Electrically Tunable Nonequilibrium Optical Response of Graphene.

The ability to tune the optical response of a material via electrostatic gating is crucial for optoelectronic applications, such as electro-optic modulators, saturable absorbers, optical limiters, photodetectors, and transparent electrodes. The band structure of single layer graphene (SLG), with zero-gap, linearly dispersive conduction and valence bands, enables an easy control of the Fermi energy, EF, and of the threshold for interband optical absorption. Here, we report the tunability of the SLG nonequilibrium optical response in the near-infrared (1000-1700 nm/0.729-1.240 eV), exploring a range of EF from -650 to 250 meV by ionic liquid gating. As EF increases from the Dirac point to the threshold for Pauli blocking of interband absorption, we observe a slow-down of the photobleaching relaxation dynamics, which we attribute to the quenching of optical phonon emission from photoexcited charge carriers. For EF exceeding the Pauli blocking threshold, photobleaching eventually turns into photoinduced absorption, because the hot electrons' excitation increases the SLG absorption. The ability to control both recovery time and sign of the nonequilibrium optical response by electrostatic gating makes SLG ideal for tunable saturable absorbers with controlled dynamics.

Introduction

Single-layer graphene (SLG) has unique optoelectronic properties,[1−3] which stem from the physics of its massless Dirac fermions. These include high electron mobility (>100 000 cm2 V–1 s–1 at room temperature (RT)[4−7]), broadband optical absorption,[8] tunability of Fermi energy, EF, via electrostatic gating,[9] resulting from the linear dispersion of conduction (CB) and valence bands (VB), and a vanishing density of states at the Dirac point.[10]

Light absorption in SLG is due to the interplay of intraband[11−13] and interband[14,15] transitions. In undoped SLG, the first ones dominate in THz[16] and microwaves,[17] the second[14] in near-infrared (NIR)[15] and visible (VIS).[14] Electrical control of EF, by exploiting the band-filling effect,[18] allows one to vary the density of electronic states available for both intraband[15] and interband transitions,[3,18] thus affecting the linear absorption of SLG over a broad range from THz[19−23] to NIR[24−27] and VIS.[28] This led to the development of SLG-based electro-optic modulators,[3,24,25,27,29−33] which can reach higher modulation speed (up to 200 GHz[34]) than LiNbO3[35] and Si[36] because of the superior mobility of SLG charge carriers, with high modulation depths both in amplitude (up to ∼60%)[3,19,21,24,26] and phase (∼65°).[3,31]

SLG also exhibits large nonlinear optical response[37−41] because of a strong coupling to light. The third-order nonlinear optical susceptibility of SLG in the NIR at 0.7 eV is[37] χ3 ∼ 5 × 10–18 m2V–2, several orders of magnitude higher than in dielectrics (e.g., ∼ 10–22m2V–2 for SiO2[42]) and atomically thin semiconductors (e.g., ∼ 6 × 10–20 m2 V–2 for single-layer WSe2[43]). Nonlinearities of higher order have been exploited for high-harmonics generation in SLG.[38,39] The strong nonlinear response results also in saturable absorption,[44] optical Kerr effect,[45] and optical bistability,[46,47] i.e., the ability to provide two stable optical outputs for a specific light input.[48]EF control via external gating allows one to tune the nonlinear optical response of SLG, resulting in gate-tunable third-harmonic generation[37,40,41] and four-wave-mixing.[49]

The EF dependence of the transient absorption properties of SLG when brought out of equilibrium remains still largely unexplored, with studies limited to the THz range,[50−53] discussing the tuning of intraband photoconductivity with EF.[50−52] The modulation of interband absorption in NIR and VIS is more challenging to study because EF needs to be ∼0.5 eV in order to cross the Pauli blocking threshold, above which the nonequilibrium optical properties have been only theoretically explored.[54]

The nonequilibrium optical response of SLG is crucial for optoelectronic applications, such as photodetectors,[55] relying on the relaxation dynamics of photoexcited charge carriers. Numerous ultrafast optical spectroscopy experiments were performed on SLG[56−60] to investigate the charge-carriers relaxation dynamics by looking at the modifications it induces on SLG absorption. In a pump–probe experiment, the system is photoexcited by an optical pulse, the pump, whose duration is to be shorter than the time scale of the relaxation processes under investigation. The relaxation of the photoexcited system is then monitored by detecting the absorption of a second optical pulse, the probe, as a function of the time delay with respect to the pump pulse.[61]

In SLG, interband absorption of the pump pulse induces out-of-equilibrium distributions of holes (h) and electrons (e) in VB and CB, respectively, peaked at ± ℏωpump/2, where ℏωpump is the pump photon energy. Carrier-carrier scattering drives the ultrafast e–h thermalization on a time-scale τth < 20 fs[60] from out of equilibrium, to hot Fermi–Dirac distributions (HFD) with defined electronic temperature, Te. The HFD can be detected in a pump–probe experiment as a photobleaching (PB) signal,[56−58] i.e., decreased probe absorption compared with equilibrium, because of Pauli blocking of interband transitions caused by the photogenerated e/h. The excess energy of the hot charge-carriers is released to the lattice via electron–phonon scattering with optical phonons,[62−64] anharmonically coupled to acoustic phonons.[62−65] Hot carriers’ cooling occurs on a few-ps time-scale[56−58,60,65] and is influenced, through the activation of additional relaxation channels, by the dielectric environment (e.g., via near-field coupling to hyperbolic optical phonons of substrate or encapsulant material[66]). Defects can also accelerate the cooling via electron–phonon interaction, by acting as scattering centers mediating the direct coupling of the hot charge carriers with finite momentum acoustic phonons.[67−69] This process, referred to as supercollision,[67−69] accelerates the cooling for increasing defect density.[70]

Here we investigate the EF dependence of the nonequilibrium optical response of SLG in the NIR range between 0.729 and 1.240 eV (1000–1700 nm), exploiting ionic liquid gating to tune EF from −650 to 250 meV, thus exceeding the Pauli blocking threshold for interband absorption, achieved when |EF| = ℏωprobe/2, where ℏωprobe is the energy of the probe beam. Applying ultrafast pump–probe spectroscopy with 100 fs time resolution, we detect the changes with EF of amplitude and sign of the differential transmission (ΔT/T), as well as of its relaxation dynamics. Starting from not intentionally doped SLG and increasing EF, we first observe a rise in PB amplitude (ΔT/T > 0) with a slow-down of its relaxation dynamics. Above the Pauli blocking threshold, photoexcitation has an opposite effect on SLG, activating additional absorption channels, as shown by the appearance of photoinduced absorption (PA) (ΔT/T < 0). The ΔT/T changes are assigned to the EF dependence of the hot carriers cooling dynamics, simulated considering relaxation through emission of optical phonons. The gate tunability of the nonequilibrium optical response is key for optoelectronic applications, such as saturable absorbers (SA) with gate-tunable response.

Results and Discussion

We modulate EF by means of the electrostatic field effect[71] using an ionic-liquid top-gated field effect transistor (FET) sketched in Figure a. The top-gate geometry, with diethylmethyl (2-methoxyethyl) ammoniumbis(trifluoromethylsulfonyl)imide (C6H20F6N2O2) as ionic liquid, is chosen to allow light measurements in transmission through a ∼1 cm2 optical window. Large area (8 mm × 8 mm) SLG is prepared by chemical vapor deposition (CVD) as for ref (72). The device fabrication follows ref (73). Figure b is a photo of the device, and Figure c an optical image of the transferred SLG, showing no macroscopic tearing nor folding.

Figure 1

(a) Schematic of device with source (s), drain (d) and gate (g) contacts used to tune the SLG EF while measuring its transmission properties. (b) Photo of representative device. Scale bar 5 mm. (c) Image of transferred SLG in a region near the drain contact (dark area on the top). Scale bar: 100 μm. (d) 514.5 nm Raman spectrum of SLG as-grown and transferred on glass.

Both as-grown and transferred SLG are characterized with a Renishaw InVia Raman spectrometer using a 50× objective, a CW laser at 514.5 nm, with power on the sample <0.5 mW to exclude heating effects. The Raman peaks are fitted with Lorentzians, with error bars derived from the standard deviation across 6 measurements and the spectrometer resolution ∼1 cm–1. The Raman spectrum of as-grown SLG on Cu is in Figure d, after Cu photoluminescence removal.[74] The 2D peak is a single Lorentzian with full-width half-maximum FWHM(2D) ∼31 ± 3 cm–1, a signature of SLG.[75] The G peak position Pos(G) is ∼1586 ± 2 cm–1, with FWHM(G) ∼ 16 ± 3 cm–1. The 2D peak position, Pos(2D), is ∼2704 ± 4 cm–1, while the 2D to G peak intensity and area ratios, I(2D)/I(G) and A(2D)/A(G), are 3.1 ± 0.4 and 6.2 ± 0.7. No D peak is observed, indicating negligible Raman active defects.[76,77]

The Raman spectrum of SLG transferred on glass is in Figure d. The 2D peak retains its single-Lorentzian line shape with FWHM(2D) ∼ 36 ± 1 cm–1. Pos(G) ∼ 1597 ± 1 cm–1, FWHM(G) ∼ 15 ± 1 cm–1, Pos(2D) ∼ 2696 ± 1 cm–1, I(2D)/I(G) ∼ 2 ± 0.2 and A(2D)/A(G) ∼ 4.9 ± 0.3, indicating p-doping with EF ∼ −230 ± 80 meV.[71,78]I(D)/I(G) ∼ 0.06 ± 0.05 corresponds[79] to a defect density ∼2.6 ± 1.9 × 1010 cm–2 for excitation energy 2.41 eV and EF = −230 ± 80 meV. Pos(G) and Pos(2D) are affected by the presence of strain.[80] For uniaxial(biaxial) strain, Pos(G) shifts by ΔPos(G)/Δε ∼ 23(60)cm–1%–1.[80,81] Pos(G) also depends on EF.[9,71] The average doping as derived from A(2D)/A(G), FWHM(G) and I(2D)/I(G), should correspond to Pos(G) ∼ 1588 ± 1 cm–1 for unstrained graphene.[9,71] However, we have Pos(G) ∼ 1597 ± 1 cm–1, which implies a contribution from uniaxial (biaxial) strain ∼0.16 ± 0.02% (0.4 ± 0.04%).[80,81]

The gate voltage, Vg, polarizes the ionic liquid leading to the formation of electrical double layers (EDLs), near SLG and Au interfaces,[71,82] that modulate the carrier density. Since the EDL thickness is ∼1 nm for ionic liquids,[83,84] the solid–liquid interfacial electric field and the induced charge densities on the surface reach values as large as[22,83] ∼10–20 MV cm–1 and 1014 cm–2 even at moderate Vg ∼ 1–2 V. The transfer characteristics of our device for source-drain bias Vds = 100 mV is in Figure a. This exhibits a typical ambipolar behavior, as seen by the V-shaped gate dependence of the source-drain current Ids. The channel resistance peaks at VCNP = 0.84 V, corresponding to the charge neutrality point (CNP), where the density of states in SLG reaches its minimum.[22,23,85]VCNP depends on EF, on the gate-metal work function,[22] and on the choice of contact materials.[86]

Figure 2

(a) Total resistance RTOT as a function of Vg-VCNP (black dots) with Drude model fitting (blue solid line) to estimate residual charge carrier density, n0, and gate capacitance C. Inset, Ids for Vds = 100 mV. (b) δT/T for different V–VCNP (indicated next to the curves) as a function of photon energy ℏω, showing the gate tunability of the absorption edge for interband transitions; (c) EF determined from δT/T as a function of V–VCNP (full dots) interpolated by f(x) = A0 + A1x + A2x2 + A3x3 with A0 = −0.03 eV, A1 = 0.37 eV/V A2 = 0.17 eV/V2A3 = 0.08 eV/V3 (gray dashed line), together with the trend (blue solid line) where n(Vg) = (Vg – VCNP)(C/e) is the gate-tunable charge carrier density

In order to determine EF as a function of Vg, we measure the static transmission T in NIR (500–1500 meV) with an Agilent Cary 7000 spectrometer. Figure b plots a selection of transmission spectra for different V, compared with that at the CNP, evaluated as . T increases with respect to the CNP, i.e., δT/T > 0, when absorption is inhibited by Pauli blocking, due to e in CB (n-doping) or h in VB (p-doping). In terms of probe photon energy, this corresponds to ℏωprobe < 2|EF|. We estimate EF considering that δT/T halves[73] for ℏωprobe = 2|EF| at values indicated by black stars in Figure b. For probe photon energies ℏωprobe < 2|EF|, interband absorption is blocked, and the sample has T ∼ 99.6–99.8%, with ∼0.2–0.4% residual absorption being attributed to intraband transitions enabled by disorder.[73] The T modulation due to the bleaching of interband absorption is ∼1.5% against the ∼2.3% expected for suspended SLG,[8] because of the presence of the glass substrate and of diethylmethyl (2-methoxyethyl) ammoniumbis(trifluoromethylsulfonyl)imide,[87] with a refractive index ∼1.418–1.420.[88]

EF extracted from the T measurements is plotted in Figure c as a function of Vg. At Vg = 0 V, there is a p-doping EF ∼ −250 meV in agreement with the Raman estimation (∼−230 ± 80 meV) without ionic liquid. From the analysis of the charge-transfer curve with the Drude model as in ref (23) (see blue line in Figure a), we evaluate a residual n0 = 5.6 × 1011cm–2, responsible for the finite conductivity at the CNP, and a gate capacitance C = 766 nF cm–2. We note this is a typical n0 for as-grown and transferred SLG.[89] A lower residual doping ∼1011 cm–2 can be achieved with cleaning techniques,[6] not used here. The finite electrical conductivity and doping at the CNP[22,23] are due to electron–hole puddles,[90] caused by charged impurities[91] located either in the dielectric, or at the SLG/dielectric interface.[91] Near the CNP, for |Vg–VCNP| < 0.6 V, the interband absorption edge is outside the spectral window of our δT/T measurements, and we evaluate[8], with vF the Fermi velocity, directly from the gate tunable charge carrier density n(Vg) = (Vg – VCNP)(C/e), with e the electron charge, see blue solid line in Figure c. At high gate voltages |Vg – VCNP| > 0.9 V, the disagreement between calculated EF and the values obtained from δT/T is attributed to the dependence of mobility on charge carrier density,[6] not included in the analysis of the transport properties used for the calculated EF, that is valid near the CNP and fails to describe the sample behavior for |Vg – VCNP| > 0.9 V. Accordingly, we assume the values extracted from δT/T (Figure c gray dashed line) in the range |Vg – VCNP| > 0.6 V, and those calculated from n(Vg) (Figure c, blue solid line), in the range |Vg – VCNP| < 0.6 V. At each Vg of Figure . We also monitor the source-drain current Ids (inset of Figure a) to determine the empirical relation with EF. The transfer curve that we obtain allows us to track EF by monitoring Ids during all subsequent measurements. We test the EF tunability of the ionic liquid top-gate device up to −800 meV, corresponding to a wide range of charge carrier densities from ∼4.5 × 1012 cm–2 (n-doping) to ∼−4.7 × 1013 cm–2 (p-doping), much wider than possible with a 285 nm SiO2 back gate, usually limited to ±6 × 1012cm–2 by the gate capacitance.[85] We got similar EF(V) in ref (73) from Raman spectra and NIR transmission.

We perform ultrafast pump–probe spectroscopy as sketched in Figure a. The pump is a 100 fs NIR pulse centered at ℏω = 0.8 eV, while the probe spectrum covers ℏωprobe = 0.729–1.240 eV (see Methods for details). The relaxation dynamics is monitored through the differential transmission evaluated from the probe transmission with (Tpump–ON) and without (Tpump–OFF) pump excitation, after a time delay t between probe and pump pulses, varied with an optical delay line. Given that the pulses duration exceeds the time-scale of carrier–carrier thermalization,[60] we can assume charge carriers thermalized to HFDs and investigate their cooling dynamics.

Figure 3

(a) Sketch of pump–probe experiment on SLG with tunable EF controlled by Vg. (b) Experimental ΔT/T at t = 150 fs as a function of |EF| for p-doping acquired at different ℏωprobe = 0.729, 0.855, 1.033, 1.127, 1.240 eV (legend of panel d) for a pump fluence F ∼ 28 μJ cm–2. Top-right inset: fluence dependence of ΔT/T amplitude above Pauli blocking for pump absorption, at ℏωprobe = 0.729 eV and EF = −590 meV (hexagonal yellow symbol in main figure), together with a superlinear power-law dependence on F (solid red line). Bottom-left inset: fluence dependence of ΔT/T amplitude, for ℏωprobe = 0.729 eV and EF = −260 meV (cyan rhombus symbol in main figure). (c) |EF0| at ΔT/T = 0, extracted from panel b, as a function of ℏωprobe. (d) Simulated ΔT/T at t = 150 fs as a function of |EF| for p-doping at the same ℏωprobe as in panel b.

Figure b plots ΔT/T at t = 150 fs, chosen as the delay at which the maximum signal amplitude is reached for Vg = 0 V. The signal is plotted as a function of EF for different probe photon energies. Since the transient response is symmetric with respect to the CNP for n- and p- doping and our SLG is p-doped at Vg = 0, we explore negative EF in order to reach higher |EF| by applying a smaller Vg. We observe a strong modulation of ΔT/T with EF, higher at the low-energy tail of the probe pulse, with the signal changing from 4 to −2 × 10–4 (see the curve at 0.729 eV in Figure b). The signal amplitude decreases for increasing ℏωprobe, as expected for a thermal distribution of carriers.[58] In all the probed range, near the CNP, we observe, as expected, a PB signal, i.e., ΔT/T(t) > 0. By increasing |EF|, first PB increases in amplitude, and then a change of sign occurs at a threshold |EF| dependent on the probe photon energy. The Fermi energy at which the sign change occurs, |EF0| in Figure c, corresponds to ℏωprobe/2, i.e., the Pauli blocking threshold for probe photons. Above this, the pump pulse, exciting e (h) to higher (lower) energy states, partially unblocks the probe interband absorption, otherwise inhibited, resulting in a PA signal, i.e., ΔT/T < 0. The PA intensity increases with EF up to a peak, whose position in terms of EF raises with probe photon energy. A constant ΔT/T ∼ −1 × 10–5 is then approached in the high |EF| limit (EF < −690 meV) in all the probed range.

We note that ref (93) reported a study of EF dependence of the transient optical properties of SLG, without reaching Pauli blocking, i.e., |EF| > ℏωprobe/2, which we investigate here, revealing the PA regime. Our results show that, by varying Vg, we can not only control the relaxation dynamics of SLG, but also change the ΔT/T sign.

Above the Pauli blocking threshold for pump interband transitions (|EF| ≥ 400 meV for ℏωpump = 800 meV), ΔT/T is expected to vanish, because the pump should not be able to photoexcite SLG. However, a finite value is observed, caused by residual pump absorption, related to both extrinsic[14,73,92] and intrinsic[14,94] effects. Among the former, charged impurities and scatterers (e.g., edge defects, cracks, vacancies) can induce residual conductivity[73,92] activating intraband absorption. Amongst the latter is the residual absorption from the tail of the carrier Fermi distribution, i.e., off-resonance absorption, which has a finite broadening at RT.[94] The fluence dependence of ΔT/T at ℏωprobe = 0.729 eV in the inset of Figure b is superlinear above the threshold for Pauli blocking of pump absorption (as measured at |EF| = 590 meV), suggesting a non-negligible contribution from two-photon absorption.[95] This could also explain the vanishing signal when approaching |EF| = 800 meV (the Pauli blocking threshold for two-photon absorption).

While height and width of PB and PA bands slightly change with ℏωprobe, we observe similar features in all the probed range upon increasing |EF|: an increase of PB, followed by a decrease, and a sign change above the Pauli blocking threshold for probe absorption. The measurements are performed using a low excitation fluence (28 μJ cm–2) to work in a perturbative regime corresponding to Te < 1000K, thus reducing the impact of the hot-phonon bottleneck and focusing on the electron-optical phonon cooling. The amplitude of ΔT/T increases by increasing the excitation fluence, as for the inset of Figure b, by almost 2 orders of magnitude for the PA signal (see top-right inset of Figure b at |EF| = 590 meV) and 1 order of magnitude for the PB signal (see bottom-left inset of Figure b at |EF| = 260 meV). We previously reported the use of ungated SLG as SA (GSA) in mode-locked lasers, in which an absorption modulation ≤1.3% is sufficient to induce and control the pulsed (mode-locked) regime.[44] The ability to electrically control amplitude, sign, and recovery time of ΔT/T in a GSA is thus of practical relevance for optimizing mode-locking for stability, pulse width, and average output power.

To understand the EF dependence of the nonequilibrium optical response of SLG, we calculate ΔT/T (see Methods for details) as a function of initial carrier density n, related[8] to EF by . ΔT/T in Figure d is computed from the changes in optical conductivity, Δσ, induced by photoexcitation as a function of EF. To evaluate ΔT/T at t = 150 fs we consider the charge carriers as distributed in energy and momentum along a HFD with a time-dependent chemical potential, μ, and T(t) > RT. Our model takes into account that, even though the pump fluence is constant, the initial Te changes with EF due to the change of pump absorption. We consider the absorption from the tail of the Fermi–Dirac distribution as source of residual pump absorption for |EF| > 400 meV. The charge carrier distribution modification with EF is sufficient to reproduce qualitatively the experimental PB signal increase, the change of sign at ℏωprobe/2, and the PA decrease for EF > 400 meV, Figure d.

Our data and model indicate that the PA signal amplitude is maximized at EF ∼ 0.4ℏωprobe, unlike claimed in ref (93), i.e., that the maximum occurs at EF = 0.5ℏωpump. We attribute this difference to the coarser sampling of EF in ref (93) (50 meV steps against our 4 meV, Figure b), to the fact that ref (93) used a reflection geometry, mixing contributions from transient reflection and transmission with opposite sign,[96] and to the larger probe photon energy at which intraband transitions contribute with opposite sign to that of interband transitions.[97] Furthermore, the EF thresholds that control the ΔT/T amplitude and sign are identified by ℏωprobe, not by ℏωpump as incorrectly stated in ref (93), since ℏωpump has no role in defining the cooling dynamics of the HFD after the first ultrafast (<100 fs) step of electron–electron thermalization.[60,63]

To examine the dependence of the cooling dynamics on EF, we monitor ΔT/T as a function of pump–probe delay. Figures a,c show the gate-dependent relaxation dynamics at ℏωprobe = 0.729, 1.033 eV, lower and higher than the pump photon energy. At both energies, the relaxation dynamics progressively slows down with increasing |EF|, evolving from a biexponential to a monoexponential decay, due to a reduction of the fast decay component. We can appreciate this slowdown by noting that, to see a signal reduction by a factor 10, we need to wait ∼1 ps at |EF| = 100 meV and ∼5 ps at 300 meV. Both signal intensity and relaxation dynamics are symmetric for n- or p-doping, as a consequence of the CB, VB symmetry.

Figure 4

Time-evolution of ΔT/T(t) for different EF at (a,b) ℏωprobe = 0.729 eV, with (a) experiment, and (b) simulations, and (c,d) ℏωprobe = 1.033 eV, with (c) experiment and (d) simulations. t = 0 corresponds to the pump arrival. The signal at negative delays indicates a finite build-up time, exceeding the pump–probe time duration at |EF| = 250 meV for ℏωprobe = 0.729 eV and 350 meV for ℏωprobe = 1.033 eV, approaching the Pauli blocking EF.

The observed gate-dependence can be qualitatively explained considering that, for increasing |EF|, the excess energy of the photoexcited charge carriers with respect to equilibrium is reduced, affecting the scattering with optical phonons that drives the cooling. To gain a deeper insight into the phenomena responsible for quenching the fast relaxation component, we solve a set of phenomenological equations of motion (EOMs)[98] for T and for the occupation of the phonon modes. We include the optical phonon modes at the K and Γ points of the SLG Brillouin zone, and we consider that they can be emitted/absorbed by e and h and decay into acoustic modes due to anharmonic coupling[62,63,65] (see Methods).

We calculate the time-evolution of the differential conductivity for several values of μ, corresponding to EF, (i.e., μ at T = 094), in the range 250 to −650 meV. The results in Figure b,d explain the observed slowdown of the dynamics with increasing EF, with the saturation of the phase space for optical phonon-emitting electronic transitions. As EF increases, there are fewer carriers with an energy high enough (>160 meV) to emit an optical phonon, and optical phonon emission is quenched. This is a fundamental process, not dependent on the SLG substrate, like supercollision cooling through defects,[99] nor on its dielectric environment, like the cooling to hyperbolic phonons in hBN-encapsulated SLG.[66] It is determined by the intrinsic coupling of e with the K and Γ phonons.[100] The initial increase of PB amplitude with EF in Figure b,c is a consequence of the quenching of relaxation via optical phonons,[60,63] which reduces the initial fast decay.

Figure also shows that |EF0| is independent of t, both in experiments and simulations. The vanishing ΔT/T does not correspond to zero absorption, but it means that the conductivity remains at its equilibrium value for all delays. For t > 0, the e system is photoexcited. This can happen only because the e distribution undergoes a time-evolution such that the conductivity remains time-independent at ℏω = 2|EF|.

Figure a shows that for EF > 340 meV and ℏωprobe = 0.729 eV, the simulations predict a further slowdown of the relaxation dynamics, not observed in our experiments. These all saturate to a similar decay trend independent of EF (see overlapping black and green dots in Figure ). Analogous behavior is found at all ℏωprobe, provided that when we increase ℏωprobe, we tune EF to higher levels to find overlapping decay dynamics (see Figure b). To understand the saturation of this slowdown, we need to consider that additional relaxation channels may start playing a role once the cooling via optical phonons gets slower. Defects can accelerate cooling,[67−70] mediating the scattering with acoustic phonons of finite momentum and energy.[69] This supercollision mechanism[67−70] may become the dominating process once optical phonon emission is quenched. Cooling times ∼4 ps are expected for supercollision cooling[67] in SLG with EF ∼ 400 meV and mobility of a few thousand cm2 V–1 s–1, as that of our device. The EF independence of the decay dynamics in the high EF limit could be explained by the lack of dependence on carrier density of the supercollision cooling time away from the Dirac point.[67] According to refs (78) and (79), the e scattering time with defects in SLG is not expected to significantly change with EF.

Figure 5

(a,b) Experimental (colored dots) and theoretical (solid lines) ΔT/T at different EF for (a) ℏωprobe = 0.729 eV and (b) ℏωprobe = 1.033 eV for pump–probe time delays between −500 fs and 5 ps (pump arrival at t = 0).

The electrical tunability of the SLG relaxation dynamics, sketched in Figure a–c, is promising for the realization of tunable SA. Saturable absorption, i.e., the quenching of optical absorbance under intense illumination,[101] can occur in SLG at low light intensity (e.g., ∼ 0.750MW cm–2 at 0.8 eV[102]). We measured a saturation intensity[44]IS = 0.5–1.7MW cm–2 for photon energies in the range ∼0.5–2.5 eV, comparable to semiconductor saturable absorber mirrors (SESAMs) (P = 0.01–0.1MW cm–2 at 0.944 eV[103]), but maintained over a much broader spectral range.[44] The modulation depth, defined as the maximum change in absorption,[101] can be optically tuned exploiting cross absorption modulation.[104] GSAs are promising for passive mode-locking,[44,105,106] Q-switching,[107] and Q-switched mode-locking.[108]

Figure 6

(a–c) Sketch of interband absorption of a NIR probe pulse (vertical blue arrow) within the SLG Dirac cones populated at equilibrium up to EF (gray filling). The pump pulse perturbs the probe absorption by promoting e from VB to CB (red filling), which then relax through emission of optical phonons (downward black arrows). The three sketches correspond to (a) EF at the Dirac point, (b,c) moderate n-doping with EF (b) below and (c) above the threshold for interband probe absorption. By increasing EF, optical phonon emission is quenched (dashed downward arrows) and relaxation becomes slower. Above the threshold for interband absorption of the probe (dashed vertical blue arrow), photoexcitation results in ΔT/T < 0, leading to reverse saturable absorption, consisting in an increased absorption upon increasing illumination.

Figure shows that the SLG equilibrium photoresponse can be electrically tuned, providing an additional knob for controlling its SA performance in terms of modulation depth and recovery dynamics. For EF ≪ ℏωprobe/2, the intrinsic biexponential-like relaxation dynamics makes SLG an ideal fast SA, Figure a. The presence of two different time scales, in analogy with SESAMs,[109] is considered an advantage for mode locking.[109] As discussed in refs (109) and (110), the longer time scale reduces the saturation intensity, facilitating self-starting mode-locking, while the fast relaxation component is efficient in shaping subps pulses. For EF ≤ ℏωprobe/2 as in Figure b, SLG can act as slow SA[111] with recovery times 10 to 30 times longer than the pulse duration,[111,112] favoring soliton shaping,[112] or the temporal shift of the pulses caused by the SA,[109] which limits the time in which noise behind the pulse can be amplified.[111] Longer recovery time also gives an increased tolerance towards instability induced by self-phase modulation.[111]

The PA at EF > ℏωprobe/2 can be exploited to operate SLG as reverse SA,[113] for which absorption increases with increasing impinging intensity, because of depletion of the final state population (see Figure c). The PA of highly doped SLG could be exploited to realize an optical limiter,[114] based on the decrease in transmittance under high-intensity or fluence illumination. An ideal optical limiter, with the functionality of protecting delicate optical elements, should strongly attenuate intense, potentially dangerous, laser beams, while exhibiting high transmittance for low-intensity light. Carbon nanotubes[115] and few-layer graphene[116] dispersions in organic solvents have been used to prepare optical limiters. However, these rely on nonlinear scattering,[117] rather than on nonlinear absorption.[118] The nonlinear scattering of graphene dispersions[116] is based on the avalanche ionization of carbon when interacting with an incident laser pulse, and subsequent bubble formation in the solvent due to the heat released by expanding microplasmas.[115,116] The 10 ps PA lifetime of the nonlinear absorption of highly doped SLG is 10 times shorter than the typical time scales for thermal effects and bubbling of graphene dispersions, which are of the order of 100 ps,[115] allowing the application to lasers with shorter pulse duration. The nonlinear absorption in SLG is not related to a specific absorption resonance. Thus, it covers a broad spectral range, as shown in Figure b, where for |EF| = 600 meV we detect PA for photon energies in the range 0.729 to 1.127 eV. The SA to reverse SA transition could be used for all-optical logic gates.[119] Gate-dependent effects on cooling dynamics are also important for the design of transceivers for data communication.[3] E.g., ref (120) showed that longer cooling times give larger photocurrent.

Conclusions

We demonstrated that electrostatic tuning of the nonequilibrium optical response of SLG results in changes of amplitude, sign, and recovery dynamics of ΔT/T. Increasing EF quenches emission of optical phonons, i.e., of the fastest intrinsic relaxation channel for SLG hot charge carriers. The ability to tune EF above the threshold for Pauli blocking of interband absorption of NIR light results in photoinduced absorption in SLG, because of pump-induced unblocking of interband transitions for the probe. Our results anticipate the use of voltage-controlled SLG for nonequilibrium optoelectronic devices as gate tunable optical elements, which can behave either as fast, slow, or reverse SA.

Methods

High-Sensitivity Transient Absorption Microscopy

The setup for pump–probe experiments comprises a mode-locked Er-doped fiber oscillator (Toptica Photonics, FemtoFiberPro), emitting 150 fs pulses at 0.8 eV (1550 nm) at 40 MHz repetition rate. The oscillator feeds two Er-doped fiber amplifiers (EDFAs) each generating 70 fs pulses at 0.8 eV with 300 mW average power. The output of the first EDFA is attenuated to obtain pump pulses with 1 mW maximum average power. The second EDFA feeds a highly nonlinear optical fiber that produces a supercontinuum tunable between 0.729 and 1.240 eV, which serves as probe pulse. The pump and probe pulses, synchronized by a computer-controlled optical delay line and collinearly recombined by a dichroic beam splitter, are focused on the sample over spots of ∼25 μm radius. The portion of the probe transmitted by the sample, spectrally selected by a monochromator with a bandwidth ∼5 nm, is detected by an amplified InGaAs photodiode (bandpass 4.5 MHz, gain 104) and analyzed by a lock-in amplifier (Zurich Instruments HF). Pump and probe pulses have perpendicular polarizations and a linear polarizer is used to filter out the pump light scattered from the sample. The pump pulse is modulated at 1 MHz by an acousto-optic modulator, resulting in a ΔT(t)/T sensitivity of the order of 10–7, for an integration time of 300 ms. From the FWHM of the instrumental response function, we estimate an overall temporal resolution ∼100 fs. The absorbed photon density is in the range 2–3 × 1012cm–2 (depending on EF), as calculated from incident fluence and sample transmission.

Simulation of Differential Transmission Dynamics

To model the time-evolution of the differential transmission we assume that, on the time-scale given by the time-resolution of the experiment (100 fs), e in both CB and VB are thermalized at the same Te, and reach a common μ(t), such that the e energy distribution is a HFD. μ(t) is calculated at each instant in time, as it depends on Te and is fixed by the condition that the carrier density, defined as[94], with ν(ε) the electronic density of states in SLG,[122] is constant.[94] As for refs (98) and (121), we can write the following EOMs for Te and phonon occupations:Here, nΓ(t) and n(t) are the occupations of the optical phonon modes at Γ and K, with energy ℏωΓ ∼ 0.196 eV and ℏω ∼ 0.161 eV,[100] respectively, as these have the strongest electron–phonon coupling.[100] The parameter τph is the finite optical phonon lifetime, via relaxation into acoustic phonons due nonlinearities of the lattice,[65] until global thermal equilibrium with densities nΓ(0), n(0) is reached. We find good agreement between theory and experiment for τph ∼ 1.2 ps, consistent with ref (65). The constant coefficients MΓ, M correspond to the number of phonon modes in an annular region between the minimum and maximum energy that can be exchanged with e.[98,121] The time-dependent parameters ce(t) and ch(t) are the heat capacities of e in CB and h in VB, respectively. The time-dependent parameters RΓ(t) and R(t) are electronic relaxation rates per unit area, due to phonon emission and absorption, proportional to a Boltzmann scattering integral.[98,121]

In line with our assumption that, on the time-scale probed by our experiments, a common μ(t) is established between CB and VB, the heat capacities are calculated separately in the two bands, (i.e., Te variations are decoupled from interband transitions) and only intraband transitions are included in the relaxation rates. The initial Te(0), following the pump pulse, is estimated as for ref (37). The initial phonon populations nΓ,(0) are evaluated at RT. The optical photoconductivity Δσ(t) = σ(t) – σ(0)[122] depends on Te(t) and μ(t). We use the Tinkham formula[123] to obtain the differential transmission.