Hybrid erbium–graphene system with dual-gate modulation

We show the design of our hybrid system schematically in Fig. 1. The central part consists of a graphene monolayer on a nanoscale layer of erbium-doped Y₂O₃ (2%). To achieve strong near-field interactions, the erbium emitters should be located at nanoscale distances from the 2D material—ideally within the sub-wavelength volume occupied by the highly confined plasmons (<15 nm). A sufficiently thin layer with erbium emitters is therefore required. However, crystals of nanoscale dimensions typically suffer from detrimental non-radiative losses due to surface defects²⁵. Removing this loss mechanism is crucial, as it constitutes a competing energy flow channel for erbium–graphene interactions that would hinder the observation of these interactions. We overcome this experimental bottleneck by using atomic layer deposition (ALD)²⁶ with optimized post-treatment—a technique that produces few-nanometer-thick rare-earth-doped Y₂O₃ films with atomic scale thickness control and emission quality as in bulk crystals (see Supplementary Note 1, Supplementary Figs. 1 and 2, and Supplementary Table 1). The results we will show are obtained with a sample containing an erbium layer of 12 nm thickness, as measured through white light interferometry (see Supplementary Note 2 and Supplementary Figs. 3 and 4 for more sample characterization).

a Schematic illustration of the hybrid erbium–graphene system when the Fermi energy of graphene is tuned to  ~0.6 eV and the erbium–graphene interaction leads to intraband transitions, mainly associated with launching of propagating graphene plasmons (red waves). The system contains, from top to bottom, a monolayer of graphene, a thin film (~12 nm) of Y₂O₃ containing erbium ions (white spheres), a 285-nm-thick SiO₂ layer, and a p-doped silicon backgate. A backgate voltage, V_bg, is applied between the backgate and a gold electrode in contact with graphene for fast modulation. The SiO₂ layer serves as electrical isolation between graphene and the p-doped silicon. On top of graphene, there is a transparent topgate made of polymer electrolyte (not shown in the image). b Sinusoidal time evolution of the Fermi energy of graphene (dashed black line, left vertical axis) and the corresponding decay-enhancement factor F_P for an erbium emitter located at z = 5 nm from graphene (solid purple line, right vertical axis). The modulation of the Fermi energy, calculated following refs. ^28,29, leads to a modulation of F_P by more than one order of magnitude—from  <100 in the intraband regime to  >1000 in the interband regime. c Schematic illustration of the hybrid erbium–graphene system for E_F ~ 0.3 eV, which corresponds to interband transitions, mainly creating electron–hole pairs (red–blue spheres).

With the aim of dynamically controlling the erbium–graphene interactions through the Fermi energy of graphene E_F, we combine a backgate of p-doped silicon with a polymer electrolyte topgate²⁷ (see Fig. 1). The backgate, with a smaller alternating current (AC) impedance than the topgate, is very suitable for high-frequency modulation. We use it to modulate E_F at high frequencies over a range of  ~0.3 eV. On the other hand, the topgate allows Fermi energy tuning over a range >1 eV, which is sufficiently high to launch plasmons in resonance with the photons emitted by erbium, whose energy is E_Er = 0.8 eV (see Fig. 1a). We use the topgate to provide a high base Fermi energy during high-frequency modulation of the backgate. The Fermi energies induced by both gates are calibrated by Hall measurements and resistance measurements (see Supplementary Note 3 and Supplementary Fig. 5). Thus, using our dual-gated, hybrid erbium–graphene system, we can modulate E_F, for example, between 0.6 and 0.3 eV, (Fig. 1b). This modulates the system between two regimes, where the erbium emitters decay by transferring energy to graphene, leading to intraband absorption (Fig. 1a) and interband absorption (Fig. 1c), respectively. The intraband regime is mainly associated with plasmon generation in graphene, whereas in the interband regime mainly electron–hole pair creation occurs.

Emission contrast and decay enhancement

The intraband and interband absorption regimes can be experimentally distinguished because they are associated with different local densities of optical states (LDOS), leading to distinct decay rates for the emitters interacting with graphene. Our calculations following refs. ^28,29 show that the decay-enhancement factor F_P for an emitter at 5 nm from graphene is  ~100 in the intraband regime (E_F = 0.6 eV) and  >1000 in the interband regime (E_F = 0.3 eV). In order to observe these regimes experimentally, we excite the erbium ions with a 532-nm laser and detect their emission at 1.54 μm in a scanning confocal microscope (see “Methods”), while varying the Fermi energy using the topgate. Figure 2a shows the measured emission contrast, defined as the emission without graphene (measured by shining the laser outside the graphene channel) divided by the emission with graphene (measured by shining the laser on graphene; see also Supplementary Fig. 3). The excitation power is sufficiently low (fluence of 10⁴ W cm⁻²) to ensure that the ion transition is not saturated (see Supplementary Fig. 4). We indeed identify two different regimes: at low Fermi energies (E_F < 0.4 eV), energy transfer is mostly caused by interband transitions in graphene. Above 0.4 eV, interband transitions become suppressed by Pauli blocking, because E_F > E_Er/2, and intraband transitions become the dominant energy decay pathway of the ions. The positive slope of the emission contrast for E_F > 0.6 eV is a clear signature of the presence of plasmons. The slope is positive because the decay length of the plasmon field is approximately the same as the plasmon wavelength, λ_pl, which scales linearly with E_F. Therefore, as E_F increases, the volume occupied by the plasmon field increases, thus increasing the number of ions interacting with plasmons, and thereby decreasing the amount of emitted light.

a Measured emission contrast, defined as the emission with the excitation laser shining outside graphene divided by the emission with the laser on graphene, as a function of the Fermi energy (black dots). The black solid line represents the calculated emission contrast from N erbium ions located at different distances z_i from graphene (see “Methods”), using the total graphene conductivity. The red (blue) dotted line represents the calculated emission contrast by only considering the conductivity of the intraband (interband) excitation, showing the different microscopic origins of the emitter–graphene interactions at high and low E_F. The interband contribution (blue) quickly drops when E_F > 0.4 eV, due to Pauli blocking, whereas the intraband contribution (red) steadily rises with E_F. b Measured decay curves of erbium ions for three cases: graphene in the interband regime (E_F = 0.3 eV, blue dots), graphene in the intraband regime (E_F = 0.8 eV, red dots), and without graphene (orange dots). The black solid lines are the best-fit stretched-exponential functions. The inset focuses on the beginning of the decay curves, illustrating the multi-exponential behavior. c Erbium density distribution (normalized to maximum) vs. erbium–graphene distance obtained from the combined analysis of the emission contrast and the decay curve of the interband regime (blue) and from TOF-SIMS measurements (black). d Cumulative distribution of the decay enhancement factor calculated for the interband (blue) and the intraband (red) regimes. These curves show that about 80% (50%) of the ions have F_P > 10 for the interband (intraband) regime, and approximately 25% of ions have F_P > 1000 for both regimes. The inset shows the total decay rate distributions directly obtained from the multi-exponential fitting of the experimental decay curves (see “Methods”). The colors are the same as in b.

To obtain more insight into the dynamics of the erbium–graphene interactions, we measure the erbium emission decay curves. We modulate the excitation laser into pulses and use single-photon counting electronics to create emission histograms (see “Methods”). Figure 2b shows the decay curves measured with the laser shining on graphene for the interband transition regime (E_F ~ 0.2 eV) and for the intraband regime (E_F ~ 0.8 eV), as well as in a region without graphene. We observe faster decay in the intraband regime (e⁻¹ time of  ~3 ms) and even faster decay in the interband regime (e⁻¹ time of  ~1 ms), compared to the decay without graphene (e⁻¹ time of  ~6 ms), in qualitative agreement with the emission contrast measurements, which show less emission in the interband regime than in the intraband regime.

Strikingly, the decay curves are multi-exponential, with a large negative slope in the beginning of the decay (see inset of Fig. 2b). This multi-exponential behavior stems from the different decay rates γ of the emitters, depending on their distance to graphene z, since γ scales with z⁻⁴ in the interband regime and with \(\exp (-4\pi z/{\lambda }_{{\rm{pl}}})\) in the intraband regime^28,30. The ions located furthest away from graphene have the lowest energy transfer rates, thus emitting more photons and extending the decay curves to long times. The strongly negative slope in the beginning of the decay curves (with an estimated decay time  <100 μs; see inset of Fig. 2b and Supplementary Fig. 10) is due to the high energy transfer rate for small z and indicates the presence of a significant fraction of ions with very strong erbium–graphene interactions. Thus the decay curves provide information of the z-dependence of the interactions, while the emission contrast measurements reflect the overall (distance-integrated) graphene-induced decay-enhancement F_P, which is  ~7 (~2.5) in the interband (intraband) regime. This indicates that an overall fraction of η ≈ (F_P − 1)/F_P ≈ 85% (60%) of the energy of excited erbium emitters flows to interband (intraband) transitions in graphene. We note that we have reproduced these decay curves with multiple erbium–graphene samples (see Supplementary Note 4 and Supplementary Fig. 9).

Quantifying erbium–graphene interaction

Given the strong z-dependence of the emitter–graphene interaction, it is crucial to determine the distribution of erbium ions in order to quantitatively understand the energy transfer efficiency of the different erbium ions in the nanolayer. We analyze the emission contrast and the decay curves (Fig. 2a, b) together because they provide complementary information. The decay curves provide with high accuracy the distribution of ions with relatively low F_P factors (large z), as these are the ions that emit the largest number of photons during lifetime measurements. On the other hand, the emission contrast measurements of Fig. 2a reveal more accurately the effect of the ions with relatively large F_P (small z), as these are the ions that transfer the highest fraction of their energy to graphene, leading to the largest decrease in emission. Together, the decay curves and emission contrast measurements yield the density of erbium ions as a function of z (see Fig. 2c).

In detail, we extract the distribution for ions with z > 7 nm directly from the decay curves of Fig. 2b, by describing each multi-exponential decay curve by a continuous sum of exponentially decaying functions, whose probability amplitudes are given by the decay-rate distribution P(γ). In terms of the ion dynamics, the distribution P(γ) can be interpreted as the likelihood that a given ion decays with a certain decay rate γ. The inset of Fig. 2d clearly shows that the decay-rate distribution P(γ) is shifted toward higher values of γ in the regions of the device with graphene, indicating that the decay rate increases due to energy transfer to graphene. From the analysis of these decay-rate distributions, we extract the distribution of decay-enhancement factors P(F_P), following the numerical procedure described in the “Methods” section. Then we convert P(F_P) into the distribution of erbium–graphene distances P(z), i.e., the density distribution, by using the theoretical relation between F_P and the emitter–graphene distance z.

It turns out that the distributions P(F_P) and P(z) obtained from the decay curves are accurate up to F_P = 10³, corresponding to z ≳ 7 nm. For higher values of F_P, the decay is so fast and has such a small amplitude that we cannot resolve it experimentally in our decay curves (see Supplementary Note 5). However, we can obtain the distribution of the ions with z < 7 nm using the emission contrast measurements of Fig. 2a. For this, we use a computational model of N ions at different distances from graphene, z_i (i = 1, ..., N), and find the ion density distribution P(z) that best reproduces the measurements of Fig. 2a, b (see “Methods”). This is how we obtain the ion density distribution in Fig. 2c.

We compare the ion distribution extracted from optical measurements P(z) with the density distribution measured by means of time-of-flight secondary ion mass spectrometry (see “Methods” and Supplementary Note 6). The similarity between the two density distributions confirms the validity of our analysis. Interestingly, our results indicate that some ions have diffused from the Y₂O₃ layer into the underlying SiO₂ layer. This has likely occurred during the annealing post-treatment of the films. These diffused ions interact less strongly with graphene and lead to the moderate overall emission contrast we observe in Fig. 2a.

Importantly, our analysis of the experimental data of Fig. 2a, b provides evidence of very strong emitter–graphene interactions at the shortest distances. Figure 2d shows the calculated cumulative distribution of P(F_P),

which describes the probability that an ion has a decay-enhancement factor larger than p. In this way, the cumulative distribution P(F_P > p) provides the fraction of ions with F_P > p. Using Eq. (1), we find that about 80% (50%) of the ions have decay enhancement factor F_P > 10 for the interband (intraband) regime, and approximately 25% of ions have F_P > 1000 for both regimes, which means that >99.9% of the energy from these ions flows to graphene.

Fast electrical modulation of near-field interactions

Having established the occurrence of highly efficient erbium–graphene interactions in our system, we now demonstrate dynamic control of these interactions on a time scale that is much shorter than the emitter’s lifetime of  ~10 ms³¹. We induce a fast temporal variation in the LDOS experienced by the emitters by modulating the Fermi energy of graphene. To this end, we apply an AC voltage to the backgate and verify the effect of this modulation on the excited state populations of erbium by measuring the temporal variations of photon emission using a single-photon counting set-up. In these experiments, we keep the excitation laser power constant. In Fig. 3, we modulate the Fermi energy between 0.3 and 0.6 eV at different modulation frequencies \({f}_{{\rm{mod}}}\) between 20 Hz and 300 kHz. This corresponds to a modulation of the erbium decay pathway between interband and intraband transitions, as in Fig. 1. In these measurements, we thus establish control over the timing of plasmon launching from erbium ions down to the microsecond range, which is remarkable for emitters with millisecond natural lifetime. We verified that there is no dynamic response outside graphene and that a backgate voltage of  <10 V is sufficient for complete modulation between the two emitter–graphene interaction regimes (see Supplementary Fig. 6).

As \({f}_{{\rm{mod}}}\) increases and becomes higher than the emitter decay rate, the internal dynamics of the ions are not able to follow the temporal variations of the environment. This results in a gradual delay of the maximum and minimum of the time-dependent emission with respect to the sinusoidal oscillation of the Fermi energy and the reduction of the emission modulation amplitude, which depends on \(1/{f}_{{\rm{mod}}}\) whenever \({f}_{{\rm{mod}}}\gg \gamma\). Interestingly, these modulation frequencies surpass not only the decay rate of the ions but also the quantum coherence decay rate of erbium in Y₂O₃ (11 kHz; measured at 2.5 K in a bulk ceramic sample and under a small external magnetic field of 0.65 T)³².

In a second series of measurements, shown in Fig. 4, we modulate the Fermi energy between 0.7 and 0.9 eV. Here we apply a higher voltage to the polymer electrolyte topgate than in the previous modulation experiment, while modulating the backgate again with a voltage of  <10 V. In this situation, the system is always in the intraband regime, where plasmon launching is the dominant energy decay pathway, and we therefore modulate the strength of the emitter–plasmon interaction. Note that an increase in backgate voltage now leads to a decrease in emission, because stronger emitter–plasmon interaction leads to less emission (see also Supplementary Fig. 7). This is in contrast with the case in Fig. 3, where an increase in backgate voltage leads to a transition from the interband absorption regime to the intraband regime, giving more emission.

We model the temporal modulation of photon emission by simulating the dynamics of N ions using the distribution of ion distances P(z) obtained from the emission contrast measurements and decay curves. For every ion i at distance z_i, we numerically solve the rate equation with the time-dependent F_P factor and for the corresponding Fermi energy modulation (see “Methods”). We find good agreement between experimental data and numerical simulations, which do not contain any freely adjustable fit parameters (see “Methods”), adding credibility to our computational approach. We note that, only in the case of Fig. 4e, we observe a larger modulation amplitude in the experiment than in the simulation. We speculate that this could be related to additional (e.g., non-local) effects in the plasmonic local field at very short distances from graphene.