservoir N (a scalar), not as a distribution function. For this reason, the 3TM itself cannot predict any nonthermal linear or nonlinear spectral behavior. Although this framework of energy flow does not account for radiative pathways, it is central to understanding how NPL can be dominated by transient nonthermal carriers despite their intrinsically short relaxation times. In gold, nonthermal carrier lifetimes are typically on the order of 5–15 fs,^19,24 whereas thermalized carriers persist for >1 ps. A commonly adopted simplifying assumption is that nonthermal carriers do not contribute appreciably to radiation due to their short lifetimes. This assumption is strictly valid only for an infinitesimally short excitation pulse. For finite pulse durations, the 3TM explicitly shows that nonthermal carriers are continuously replenished during the pulse, leading to an effective lifetime set by the pulse envelope itself. As a result, the carrier distribution during excitation is a mixture of nonthermal and thermal populations, rather than purely thermal. This is highlighted in the diagram.

Modeling and analysis

In this section, we introduce a straightforward analytical framework for interpreting NPL measurements through their spectrally resolved PLEs. We first define wavelength-dependent PLEs and discuss how multiple nonlinear emission pathways can coexist and combine within a single spectrum. We then derive the characteristic PLE behavior expected for purely thermal carrier distributions, purely nonthermal carrier populations, and hybrid mixtures of the two. This approach allows for direct comparison across different experimental systems without requiring detailed knowledge of local field enhancements or photonic density of states. The framework provides a clear connection between measured PLE lineshapes and the underlying carrier distributions and serves as the basis for the experimental analyses that follow.

For consistency, we refer to the initial transient carrier distribution as the “nonthermal” and the thermalized high-temperature one as “thermal” throughout this article (Box 1). We note that the initial distribution has been termed “nonthermal,”¹⁶ “non-Fermi,”¹⁸ and “athermal”²⁴ in the literature. Likewise, the thermalized elevated temperature distribution is commonly referred to in the literature as the “blackbody,”¹⁶ “hot-electron,”¹ or “thermal.”¹² Both carrier distributions have sometimes been described as “hot-electron”^2,7,25–27, but here we reserve that label for a thermal Fermi–Dirac distribution at elevated temperature if any ambiguities arise.

Analysis of nonlinear photoluminescence lineshapes

An arbitrary optical emission signal, ϕ(ϵ), with an unknown nonlinear response to an excitation power P, can be expressed as

$$\varphi \left(ϵ\right)\propto P^{p\left(ϵ\right)},$$ 1

where p(ϵ) is the PLE associated with photon energy, ϵ.

In principle, multiple linear and nonlinear sources may contribute simultaneously, overlapping at the same frequency. A common occurrence of this in the literature manifests as narrowband, discrete nonlinear signals superimposed on broadband continuum backgrounds. Non-exhaustive reported examples of this include harmonic generation over broadband emission^13,28–31, χ⁽²⁾ and χ⁽³⁾ signal amalgams over frequency combs or continuum generation in fibers and photonic integrated circuit elements^32–38, and linear Raman peaks over a nonlinear background^1,18,39. In such cases, the effective PLE represents a weighted average of all underlying contributions at that frequency.

For a given reference spectrum, ϕ₀, at excitation power, P₀, the most general expression to extract the frequency-resolved PLE is given by ref. ¹¹

$$p\left(ϵ\right)=\frac{\mathrm{d}}{\mathrm{d}\log \left(\frac{P}{P_0}\right)}\log \frac{\varphi \left(ϵ\right)}{{\varphi }_0\left(ϵ\right)}~\frac{\mathrm{d}\log \varphi \left(ϵ\right)}{\mathrm{d}\log P}.$$ 2

Qualitatively, a helpful analogy is to think of the PLE as the local power-scaling behavior, or power-dependent emission efficiency, at a given wavelength (Box 2). A PLE of 1 indicates approximately linear scaling with excitation power, while a PLE of 2 indicates approximately quadratic scaling, meaning that doubling the excitation power increases the emission by a factor of four. More generally, the PLE need not be an integer. For example, a PLE of 1.5 indicates an intermediate power-law response, where the emission grows faster than linearly but more slowly than quadratically. Thus, larger PLE values indicate stronger nonlinear power scaling rather than a larger additive increase in emission. Each individual mechanism, along with the associated carrier dynamics, can have a characteristic PLE which can be extracted numerically or analytically using Eq. 2. Excellent references for PLE lineshapes of various mechanisms can be found in the main text and supplementary information of ref. ¹¹ and in the supplementary information of ref. ¹⁸.

Box 2: nonlinear photoluminescence measurements.

NPL measurements are performed by recording emission spectra while systematically varying the excitation intensity over a wide dynamic range. Rather than fitting a single emission spectrum, the emission at each wavelength is analyzed as a function of input power, yielding a wavelength-resolved PLE that directly encodes carrier dynamics.

In linear emission measurements, both thermal and nonthermal photoluminescence are strongly weighted by the photonic density of states, ρ(ϵ), commonly approximated by the scattering spectrum of the resonant mode. For a single resonance, this is often well described by a Lorentzian,

$$\rho \left(ϵ\right)=\frac{1}{1+{\left(\frac{ϵ-{ϵ}_r}{\mathrm{\Gamma }/2}\right)}^2},$$

where ϵ_r is the resonance energy and Γ the linewidth. The observed emission spectra are therefore convolutions of ρ(ϵ) with the carrier-dependent electron-hole overlap, making separation of thermal and nonthermal contributions contingent on high-fidelity knowledge of the scattering response. In practice, noise, linewidth uncertainty, and sample-specific imperfections can obscure these distinctions. To reflect realistic experimental conditions, all simulated spectra shown here include artificial noise corresponding to a minimum signal-to-noise ratio of ~13 dB.

Simulated spectra illustrate that changes in linewidth (Γ = 0.091 and 0.045 eV) alone can strongly alter emission lineshapes. Nonthermal emission can appear red-shifted and skewed relative to the plasmon resonance, effects that may be misattributed to structural disorder or inhomogeneity. Even with accurate scattering information, thermal and nonthermal spectra can become nearly indistinguishable for narrow resonances, despite substantial effective temperatures (T_eff ~ 2200 K) and finite nonthermal populations (Δρ_nt ~ 1%).

By contrast, nonlinear PLE lineshapes provide a robust, scattering-independent diagnostic. Because they depend only on how emission scales with excitation intensity, static linear weighting by ρ(ϵ) is averaged out. As shown, the PLE uniquely identifies thermal, nonthermal, and hybrid carrier distributions across resonance linewidths, enabling direct extraction of internal state parameters such as electron temperature and nonthermal carrier amplitude.

Thermal carrier emission spectra and nonlinearity

One of the simplest models for estimating PL spectra emitted from excited carriers is given by the overlap integral between electron and hole occupancies in states separated by the photon energy^1,11,

$${\varphi }_{\mathrm{th}}\left(ϵ\right)\propto {\int }_{-\infty }^{\infty }{f}_e\left({ϵ}^{\prime },T_{\mathrm{eff}}\right)\rho \left(ϵ\right)f_h\left({ϵ}^{\prime }-ϵ,T_{\mathrm{eff}}\right)\mathrm{d}{ϵ}^{\prime },$$ 3

where f_e is the electronic Fermi distribution, f_h = 1 − f_e is the hole distribution, ρ is the photonic density of states and T_eff is the effective electronic temperature.

A key assumption in Eq. 3 is the lifting of wavevector restrictions, a commonly invoked and experimentally validated assumption when strong optical confinement or sharp field gradients are present^1,4,11,29,40. In nanoscale metallic structures, optical absorption can be strongly assisted by metal-dielectric interfaces and pronounced spatial nonuniformities of the electromagnetic field. On a quantum-mechanical level, this mechanism is commonly described as Landau damping, which broadly refers to optically induced intraband transitions in which momentum conservation is enabled by large spatial-frequency components of the optical mode. Such high wavevector components arise naturally from field discontinuities, interfaces, and strongly confined geometries, thereby relaxing momentum selection rules that would otherwise suppress IBPL in bulk metals^41–45. Under these conditions, wavevector (momentum) restrictions on indirect transitions within the conduction band can be lifted to a good approximation, providing a physical basis for the assumption employed in Eq. 3.

For a simple Fermi–Dirac distribution, Eq. 3 reduces to ϵρ(ϵ)f_B(ϵ, T_eff), where f_B is a Bose–Einstein distribution. For a thermal distribution with a large effective temperature, T_eff scales with the excitation power as

$$T_{\mathrm{eff}}=T_0{\left(\frac{P}{P_0}\right)}^{1/a},$$ 4

where the subscript ₀ denotes the reference temperature (room temperature) and power, respectively, and a is the effective thermal power coefficient, typically in the range a = 2–5, depending on the type of plasmonic metal and nanostructure geometry. In the simplest equilibrium picture of a free-electron gas with linear electronic heat capacity, standard solid-state theory predicts a value a = 2⁴⁶, and this value can serve as a fixed assumption in data analysis¹¹. In practice, nanoscale corrections, nonlinear absorption, and energy flow during the pulse result in modifications to a, and it should be treated as an effective parameter. Nanoparticle silver films display an average a = 4.57 with values ranging from 2 to 7 across the film due to random surface feature variations¹. Likewise, gold films have had reported values of a = 3.2⁴⁰ and a = 3.3⁴, and aluminum nanorods have shown a = 3.85⁴⁷. Patterned thin films of transparent conductive oxides, such as indium tin oxide (ITO), exhibit a ≈ 3.5–5, showing correlation with the nominal FIB milling dose and attributed to an increased level of modification of the electronic structure of pristine ITO⁴.

Substituting Eq. 3 into Eq. 2 gives

$$p_{\mathrm{th}}\left(ϵ\right)=\frac{ϵ}{a{k}_B{T}_{\mathrm{eff}}}\left[1+f_B\left(ϵ,T_{\mathrm{eff}}\right)\right]\approx \frac{ϵ}{a{k}_B{T}_{\mathrm{eff}}},$$ 5

which holds in the limit of k_BT_eff ≪ ϵ. The carrier distribution and characteristic PLE lineshape for thermal carriers are shown in Fig. 1a, b. The physical origin of the thermal PLE lineshape can be understood from the carrier distributions that generate the emission. In the thermal case, the emission is governed by the high-energy tail of a thermalized Fermi–Dirac distribution. Increasing T_eff broadens the thermal carrier distribution, increases the range of available carrier occupancies and hole vacancies, and therefore lowers the effective threshold for higher-energy photon emission upon recombination. Lower- to mid-energy photons are also emitted more efficiently for the same reason. These high-energy tails scale approximately exponentially with temperature¹⁹; since the temperature increases with excitation power, the resulting PLE varies approximately linearly with photon energy, as described by Eq. 5. The characteristic flattening of the thermal PLE slope with increasing T_eff represents an overall increase in emission efficiency and yield.

Fig. 1. Carrier distributions and PLE lineshape dependence on hybrid-carrier parameters.

a Occupancy probability and b PLEs of purely thermal carrier distributions with T_eff = 2000–3000 K, and purely nonthermal carrier distributions with $\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0=0.01-0.1$. c Occupancy probability and d PLEs for purely nonthermal spectral range dependence on excitation laser energy for 650–930 nm. e Occupancy probability and f PLEs for varying thermal to nonthermal spectral contribution weights.

This PLE framework has been applied to emission from thin films, particle aggregates, and roughened surfaces where extreme field localization relaxes momentum selection rules^{1,7,10,25–27,48}. A key strength of this approach is its independence from the local photonic density of states. In practice, this is highly advantageous because extracting the local temperature of a system from a single emission spectrum typically requires additional absorption or dark-field measurements, or estimates of local field enhancement, all of which can vary significantly between systems. Nonlinear measurements circumvent these requirements, relying only on controlled variation of the input excitation field. In fact, merely fitting PLEs to aT_eff has been very successful in capturing this telltale nonlinear signature^1,4,11,29.

Nonthermal carrier emission spectra and nonlinearity

Nonthermal contributions to the PLE arise from ultrafast excitation processes that populate carriers far from equilibrium, producing emission signatures that cannot be captured by a single effective temperature and instead manifest as distinct spectral PLE lineshapes. To describe this regime, we adopt the standard nonthermal carrier distribution introduced in ref. ¹⁹:

$$f_{\mathrm{nt}}\left(ϵ\right)=f_0\left(ϵ\right)+\mathrm{\Delta }{\rho }_{\mathrm{nt}}\left(ϵ\right),$$ 6

where the second term accounts for the ultrafast initial response and produces a step-like transient carrier distribution. The nonthermal correction is expressed as

$$\mathrm{\Delta }{\rho }_{\mathrm{nt}}\left(ϵ\right)=\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0\left\{f_0\left(ϵ-\ell \right)\left[1-f_0\left(ϵ\right)\right]-f_0\left(ϵ\right)\left[1-f_0\left(ϵ+\ell \right)\right]\right\},$$ 7

where f₀ is the equilibrium distribution prior to absorption (T = T₀), $\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0$ is the amplitude of the population change determined by the excitation power and ℓ is the photon energy of the pump laser. Energies are measured from the Fermi level, ϵ_F = 0. The shape of the nonthermal carrier distribution and its dependence on $\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0$ and ℓ are shown in Fig. 1a, c, respectively. Equation 7 describes the leading-order nonthermal contribution to the emission process and does not explicitly include sequential MPA pathways. At emitted photon energies approaching and exceeding approximately three times the excitation photon energy, higher-order nonthermal processes can become significant, leading to integer-valued PLEs (n = 3, 4,...) and comparatively flat PLE spectra, as shown in ref. ¹⁶. A complete treatment of these sequential absorption processes lies beyond the scope of the present perspective, but they represent an important extension of the nonlinear nonthermal framework discussed here.

Equation 7 can be rewritten in terms of hyperbolic functions (see Supplementary Information) and inserting Eq. 6 into Eq. 3 yields

$${\varphi }_{\mathrm{nt}}\left(ϵ\right)\propto \rho \left(ϵ\right)\left[A\left(ϵ\right)+B\left(ϵ\right)\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0+C\left(ϵ\right){\left(\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0\right)}^2\right],$$ 8

where, A, B, and C are weighting functions. These functions depend only on experimentally controllable parameters: A(ϵ; T₀), B(ϵ; ℓ, T₀), and C(ϵ; ℓ, T₀). Greater discussion and explicit forms are provided in the Supplementary Information Section “Full mathematical expressions for nonthermal emission spectra.” It is important to note that nonthermal emission spectra, like their thermal counterparts, are weighted by the local photonic density of states, ρ(ϵ). Because ρ(ϵ) is highly system-dependent, raw emission spectra alone cannot be used to reliably identify thermal or nonthermal signatures without detailed knowledge of the specific photonic environment. Instead, the photonic density of states should be viewed as an effective coupling channel through which the underlying carrier distributions optically transition, rather than as a direct fingerprint of the carrier statistics themselves.

For any experimentally chosen laser energy ℓ and ambient temperature T₀, which remain fixed during an NPL experiment, the spectral weights A(ϵ), B(ϵ), and C(ϵ) are unique functions of ϵ. Importantly, they are independent of the thermal and nonthermal carrier state variables T_eff and $\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0$, since they are determined solely by ℓ and T₀, and therefore have no intrinsic power dependence. Consequently, the characteristic PLE for a nonthermal distribution depends only on the amplitude term $\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0$. The nonthermal amplitude $\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0$ introduces power dependence, and scales linearly with excitation power¹⁹,

$$\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0\propto P,$$ 9

where the proportionality is related to the absorbance. Substituting Eq. 8 into Eq. 2 yields the PLE for purely nonthermal emission:

$$p_{\mathrm{nt}}\left(ϵ\right)=\frac{\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0}{{\varphi }_{\mathrm{nt}}\left(ϵ\right)}\frac{\mathrm{d}{\varphi }_{\mathrm{nt}}\left(ϵ\right)}{\mathrm{d}\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0}\approx \frac{1+2R\left(ϵ\right)\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0}{1+R\left(ϵ\right)\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0},$$ 10

where R(ϵ) = C(ϵ)/B(ϵ), and we have used ∣A(ϵ)∣ ≪ ∣B(ϵ) + C(ϵ)∣ outside the low-energy limit for simplification. The characteristic lineshape described by Eq. 10 reproduces key features observed across numerous studies of NPL (Fig. 1b, d). This is discussed in greater detail in Section “Experimental validation.”

Compared to the thermal PLE, the nonthermal case is qualitatively different. Here, absorption of a pump photon with energy ℓ creates a step-like perturbation to the equilibrium carrier distribution over an energy window of 2ℓ (Fig. 1c), with an amplitude controlled by $\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0$ (Fig. 1a). Emission from this structured nonequilibrium distribution produces the step-like PLE shown in Fig. 1b, which approaches p ≈ 1 and p ≈ 2 at the low- and high-energy ends, respectively. At lower emitted photon energies, the available electron-hole overlap samples an approximately uniform portion of the nonthermal population, so the emission scales nearly linearly with excitation power and approaches p ≈ 1. At higher emitted photon energies, approaching the maximum electron-hole separation enabled by the nonthermal distribution, emission requires two pump-driven population changes and therefore approaches p ≈ 2. The strongest variation occurs near the excitation energy, where the electron-hole overlap changes most rapidly between these two limiting regimes, and this region is most sensitive to changes in the nonthermal amplitude. For the same reason, the characteristic PLE spectral range follows the excitation energy (Fig. 1d).

Hybrid carrier emission spectra and nonlinearity

For intermediate cases, such as single particles under pulsed excitation, the observed emission falls between the PLE behaviors described by Eqs. 5 and 10. This suggests that the spectra originate from a hybrid carrier distribution containing both thermal and nonthermal components, which can be approximated as¹⁸

$${\varphi }_{\mathrm{th}+\mathrm{nt}}\left(ϵ\right)\propto \alpha {\varphi }_{\mathrm{th}}\left(ϵ\right)+\left(1-\alpha \right){\varphi }_{\mathrm{nt}}\left(ϵ\right),$$ 11

where 0 ≤ α ≤ 1 defines the relative weight of the thermal contribution and 1 − α corresponds to the nonthermal contribution.

It is important to note that the PLE of such a hybrid distribution cannot be obtained simply by combining Eqs. 5 and 10 with weights α and 1 − α. Because p(ϵ) is defined on a logarithmic scale, the contributions of each term are not preserved by linear weighting. To correctly determine the PLE, Eq. 11 must be substituted directly into Eq. 2 giving

$$p_{\mathrm{th}+\mathrm{nt}}\left(ϵ\right)=\frac{\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0}{{\varphi }_{\mathrm{th}+\mathrm{nt}}\left(ϵ\right)}\left[\alpha \frac{\mathrm{d}{\varphi }_{\mathrm{th}}\left(ϵ\right)}{\mathrm{d}\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0}+\left(1-\alpha \right)\frac{\mathrm{d}{\varphi }_{\mathrm{nt}}\left(ϵ\right)}{\mathrm{d}\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0}\right].$$ 12

The nonthermal contribution has already been derived in the previous section, and its derivative term in Eq. 12 follows directly. However, the thermal contribution requires a relationship between T_eff and $\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0$. Using Eqs. 9 and 4, this connection can be expressed as

$$T_{\mathrm{eff}}=\gamma {\left(\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0\right)}^{1/a},$$ 13

where the constants have been combined into a single parameter, $\gamma =T_{\mathrm{eff,0}}{\left(\beta /P_0\right)}^{1/a}$. Substituting this relation, the thermal term in Eq. 12 becomes

$$\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0\alpha \frac{\mathrm{d}{\varphi }_{\mathrm{th}}\left(ϵ\right)}{\mathrm{d}\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0}=\alpha \rho \left(ϵ\right)\frac{{ϵ}^2}{4a{T}_{\mathrm{eff}}}{\mathrm{csch}}^2\left(\frac{ϵ}{2T_{\mathrm{eff}}}\right).$$ 14

Thus, fitting a hybrid distribution requires only three parameters: the relative weight α, and the two state variables T_eff and $\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0$. The full expression for the PLE of a hybrid carrier distribution, which may be easily implemented for fitting experimental data, is given in the Supplementary Information Section “Full PLE for hybrid thermal and nonthermal carrier distributions.”

The emission depends on the relative spectral weights of the thermal and nonthermal distributions (Fig. 1e), but the PLE is defined through a logarithmic derivative of the total emission intensity and therefore does not combine in a simple way analogous to Eq. 11. The hybrid case interpolates between these two limiting behaviors, but not through a simple linear average of the thermal and nonthermal PLEs. As a result, a hybrid distribution can appear nearly thermal over one spectral range while retaining nonthermal curvature or asymmetry elsewhere (Fig. 1f). This is particularly important when the thermal component has a large T_eff, since the thermal PLE becomes flatter and can partially mask the step-like structure of the nonthermal contribution. The parameter α helps address this overlap by capturing the relative thermal-to-nonthermal spectral weight, while T_eff and $\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0$ retain their roles as the characteristic thermal and nonthermal state parameters, respectively. In practice, this can be problematic for datasets that do not cover the full PLE spectral range. This highlights the importance of analyzing the complete PLE lineshape, rather than relying on limited spectral windows, such as only the up- or down-converted region. This behavior is illustrated in Fig. 1f, where the hybrid PLE transitions nonuniformly from the linear thermal limit to the step-like nonthermal limit as the relative nonthermal contribution increases. If only up-converted emission is recorded, a true value of α = 0.95 could easily be mistaken for purely thermal emission with no nonthermal contribution at all.

Experimental validation

In this section, we apply the analytical framework developed above to a range of experimental NPL measurements in well-defined systems. We begin by examining geometries where carrier generation and relaxation can be systematically controlled, enabling direct comparison between expected and observed PLE behavior. We then extend the analysis to previously reported experiments from the literature, highlighting cases where the framework captures both thermal and nonthermal contributions. Finally, we discuss electrically driven and hybrid optoelectronic platforms in which related analyses have been applied, primarily within thermal carrier models, and comment on the limited but emerging evidence for nonthermal contributions in these systems. Together, these examples illustrate the applicability and current limitations of the approach across diverse material systems and excitation conditions.

Gap-mode control of nonlinear photoluminescence

In our recent work¹⁸, we investigated the origin of NPL from plasmonic metal-dielectric-metal nanogaps with nanometer-scale confinement. By precisely controlling the gap thickness and tuning the plasmonic resonance (Fig. 2a), we showed that strong field confinement in ultra-narrow gaps leads to a breakdown of the dipole approximation, enabling large-momentum electronic transitions within the metal. The resulting sharp field gradients greatly enhance intraband absorption and efficiently generate nonthermal carrier distributions. These populations dominate the emission process, particularly under resonant excitation, in direct contrast to conventional interpretations based on hot-electron thermalization. The enhanced Landau damping associated with these transitions provides a natural explanation for the observed PLEs and their spectral dependence, which cannot be reproduced by two-photon photoluminescence, Raman processes, or purely thermal hot-electron emission.

Fig. 2. Results for gap-dependent NPL as presented in ref. ¹⁸.

a Gap-plasmon mode resonance matching the excitation field. b Robust fitting of hybrid carrier distribution. c Fit parameters with robust fitting show excellent agreement with the previous numerically optimized fitting approach; increased error bars at the smallest gap reflect a lower signal-to-noise ratio.

To interpret this behavior, our earlier analysis of individual experimental PLE lineshapes (Fig. 2b), revealed consistent trends: both T_eff and $\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0$ increase as the gap thickness decreases, while the thermal component becomes negligible under resonant excitation (Fig. 2c). These observations underscore the importance of enhanced Landau damping and nonthermal carrier populations in gap-mode emission^41–45.

The gap-mode plasmon data shown in Fig. 2 highlight an important limitation of qualitative analysis of PLE lineshapes. The smallest-gap PLE can appear nearly linear by visual inspection, which would ordinarily suggest a predominantly thermal carrier distribution. Its relatively flat slope could also be interpreted as a thermal contribution with a large effective temperature. However, closer inspection shows subtle curvature and asymmetry in the data, indicating that the lineshape is not purely thermal. This illustrates a central motivation for the hybrid fitting approach: visual inspection alone can be insufficient for distinguishing thermal and nonthermal contributions.

In this case, the best fit is obtained from a mixture of a high-temperature thermal component and a significant nonthermal component. The apparent linearity, therefore, arises from the combined response of both carrier populations rather than from a purely thermal distribution. The extracted nonthermal contribution increases with decreasing gap thickness, even when the resulting PLE appears superficially thermal-like over a limited spectral range. What is needed now is a common analytical framework that allows such results to be compared across systems and conditions, making it possible to identify where nonthermal contributions are most pronounced.

In the previous sections, we recast the hybrid model into a fully analytical form that allows robust and reproducible fitting of experimental data with standard optimization routines. Although the model is nonlinear and involves some parameter correlations, the resulting fits reproduce experimental PLE lineshapes with high fidelity (Fig. 2c). Just as importantly, the extracted parameters remain consistent with values reported previously, which builds confidence in this approach and opens the door to systematic comparisons across different materials, geometries, and excitation conditions.

These results reinforce the original conclusions through a more rigorous and transparent fitting framework. The hybrid description captures the essential physics of the system while enabling broader application and comparison across platforms. As discussed in the next “Literature survey” Section, its consistency with previously published data strengthens the physical interpretation proposed earlier and, at the same time, enhances reproducibility and accessibility for future studies. Just as importantly, this framework provides a practical tool for designing experiments that can isolate and probe nonthermal carriers with greater accuracy.

Literature survey

Our theoretical framework was applied to gap-mode plasmonic systems under varying confinement, resonance, and fluence conditions, showing excellent agreement with experimental observations. Considerable effort has gone into interpreting PLE lineshapes and the physical mechanisms they imply. A central open question remains whether a universal mechanism governs these behaviors or whether they arise from distinct case-specific processes that require separate explanations.

In practice, multiple mechanisms often coexist and overlap, which is entirely plausible. The challenge is that, without careful disentangling, one mechanism can mimic the signature of another, obscuring the dominant contributor. This difficulty is compounded by the common practice of binning intensity counts before extracting PLEs, either because of low signal counts or because traditional analysis assumes the PLE spectrum is wavelength-independent, as in harmonic generation or multiphoton processes.

Closer scrutiny of prior work shows that binning can mask critical features, leading to apparent deviations from integer scaling exponents^1,10. When analyzed spectrally, a distinct PLE lineshape consistently emerges. The framework presented here unifies thermal and nonthermal contributions and provides a consistent basis for interpreting these results. It is also important to emphasize that the absence of a linear PLE lineshape is not definitive evidence against IBPL, an assumption often made in favor of Raman-based interpretations¹⁴. In reality, RS and IBPL can produce very similar emission profiles, making it essential to rely on multiple signatures to separate their contributions.

Consequently, we compare our hybridized carrier PLE framework in Section “Hybrid carrier emission spectra and nonlinearity” with a wide range of PLE spectra reported in the literature^{1,4,11,12,14,15,29,31,40,47,49,50}. For studies reporting power-dependent emission spectra or calculated PLEs, we accessed the published figures and extracted the underlying spectral data using calibrated digitization of the axes scales, allowing us to apply the proposed PLE analysis directly to the reported measurements. While this approach is necessarily limited by the resolution and presentation of the published data, it is sufficient to assess the qualitative and semi-quantitative PLE behavior discussed here.

As shown in Fig. 3 (top panel), there is strong agreement between our predictions and previously published data, underscoring both the relevance and generality of the approach. For clarity, the fits are grouped into thermal-dominant, nonthermal-dominant, and intermediate cases. In thermal-dominant cases (α ≈ 1), a linear slope is the most prominent feature by visual inspection, consistent with expectations for predominantly thermal emission. This trend is further illustrated in Fig. 1f, where the hybrid PLE approaches the characteristic linear thermal limit as α → 1, indicating that emission is dominated by thermal carriers. Nevertheless, even in these thermal-dominant regimes, small but finite nonthermal contributions remain, as reflected by fitted values α ≲ 1. Qualitatively, this appears as a slight asymmetry between the up- and down-converted regions relative to an idealized linear slope. Conversely, in nonthermal-dominant cases (α ≈ 0), the step-like PLE lineshapes are sufficiently pronounced that hybrid modeling becomes unnecessary and a purely nonthermal description is adequate. Across the majority of studies surveyed, substantial nonthermal contributions are evident, and at a minimum, a nonnegligible nonthermal component can be consistently identified.

Fig. 3. Hybrid thermal and nonthermal PLE fits applied to various NPL experiments in the literature.

The results are grouped into three categories: thermal dominant (α ≈ 1), nonthermal dominant (α ≈ 0), and intermediate (α closer to the midpoint, or spectrally limited data sets), corresponding to each of the three columns of subplots, respectively. Raw or extracted data and the corresponding fits are shown in the top panel as labeled. The corresponding hybrid PLE fitting parameters for thermal dominant (refs. ^1,4,11), nonthermal dominant (refs. ^{11,12,71,14,50}), and intermediate (refs. ^{40,15,31,47,29,50}) are given in the lower panels. The references listed in the corresponding categories are given from left to right as they appear in the figure.

Several key observations emerge from this comparison. The step-like transition from p ≈ 1 to p ≈ 2, widely reported in nanoparticle studies, is reproduced with accuracy, including not only the asymptotic values but also the intermediate slope near the laser energy. For example, in ref. ¹⁴, a flat PLE of p ≈ 2 was observed in the up-converted region, with no simultaneous down-converted data. While this was interpreted as evidence for a Raman-based mechanism, our analysis and existing theoretical frameworks¹⁶ show that such a slope is also consistent with nonthermal IBPL (Fig. 1b, f). Only full spectral analysis across both regions can definitively rule out IBPL when nonthermal carriers are considered, as each region has a distinct lineshape and asymptotic behavior. Even partial access to the down-converted side, as in refs. ^40,29, reveals this contrast: the up-converted region appears linear, while the down-converted side clearly deviates. This emphasizes a critical point for future experiments: both the up- and down-converted regions must be measured, either with a notch filter or by sequential sampling. Without both regions, essential information about the carrier distribution is lost, leading to misinterpretations.

Continuous-wave excitation provides another instructive case. In refs. ^12,15, PLE spectra show pronounced asymmetry between the up- and down-converted regions. Although the down-converted side approaches p ≈ 1, the up-converted side begins to transition but does not reach p ≈ 2, likely due to low signal counts and restricted spectral range. Hot RS has been proposed as an alternative explanation, as it can mimic similar lineshapes¹¹. However, when nonthermal carriers are included, IBPL provides a comparably good description of the CW spectra. Why a process usually associated with intense ultrafast fields should reproduce quasi-steady-state CW data remains an open question, although recent theoretical efforts have begun to address this connection in terms of nonthermal carriers⁹.

Electrical control of nonlinear photoluminescence

The emergence of on-chip nanoscale devices and applications has created a growing need to control and directly drive nanoscale light emission using electrical methods. For instance, optically excited NPL emission from Au nanowires⁵⁰, which we analyzed using our PLE framework to gain insight into carrier distributions (see Supplementary Information Section “Electrical control of nonthermal NPL emission case study”), can be electrically modulated by integrating the nanowire with electrodes^5,51,52. In this experiment, excitation at one end of the nanowire launches a propagating surface plasmon that generates NPL along the wire length. Applying the hybrid carrier model to the position-dependent PLE spectra provides a dynamic carrier analysis that reveals that the nonthermal contribution is not spatially uniform. The extracted hybrid carrier PLE fits are shown in Fig. 3 for three selected points: excitation input, surface plasmon propagation midway, and termination output positions (see Supplementary Information Fig. 1 for full PLE analysis of this study). The fitted nonthermal amplitude, $\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0$, increases toward the nanowire termination, where the strongest nonthermal amplitude is observed in addition to strong thermal components. This case study highlights how spatial mapping of NPL via PLE analysis can provide a route to controlling and identifying nonthermal emission channels.

Beyond electrically modulating optically excited emission, electrically excited nanoscale light emission has attracted significant attention for ultra-compact devices. Notably, plasmonic modes can be driven electrically^53,54. Inelastic tunneling electrons in biased junctions can induce surface plasmons^8,55,56, making plasmonic tunnel junctions an attractive platform for nanoscale light-matter interactions^57,58. These junctions can function as compact light sources with ultrafast temporal response and broadly tunable emission spectra^59–62, and they naturally generate plasmon-induced hot carriers. Hot-carrier generation in tunnel junctions is often accompanied by above-threshold light emission, where the emitted photon energy exceeds the inelastic electron energy loss, providing a pathway for electroluminescence (EL) beyond the bias voltage limit. Multiple mechanisms have been proposed to explain this emission, including blackbody-like radiation from hot electron gases heated by collisions⁶³, recombination of hot carriers generated by nonradiative plasmon decay^64,65, and emission from thermalized electrons produced by plasmon decay⁶⁶. These distinct processes imply differences in the underlying carrier distributions. Furthermore, combining electrical and optical excitation can significantly enhance EL intensities⁶⁷, and electrical control has been shown to modulate NPL associated with hot electrons^5,49.

Although emission spectra can provide an estimate of the effective electronic temperature, the equilibrium or nonequilibrium nature of the carriers remains poorly understood, which makes it difficult to distinguish between thermal and nonthermal contributions in tunneling systems and electrically controlled systems. Advances in junction design and measurement have begun to address this challenge by revealing hot-electron dynamics⁶⁸, mapping carrier distributions⁶⁹, and even proposing criteria to separate thermal and nonthermal populations³. Yet these approaches still fall short of directly correlating EL signals with the underlying carrier distributions. First, such electrical measurements typically require sophisticated instrumentation or carefully engineered junctions to access carrier-distribution information, which may be incompatible with junctions optimized for optical performance. Second, hot carriers generated through plasmon decay cannot be fully captured by electrical probes due to their broad energy spectra as well as the complex propagation and distribution of plasmons. However, thermal and nonthermal hot carriers exhibit distinct mechanisms and properties, leading to fundamentally different functionalities in practical applications. For example, nonthermal hot carriers possess significantly shorter lifetimes compared to their thermal counterparts, enabling them to support ultrafast modulation rates. Furthermore, in plasmonic photocatalysis, thermal and nonthermal carriers can drive distinct reaction pathways⁷⁰. Many potential applications relying on electrically excited or controlled hot carriers depend intrinsically on these unique carrier distributions. Thus, a more direct and quantitative optical approach is required to accurately characterize the underlying carrier distributions. Our PLE analysis framework offers a promising route to infer hot-carrier signatures from emission spectra and to identify the distinct roles of thermal and nonthermal populations. Analyzing these parameters is essential for understanding NPL, EL, and above-threshold light emission, as well as for leveraging these emitters as on-chip nanoscale light sources. At the same time, applying this framework to electrically controlled systems introduces new challenges. Both plasmon intensity and energy can depend on the applied bias, complicating the extraction of spectral exponents. Determining the effective excitation power requires careful consideration because multiple mechanisms and intermediate processes can contribute to the observed response. Moreover, disentangling EL contributions from inelastic electron tunneling and those arising from hot carriers remains difficult. Addressing these challenges will require further optimization and adaptation of the PLE approach to electrically driven systems.

Discussion

In this perspective, we have outlined a simple yet versatile analytical framework for extracting the PLE of NPL associated with both thermal and nonthermal carrier distributions. In contrast to approaches that require complex dynamical simulations, such as 3TMs, this formulation offers an experimentally accessible way to fit PLE lineshapes and directly extract state variables and relative contributions. Even without time-resolved measurements, it enables robust and intuitive interpretation of carrier populations.

The framework aligns with physical intuition regarding enhanced Landau damping and momentum-lifting transitions, and its generality extends across a broad survey of films, nanoparticles, conducting oxides, and semiconductors. In some systems, nonthermal contributions appear small but consistently detectable. In others, they dominate to such an extent that hybrid modeling becomes unnecessary. Even in cases long attributed to purely thermal processes, closer inspection often reveals subtle nonthermal components.

From this perspective, several guidelines for future experiments become clear (Box 3). First, full PLE lineshapes must be measured and analyzed, since they average over photonic density of states, local field effects, and resonance complexity that otherwise obscure interpretation. Second, spectra should be analyzed wavelength by wavelength, without binning, to preserve spectral variation in the nonlinear response. Third, both the up- and down-converted regions must be sampled together, as only the joint asymptotic behavior provides conclusive evidence for or against nonthermal carriers. A flat slope in the up-converted region alone is not sufficient to exclude IBPL, nor is a linear slope enough to conclude a purely thermal origin.

Beyond these methodological guidelines, the framework opens specific scientific and technological directions that the community is now positioned to pursue.

Material systems beyond noble metals. The vast majority of NPL studies, including most entries in our survey, focus on gold and silver. Aluminum, transparent conducting oxides, transition metal nitrides, and doped semiconductors have markedly different band structures, electron-phonon coupling strengths, and screening properties. These differences should produce quantitatively distinct PLE lineshapes within the same hybrid framework, providing a route to benchmark material-specific carrier dynamics on a common footing. Systematic PLE surveys across these platforms would test whether the hybrid description remains predictive when the underlying density of states departs strongly from the free-electron approximation.

Connection to time-resolved measurements. The hybrid PLE extracts state parameters from steady-state nonlinear scaling but does not measure carrier lifetimes directly. Pairing PLE analysis with pump-probe spectroscopy on the same nanostructure would allow the extracted nonthermal amplitude $\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0$ and effective temperature T_eff to be cross-validated against directly measured relaxation times. This combination would also test the boundary where the leading-order nonthermal correction ceases to describe the data and where higher-order or sequential absorption pathways must be added explicitly.

CW-pulsed correspondence. The similarity between CW and pulsed PLE lineshapes remains one of the most striking open observations in the literature survey. The present framework complements recent treatments of steady-state NPL^9,16 by providing a compact way to quantify the thermal-to-nonthermal balance directly from measured PLE spectra. Here, $\mathrm{\Delta }{\rho }_{\mathrm{nt}}^0$ captures the nonthermal amplitude, while α captures the time-integrated relative nonthermal emission weight. Because the instantaneous nonthermal amplitude and its integrated radiative contribution need not scale together, the two parameters can be applied independently to compare CW and pulsed excitation, or to study pulse-duration effects at fixed peak power. Tunable CW measurements and controlled pulse-duration studies offer a direct experimental test.

Geometric design for selective carrier generation. Our earlier work on gap-mode plasmons demonstrated that confinement controls the relative weight of thermal and nonthermal carriers. The hybrid PLE framework now makes it possible to use this control as an engineering parameter rather than a curiosity. Metasurfaces, tip-enhanced geometries, and patterned films could be designed with PLE lineshapes as the target observable, enabling inverse design of nanostructures that selectively maximize nonthermal carrier output for a chosen photon energy. This is particularly relevant for hot-carrier photochemistry, where nonthermal and thermal carriers drive distinct reaction pathways and where the ability to bias one population over the other has direct functional consequences.

Electrically driven and hybrid optoelectronic platforms. Plasmonic tunnel junctions and gate-controlled plasmonic gaps have shown that nonthermal carrier emissivity can be modulated electrically. Extending PLE analysis to these systems requires care because bias voltage modifies both the excitation amplitude and the local density of states, and because EL contributions from inelastic tunneling overlap with hot-carrier emission. A bias-resolved PLE methodology, in which the effective excitation is calibrated against the tunneling current and the spectral exponent is extracted at fixed photon energy as a function of bias, would provide a direct optical readout of the carrier distribution under electrical drive. This is, in our view, the most consequential near-term application of the framework, because it connects nonthermal carrier diagnostics to functional on-chip devices.

We emphasize that the analytical framework presented here is not a substitute for microscopic theory. Kinetic descriptions based on the BTE, ab initio treatments of band-resolved scattering, and the full 3TM remain essential for predictive modeling of carrier dynamics. What the hybrid PLE provides is complementary: a low-overhead diagnostic that converts a standard power-dependent measurement into quantitative carrier-state information in a form that can be applied uniformly across the literature. The agreement between this simple description and a wide range of published data, including CW measurements that have resisted conventional interpretation, suggests that PLE lineshapes report on a more universal feature of nonequilibrium emission than has been generally recognized.

The broader implication is that nonthermal carriers are not a curiosity confined to specific geometries or excitation conditions. They appear, often subtly, across the full range of plasmonic systems where NPL has been measured. The question to ask of every NPL dataset is therefore not whether the carriers are thermal or nonthermal, but in what ratio. Framed this way, the next generation of NPL experiments has the opportunity to move from cataloguing emission to controlling it.

Box 3: guidelines for generalizable NPL experiments.

Below, we highlight three key points for the advantages of NPL measurements and guidelines for robust and conclusive analysis:

1. Circumventing system-specific idiosyncrasies—A single emission spectrum depends crucially on the local photonic density of states when fitting to extract internal state parameters of the system. This typically requires additional absorption or dark-field measurements, or estimates of local field enhancement, all of which can vary significantly between systems. Nonlinear measurements bypass these requirements, relying only on controlled variation of the input excitation field.

2. Discriminating nonlinear contributions—In general, a plethora of nonlinear sources are capable of being present in PL spectra of a given experiment. When these signals overlap at the same frequency, their nonlinear scaling adds in non-trivial ways. Spectral binning over a large frequency range compounds this issue further and can even mask nonlinear behavior altogether. When sufficient emission counts permit, spectra should be analyzed wavelength by wavelength to recover a PLE, without binning, to preserve spectral variation in the nonlinear response.

3. Towards ambiguity-free diagnosing—Careful attention to the full PLE lineshape, including both up- and down-converted regions, is critical for distinguishing between thermal and nonthermal contributions. Both the up- and down-converted regions must be sampled together, as only the joint asymptotic behavior provides conclusive evidence for or against nonthermal carriers. A flat slope in the up-converted region alone is not sufficient to exclude IBPL, nor is a linear slope enough to conclude a purely thermal origin.

Author contributions

All authors contributed to the conceptual framing, discussion of results, and manuscript revision. R.L. led the discussion and development of the analytical framework and literature survey, and performed the fitting analysis. Y.T. led the discussion of electrically controlled plasmonic systems. R.L. and Y.T. prepared and revised the figures. H.H. supervised the project and contributed to the interpretation and outlook discussion. All authors approved the final version.

Peer review

Peer review information

Nature Communications thanks Alexandre Bouhelier and the other anonymous reviewers for their contribution to the peer review of this work

Data availability

The data analyzed in this perspective were extracted from previously published literature sources cited in the manuscript. Digitized data and fitting results are available from the corresponding author upon reasonable request.

Code availability

The analytical expressions required to reproduce the PLE fitting procedure are provided in the main text and in the Supplementary Information. The fits were performed using standard nonlinear least-squares routines in MATLAB. Additional fitting scripts used to generate the plotted results are available from the corresponding author upon reasonable request.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains Supplementary material available at 10.1038/s41467-026-75070-0.