Quantifying the role of surface plasmon excitation and hot carrier transport in plasmonic devices

Harnessing photoexcited “hot” carriers in metallic nanostructures could define a new phase of non-equilibrium optoelectronics for photodetection and photocatalysis. Surface plasmons are considered pivotal for enabling efficient operation of hot carrier devices. Clarifying the fundamental role of plasmon excitation is therefore critical for exploiting their full potential. Here, we measure the internal quantum efficiency in photoexcited gold (Au)–gallium nitride (GaN) Schottky diodes to elucidate and quantify the distinct roles of surface plasmon excitation, hot carrier transport, and carrier injection in device performance. We show that plasmon excitation does not influence the electronic processes occurring within the hot carrier device. Instead, the metal band structure and carrier transport processes dictate the observed hot carrier photocurrent distribution. The excellent agreement with parameter-free calculations indicates that photoexcited electrons generated in ultra-thin Au nanostructures impinge ballistically on the Au–GaN interface, suggesting the possibility for hot carrier collection without substantial energy losses via thermalization.


Supplementary Note 3: Interplay of the metal band structure and interface properties on IQE
In Figure 3 of the manuscript, we reported the energy distributions of hot-electrons generated in Au for different photon energies (Figure 3a,b), as obtained from ab-initio calculations, and we discussed how the interplay with a Schottky barrier of ~1.2 eV determines the salient features of the experimentally observed IQE spectra (Figure 3c). Here we extend our analysis to different values of the Schottky barrier.
Supplementary Figure 3a shows the calculated IQE for an Au/semiconductor interface (electron effective mass of GaN ~0.27 m 0e 1 as a function of the interfacial Schottky barrier height. Beyond the specificity of each curve we observe that some features are common to all the spectra: • At low photon energies (close to the Schottky barrier) the IQE grows rapidly, roughly following the expected free electron-like Fowler behavior • Around 1.7eV, the IQE growth is suppressed and the curve deviates significantly from the Fowler model • Beyond ~2.3-2.4 eV the IQE exhibits a peak and then a pronounced drop These spectral features of the IQE are directly associated with the Au electronic band structure: • Up to ~1.6 eV, intraband transitions dominate hot carrier excitation. For this reason the IQE spectrum follows closely the Fowler model, which accounts only for free-electron energy distributions; • Between 1.6 eV and 1.8 eV, interband transitions start to contribute significantly and the IQE deviates more prominently from the free-electron like Fowler model. The suppression of IQE can be understood by looking at Supplementary Figure 3. Indeed, starting at around 1.7 eV, interband transitions induce a redistribution of the energy of the hot-electrons towards lower energies, close to the metal Fermi level. This reduces the number of high-energy electrons that can be collected with better efficiency than lowenergy electrons and suppresses the IQE compared to expectations based solely on the Fowler model. • Around ~2.3-2.4 eV, interband transitions finally become the dominant mechanism for hot-electrons generation leading to a sharper drop in IQE, which then reaches a minimum around 2.6 eV.
The effect of the interplay between intra and interband transitions can be entirely explained by considering the metal electronic band structure. On the other hand, the energy filtering effect of the metal/semiconductor interface, which combines the energetics of the Schottky barrier and momentum matching conditions, modulates the relative prominence of these three different regimes (intraband regime up to 1.6 eV, mixed regime 1.7-2.3 eV, interband regime above 2.4 eV). In fact, we observe that: • For a Schottky barrier of 0.9 eV, effective collection of high-energy electrons from intraband transitions leads to a peak in IQE around 1.7 eV, right before the onset of interband transitions. A slow decrease in IQE is observed in the mixed regime before a sharp drop-off occurs around 2.4 eV; • For a Schottky barrier of 1.2 eV, the collection of hot electrons in the intraband regime is reduced and therefore the mixed regime between 1.7 eV and 2.3 eV still exhibits a slow growth of IQE. Indeed, the increased injection probability of the highest energy electrons compensates the overall reduced probability of intraband generated hot-electrons. Only the complete transition to the interband regime around 2.4 eV suppresses the IQE; • For an even larger Schottky barrier of 1.5 eV, carrier collection is negligible across the entire intraband regime. Therefore the transition between the Fowler-like rapid growth in IQE and the IQE reduction due to interband transitions cannot be observed. Instead, throughout the mixed regime, IQE slowly grows thanks to the highly energetic fraction of hot-electrons generated through intraband transitions. Transition into the interband regime around 2.4 eV, however, halts this growth. From this analysis we observe that due to the combined effect of the metal band structure and the interface, the IQE spectrum can exhibit a peak in the photon energy range anywhere between ~ 1.8 eV to 2.4 eV. Although small gold nanoparticles or thin Au films can exhibit a plasmon resonance in this very same energy range, these two aspects should not be confounded. Indeed, plasmon excitation will affect the optical properties of the system and manifest itself in an enhanced external quantum efficiency of the device. However the peak in IQE originates solely from the electronic properties of the metal, with modulations induced by the characteristics of the metal/semiconductor interface.

Supplementary Note 4: Detailed analysis of IQE -Injection Probability
Supplementary Figure 4: Injection probability. Calculated injection probability including momentum matching and tangential momentum conservation across the metal-semiconductor interface shown for an increasing value of the barrier height, Φ B . The low values originate from the low effective electron mass (~0.27m 0e ) in the close-to-ideal, parabolic conduction band of GaN and from the smooth metal-semiconductor interface imposing tangential momentum conservation on hot electron injection.

Supplementary Note 5: Detailed analysis of IQE -Transport vs Barrier Height
Supplementary Figure 5: IQE as a function of Schottky barrier height and transport for Au/n-GaN interfaces. a) Φ B = 0.05 eV : purple curve -expected IQE accounting for momentum matching (P inj ) but not for transport; blue curve IQE accounting for transport (solid -ballistic injection; dashed -N=3); b) Φ B = 1.2 eV : purple curve -expected IQE accounting for momentum matching but not for transport; green curve IQE accounting for transport (solid -ballistic injection; dashed -N=3); Supplementary Figure 5a shows that even with a negligible Schottky barrier of 50 meV, momentum-matching conditions set a very tight restriction on IQE, limiting the efficiency to less than 1% across the entire visible spectrum (purple curve). Transport effects, meaning scattering and recombination of carriers in the structure with finite dimensions, further reduce IQE by one to two orders of magnitude (blue solid line). In particular, in the case of Au, IQE decreases with the onset of interband transitions (~1.8eV) due to the lower probability of injection across the Au/GaN interface for carriers with smaller energies (Figure 3 and Supplementary Figure 4). Collection of scattered carriers offers only a marginal improvement of IQE (blue dashed line) for the same reason. A larger barrier (Supplementary Figure 5b) exacerbates these mechanisms. Therefore improving the injection efficiency across the interface and ensuring ballistic collection are of primary importance, irrespective of the barrier height of a given metal-semiconductor junction. In this respect, roughening of the interface and choice of semiconductors with high density of states in the conduction band (e.g. TiO 2 ) would be highly beneficial.

Supplementary Note 7: Fabry-Perot modelling
In a Fabry-Perot etalon the peaks in transmission correspond to dips in reflection and vice versa. The peaks occur when the phase accumulation of the light wave during a round-trip across the high-index region (GaN in our case) is equal to 2π (constructive interference). If the refractive index and thickness of the GaN layer are n and t, respectively, we can calculate the phase accumulation as: = 2 • 2 cos where θ is the angle of incidence of the incident light beam. Therefore, the peak wavelengths can be identified by the condition: where m must be an integer number and we have also included the wavelength-dependence of the refractive index, n(λ).
From our measurements we actually have a series of peak,exp values, and can therefore find a thickness t that correctly predicts our experimental peak series. As can be observed in the Supplementary Figure 7.a, when t = 4228 nm ≈ 4.23 μm we can match very well the calculated Fabry-Perot transmission peaks (vertical dashed lines) with the peaks in the experimental transmission spectrum (red curve) or the dips in the experimental reflection spectrum (blue curve). According to the manufacturer, our GaN substrates have a nominal thickness of t nominal = 4 ± 1 μm and therefore the estimated thickness t is in excellent agreement with the expected one. Small deviations can be attributed to i) minor variations between the used values of n(λ), that we took from the literature, and the refractive index of our specific substrate and ii) the finite resolution of our experimental spectra (acquired with 2 nm resolution).
In order to reduce computation time as well as improve clarity of data display, in Figure 2e and Supplementary  Figure 2c we have reported the absorption calculated using a thin GaN substrate (500 nm thick), which does not display Fabry-Perot oscillations. In Supplementary Figure 7.b we show the result of a numerical simulation where we have included the full GaN substrate: as expected we observe Fabry-Perot oscillations in the absorption spectrum.
Lastly we note that Fabry-Perot interference is extremely sensitive to the light path within the high refractive index material (in our case GaN). The incomplete cancellation of the Fabry-Perot fringes in the IQE spectra (Figure 2g) is attributable to differences in the angular alignment of ±3 degrees between the IQE and absorption measurements, which were done in separate experimental setups. The plasmonic absorption is instead insensitive to angular a) b) variations of this magnitude. Therefore, the small discrepancy in Fabry-Perot fringes, which leads to their incomplete cancellation, has no bearing on the overall IQE measurement.