Nanoscale-femtosecond dielectric response of Mott insulators captured by two-color near-field ultrafast electron microscopy

Characterizing and controlling the out-of-equilibrium state of nanostructured Mott insulators hold great promises for emerging quantum technologies while providing an exciting playground for investigating fundamental physics of strongly-correlated systems. Here, we use two-color near-field ultrafast electron microscopy to photo-induce the insulator-to-metal transition in a single VO2 nanowire and probe the ensuing electronic dynamics with combined nanometer-femtosecond resolution (10−21 m ∙ s). We take advantage of a femtosecond temporal gating of the electron pulse mediated by an infrared laser pulse, and exploit the sensitivity of inelastic electron-light scattering to changes in the material dielectric function. By spatially mapping the near-field dynamics of an individual nanowire of VO2, we observe that ultrafast photo-doping drives the system into a metallic state on a timescale of ~150 fs without yet perturbing the crystalline lattice. Due to the high versatility and sensitivity of the electron probe, our method would allow capturing the electronic dynamics of a wide range of nanoscale materials with ultimate spatiotemporal resolution.

applications in emerging technologies, such as volatile memories and neuromorphic computating 1,4,5 . This scenario becomes even more intriguing when the physical dimensions of the Mott systems shrink down to nanometer length scales. This is, in fact, the typical dimensions of the basic building blocks for most Mottronic devices, where quantum confinement and surface effects may lead to substantial modifications of the structural and electronic dynamics during the Mott transition process. Therefore, it is crucial to provide a deeper understanding of the out-of-equilibrium interplay between electronic and structural degrees of freedom in individual nanoscale Mott systems, which can be achieved only when both spatially and temporally resolved information is simultaneously retrieved.
So far, the underlying mechanism and structural dynamics of the IMT in the VO2 have been intensively studied by ultrafast X-rays diffraction [6][7][8] and ultrafast electron diffraction [9][10][11] , which are indeed able to investigate the material's behaviour with combined fs/atomic-scale resolution and have been shown to provide crucial insights into the structural mechanisms governing the Mott transitions. However, these approaches are not able to give direct information on the electronic degrees of freedom and are mainly limited to bulk crystals, thin films, or clusters of many nanostructures. On the other hand, time-resolved optical pump-probe techniques, such as reflectivity, ellipsometry, and photoemission, can provide access to detailed information on the fs dynamics of the initial electronic processes in the phase transitions by measuring the dielectric response [12][13][14] . However, such methods are inherently limited to a spatial resolution of micrometer scale, because of the long wavelength of the optical probes. Spatial information at smaller length scales can usually be obtained using conventional electron microscopy techniques 15,16 or coherent x-ray imaging methods 17,18 , which can investigate single nanostructures with high spatial resolution, but without temporal resolution. Therefore, hitherto, investigation of the out-of-equilibrium interplay between electronic and structural degrees of freedom in VO2 has been mainly devoted to bulk crystals and thin films, whereas it has been challenging on individual nanostructures for which there is a substantial deficiency of relevant experimental data.
Ultrafast electron microscopy (UEM) is the most promising choice to address such challenge due to its unique high spatiotemporal imaging capabilities [19][20][21][22][23][24][25][26][27][28][29] . The compelling aspect of the UEM technique is the wide range of information provided by the analysis of the transmitted electrons coupled with high temporal resolution not limited by the detector response. It is possible to perform real space imaging at the nanometer scale, record diffraction patterns for structural and lattice dynamics investigations, or acquire electron spectra with sub-eV resolution [19][20][21][22][23][24][25][26][27][28][29] . The latter case is generally referred to as electron energy loss spectroscopy (EELS) and provides a measure of the loss function of electrons scattered from the material (ℑ{ −1 ⁄ }). In an ideal case, from the knowledge of the loss function and using Kramers-Kronig relations, one can retrieve the complex dielectric function of the material under investigation 30 . As the dielectric function directly correlates with the electronic degrees of freedom in materials, it would be possible to access the electronic dynamics across the IMT in the Mott systems via measuring the dielectric function change. However, such approach works well only when the loss function is known over the whole energy range. In a real situation, such constraint is generally hard to achieve, especially in the low-energy region. Moreover, the typical temporal resolution of UEM experiments is on the order of several hundreds of fs even with a single electron per pulse in the absence of space charge effects 22,23,[27][28][29] . This value is determined by the statistical distribution of the time of arrival of electrons photoemitted from the photocathode with slightly different velocities, and it is generally much larger than the typical time scales of the electronic motion in materials, which tend to be on a few tens of fs or below. Although "photon gating" 31,32 and several electron pulse compression schemes such as microwave compression 33 and terahertz compression 34 , have been proposed to improve the temporal resolution of UEM into the time scale of electronic motion, implementation of pump-probe nanoscale imaging of material electronic dynamics only lasting few tens of fs have not been experimentally observed yet.
Here, we overcame these issues by exploiting a well-established variant of the UEM technique named photon-induced near-field electron microscopy (PINEM). In PINEM, inelastic electron-light interaction takes place in the presence of nanostructures when the energy-momentum conservation condition is satisfied 35 . As a result of such interaction, electrons inelastically exchange multiple photon quanta that can be resolved in the electron energy spectrum as a series of discrete peaks, spectrally spaced by multiples of the photon energy ( ℏ ) on both sides of the zero-loss peak (ZLP) [35][36][37] . The largest electron-light interaction is achieved when the optical pulse and electron pulse arrive simultaneously at the specimen. As shown in the theory 38,39 , the fundamental quantity to describe the PINEM process is the scattered field integral ( , ) defined as: where Ez is the component of the electric field along z (electrons propagation direction), the frequency of the laser and the speed of electrons. The probability of interaction between the near-field and the electron is determined by: where Jn is the Bessel function of the first kind of order n, and it represents the probability of exchanging n quanta of light. The PINEM spectrum is then given by the sum over all the contributions: In the case of weak interaction, one can write that ( ) ~ 2 , and thus the expression for the PINEM spectrum becomes: For a cylindrical nanowire (NW) and within the weak interaction approximation the field integral can be approximated at first order as 40 : where is the azimuth angle to the polarization direction, = 2 ( )−1 ( )+1 is the cylindrical susceptibility, a is the NW radius and b is the impact parameter representing the in-plane projected distance between the electron and the NW central axis 40,41 . From Eq. (4) and (5), we note that the dielectric function ( ) at the optical pump frequency ω is directly encoded in the PINEM signal through the localized near-field generated around the nanostructure.
As the lifetime of such near-field is defined by the optical pulse, which is considerably smaller than the electron pulse duration, PINEM is inherently a stationary method. To measure the temporal evolution of the dielectric response and the related electronic dynamics of materials, in addition to the infrared optical pulse (P1) that creates the PINEM, we have introduced an additional visible optical pulse (P2) to excite the sample transiently (Fig. 1a). P1 is synchronous with the electron pulse at the specimen to create PINEM for probe, while P2 is delayed with respect to P1 by a variable delay time for photon pump: a scheme that could be concisely referred to as two-colour photon-pump/PINEM-probe experiment 31 . Such a method takes both advantages of the direct relation between localized near-field intensity and dielectric function, and the strong enhancement of the localized near-field excitation. It can potentially retrieve ε(ω) with an energy resolution entirely determined by the laser bandwidth (about 20 meV) rather than the electron bandwidth (about 1 eV) as in the regular EELS case 42 . Another crucial advantage with respect to regular EELS or PINEM regards the temporal resolution. In our two-colour PINEM approach, the probe is no longer the initial, primary photoelectron pulse, whose temporal duration is typically on the order of several hundreds of fs. The probe in this approach is actually given by the inelastically scattered electrons, whose temporal duration is instead determined by the infrared laser pulse (P1) that drives the electron-light coupling. Thus, the infrared pulse P1 acts as a 'temporal gate' of the electron pulse 31,32 . A two-colour PINEM experiment, where such gated electrons are measured as a function of the delay time between the two optical pulses (P2 and P1), thus has a temporal resolution only determined by the laser pulse (50 fs in our case), which almost an order of magnitude better than a normal UEM experiment where the gate pulse P1 is absent.
To demonstrate the capabilities of our approach, we have investigated the photoinduced IMT occurring in a single VO2 NW. The two-colour PINEM imaging allowed to retrieve the delectric response of the VO2 NW with combined nm-fs resolutions. We reveal that ultrafast photo-doping drives the NW into a metallic state on a time scale of ~ 150 fs, as a result of a photocarrier-driven change of the interatomic potential, without yet perturbing the crystalline lattice. This is then followed by an ensuing recovery to the electronic equilibrium on a tens of piecosecond (ps) timescale related to the anharmonic excitation of transverse acoustic phonons. Such observations elucidate the crucial role of the electronic dynamics for initiating the IMT in the VO2 NW.

One-colour PINEM of a single VO 2 NW
First, we performed a one-colour PINEM experiment on a single VO2 NW to characterize the features of its PINEM spectrum at both insulating and metallic phases. The single-crystal VO2 NWs were synthesized by chemical vapour deposition (see Methods) and were directly transferred to an amorphous silicon nitride membrane (Si3N4, 20-nm thick) window for measurements. The left panel of Fig. 2a shows a bright-field image of the NW (diameter of ~350 nm). In this experiment, only the first optical pulse P1 (duration of 50 fs, = 800 nm, fluence of ~4.1 mJ/cm 2 ) was used to excite the sample with a polarization perpendicular to the NW axis to maximize the near-field excitation. Electron energy spectra were measured as a function of time delay (t1) between the optical pulse P1 and the electron pulse (Fig. 1a). The recorded PINEM spectra are presented in Fig. 1c. Discrete peaks at integer multiples of ℏ appear on both sides of the ZLP (the latter is shown as a shaded area in Fig. 1b at t1 = 1.0 ps and -1.0 ps), and exhibit a maximum intensity at t1 = 0 fs (the shaded pink curve). As the P1 optical pulse is much shorter than the electron pulse, the temporal profile of the PINEM intensity shown in Fig. 1d is mainly determined by the electron pulse duration (~650 fs via a Gaussian fitting), which represents the typical temporal resolution of regular UEM experiments. As a result of the "photon gating" in a two-colour PINEM experiment, the temporal duration of the inelastically-scattered electrons (PINEM electrons) is instead on the order of ~50 fs (given by the gating optical pulse duration 31,32 ), thus improving the temporal resolution of about one order of magnitude.
To quantitatively characterize the localized near-field and the dielectric function of the VO2 NW in insulating and metallic phases, we acquired energy-filtered images while thermally heating the sample across the IMT. A real-space map of the localized near-field is retrieved by selecting only those electrons that have acquired photon energy quanta (see Methods). In these measurements, the time delay between the P1 optical pulse and the electron pulse is fixed at t1 = 0 fs to attain the maximum electron-photon coupling. Typical room temperature bright-field image (left panel) and energy-filtered PINEM image This effect is further supported in Fig. 2c, which depicts the temperature dependence of the integrated intensity. An abrupt drop is observed when crossing the transition temperature, where the NW transforms from the monoclinic insulating phase into the rutile metallic phase (see insets in Fig. 2c). Such behaviour can be attributed to the sudden decrease of the VO2 dielectric function at the photon energy 1.55 eV 43 , typical of a first-order transition from the insulating to the metallic phase, which results in a substantially smaller susceptibility. Such weaker dielectric response is thus responsible for a weaker localized nearfield, and thus a reduced PINEM coupling.
To further confirm the dielectric origin of the observed behaviour, we have performed numerical simulations using a finite element method, where we calculate the scattered field from a single VO2 NW illuminated by an 800-nm optical field (see Methods). In the calculations, the incident field is chosen to be linearly polarized along the y-axis (perpendicular to the NW axis) and propagating along z negative direction (Fig. 3a). The transition is modelled as a variation of the dielectric function from εins = 5.  44 . Fig. 3b represents the computed spatial distribution of the field integral projected on the NW plane (x-y plane) for the insulating and metallic cases, together with their difference map. In Fig. 3c, we plot instead the absolute value of for the two phases integrated along the x-direction and shown as a function of the spatial y-coordinate across the NW. The simulations clearly confirm that the reduced permittivity in the metallic phase is responsible for a weaker scattered field at the NW/vacuum interface, which results in the lower intensity of the PINEM signal as observed in our one-colour experiment ( Fig. 2b and 2c).
Such pronounced contrast in the PINEM signal and the qualitative agreement between experiments and simulations attest to the sensitivity of our approach to probe modifications of the dielectric function crossing the IMT and thus the ability to monitor its ultrafast dynamics with a two-colour PINEM approach.
Such high sensitivity of the technique is obviously not limited to the 800-nm light used here, but it extends to a wide range of light wavelengths.

Two-colour PINEM of a single VO 2 NW
We present here the two-colour PINEM experiment performed on a single VO2 NW. As we will describe below, such a method allowed us to monitor the transient change of the dielectric function of the NW accessing the IMT dynamics with combined fs and nm resolutions. For a conceptually clean experiment (Fig. 1a), two essential requirements need to be fulfilled: i) the pump optical pulse P2 driving the Mott transition must photo-excite the VO2 NW into the metallic phase without producing any appreciable nearfield (i.e., PINEM signal), and ii) the optical gating pulse P1 must have sufficient fluence to produce PINEM with an intense near-field signal but be below the threshold to trigger the Mott transition. In our case, an optical pulse (P2) with a central wavelength of 400 nm (duration of 50 fs) drives the transition, at which VO2 exhibits a significant optical absorption. This P2 optical pulse was set at a fluence of ~15.3 mJ/cm 2 and was characterized by a polarization parallel to the NW axis, which minimizes the near-field excitation as the near-field only occurs at the end of the NW in such configuration (see the right panel of optical pulse was polarized along the direction perpendicular to the NW axis to maximize the opticallyinduced near-field, and its time delay relative to the electron pulse was fixed at t1 = 0 fs to maximize the PINEM coupling (see Movie S1).
In VO2, a density-driven photoinduced IMT has been proven to occur with a critical absorbed photon density of 2 nm −3 at room temperature 45 . The absorbed photon density nexc for P1 and P2, can be evaluated as: where R and are the reflectivity and the absorption coefficient at a specific photon wavelength, respectively, is the incident fluence and ℎ is the photon energy. From the calculations we obtain an excited electron density of 0.9 nm −3 and 4.2 nm −3 for the 800-nm gating pulse and the 400-nm pump pulse, respectively, confirming that P2 is able to trigger the ultrafast IMT while P1 acts only as PINEM probe. Note that, electron beam may also take effects on the IMT in VO2, such as lowering the IMT temperature by creating oxygen vacancies 46 . However, the dose of the electron pulse in our experiment is several orders of magnitude lower than the regular electron beam and its effect is negligible.
As mentioned above, the amplitude of the localized near-field created around the NW by the P1 optical pulse (800 nm) strongly depends on the susceptibility and thus on ε(ω). Besides the amplitude, also the decay length of such near-field can be directly connected to the material permittivity. In fact, according to Mie scattering theory for an infinite long cylinder the complex refractive index exhibits a direct spatial dependence 47 . This means that a change in the dielectric response induced by the P2 optical pulse (400 nm) can lead both to a modification of the PINEM intensity and, at the same time, to a different decay length of the PINEM signal at the NW/vacuum interface. To combine spatial and spectroscopic information, we acquired energy-filtered PINEM images at different delay times of t2 (see Methods) [48][49][50] .  Fig. 4b for two different delay times, t2 = 0 ps and t2 = 1.28ps (t1 = 0 ps). For a more clearer comparison of their lateral decay length, we also plot them in Fig.   4c with each curve normalized to its own maximum. The PINEM intensity shows a substantial decrease at 1.28 ps, while the lateral decay length shows a simultaneous increase. The variation of the integrated PINEM intensity as a function of the delay time t2 is plotted in Fig. 4d. Upon the 400 nm pump, the PINEM intensity shows an initial ultrafast decrease of ~30% with a time constant of ~145 fs, followed by a slower recovery on tens of ps time scale. Such behaviour is consistent with a transformation to a metallic phase with a smaller permittivity, which results in a weaker PINEM coupling (consistently with the observations of one-colour PINEM at high temperatures). Fig. 4e shows the temporal evolution of the lateral width (measured as the full width at half maximum, FWHM) of the integrated PINEM signal surrounding the NW. We observed an ultrafast lateral increase of the localized near-field profile of about 40-60 nm with a time constant of ~155 fs, followed by a slower recovery on a ~10 ps time scale, whose dynamics is consistent with the intensity variation. Therefore, using our approach, it is also possible to follow the transition by looking at the change of the nanoscale spatial decay of the localized near-field. In both Fig.   4d and 4e, the red curves represent the best fit of the experimental data with a bi-exponential model where the temporal duration of the gated PINEM electrons and pump optical pulse P2 is explicitly taken into account. Importantly, it is also worth noting that the observed experimental behaviour is well confirmed by the numerical simulations obtained for the two phases (see Fig. 2b), which show a lower interaction strength and a longer spatial decay in the metallic case with respect to the initial insulating state (see Fig.   3c). As shown in Fig. 4b, and Fig. 4d, the localized PINEM signal from the VO2 NW observed in our experiments exhibits an initial ultrafast dynamics with a time constant of ~150 fs followed by a slower recovery process, which indicates the transition from the insulating to metallic phase without evidence of passing through any intermediate states previously reported on bulk and film counterparts by ultrafast electron diffraction studies 9,10 . Futhermore, this value is nearly twice shorter than the ~300 fs of the coherent V-V displacement motions observed from previous structural probes 9,10 , while it is consistent with the time scale for photo-doping-induced lattice potential change, which could abruptly unlock the V dimers and yield large-amplitude uncorrelated motions, as also observed from the ultrafast optical studies on the bulk crystals 7 . Thus, with our approach, we can capture the transient state in which the NW has become metallic as induced by the photocarrier-induced change of the interatomic potential, but before the crystalline lattice perturbations occur and so the structure remains in the monoclinic phase, namely, the transient quasi-rutile metallic phase 7 . Since both the PINEM intensity and its spatial distribution can be directly related to the dielectric function, our method is inherently sensitive to the electronic dynamics of the VO2 NW. The advantage of our approach is the ability to retrieve such electronic dynamic information on a single nanostructure with combined nm-fs spatiotemporal resolution, which is particularly relevant, especially when nanoscale inhomogeneity plays a decisive role in the transition process 18 . Following the fs dynamics, we also observe a slower recovery toward the electronic equilibrium through the electron-lattice coupling on a time scale of tens of ps, which can be readily associated with anharmonic excitations of acoustic phonons. Thus, a themodynamically-stable metallic rutile phase is not fully reached and then the system relaxes back to the insulating monoclinic phase.

Microscopic mechanism for the IMT dynamics
It is worth noting that, because the photon-pump/PINEM-probe experiment was carried out in an ultrafast electron microscope, it would be possible to interrogate similar nanostructured materials under the same experimental conditions by ultrafast dark-field imaging or ultrafast diffraction using the temporally-gated PINEM electrons. The latter would provide structural information with similar enhanced temporal and spatial resolutions and enable the possibility to simultaneously explore both the electronic (dielectric) and structural dynamics of the investigated individual nanostructures on a few tens of fs time scale.

CONCLUSIONS
In this work, we have implemented a two-colour near-field ultrafast electron microscopy method and demonstrated its ability to access the intial ultrafast electronic process in the optically-induced IMT in an individual VO2 NW. We observed the temporal evolution of its dielectric response with PINEM imaging on nm and fs scales, achieving a combined spatiotemporal resolution several orders of magnitude larger than the conventional optical probes and static imaging. The high sensitivity of PINEM to the ultrafast dielectric response driven by laser photoexcitation attests to the high versatility of our approach for spatially-resolved investigation of electronic dynamics and phase transitions that last a few tens of fs.
Futhermore, incorporating with the advanced attosecond optical pulse generation techniques 54,55 , it is feasible to achieve sub-fs and even attosecond temporal resolution in UEM via our approach. Therefore, this demonstration would be an important step towards the ultimate establishment of sub-fs/as resolution in electron microscopy for capturing electron motion in nanomaterials in real space and time.

Synthesis of VO 2 NWs
The single-crystal VO2 NWs were synthesized in a low-pressure horizontal quartz tube furnace by a chemical vapor transport deposition (CVD) method. In brief, V2O5 powder was placed in a quartz boat at the center of a horizontal quartz tube furnace, and a ~1.5 cm downstream unpolished quartz (~1 cm×0.6 cm) was used as product collecting substrate. The furnace was heated to 950 ℃ to evaporate the V2O5 powder. Then the evaporated V-related species were transported by Ar carrier gas (6.8 sccm, 4 Torr) to the quartz substrate, and free-standing VO2 NWs grew on the substrate surface. After 15 min, the CVD system naturally cooled down to room temperature.

One-and two-colour PINEM experiments
A sketch of our one-and two-colour PINEM experiments is depicted in Fig. 1a. The UEM used in this work, which is detailed in Ref. 20  to the NW axis to maximize the localized near-field excitation and PINEM coupling. For the two-colour PINEM experiment, both 400 nm (P2) and 800 nm (P1) optical pulses were used to illuminate the specimen: the 800 nm (P1) optical pulse is linearly polarized perpendicular to the NW axis and fixed at the zero-time delay (t1 = 0 fs) relative to the electron pulse to maximize the localized near-field excitation and PINEM coupling, whereas the 400 nm (P2) optical pulse for pump is linearly polarized parallel to the NW axis to minimize the localized near-field excitation and PINEM coupling.

Equilibrium heating model
For the static heating experiment it is important to estimate the contribution to the nanowire equilibrium temperature from the 800-nm light pulse. In first approximation, such temperature change can be obtained following a phononic heat capacity approach 56 . In this model the absorbed flux fabs is given by: where is the absorption coefficient, ℎ the photon energy, the energy gap, R the reflectivity, the laser fluence, and Cph is the phononic heat capacity. The latter can be written as: where na is the atomic density, and θD is the Debye temperature. When solving the integral equation Eq.
(7) with the parameters specified in Table I 43,44,57 , we find that the temperature jump associated with the 800 nm infrared pulse is 26 K, thus below the critical transition temperature and consistent with the observation of the thermally induced IMT. For the Si3N4 substrate, following the same approach, the temperature variation induced by the laser is smaller than 1 K and thus negligible.

Numerical simulations
The experimental geometry has been replicated to perform a finite element method simulation using COMSOL Multiphysics. The 3D model is composed of the VO2 NW and a thin Si3N4 substrate. In the outer part, a perfect matching layer (PML) was added to confine the solution. The Si3N4 substrate passes through the PML and can be considered infinite. The incident light wave is chosen to be linearly polarized along the y-axis (perpendicular to the NW axis) and propagating along z negative direction. A parametric sweep of the real and imaginary part of the VO2 complex refractive index ñ, using the values reported in Ref. 43 , replicates the insulator-to-metal transition. The dielectric permittivity of the Si3N4 layer remains constant since it exhibits a negligible variation over a temperature range of several hundreds of • C 44 .
Moreover, the temperature change induced by the pump optical pulse on the Si3N4 membrane is much smaller than 1 K. The study is solved at a frequency of ν = 374 THz (1.55 eV, 800 nm), considering a NW with dimensions 3 μm as length and 300 nm as diameter, placed on a 20 nm thick Si3N4 substrate. By taking advantage of the symmetry of both geometry and field source, the simulation volume has been reduced to one quarter, placing PMC and PEC surfaces in the yz and xz planes, respectively. The PINEM field and its spatial integral has been performed over a 3 μm region along z and of 4 μm along x, i.e. a smaller region compared to the actual size of the simulation domain in order to minimize possible residual reflections from the boundary conditions.
Supplementary Movie S1: Temporal elvolution of one-colour PINEM images of a single VO2 NW with P1 optical pulse (duration of 50 fs, = 800 nm, fluence of ~4.1 mJ/cm 2 ) polarized perpendicularly to the NW axis.
Data availability. The data that support the findings of this study are available from the corresponding authors upon reasonable request. mJ/cm 2 ) were used and the time delay was set at t1 = 0 fs to maximize the PINEM coupling. c Integrated PINEM intensity of the NW as a function of temperature. Tset is the set temperature of the sample on the heating holder, and Teff is the effective temperature, which differs from the Tset due to an additional