Yaniv Kurman

Raphael Dahan

Hanan Herzig Shenfux

Kangpeng Wang

Michael Yannai

Yuval Adiv

Ori Reinhardt

Luiz H G Tizei

Steffi Y Woo

Jiahan Li

James H Edgar

Mathieu Kociak

Frank H L Koppens
email: frank.koppens@icfo.eu

Ido Kaminer
email: kaminer@technion.ac.il

Spatiotemporal imaging of 2D polariton wavepacket dynamics using free electrons

Coherent optical excitations in 2D materials -2D polaritons -create a plethora of possibilities for unique optical phenomena that arise from extraordinary dispersion relations which do not exist in regular materials. However, to date, all dynamical phenomena of 2D polaritons have remained out of reach -as they require simultaneous spatial & temporal imaging capabilities. These capabilities could reveal unknown coherent optical phenomena in 2D materials, where the extraordinary dispersion promise exotic wavepacket dynamics, such as multi-branch evolution and splitting. Here we present the first spatiotemporal measurement of 2D wavepacket dynamics, from its formation to its decay. We use an ultrafast transmission electron microscope driven by femtosecond mid-infrared pulses to coherently-excite phonon-polariton wavepackets and probe their evolution in a non-destructive manner. Our approach reveals intriguing dispersion-dependent dynamics that includes splitting of multi-branch wavepackets, and unexpectedly, wavepacket deceleration and acceleration. Accessing the full spatiotemporal dynamics of 2D wavepackets can shed light on puzzles in topological polaritons and discover novel nonlinear optical phenomena in 2D materials.

Over the past decade, research on propagating optical polaritons in 2D materials progressed from a promising concept (1, 2) to a platform for demonstrating rich physical phenomena [START_REF] Grigorenko | Graphene plasmonics[END_REF][START_REF] Dai | Tunable Phonon Polaritons in Atomically Thin van der Waals Crystals of Boron Nitride[END_REF][START_REF] Iranzo | Probing the ultimate plasmon confinement limits with a van der Waals heterostructure[END_REF][START_REF] Basov | Polariton panorama[END_REF][START_REF] Hu | Topological polaritons and photonic magic angles in twisted α-MoO3 bilayers[END_REF], now showing an impact on emerging opto-electronics [START_REF] Brar | Emerging photonic architectures in two-dimensional optoelectronics[END_REF][START_REF] Long | Progress, Challenges, and Opportunities for 2D Material Based Photodetectors[END_REF] and nanophotonic technologies [START_REF] Chaudhary | Engineering phonon polaritons in van der Waals heterostructures to enhance in-plane optical anisotropy[END_REF]. These polaritons exhibit relatively low loss and long propagation distances, simultaneous with extreme confinement factors [START_REF] Iranzo | Probing the ultimate plasmon confinement limits with a van der Waals heterostructure[END_REF][START_REF] Giles | Ultralow-loss polaritons in isotopically pure boron nitride[END_REF][START_REF] Ni | Fundamental limits to graphene plasmonics[END_REF][START_REF] Lee | Image polaritons in boron nitride for extreme polariton confinement with low losses[END_REF][START_REF] Dubrovkin | Resonant nanostructures for highly confined and ultra-sensitive surface phonon-polaritons[END_REF], facilitating their unique light-matter interactions [START_REF] Koppens | Graphene plasmonics: A platform for strong light-matter interactions[END_REF][START_REF] Grigorenko | Graphene plasmonics[END_REF][START_REF] Kumar | Tunable light-matter interaction and the role of hyperbolicity in graphene-hbn system[END_REF][START_REF] Lee | Graphene acoustic plasmon resonator for ultrasensitive infrared spectroscopy[END_REF][START_REF] Kurman | Tunable bandgap renormalization by nonlocal ultra-strong coupling in nanophotonics[END_REF][START_REF] Epstein | Far-field excitation of single graphene plasmon cavities with ultracompressed mode volumes[END_REF][START_REF] Rivera | Light-matter interactions with photonic quasiparticles[END_REF].

Strong motivation exists to utilize the novel properties of 2D polaritons and integrate them into ultrafast optical technologies that rely on the spatiotemporal control of light.

Conventional areas of ultrafast optics achieve such control using pulse-shaping [START_REF] Stix | Pulse shaping in passively mode-locked ring dye lasers[END_REF][START_REF] Weiner | Femtosecond pulse shaping using spatial light modulators[END_REF][START_REF] Chong | Airy-Bessel wave packets as versatile linear light bullets[END_REF] and dispersion engineering [START_REF] Shumakher | On the balance between delay, bandwidth and signal distortion in slow light systems based on stimulated Brillouin scattering in optical fibers[END_REF][START_REF] Colman | Temporal solitons and pulse compression in photonic crystal waveguides[END_REF][START_REF] Schulz | Dispersion engineered slow light in photonic crystals: a comparison[END_REF], which are instrumental, for example, in photonic waveguides [START_REF] Colman | Temporal solitons and pulse compression in photonic crystal waveguides[END_REF][START_REF] Schulz | Dispersion engineered slow light in photonic crystals: a comparison[END_REF]. The attainment of similar control with 2D polaritons could promote the integration of 2D polaritonic materials in mature areas of science and technology and contribute to their fundamental understanding. However, the spatiotemporal control of 2D polariton wavepackets has remained out of reach for exactly the same reasons that make their potential applications exciting: they have extremely small wavelengths and are strongly confined inside the material.

New capabilities are necessary for accessing the spatiotemporal dynamics of 2D polaritons and their wavepackets, with nanometric spatial resolution and femtosecond temporal resolution.

In this regard, it is particularly interesting to consider wavepackets in materials exhibiting hyperbolic dispersion [START_REF] Noginov | Bulk photonic metamaterial with hyperbolic dispersion[END_REF][START_REF] Poddubny | Hyperbolic metamaterials[END_REF]. Owing to this unique dispersion, polaritons in hyperbolic materials show rich physical behavior, ranging from negative refraction [START_REF] Sreekanth | Negative refraction in graphene-based hyperbolic metamaterials[END_REF] and subdiffraction imaging [START_REF] Noginov | Bulk photonic metamaterial with hyperbolic dispersion[END_REF][START_REF] Jacob | Optical Hyperlens: Far-field imaging beyond the diffraction limit[END_REF][START_REF] Li | Hyperbolic phonon-polaritons in boron nitride for near-field optical imaging and focusing[END_REF] to effective Hawking radiation [START_REF] Smolyaninov | Hyperbolic metamaterial interfaces: Hawking radiation from Rindler horizons and spacetime signature transitions[END_REF]. While this is most famously seen in metamaterials, phonon polariton (PhP) excitations in 2D materials also exhibit hyperbolic dispersion [START_REF] Dai | Tunable Phonon Polaritons in Atomically Thin van der Waals Crystals of Boron Nitride[END_REF][START_REF] Hillenbrand | Phonon-enhanced light-matter interaction at the nanometre scale[END_REF][START_REF] Taubner | Near-field microscopy through a SiC superlens[END_REF][START_REF] Caldwell | Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride[END_REF]. There, the phononic resonance creates a dispersion relation that contains multiple branches (see Fig. 1(b)), which were shown to be tunable by the 2D material geometry, thickness, and surrounding environment [START_REF] Dai | Tunable Phonon Polaritons in Atomically Thin van der Waals Crystals of Boron Nitride[END_REF][START_REF] Chaudhary | Engineering phonon polaritons in van der Waals heterostructures to enhance in-plane optical anisotropy[END_REF][START_REF] Kumar | Tunable light-matter interaction and the role of hyperbolicity in graphene-hbn system[END_REF][START_REF] Li | Hyperbolic phonon-polaritons in boron nitride for near-field optical imaging and focusing[END_REF][START_REF] Caldwell | Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride[END_REF], reaching high confinement with relatively low losses, even in room temperature [START_REF] Giles | Ultralow-loss polaritons in isotopically pure boron nitride[END_REF][START_REF] Lee | Image polaritons in boron nitride for extreme polariton confinement with low losses[END_REF]. These advances exposed the intriguing spatial behavior of hyperbolic polaritons, and yet, their rich temporal dynamics -creating hyperbolic wavepackets -has remained unexplored.

The novel physics and surprising dynamics of hyperbolic wavepackets can be seen by revisiting the basic concept of the group velocity: the derivative of the dispersion, 𝜕𝜔/𝜕𝑘, approaches zero in an hyperbolic medium, since the medium supports excitations with (in principle) arbitrarily large momenta. In practicality, the group velocity of PhPs is well defined, given an excitation of a finite bandwidth exciting a specific branch. However, in 2D hyperbolic materials, any coherent excitation is expected to simultaneously stimulate multiple PhP branches, each having different dynamics. Moreover, even a small variation in the excitation's bandwidth/frequency results in a large change of the polariton group velocity and the entire wavepacket propagation dynamics. These prospects are especially intriguing in 2D hyperbolic materials such as thin flakes of hexagonal boron nitride (hBN), where the group velocity of polaritons was shown be as low as c/500 over a relatively wide bandwidth [START_REF] Yoxall | Direct observation of ultraslow hyperbolic polariton propagation with negative phase velocity[END_REF]. To explore these prospects of hyperbolic wavepackets and reveal their novel physics, we need access to the field comprising the wavepacket during its evolution inside the 2D material, which necessitates hitherto nonexistent microscopy capabilities.

In this study, we present the first observation of coherent wavepacket dynamics in 2D materials. Specifically, our approach measures the spatiotemporal dynamics of hyperbolic PhP wavepackets inside isotopically pure hBN ( 11 B) flakes [START_REF] Giles | Ultralow-loss polaritons in isotopically pure boron nitride[END_REF]. Our measurement is made possible by exploiting a fundamental interaction: between free electrons and PhP wavepackets. The temporal dynamics is obtained using a pump-probe technique with a mid-infrared (IR) laser pump and a free electron probe in an ultrafast transmission electron microscope (UTEM) [START_REF] Barwick | Photon-induced near-field electron microscopy[END_REF][START_REF] Park | Photon-induced near-field electron microscopy (PINEM): theoretical and experimental[END_REF][START_REF] Flannigan | 4D electron microscopy: Principles and applications[END_REF][START_REF] Lahme | Femtosecond single-electron diffraction[END_REF][START_REF] Piazza | Simultaneous observation of the quantization and the interference pattern of a plasmonic near-field[END_REF][START_REF] Cremons | Femtosecond electron imaging of defect-modulated phonon dynamics[END_REF][START_REF] Valley | Imaging Intra-and Interparticle Acousto-plasmonic Vibrational Dynamics with Ultrafast Electron Microscopy[END_REF][START_REF] Liu | Infrared PINEM developed by diffraction in 4D UEM[END_REF][START_REF] Lummen | Imaging and controlling plasmonic interference fields at buried interfaces[END_REF][START_REF] Priebe | Attosecond electron pulse trains and quantum state reconstruction in ultrafast transmission electron microscopy[END_REF][START_REF] Vanacore | Attosecond coherent control of free-electron wave functions using semi-infinite light fields[END_REF][START_REF] Madan | Holographic imaging of electromagnetic fields via electron-light quantum interference[END_REF][START_REF] Vanacore | Ultrafast generation and control of an electron vortex beam via chiral plasmonic near fields[END_REF][START_REF] Wang | Coherent interaction between free electrons and a photonic cavity[END_REF][START_REF] Kfir | Controlling free electrons with optical whispering-gallery modes[END_REF], illustrated in Fig. 1(a). The pulsed free electron penetrates the sample and consequently changes its energy spectrum according to the integrated electric field along its path (inset of Fig. 1(c)). By energy filtering of the post-interaction electron, we reconstruct the image of the PhP wavepacket. The use of laser-driven energy-filtered transmission electron microscopy (EFTEM) was first demonstrated in photon-induced nearfield electron microscopy (PINEM) [START_REF] Barwick | Photon-induced near-field electron microscopy[END_REF][START_REF] Flannigan | 4D electron microscopy: Principles and applications[END_REF][START_REF] Piazza | Simultaneous observation of the quantization and the interference pattern of a plasmonic near-field[END_REF][START_REF] Lummen | Imaging and controlling plasmonic interference fields at buried interfaces[END_REF]. Our approach takes PINEM to the mid-IR range and combines this emerging field with 2D materials for the first time.

We record the PhP wavepacket creation and propagation (Fig. 1(d), and Movies S1 and S2), revealing unforeseen physical behaviors, such as multi-branch wavepacket splitting, acceleration and deceleration. The measured acceleration and deceleration dynamics are especially surprising, as wavepackets conventionally have a fixed group velocity. In certain cases, as in the thin hyperbolic hBN flakes in our case, the group velocity is expected to be very slow -indeed, we observed group velocities from c/45 to c/850 -and still, each wavepacket is naively expected to propagate with a fixed group velocity. In contrast with this expectation, we show that the dispersive nature of PhPs (i.e., their momenta change rapidly in frequency) facilitates acceleration and deceleration dynamics. This result serves as a key example for the rich physical phenomena that can be found when probing non-destructively the spatiotemporal dynamics of 2D polaritons using the unique combination of femtosecond temporal and nanometer spatial resolution of the UTEM. Among the various experimental techniques used in the field of 2D polaritons [START_REF] Luo | Probing Polaritons in 2D Materials[END_REF], the scanning nearfield optical microscopy (SNOM) and its unique variants have had the most impact so far on the direct nearfield imaging of 2D polaritons [START_REF] Hu | Topological polaritons and photonic magic angles in twisted α-MoO3 bilayers[END_REF][START_REF] Giles | Ultralow-loss polaritons in isotopically pure boron nitride[END_REF][START_REF] Ni | Fundamental limits to graphene plasmonics[END_REF][START_REF] Dubrovkin | Resonant nanostructures for highly confined and ultra-sensitive surface phonon-polaritons[END_REF][START_REF] Li | Hyperbolic phonon-polaritons in boron nitride for near-field optical imaging and focusing[END_REF][START_REF] Hillenbrand | Phonon-enhanced light-matter interaction at the nanometre scale[END_REF][START_REF] Taubner | Near-field microscopy through a SiC superlens[END_REF][START_REF] Shi | Amplitude-and Phase-Resolved Nanospectral Imaging of Phonon Polaritons in Hexagonal Boron Nitride[END_REF].

Recent advances in time-resolved SNOM also allowed the polariton's group velocity to be extracted from the interference of scattered polaritons with different time delays [START_REF] Yoxall | Direct observation of ultraslow hyperbolic polariton propagation with negative phase velocity[END_REF][START_REF] Mrejen | Transient exciton-polariton dynamics in WSe 2 by ultrafast near-field imaging[END_REF][START_REF] Sternbach | Femtosecond exciton dynamics in WSe2 optical waveguides[END_REF].

However, this interferometric technique cannot image the wavepacket dynamics, as it does not provide the spectral phase, i.e., the phase difference between photons of different wavelengths.

Other important experimental approaches, such as photo-emission electron microscopy (PEEM) [START_REF] Gong | Ultrafast imaging of surface plasmons propagating on a gold surface[END_REF], are also used for nearfield imaging in plasmonics; these approaches thus far did not access the mid-and far-IR region. We discuss the different experimental approaches in the Supplementary materials (SM), section S.5. Importantly, techniques based on transmission electron microscopy stand out from all the above, since the electron penetrates through the sample and becomes sensitive also to the buried field rather than only to the field on the surface [START_REF] Lummen | Imaging and controlling plasmonic interference fields at buried interfaces[END_REF], an advantage for probing the highly confined 2D polaritons.

Our approach for the observation of PhP wavepacket dynamics contributes to the expanding toolbox of electron-beam spectroscopy and microscopy [START_REF] Polman | Electron-beam spectroscopy for nanophotonics[END_REF]. Of particular importance for our approach are the advances in the imaging of polaritons using electron energy loss spectroscopy (EELS) [START_REF] Ibach | Electron Energy Loss Spectroscopy and Surface Vibrations[END_REF][START_REF] Egerton | Electron Energy-Loss Spectroscopy in the Electron Microscope[END_REF], which reached the ability to measure IR excitations in vibrational electron spectroscopy [START_REF] Krivanek | Vibrational spectroscopy in the electron microscope[END_REF]. EELS enabled the measurement of purely vibrational modes (phonons) in bulk media and on surfaces [START_REF] Lourenço-Martins | Vibrational Surface Electron-Energy-Loss Spectroscopy Probes Confined Surface-Phonon Modes[END_REF][START_REF] Lagos | Mapping vibrational surface and bulk modes in a single nanocube[END_REF] and the extraction of their dispersion relations using electron imaging and diffraction (62,[START_REF] Senga | Position and momentum mapping of vibrations in graphene nanostructures[END_REF]. In recent years, the improvements in the energy resolution of EELS (59) enabled the imaging of hBN PhP resonances [START_REF] Govyadinov | Probing low-energy hyperbolic polaritons in van der Waals crystals with an electron microscope[END_REF] and recently the extraction of the PhP dispersion in extremely thin samples [START_REF] Li | Direct observation of highly confined phonon polaritons in suspended monolayer hexagonal boron nitride[END_REF]. Such experiments translated methodologies in electron microscopy, once applied to plasmons in the visible range (e.g., Refs. [START_REF] Nelayah | Mapping surface plasmons on a single metallic nanoparticle[END_REF][START_REF] Liou | Plasmons dispersion and nonvertical interband transitions in single crystal Bi2Se3 investigated by electron energy-loss spectroscopy[END_REF]), to phonons in the mid-IR range. Similar methodologies are key to understanding the nanoscale behavior of mixed photon-matter excitations in different strongly coupled systems [START_REF] Bitton | Vacuum Rabi splitting of a dark plasmonic cavity mode revealed by fast electrons[END_REF][START_REF] Tizei | Tailored Nanoscale Plasmon-Enhanced Vibrational Electron Spectroscopy[END_REF].

In all these EELS experiments, one obtains static, time-independent information on the polaritonic modes and other excitations of the sample, all of which are triggered by the free electron. In contrast, PINEM-based techniques such as ours use the electron only as a timedependent probe (and not as a trigger of the excitation); thus, PINEM allows the extraction of time-dependent information on the temporal dynamics of the polaritons that are excited by a separate laser pulse.

Results

Fig. 1(d) and Movie S1 present an example of our experimental results: measurements of the wavepacket during its creation and propagation inside the flake. The video is created by repeating the measurement for a range of time delays between the laser pump and the electron probe. Such measurements of wavepacket dynamics rely on stimulated free-electron-PhP interactions. For the wavepacket, the electron is a non-destructive probe: the interaction alters the wavepacket in a negligible manner and hence does not interfere with the wavepacket evolution dynamics. Thus, the wavepacket propagates across the sample uninterrupted, starting from a single edge (chosen by optimizing the laser coupling -SM section S.3). At each time delay, the wavepacket profile is reconstructed from the electron energy distribution: at points where the PhP's out-of-plane electric field is stronger, there is a larger probability for the probing electron to gain energy and pass an energy filter. The energy filter is chosen to maximize the signal (see Fig. S4); it creates a threshold for the detectable PhP field and thus reduces the signal-to-noise ratio. Consequently, as we describe in the theory below, the connection between the number of counts and the electric field is non-linear.

To model the free-electron-PhP interaction, we find it essential to generalize the singlefrequency theory of conventional PINEM [START_REF] Park | Photon-induced near-field electron microscopy (PINEM): theoretical and experimental[END_REF][START_REF] Garcia De Abajo | Optical excitations in electron microscopy[END_REF] that was used to describe most PINEM experiments to date. The need to go beyond the successful PINEM theory lies in the finite bandwidth of the excitation laser (necessary for creating the pulsed PhP wavepacket) that excites the highly dispersive hBN PhPs. To capture the spectral bandwidth, we describe the free-electron-PhP coupling through a generalized coupling function 𝑔(𝑥, 𝑦, 𝜔) (71). This coupling function 𝑔 quantifies the strength of the interaction for each in-plane coordinate (𝑥, 𝑦) and each frequency 𝜔. According to this continuous-PINEM theory [START_REF] Reinhardt | Theory of Shaping Electron Wavepackets with Light[END_REF],

𝑔(𝑥, 𝑦, 𝜔) = 𝑒 ℏ𝜔 𝑑𝑧 𝐸 (𝑥, 𝑦, 𝑧, 𝜔)𝑒 , ( 1
)
where 𝑒 and 𝑣 are the electron charge and velocity, respectively, and ℏ is the reduced Planck's constant. The integral is performed along the electron propagation direction 𝑧, on the 𝑧 component of the electric field phasor 𝐸 (𝜔) = ∫ 𝐸 (𝑡)𝑒 𝑑𝑡, which includes all the PhPs modes due to their TM polarization.

The PhP wavepacket is imprinted on the electron as a time-dependent phase modulation:

exp 2𝑖 𝑑𝜔 |𝑔(𝑥, 𝑦, 𝜔)| sin 𝜔𝑡 -arg 𝑔(𝑥, 𝑦, 𝜔) . ( 2
)
This phase modulation multiplies the initial electron wavefunction with a time delay (𝜏 ) relative to the PhP wavepacket excitation; 𝜏 is shifted for recording a video of the dynamics.

Our theory predicts the measured electron energy spectra as the Fourier transform (time→energy) of the resulting electron wavefunction. Eq. ( 2) shows how larger 𝑔 values imply stronger modulation in the phase of the free-electron wavefunction, equivalent to the electron gaining and losing more energy (see SM section S.4). The energy required for a detectable signal is related to the incoherent energy width of the pre-interaction electron (also called zero-loss peak). Because the width (0.9 eV in our system, Fig. 1(c), right inset) is larger than the energy of a single PhP quanta, the post-interaction EELS (Fig. 1(c), left inset) does not have discrete peaks as in PINEM experiments in the visible or near-IR range [START_REF] Liu | Infrared PINEM developed by diffraction in 4D UEM[END_REF][START_REF] Madan | Holographic imaging of electromagnetic fields via electron-light quantum interference[END_REF].

Nevertheless, the change in the electron's energy is sufficient for probing the PhP wavepacket:

the electron image in the x-y plane is filtered by energy for different time delays 𝜏 to extract the PhP spatiotemporal dynamics.

In the analysis of the measured PhP wavepacket dynamics, we first extract the field profile along the direction of propagation (Fig. 2(a)). When averaging the signal along the sample's excited edge, we reduce the signal-to-noise ratio (described in SM section S.3) and find that a Gaussian fitting describes the measured wavepacket quite accurately. This analysis reveals the formation of the wavepacket during the arrival of the excitation pulse. Fig. 2(b)

shows an intriguing phenomenon: the wavepacket appears to remain stuck at the edge for a certain time duration and does not immediately propagate away from the boundary. Instead, the wavepacket's width and amplitude gradually increase. Thus, the gradually forming wavepacket increases in amplitude while remaining near the edge. Only toward the end of the excitation pulse (when its tail arrives) does the wavepacket start to move away from the edge more quickly, exhibiting phenomena of acceleration and deceleration that vary between samples and excitation wavelengths. Once the excitation pulse is ended, we can extract the stable group velocities reported below. To corroborate these effects of wavepacket dynamics, we present in Fig. 3 the wavepacket properties during its formation, propagation, and gradual decay. Fig. 3 (a-c) summarizes the wavepacket dynamics in three different samples and a range of excitation pulses. As expected, the group velocities become smaller as the sample thickness decreases.

The lowest measured group velocity (bright blue in (a)) is 0.35 𝜇m/ps, which is 860 times lower than the speed of light in vacuum. In a thicker sample, the fastest recorded group velocity is 6.7 𝜇m/ps, which is 45 times lower than the speed of light in vacuum, but still sufficiently low for our free-electron probing technique to record the dynamics. An additional notable measurement includes the propagation over a duration of more than 2.5 ps in a 7.5 nm thick sample and propagation lengths over distances of more than 12 𝜇𝑚 for a 55 nm thick sample, crossing the entire length of the sample. These extensive propagation distances and durations are a merit of the isotopically pure hBN ( 11 B) [START_REF] Liu | Single Crystal Growth of Millimeter-Sized Monoisotopic Hexagonal Boron Nitride[END_REF], which encounters smaller losses than normal hBN flakes [START_REF] Giles | Ultralow-loss polaritons in isotopically pure boron nitride[END_REF]. The data show a first demonstration of a dramatic change in the group velocities of PhP wavepackets during their propagation, for which we use the terminology wavepacket acceleration and deceleration. We observed this effect in all sample thicknesses (Fig. 3(a-c)).

For example, in the 7.5 nm sample, the group velocity decreased by a factor of 5 (Fig. 3(a)).

This effect is surprising, because the PhP dispersion is expected to cause the slower modes to decay faster, which would result in acceleration rather than deceleration.

We find that this effect reveals a more general S-shaped trajectory for the PhP wavepackets: during the pulse excitation (bright colored background in the panels of Fig. 3), the wavepacket first remains near the edge of the sample (extraordinarily slow group velocity, as explained above), then experiences fast propagation, and finally decelerates to reach its final Our experiment identifies intriguing features of the PhP wavepackets during the wavepacket formation and free propagation. We integrate over the entire observable signal at each time delay to analyze the competition between the laser pumping to the dispersion and decay. We identify two intriguing features in the PhP wavepacket dynamics: (1) We measure the gradual buildup of the wavepacket when the pump overcomes the PhP wavepacket dispersion and intrinsic losses (shown as bright background in Fig. 3(e)). Remarkably, we identify cases for which the integrated signal continues to grow even when the wavepacket peak is already 10 𝜇𝑚 from the edge and is propagating at a stable group velocity (green curves). ( 2) We show that the measured wavepackets disperse and decay at different rates (Fig.

3(e-f)), quantified by a dispersion time 𝜏 (Fig. 3(e)). As expected from theory, shorter dispersion times (i.e., larger dispersion) occur for slower group velocities.

The decay in the integrated signal is caused mostly by the wavepacket dispersion (subps timescale) and not by intrinsic loss (few ps timescale). The dispersion-driven broadening reduces the portion of the wavepacket that has a sufficiently large electric field, above the threshold field needed for the electron to pass the energy filter. This measurement threshold (see SM section S.1) improves our signal-to-noise ratio. Our measurement approach allows us to isolate the competition of wavepacket dispersion broadening and decay, which occur simultaneously with the laser pumping.

The efficient electron-polariton interaction enables measurement of the propagation of multi-branch wavepackets that split over time into distinguishable Gaussian-like wavepackets of different group velocities (Fig. 4 and Movie S2). The multi-branch wavepackets are created because the excitation at each wavelength can couple into more than a single branch in the PhP dispersion (Fig. 1(b)). Fig. 4(a) shows the propagation of a multi-branch PhP wavepacket that splits into two single-branch wavepackets, propagating as a double Gaussian (Fig. 4(b)). From the location of the double Gaussian peaks (shown in Fig. 4(c)), we extract two different group velocities for the first-and second-branch wavepackets (6 and 1.1 𝜇m/ps, respectively). We enhanced the observation of this effect by changing the electron energy slit to be sensitive to a smaller threshold electric field. As confirmed by the FDTD simulations in Fig. 4(d-g), at short times the two wavepackets completely overlap. Then, at longer times the wavepackets gradually split from one another. Interestingly, as also confirmed by the FDTD simulation, the PINEM-type measurement shows a clear spatial separation between the wavepackets while their fields still appear to overlap. It is the electron's sensitivity to the field inside the hBN that enables one to distinguish the individual profiles of the two partially-overlapping wavepackets. The energy-filtered electron signal, calculated using the fields from (d-f) and thus extracting the wavepacket profiles using PINEM. The PINEM signal shows a clear distinction between the wavepacket of the first mode, second mode, and higher modes, similar to the experimental results. Noticeably, the PINEM technique remains sensitive to the higher-order branches, demonstrating its potential for probing polaritons of extreme confinement.

We found that the electron-polariton PINEM interaction remains efficient for polariton wavepackets at higher-order branches. This feature is seen in the experimental data: the firstand second-order PhP branches provide a signal of a similar magnitude (Fig. 4(b)). Our numerical simulations confirmed this surprising result: despite the second-branch having an overall smaller energy (due to less efficient coupling), the PINEM signal is of comparable strength. This result arises from the nature of the interaction, whereby the electron integrates along its trajectory and thus includes the field inside the hBN, which is larger as the field is more confined. Higher-branch modes have larger confinement and thus higher field amplitudes and yet they decay faster along the z direction; thus, the integrated signal remains comparable.

Most nearfield imaging techniques probe the surface of the material and are thus essentially less efficient when the goal is to image wavepackets in higher-order branches, since they decay more rapidly along the z direction (the inefficient light out-coupling of the higher-branches reduces the signal further). By using the penetration of free electrons, PINEM-type techniques such as ours bypass these limits, becoming especially advantageous for imaging higher-branch polaritonic modes and highly confined polaritons in general.

Summary and Outlook

Our measurements provided a first observation of the spatiotemporal dynamics of a coherent excitation in a 2D material. By virtue of the direct access to the wavepacket dynamics, we extract several unique phenomena that can be further explored. Our experiments rely on PINEM in the mid-IR range and successfully resolve the PhP wavepackets. The success of the measurement is due to the extremely slow group velocities of PhPs, tens-to-hundreds of times slower than the speed of light, which enables relatively small spatial extents, where the entire wavepacket extends over a few microns along the direction of propagation.

Our work builds on key advances over the past decade in the areas of PINEM, ultrafast electron microscopy, and ultrafast electron diffraction [START_REF] Lahme | Femtosecond single-electron diffraction[END_REF][START_REF] Sciaini | Femtosecond electron diffraction: heralding the era of atomically resolved dynamics[END_REF]. To date, PINEM techniques examined the temporal dynamics of optical excitations by interference patterns of plasmon polaritons [START_REF] Madan | Holographic imaging of electromagnetic fields via electron-light quantum interference[END_REF][START_REF] Vanacore | Ultrafast generation and control of an electron vortex beam via chiral plasmonic near fields[END_REF] and decay of photonic cavity modes [START_REF] Wang | Coherent interaction between free electrons and a photonic cavity[END_REF][START_REF] Kfir | Controlling free electrons with optical whispering-gallery modes[END_REF]. In contrast, our experiments resolve the dynamics of the entire optical wavepacket due to the slower group velocities and shorter wavepacket extents of PhPs. The slow group velocity causes the wavepacket envelope to remain almost static within the duration of the electron interaction (the wavepacket moves only a fraction of a micron during the electron's duration, 0.13 ps standard deviation); thus, we can record the wavepacket dynamics in a way that has not been possible so far. In this manner, our experiment is operated at frequencies that are in between the optical range and the range of mechanical vibrations (typically at GHz frequencies), where the dynamics of phonons was observed thanks to their slow oscillation relative to the duration of the electron probe [START_REF] Cremons | Femtosecond electron imaging of defect-modulated phonon dynamics[END_REF][START_REF] Valley | Imaging Intra-and Interparticle Acousto-plasmonic Vibrational Dynamics with Ultrafast Electron Microscopy[END_REF].

The PhPs in our experiment differ from these mechanical vibrations by being coherent optical excitations, which are in fact hybrids of the mechanical vibrations (optical phonons) and photons in the IR region. Consequently, the imaging of polaritons in 2D materials utilize advantages from both plasmon imaging [START_REF] Madan | Holographic imaging of electromagnetic fields via electron-light quantum interference[END_REF][START_REF] Vanacore | Ultrafast generation and control of an electron vortex beam via chiral plasmonic near fields[END_REF] and phonon imaging [START_REF] Cremons | Femtosecond electron imaging of defect-modulated phonon dynamics[END_REF][START_REF] Valley | Imaging Intra-and Interparticle Acousto-plasmonic Vibrational Dynamics with Ultrafast Electron Microscopy[END_REF].

We envision future experiments in which free electron probes are used for record-high spatial resolution in nearfield optical microscopy, reaching single-nm resolutions and below (considered beyond the reach of conventional nearfield imaging tips used in SNOMs). Singlenm resolutions are of great importance for imaging polaritons of the highest confinement (e.g.,

Ref. ( 65)), where new physics can emerge, such as the effects of optical nonlocality. The features in the PhP wavepacket in our experiment were not sufficiently small to test this limit.

A disadvantage of the current PINEM approach is that the pulsed free electron does not sense the PhP phase since it is averaged-out over the electron pulse duration (~300 fs). To measure the propagating wavepacket phase as in SNOM, one could consider interference between two polaritons [START_REF] Madan | Holographic imaging of electromagnetic fields via electron-light quantum interference[END_REF], shorter electron pulse durations, or pre-bunched electrons, which were already shown to reach attosecond timescales in electron microscopes [START_REF] Priebe | Attosecond electron pulse trains and quantum state reconstruction in ultrafast transmission electron microscopy[END_REF][START_REF] Vanacore | Attosecond coherent control of free-electron wave functions using semi-infinite light fields[END_REF]. Using attosecond electron probes would open the possibility to measure the dynamics of plasmon polariton wavepackets in the visible range.

Looking forward, the unique optical properties of 2D polaritons can enable many novel phenomena concealed in their dynamics. For example, we could explore diffractionless polariton wavepackets [START_REF] Hu | Topological polaritons and photonic magic angles in twisted α-MoO3 bilayers[END_REF] or polariton focusing and continuous wavepacket deceleration in cases where the polariton propagates in a flake with a gradually reducing thickness [START_REF] Voronin | Nanofocusing of acoustic graphene plasmon polaritons for enhancing mid-infrared molecular fingerprints[END_REF]. The current challenge is to improve the in-coupling of the external laser excitation into the polaritonic mode, which can be overcome by using metallic nano-antennas [START_REF] Pons-Valencia | Launching of hyperbolic phonon-polaritons in h-BN slabs by resonant metal plasmonic antennas[END_REF] or nanoparticles [START_REF] Dias | Fundamental Limits to the Coupling between Light and 2D Polaritons by Small Scatterers[END_REF] instead of only the sample edge, as in our study. Improved coupling efficiency would also enable the investigation of optical nonlinearities in 2D materials [START_REF] Jadidi | Phonon-polariton-enhanced nonlinearity in hexagonal boron nitride[END_REF]. A Kerr-like nonlinearity, as found in hBN, incorporates a potential for the observation of spatiotemporal soliton-wavepackets of 2D polaritons and novel forms of 2D shockwaves and light bullets. Spatial optical solitons in Moiré lattices of 2D materials [START_REF] Sunku | Photonic crystals for nano-light in moiré graphene superlattices[END_REF][START_REF] Ni | Soliton superlattices in twisted hexagonal boron nitride[END_REF] have also recently become a topic of great interest. The observation of the dynamics of polaritons in such materials can unveil their intrinsic topological nature, such as the one-way dynamics of polaritons that propagate only along the supercell edges. Looking further ahead, the spatial and temporal resolution of the approach we developed, combined with its flexibility in mixing excitation wavelengths, could promote the observation of out-of-equilibrium polaritons and their topological properties [START_REF] Karzig | Topological polaritons[END_REF][START_REF] Song | Chiral plasmons without magnetic field[END_REF].

As future free-electron-PhP research avenues, the free electrons can be brought to interact with propagating polaritons in a phase-matched manner by aligning the electron to propagate parallel to the surface [START_REF] Dahan | Resonant phase-matching between a light wave and a free-electron wavefunction[END_REF]. In such ("aloof") configurations, we expect exceptionally strong interactions and the emergence of the yet unobserved Cherenkov-type phenomena in 2D materials [START_REF] García De Abajo | Boundary effects in Cherenkov radiation[END_REF], potentially reaching nonlocal ultrastrong coupling between polaritons and electrons [START_REF] Kurman | Tunable bandgap renormalization by nonlocal ultra-strong coupling in nanophotonics[END_REF].

( 84), ( 85) [START_REF] Frisenda | Recent progress in the assembly of nanodevices and van der Waals heterostructures by deterministic placement of 2D materials[END_REF][START_REF] Castellanos-Gomez | Deterministic transfer of two-dimensional materials by all-dry viscoelastic stamping[END_REF]. [START_REF] Wang | Coherent interaction between free electrons and a photonic cavity[END_REF][START_REF] Polman | Electron-beam spectroscopy for nanophotonics[END_REF][START_REF] Talebi | Near-Field-Mediated Photon-Electron Interactions[END_REF]. (89) (55, 90), [START_REF] Razinskas | Normal-Incidence PEEM Imaging of Propagating Modes in a Plasmonic Nanocircuit[END_REF]. [START_REF] Ulbrich | Polariton wave packet propagation in the exciton resonance of a semiconductor[END_REF][START_REF] Cheung | Excitation of coherent phonon polaritons with femtosecond optical pulses[END_REF]. [START_REF] Lanin | Time-domain spectroscopy in the midinfrared[END_REF]

Fig. 1 .

 1 Fig. 1. Direct observation of 2D polariton wavepackets using ultrafast transmission electron microscopy (UTEM). (a) Experimental setup. A femtosecond laser (orange) splits into two branches. The bottom branch is converted into a mid-infrared (IR; red) pulse using difference frequency generation (DFG) and excites the phonon-polariton (PhP) wavepacket in the isotopically pure hBN ( 11 B) sample. The top branch is converted into an ultraviolet (UV; purple) pulse using fourth harmonic generation (FHG) and excites photo-electrically the electron pulse (cyan). The electron images the hBN sample when counting electrons at all energies (unfiltered image) and images the PhP wavepacket when counting only electrons that gained energy (filtered image). (b) Dispersion relation of a 55 nm thick isotopically pure hBN ( 11 B) flake on a 20 nm thick Si3N4 membrane (for sample-preparation, see Supplementary Materials, SM, section S.2). The hyperbolic nature of hBN creates multiple dispersion branches in each frequency, each creating a different propagating wavepacket. The spectral bandwidth of the IR pulse (pink) excites a range of polariton modes (circled in green). (c) Free electron probing the (TM polarized) propagating PhP wavepacket inside the hBN. Insets: electron energy loss spectra (EELS) with the laser on (left) and off (right). (d) Measurement of the energy-filtered electrons for different time delays 𝜏 between the laser pulse and the electron probe, showing the propagation dynamics of the PhP wavepacket (shown in top-view in Fig. S1, and in Movie S1). Note that the definition of 𝜏 is up to arbitrary shifts in all figures and movies.

Fig. 2 .

 2 Fig. 2. Direct observation of the phonon-polariton (PhP) wavepacket's spatiotemporal dynamics. (a) Measured signal when averaged along the edge direction (dots) with the fit of a Gaussian profile for each measurement (blue-green curves). Inset: unfiltered image of the hBN flake; the scale bar denotes 5 𝜇m. The times are in units of ps. (b) Map presenting the measured wavepacket as a function of time and distance from the edge; blue represents the Gaussian wavepacket peak and gold the laser excitation intensity. At short times, while the laser is still on, the group velocity undergoes changes due to different spectral components arriving at different delays (due to a chirp excitation). In this figure, the time delay 𝜏 = 0 is related to the peak of the excitation intensity. This measurement uses a 55 nm thick hBN flake, excited by a 6470 nm laser with a bandwidth of 175 nm.

Fig. 3 .

 3 Fig. 3. Analysis of the PhP wavepacket formation, propagation, and decay -extracting the group velocities and wavepacket behavior. (a-c) The fitted Gaussian peak locations as a function of time with the extracted group velocities in units of μm/ps. We compare three h 11 BN samples having a thickness of (a) 7.5 nm, (b) 12 nm, and (c) 55 nm. The colored background represents the times at which the wavepacket is still being pumped. For clarity, the measurements are shifted in time from one another; also, the dashed horizontal lines in (c) represents that the blue and red curves are shifted in space by 2.5 𝜇m and the yellow curve by 5 𝜇𝑚. Top inset in (b): unfiltered image of the 12 nm sample (dashed lines represent sample edges); scale bar is 5 𝜇m. Bottom inset in (b): an electron energy loss filtered image showing the group at a specific time. (d) Simulation results of the wavepacket peak location as a function of time in a 55 nm h 11 BN sample, comparing different levels of chirp (the excitations' temporal profiles are brought above the curves). The simulations show the acceleration and deceleration observed experimentally. Top inset: the simulated field. Bottom inset: the energy-filtered electron signal calculated using the field from the top inset, showing that the PINEM technique indeed extracts the wavepacket profile. Further simulations of the velocity change can be seen in Fig. S5 and movies S3 and S4. (e) Wavepacket integrated signal (derivation in the SM section S.3), which helps identify the transition from a formation stage, in which the pump overcomes the dispersion and intrinsic PhP losses, to the eventual decay (the effective timescales are denoted in ps). (f) There is a clear correlation between the dispersion rate and the wavepacket's group velocity, as expected by theory: wavepackets with higher group velocities are less dispersive.

  steady propagation velocity. The first two regimes are related to the chirp of the excitation pulse, which pumps the PhP with different spectral components at different times. The measured phenomena of wavepacket deceleration and acceleration are consistent with the continuous-PINEM analysis of a set of finite-difference time-domain (FDTD) simulations, as shown in Fig.3(d), Fig.S5, and Movies S3 and S4. The simulations show that when the pulse is longer, and/or when the chirp is stronger, the change in velocity becomes more dominant, in agreement with our measurements.

Figure 4 :

 4 Figure 4: Observation of a multi-branch PhP wavepacket that splits into two distinct wavepackets of different branches and different group velocities. (a) Snapshots at different times of the two wavepackets propagating through the sample, with small blue and red arrows marking the peaks of each wavepacket. (b) Measured signal when averaged along the edge (dotted curves) with the double-Gaussian fit for each measurement (solid curves). The data show two distinct wavepackets with different group velocities. The time steps between the curves are 0.1 ps, starting at 𝜏 = 0.2 ps. (c) Locations of the two wavepackets as a function of time, extracting the group velocities in units of μm/ps. The colored background represents times at which the wavepacket is still being pumped, as the excitation pulse has not ended. (df) Simulated z-component of the electric field at specific times 𝜏 , showing an example of wavepacket evolution as it splits into multiple individual wavepackets of different orders. (g) The energy-filtered electron signal, calculated using the fields from (d-f) and thus extracting


Sven Oras

Sergei Vlassov
email: sergei.vlassov@ut.ee

Simon Vigonski

Boris Polyakov

Mikk Antsov

Vahur Zadin

Rünno Lõhmus

Karine Mougin

The effect of heat treatment on the morphology and mobility of Au nanoparticles

Keywords: annealing, atomic force microscopy (AFM), Au nanoparticles, manipulation, melting, nanotribology

In the present paper, we investigate the effect of heat treatment on the geometry and mobility of Au nanoparticles (NPs) on a Si substrate. Chemically synthesized Au NPs of diameter ranging from 5 to 27 nm were annealed at 200, 400, 600 and 800 °C for 1 h. A change in the geometry from faceted to more rounded shapes were observed with increasing annealing temperature. Kinetic Monte Carlo simulations indicate that the NPs become rounded due to the minimization of the surface area and the transition to lower energy surface types {111} and {100}. The NPs were manipulated on a silica substrate with an atomic force microscope (AFM) in tapping mode. Initially, the NPs were immovable by AFM energy dissipation. However, annealed NPs became movable, and less energy was required to displace the NPs annealed at higher temperature. However, after annealing at 800 °C, the particles became immovable again. This effect was attributed to the diffusion of Au into the Si substrate and to the growth of the SiO 2 layer.

Introduction

Gold is one of the most prominent materials used in studies related to nanostructures. The small size and the enhanced properties of Au nanoparticles (NPs) compared to bulk gold make them important for the development of novel applications, for example, in the field of drug delivery [1], sensor technology [2], printing [3] and catalysis [4]. Due to their inert state, geometrical diversity and convenient synthesis, Au NPs are an attractive model system for nanotribological manipulation experiments [5][6][7][8][9][10][11][12] and simulations [13]. Additional flexibility is provided by the ability to tune the properties of the NPs by varying the size [14] , shape [14] and chemical composition of the outer layer covering the NP [5]. These modifications become especially important for tribological applications since they can affect the interfacial behavior during contact.

The shape of a NP is known to have a significant impact on the contact area between the NP and a substrate [14]. The morphology, and thus the contact area of the NP, can be tuned by changing the synthesis parameters [15]. For readily synthesized NPs, the morphology can be changed by adding energy to the NPs by laser treatment [16] or by heating the NPs in an oven. Partial melting can be achieved at a significantly lower temperature than the melting temperature of bulk Au [17].

In this study, chemically synthesized faceted Au NPs were annealed at different temperatures between 200 and 800 °C, which resulted in rounding of the NPs. The particles were then displaced with an atomic force microscope (AFM) in order to study the effect of annealing on their tribological behavior.

Experimental Nanoparticle synthesis

A colloidal suspension of Au NPs was produced by reducing an aqueous solution of HAuCl 4 •H 2 O. The procedure consisted of adding 3 mL of 1% aqueous HAuCl 4 (Sigma-Aldrich) to 150 mL of pure H 2 O at 90 °C under vigorous stirring. After 1 min, 2 mL of a 1% aqueous solution of sodium citrate (C 6 H 5 Na 3 O 7 , Sigma-Aldrich) were added to stabilize the suspension. By reducing HAuCl 4 , sodium citrate transfers the negative charge of the citrate ions to the Au NP surface [18,19]. The solution was stirred for 5 min and then stored at 4 °C until needed [10].

Annealing

The silicon wafers (100, n/phosphorus-doped, 3-6 ohm•cm, Mat-Technology) were cleaned with ethanol and the NPs were deposited by drop-casting. The drops were dried at ambient conditions. Five samples of Au NPs on Si wafers were prepared, and four of them were then separately annealed in an oven for 1 h at 200, 400, 600 and 800 °C while the fifth was left untreated. Separate samples were prepared on heat-resistant silicon nitride support films of 50 nm membrane thickness (Pelco, Ted Pella) to be used for transmission electron microscopy (TEM).

Characterization

The morphology of the NPs annealed at different temperatures was studied by TEM (ARM200, JEOL). A null ellipsometer (Multiskop, Optrel, Germany) equipped with a 532 nm Nd:YAG laser was used for measuring the thickness of the SiO 2 layers.

AFM setup

The manipulation of the NPs was performed with a Bruker Multimode 8 AFM in the PeakForce quantitative nanoscale mechanical characteriztion (PeakForce QNM) mode using a rectangular AFM cantilever (Bruker, RTESPA-300, k = 40 N/m) with a resonance frequency of around 300 kHz.

Prior to each manipulation, the samples were heated to 100 °C to remove excess water. An image was first taken in the highresolution QNM mode to find the Au particles. Then, the operation mode was switched to tapping mode. The oscillation amplitude was kept constant with a feedback loop on, and the power dissipated during tapping was calculated with the following equation [20]:

where k is the cantilever spring constant, f 0 is the resonance frequency of the cantilever, A set is the setpoint amplitude, A piezo is the drive amplitude, θ is the phase signal and Q is the quality factor of the AFM cantilever. The dissipated power was used as a measure of the mobility of the NPs.

The calibration of the cantilevers was performed by the thermal tuning method. The oscillation amplitudes ranged from 0.05 to 0.6 V with a sensitivity of 25 to 31 nm/V. The sensitivity was measured individually for each cantilever by means of damping the AFM tip against a silicon substrate. The phase values were extracted from the tapping mode phase images for each manipulated NP and processed using the Gwyddion software. The phase shift values were extracted from the profile of the trace on the phase shift image by averaging ten lines using the Gwyddion program.

After each scan, the force applied on the NPs was increased until all the particles in the area were displaced or the force was high enough to start severely damaging the tip. If the tip was too dull for performing manipulations, the tip was replaced with a fresh and sharp AFM tip.

Simulation setup

Simulations of the NP rounding were carried out using the kinetic Monte Carlo (KMC) code Kimocs [21], which is specifically designed for metal surfaces. The parameters for Au, developed by Vigonski et al. [17], use the tethering method for stabilization [22].

Initially cubic NPs of two sizes with side lengths of 2 and 5 nm were chosen. The simulations were performed at 726.85 °C (1000 K) in order to obtain results within a reasonable computa- tion time. This simplification is expected to have only minor influence, especially in the case of a NP diameter of 5 nm and larger [17]. The parametrization of Au implemented by Vigonski et al. [17] utilizes only nearest-neighbor jumps, which lead to a rather uniform behavior of larger NPs. In the diffusion model used in Kimocs, the computational time increases exponentially with decreasing temperature, making simulations at experimental temperatures unfeasible. Nevertheless, the temperature used in the simulations is significantly below the melting temperature of Au. Furthermore, in the computational model, the dynamics of diffusion are unaffected by temperature, making comparisons to experimental results possible [21]. Visualizations of the results of the simulation were generated with Ovito [23].

Results and Discussion

As observed in the TEM images, most of the untreated Au NPs had irregular faceted shapes with a diameter ranging from 5 to 27 nm with a median diameter of 13 ± 3.96 nm (measured on 90 particles). The term "diameter" in the present study was defined as the average value of the largest and the smallest dimension of the NPs visible on the TEM image. A typical TEM image of the synthesized Au NPs is presented in Figure 1.

Influence of temperature -change in morphology

The TEM analysis revealed a considerable change in the geometry of the NPs to a more rounded shape as a result of annealing, although the temperatures used were far below the melting point of bulk Au (1064 °C). Figure 2 presents typical TEM images of the NPs annealed at different temperatures. Some tendency for rounding can be observed even for the sam-ple annealed at 200 °C. The effect becomes clearly visible at 400 °C and increases further for higher temperatures (Table 1). To evaluate the morphological changes more quantitatively, we introduced a criterion based on the length of the facet: if the NP is smaller than 10% of its total visible perimeter, then it is classified as rounded. The reduced melting temperature of NPs compared to their bulk counterparts is not a new phenomenon. In 1976, Buffat and Borel [24] demonstrated that the melting temperature of Au NPs is size-dependent. However, the decrease of the melting temperature presented in their work was only up to a few hundred Kelvin for particles of diameter around 5 nm and was even less pronounced for larger particles, approaching the melting point of the bulk above 20 nm. In contrast, we observed rounding at much lower temperatures, even for particles exceeding 20 nm in size. Moreover, rounding of Au NPs at just 500 °C has been reported even for particles exceeding 100 nm in diameter [6]. To explain the rounding, it is probably more appropriate to consider the diffusion of surface atoms rather than complete melting. Another important factor for heat-induced morphological changes, overlooked by many other studies, is annealing time. We believe that the surface effects resulting in the rounding of the NPs are due to energy minimization via a rearrangement of surface atoms, similar to the effect demonstrated recently by Vigonski et al. [17] for heat-induced segmentation of Au nanowires at temperatures much below the melting point of bulk Au. We observed that the effect of rounding was more pronounced for NPs with a five-fold twinned inner structure compared to single-crystalline particles in the form of truncated triangles. This can be explained by the presence of inner stresses stored inside the five-fold twinned NPs and the tendency of such particles to seek mechanisms of stress relaxation [25]. The rearrangement of surface atoms into more rounded outer geometries can be a way of energy minimization while preserving the five-fold twinned inner structure of the NP, as shown in Figure 3 for a Au NP heated to 800 °C.

KMC simulations at 726.85 °C (1000 K) show that the NPs become rounded through a diffusion process which combines the minimization of the surface area and the transition to lower energy surface types (Figure 4). The preferred surface of face- centered cubic (FCC) materials such as Au is {111}, and the simulations clearly show that surfaces of this type grow as the NP relaxes. The KMC model used here describes perfect crystals. Therefore, the resulting shape resembles a dodecahedron bounded by the energetically favorable surfaces {111} and {100}. Our simulations do not include five-fold twinned particles and lower temperatures due to limited computational power and time. Advanced modelling and in-depth theoretical analysis of the rounding process lies outside the scope of the present study and should be addressed separately. Nevertheless, this relatively simple model agrees well with the results of our experiments and sheds light on the processes taking place in small metal particles at elevated temperatures.

Manipulation of Au nanoparticles

The main goal of this paper was to investigate the effect of the heat-induced rounding of Au NPs on their mobility. Rounding of NPs should decrease the contact area compared to faceted particles, and hence reduce the friction forces in accordance with the known relation τ = F/A [6], where τ is the contact strength, F is the friction force and A is the contact area. For a round particle, the contact area is determined by contact mechanics [14,26]. The contact area of a perfect sphere can be two orders of magnitude smaller than that of a polyhedron-like NP, as was shown by Vlassov and co-workers [6].

The mobility of the Au NPs was evaluated by means of the power dissipated in tapping-mode AFM, which has previously been shown to be an effective technique for the evaluation of NP mobility [10]. An example of the manipulation sequences for the sample annealed at 400 °C is given in Figure 5. A selected area is scanned in tapping mode, and the driving amplitude is increased until the displacement of NPs is observed. The dissipated power is then calculated.

In total, 34 data points were collected: 12 measurements for NPs annealed at 600 °C, 16 for NPs annealed at 400 °C and four for NPs annealed at 200 °C (Figure 6). The low amount of data points for the NPs annealed at 200 °C is due to the fact that most of the NPs required more power for a displacement than the experimental setup could provide. Therefore, the actual average friction determined for the NPs annealed at 200 °C is even higher. The power required to displace NPs is the highest for particles annealed at 200 °C and the lowest for NPs annealed at 600 °C. This finding agrees well with the rounding observed by TEM. Moreover, the amount of dissipated power divided by the radius of the NP (power per NP radius) decreases with increasing size of the NP annealed at 600 °C series, which may be attributed to a rolling motion as was demonstrated by

Darwich and co-workers [8]. They studied the rolling and sliding motion of NPs as a function of NP size and showed that the dissipated power necessary for sliding has little or no dependence on the NP radius, while the dissipated power needed to provoke rolling decreased clearly with increasing NP radius.

The indirect indication of a rolling motion in the present study may serve as additional confirmation of the rounded geometry of the NPs annealed at higher temperatures.

A displacement of both the unannealed NPs and those annealed at 800 °C not possible. The immobility of unannealed NPs can be linked to their more pronounced facets (and hence higher surface area) as well as the presence of surfactants that can act as a glue. The immobility of the NPs heated to 800 °C was surprising, especially considering the fact that they look the most round in the TEM images. One possible explanation is related to the partial diffusion of Au into the Si substrate. It is known that gold diffuses into silicon at higher temperatures by the so-called kick-out mechanism, in which self-interstitials present at thermal equilibrium displace substitutionally dissolved Au atoms into interstices, such that these may undergo rapid interstitial diffusion [27]. The process is highly temperature-dependent, and diffusion at 800 °C is four orders of magnitude faster than at 400 °C (see Table 7 in Fisher [START_REF] Fisher | Diffusion in Silicon -10 Years of Research[END_REF]). Another reason may be the temperature-sensitive growth of the SiO 2 layer [START_REF] Gorantla | [END_REF], which escalates rapidly above 600 °C, as can be seen from the ellipsometry measurements presented in Figure 7.

How exactly the rapid growth of the SiO 2 layer may be related to the drastic increase in friction remains unclear and can be the subject for future studies. Overall, we demonstrated that heat treatment, which is widely used as a surfactant removal step prior to nanomanipulation experiments, can have extensive effects on the mobility of Au NPs.

Conclusion

Chemically synthesized Au NPs with a medium diameter of 14 nm were annealed at 200, 400, 600 and 800 °C for 1 h. Untreated particles had irregular faceted shapes. Annealing resulted in geometry changes to a more rounded shape. A slight tendency towards rounding was noticed at 200 °C. The rounding effect became clearly prominent at 400 °C and increased further for higher temperatures. After annealing at 800 °C, most of the particles had sphere-like shapes. KMC simulations at 726.85 °C (1000 K) show that the NPs become rounded by a diffusion process that minimizes the surface energy. The process combines minimizing the surface area and transitioning to the lower energy surface types {111} and {100}. In AFM manipulation experiments, it was found that the higher the annealing temperature, the lower the power necessary to displace the NPs, indicating that the mobility of the particles increases at elevated annealing temperature. However, after treatment at 800 °C, the particles became immovable. We attributed this surprising effect to the diffusion of Au into Si and to the growth of the SiO 2 layer. Both processes are highly temperature-dependent, exhibiting drastic enhancement above 700 °C. Overall, we demonstrated that heat treatment, which is widely used for surfactant removal prior to nanomanipulation experiments, can have an extensive effect on the mobility of Au NPs.

Figure 2 :

 2 Figure 2: Morphology of the Au NPs and after annealing at (a) 200 (b), 400 (c) 600 (d) and 800 °C. Slight rounding of the NPs can be seen at higher temperatures.

Figure 1 :

 1 Figure 1: Typical TEM image of Au NPs before thermal treatment.

Figure 3 :

 3 Figure 3: Au NP annealed at 800 °C with well-pronounced five-fold twinned inner structure.

Figure 4 :

 4 Figure 4: The transition of 5 and 2 nm NPs into rounded shapes bounded by energetically favorable surfaces. Atoms are colored according to their crystallographic surface type.

Figure 5 :

 5 Figure 5: (a) Profile of a Au NP and (b) topography image of NPs on the Si substrate prior to manipulation. (c and d) Phase images recorded during the manipulation.

Figure 6 :

 6 Figure 6: Dissipated power per NP radius as a function of the NP diameter. The diameter is defined as the height of the NP in the AFM image.

Figure 7 :

 7 Figure 7: Evolution of the SiO 2 layer thickness as a function of the annealing temperature.

Table 1 :

 1 Ratio of truncated to rounded particles depending on the annealing temperature.

	Temperature (°C) Truncated	Rounded	Ratio
	200	105	10	10.5
	400	64	43	1.5
	600	37	42	0.9
	800	9	57	0.2

Funding

This work was supported by The Centre National de la Recherche Scientifique (CNRS) of France and the French Embassy Program. The authors are also grateful for partial support by COST Action CA15216, the Estonian Science Foundation (grants PUT1689 and PUT1372), the Estonian Centre of Excellence in Zero Energy and Resource Efficient Smart Buildings and Districts, ZEBE, grant 2014-2020.4.01.15.0016 and Latvian Science Council grant lzp-2018/2-0083. ORCID ® iDs Sven Oras -https://orcid.org/0000-0002-8313-8608 Sergei Vlassov -https://orcid.org/0000-0001-9396-4252 Mikk Antsov -https://orcid.org/0000-0003-1291-3520 Preprint A non-peer-reviewed version of this article has been previously published as a preprint doi:10.3762/bxiv.2019.68.v1