Nanoscale diffractive probing of strain dynamics in ultrafast transmission electron microscopy Open Access

Controlling confined phononic modes in the giga- to terahertz frequency range offers new approaches to steer the flow of heat in nanoscale structures¹ with a broad field of potential applications, ranging from advanced thermoelectric devices² to the heat management in dense semiconductor circuits.³ Furthermore, coupled to tailored light fields, phononic modes with mega- to gigahertz resonance frequencies already developed into essential building blocks in nanometrology.^4,5

Nanophononics based on tailored multilayer structures has made great progress in recent years, achieving, for example, phonon filtering⁶ and phonon amplification.⁷ Beyond layered systems, three-dimensionally nanostructured materials facilitate thermally rectifying behavior,⁸ highly efficient channeled thermal transport across nanoscale vacuum gaps,^9–11 enhanced light matter interactions in combined phononic-photonic resonators,¹² and phonon lasing.^13,14 Optical methodologies, such as ultrafast optical spectroscopy¹⁵ and Brillouin scattering,^16–18 allowed for experimental access to the spectral and temporal properties of nanophononic systems, including resonance frequencies, dissipation times,¹⁹ and nonlinear couplings.²⁰ However, extracting quantitative information on the structural distortion in nanophononic structures often requires elaborate theoretical modeling. Knowledge of the strain field is essential for tailoring the interaction between phononic fields and other degrees of freedom, such as the coupling of lattice distortions to the electronic²¹ and magnetic^22,23 subsystems, interaction with confined light fields,¹² and phase-transitions driven by acoustic²⁴ and optical²⁵ phonon fields.

In laterally homogenous samples, ultrafast electron^26–34 and X-ray^35–39 diffraction allows for quantitative access to collective transient lattice distortions. Extending these approaches to three-dimensionally nanostructured geometries remains challenging, despite recent progress in ultrafast coherent diffractive dark-field imaging^40,41 utilizing intense X-ray pulses at free-electron laser facilities.⁴² In a table-top approach, ultrafast transmission electron microscopy (UTEM)^43–51 provides a visualization of nanophononic modes by time-resolved bright-field imaging,^52–55 with first steps towards local diffractive probing.^56–58 However, the full capabilities of conventional transmission electron microscopy^59–65 for the quantitative mapping of strain fields have not been harnessed in UTEM.

Here, we demonstrate the quantitative nanoscale probing of optically triggered ultrafast strain dynamics in UTEM, employing ultrashort electron pulses in convergent beam electron diffraction (CBED). We achieve a quantitative three-dimensional spatio-temporal reconstruction of the ultrafast lattice distortions in nanoscopic volumes close to the edge of a single crystalline graphite membrane. High-amplitude coherent expansional and shear acoustic waves are launched at the symmetry-breaking sample boundaries, and we track their ballistic propagation and dephasing on nanometer length scales.

In the experiments, we generate low-emittance ultrashort electron pulses by localized photoemission from a tip-shaped field emitter.^46,51,66 The femtosecond electron pulses are accelerated to an electron energy of 120 keV, coupled into the electron optics of a transmission electron microscope and tightly focused (28-nm focal spot size) onto a 120-nm thick graphite membrane. For varying probing positions relative to the edge of the membrane, electron diffraction patterns are detected in the far-field [Fig. 1(a)]. The sample is optically excited by femtosecond laser pulses focused to a 50-μm focal spot diameter (centered at the graphite edge, 800-nm central wavelength, 50-fs pulse duration, 16-mJ/cm² fluence). Inhomogeneous structural dynamics are induced on length scales much smaller than the optical focal spot size due to the broken translation symmetry at the nearby sample edge. At an adjustable delay time Δt relative to the electron pulse arrival, local structural dynamics are stroboscopically mapped at the electron focal spot position. See supplementary material for further details on the experimental setup and the graphite sample system.

Figure 1(b) displays a typical ultrafast large-angle convergent beam electron diffraction pattern recorded with femtosecond electron pulses before optical excitation (Δt < 0). In the pattern, the central intense disc-like feature represents the angular distribution of the illuminating electron pulses. Bragg scattering conditions for the graphite lattice planes (hkl) are fulfilled along specific lines in momentum space.⁶⁷ At their intersection with the central disc, efficient scattering occurs, forming deficit intensity lines within the disc, and excess lines, which are radially displaced by Bragg angles θ_B [Figs. 1(e) and 1(f)].

The angular displacement of each line encodes the length and orientation of a specific reciprocal lattice vector G_hkl and the scattering efficiency encodes the corrugation of the scattering potential.^62,68 Thereby, U-CBED gives access to the ultrafast temporal change of local lattice periodicities d_hkl and atomic mean-square displacements $u2$ (supplementary material, SI 4). The broad angular range of the incident electron beam (50 mrad full convergence angle) and the chosen sample orientation enable the simultaneous observation of multiple independent Bragg scattering conditions and the corresponding rocking curves,⁶⁹ providing direct experimental access to the local structural distortion and its temporal evolution.

After optical excitation, we observe pronounced delay-dependent radial Bragg line shifts Δθ (by up to 6 mrad) in the CBED pattern. For a series of delay-dependent diffraction patterns, see supplementary material movies, M1 and M2. The induced strain dynamics results in no significant azimuthal rotation of Bragg lines for the chosen sample orientation. In the following, we therefore consider the transient changes of Bragg line profiles, obtained by integrating the diffracted intensity along the individual line directions.

In Fig. 2, we show the delay-dependent profiles of selected excess Bragg lines for two different probing positions. With the electron focal spot placed at a distance of r = 500 nm from the edge of the graphite membrane [Fig. 2(b)], the (422) and (⁠$3¯2¯1¯$⁠) lines display a strong multi-frequency oscillatory behavior of the average line position and a modulation of the line profile, even including line splittings into multiple components. Other Bragg lines show a different temporal characteristic [e.g., (⁠$2¯4¯0$⁠)] or only very weak overall changes [e.g., (020)]. Remarkably, the recorded transient changes are strongly influenced by the nearby membrane edge. In a continuous part of the film, a much simpler dynamic behavior of the line profiles is observed, as is evident by comparing the transient (422) profiles in Figs. 2(a) and 2(b).

Disentangling the complex structural dynamics encoded in the ultrafast Bragg line shifts requires a quantitative description of the CBED pattern and its dependence on the distortion of the graphite film. Microscopically, the evolution of the local structural deformation of the membrane is described by the time-dependent tensor field of the deformation gradient F(r, Δt)= ε + ω + Ι₃, which can be decomposed into a symmetric strain tensor ε and an antisymmetric rotation tensor ω (I₃: unit tensor).⁶² 

Calculating the position of deficit and excess Bragg lines in the CBED pattern requires an adequate description of the Bragg scattering conditions in reciprocal space, which we derive from the graphite unit cell⁷⁰ defined by $a1=a[1, 0, 0]T, a2=a [1/2, 3/2, 0]T, a3=c[0, 0, 1]T$⁠, with lattice constants $a=2.46 Å$ and $c=6.71 Å$⁠. To account for an arbitrary sample orientation, the real space basis, represented by the matrix $Bu=[a1 a2 a3]$ in the three-dimensional coordinate system (x, y, and z), is rotated [see Fig. 1(a)] by applying a matrix $R=RZ(γ)RY(β)RX(α)$⁠, with rotation matrices $RX,Y,Z$ around a laboratory-fixed coordinate system, X, Y, and Z axes, respectively. Taking the planar sample orientation into account, the angles $α$ and $β$ correspond to the angular degrees-of-freedom of the double-tilt sample holder, and $γ$ is related to the azimuthal orientation of the graphite flake. The reciprocal basis in the laboratory-fixed coordinate system is given by $Gu=(RBu)−1$⁠, so that the reciprocal lattice vector with Miller indices h, k, and l is expressed as $G=Gu[h, k, l]T$⁠. For scattered and incident wave vectors k and k₀, allowed scattering conditions are obtained from $G2+2k0⋅G=0$ by considering the Laue equation $G=Δk=k−k0$ (conservation of momentum) and elastic scattering $(G+k0)2=k2$ (conservation of energy).⁶⁹ In the paraxial approximation, i.e., $k0X, k0Y⪡k0$ and $kX, kY⪡k$ (for the optical axis chosen along Z), the scattering conditions can be simplified to

which describes straight lines $[kX, kY]$ in transverse k-space, for each reciprocal lattice vector G. In CBED, the incidence electron spot covers a circular region in the diffraction pattern, and, with the detector plane perpendicular to the optical axis, the allowed scattering conditions are visible as deficit lines with a distance to the origin of $rdeficit=(G2/2−k0GZ)/GX2+GY2$ and an inclination angle of $tan (φ)=GX/GY$⁠.

The electrons are scattered into excess lines, which are displaced from the corresponding deficit line by the projected radial scattering vectors $[GX, GY]$⁠, so that their radial distance becomes

Bragg line shifts are evaluated by considering changes in their center-of-mass. Additional broadening in the Bragg line profiles due to the inhomogeneous strain distributions is analyzed in Sec. VI. For the current sample system, diffusively large-angle scattered electrons only give a minor contribution to the diffraction intensity, so that only a few Kikuchi lines are visible [e.g., the deficit (010) Kikuchi line].

Considering rotation angles $[α, β, γ]= [1.46°, 8.05°, 22.9°]$ of the graphite crystal and an initial convergence angle of 25 mrad (half angle), the precise position (radius and inclination) of deficit and excess Bragg lines in the diffractograms are reproduced, allowing for an assignment of the indices h, k, and l [cf. Fig. 1(f)] and further validating the use of the paraxial approximation (all utilized scattering angles are smaller than 80 mrad from the electron optical axis).

A time-dependent distortion of the unit cell can be described by applying the deformation gradient tensor $F(Δt)$ to the undistorted real-space basis of the graphite lattice $Bt(Δt)=F(Δt)⋅Bu$⁠. Extracting lattice deformations from CBED patterns is a well-established procedure in electron microscopy with continuous beams,^62,71 which we now apply to time-resolved diffraction data. Generally, the average unit cell deformation within the electron beam probing volume (cf. Figs. 3 and 4) can be extracted by applying a forward least squares regression analysis,^72,73 fitting the absolute change in Bragg line positions $Δr(hkl)=rexcess,(hkl), exp −rexcess,(hkl),calc(F)$ and inclination angles $Δφ(hkl)=φ(hkl), exp −φ(hkl),calc(F)$ of the most intense lines with the components of the tensor F as free parameters

No change of the inclination angles $φ(hkl), exp$ is observed, and we therefore adopt $Δφ(hkl)=0$ in the fitting procedure.

In our experiment, the radial position of the excess Bragg lines exhibits a high sensitivity to changes of the displacement field u along the Z direction, mainly related to the components F_xz = (ε+ω)_xz, F_yz = (ε+ω)_yz, and F_zz = 1 + ε_zz of the deformation gradient tensor.⁷⁴ Adapting these tensor components, we can quantitatively reproduce the center-of-mass shift of the selected excess lines [cf. reconstructed line positions in Fig. 3(c)]. We note that a pure membrane shear along the x-axis is described by a finite value of F_xz and F_zx = 0, so that F_xz/2 = ε_xz = ω_xz. The full tensor F can in principle be determined by additionally analyzing deficit Bragg lines^71,73 and by recording diffraction patterns along different crystal directions.⁷⁴ 

Generally, in electron diffraction, rocking curves can be strongly affected by multiple scattering processes. However, for the deformation gradient tensor analysis, we only evaluated Bragg lines with extinction lengths ξ > 420 nm, which is significantly larger than the membrane thickness, so that the influence of multiple scattering effects can be neglected and kinematic scattering theory yields a good approximation.

A quantitative analysis of the Bragg line shifts in Fig. 2 allows us to identify the complex superposition of the acoustic lattice distortions involved in the optically driven dynamics at a homogeneous part of the membrane and at its edge. The local distortion alters the spacing and orientation of crystal lattice planes, resulting in characteristic shifts of Bragg conditions in momentum space [Fig. 3(a)]. We extract the components of the local deformation gradient tensor F for each delay time considering the center-of-mass of multiple experimental Bragg line positions [selected traces shown in Fig. 3(c), see supplementary material, SI 3).

The temporal evolution of the tensor F near the membrane edge is dominated by two components F_zz(Δt) = 1 + ε_zz and F_xz(Δt) = (ε+ω)_xz [Figs. 3(d) and 3(e), red curves], corresponding to an expansional strain along the graphite out-of-plane z-axis [for the coordinate system, see Figs. 1(a) and 1(d)] and a shear-rotation in the xz-plane (perpendicular to the membrane edge), respectively. Both deformations leave the (0k0) lattice planes unchanged, consistent with the experimentally found negligible transient changes of the (020) line profiles [cf. Fig. 2(b)]. Remarkably, the deformation gradient tensor analysis disentangles the multi-frequency temporal behavior of individual Bragg line shifts. The components ε_zz and (ε+ω)_xz each exhibit damped oscillations at a single frequency, with periods of T_expansion = 56.5 ± 1.6 ps and T_shear-rot = 154 ± 5 ps [central frequencies of 17.7 ± 0.5 and 6.5 ± 0.2 GHz, see Figs. 3(d) and 3(e)] for the expansional and shear motion, respectively. Far from the membrane edge (150-μm distance), the optically induced deformation [Figs. 3(d) and 3(e), black curves] is primarily governed by the expansional out-of-plane motion, and no significant amplitude in the xz-component of F is observed.⁷⁵ 

The periods of the expansional and shear-rotational distortion, T_expansion and T_shear-rot, are given by the roundtrip time of the acoustic waves propagating between the two faces of the membrane. The ratio T_shear-rot/T_expansion = 2.73 ± 0.16 is in excellent agreement with the relative magnitude of the corresponding longitudinal and transverse acoustic sound velocities in single crystalline graphite v(LA[001])/v(TA[001]) = (4140 m/s)/(1480 m/s) = 2.80.⁷⁶ Furthermore, the periods T = 2 l/v yield a membrane thickness of l = 117 nm, which matches the value of 120 nm derived by evaluating the thickness fringes^67,77 of the (010) Bragg line.

At long delay times (Δt > 800 ps), the oscillatory membrane expansion becomes strongly damped, approaching an average graphite interlayer distance increase of about 1.5% at the membrane edge (continuous membrane: 0.9%). In order to compare these strain values to a thermal expansion model, we extract the local temperatures from the integrated diffracted intensity change of the (452) Bragg-line after optical excitation [Fig. 3(b)]. For an equilibrated phonon distribution (Δt > 100 ps), a thermal Debye-Waller behavior is reached and we extract an optically induced temperature rise of ΔT_cont = 270 K at a continuous part of the membrane and ΔT_edge = 480 K at the membrane edge, which corresponds to thermal film expansions of 0.93% and 1.65%, respectively (see supplementary material, SI 4). Importantly, ultrafast CBED directly yields full transient rocking curves, so that an acoustic lattice distortion (line shift) and a change in the atomic mean square displacement (line intensity) can be distinguished.

At early delay times, a biexponential drop of diffracted intensity is observed, which is attributed to the previously reported initial non-thermal phonon distribution after optical excitation.^78–80 This delayed increase in atomic mean square displacement is also reflected in a phase shift of the out-of-plane breathing oscillation. Specifically, we observe the first maximum of ε_zz at 36 ps, corresponding to a considerable time lag of about 7 ps relative to a cosine-like transient. The quantitative relation between the non-equilibrium atomic mean square displacement and the resulting stress in the in-plane and out-of-plane directions requires further study, potentially contributing to elucidate the complex hierarchy of energy dissipation in graphite.^{30,34,78–84}

The out-of-plane expansional breathing modes, visible in ε_zz, are universal features observed in laser excited thin films as a result of a transient stress gradient σ(z) in the depth of the film, with electronic and lattice contributions.^{15,30,34,85–89} For the generation of shear modes, as mapped in F_xz, a symmetry breaking in the lateral direction is required, such as in anisotropic or strained crystal lattices or by local light fields.^90–93 In the following, we will further analyze the mechanism responsible for the coherent generation of these acoustic shear wave components.

In our sample geometry, the structural symmetry is locally broken on mesoscopic length scales due to the presence of the membrane edge. Ultrafast CBED now allows for a local mapping of the evolving deformation gradient tensor field and the sources of the corresponding acoustic waves. To this end, we record time-resolved local diffraction patterns with the focused electron pulses placed at varying distances r from the membrane edge. Figure 4(a) exemplarily shows the angular shift of the (201) Bragg line as a function of the delay time Δt and the probing position r, together with the extracted tensor components ε_zz(r, Δt) and F_xz(r, Δt) [Figs. 4(b) and 4(e)].

The expansional mode is observed at all probing positions with an equal phase. Its amplitude is spatially modulated and in particular at r = 500 nm is increased by about 70% compared to the value found at a larger distance from the graphite edge. This ratio agrees well with the larger temperature rise at this probing position, as observed by the transient Debye-Waller behavior [see Fig. 3(b)]. The locally increased sample excitation can be attributed to an interference pattern formed by the optical excitation close to the membrane edge, which is also observable in optically driven inelastic electron scattering, utilizing scanning photon-induced near-field electron microscopy (S-PINEM)^46,94–96 [Fig. 4(d), supplementary material, SI 6].

In contrast to the film breathing mode, the shear-rotation component F_xz shown in Fig. 4(e) exhibits a pronounced spatial dependence. In particular, the onset time of F_xz scales linearly with the distance from the membrane edge, with a slope corresponding to a phase velocity of ∼22 km/s.

To further analyze the peculiar spatio-temporal strain dynamics, we numerically solve the elastodynamic wave equation for our sample geometry, considering a thermal stress model, a laterally homogeneous sample excitation profile, and graphite bulk properties for the elasticity tensor. The temperature field is obtained by taking into account the inhomogeneously deposited optical excitation and the graphite heat capacity.⁹⁷ In addition, diffusional heat transport was included in the model using an anisotropic heat conductivity.⁹⁷ For further details on the numerical simulations, see supplementary material.

For the ε_zz(r, Δt) component, we obtain an r-independent temporal evolution [Fig. 4(c), top], in agreement with the breathing mode of a continuous membrane. In addition, optical excitation results in an in-plane thermal stress σ_x of the graphite lattice, which launches an expansional shock wave in ε_xx(r, Δt) from the membrane edge [Fig. 4(c), center], propagating perpendicular to the edge with the longitudinal in-plane sound velocity LA[100] = 22.16 km/s.⁷⁶ Due to the optical excitation profile, the expansional in-plane shock wave is localized to the top of the membrane and thereby induces shearing of the thin film sample. The ultrafast build-up of shear strain at the top initiates the shear wave travelling back and forth between the membrane faces.⁹⁸ This model readily explains the local excitation of the experimentally observed shear wave with its onset time scaling linear with the distance to the graphite edge.

Up to here, Bragg line shifts in scanning U-CBED yielded a spatio-temporal map of the lateral structural distortion of the photo-excited graphite membrane. In addition, rich experimental information on the inhomogeneous strain within the depth of the membrane is contained in the profiles of the Bragg lines, which we analyze in the following. Within kinematic scattering theory, a strained crystal imprints a phase modulation onto the diffracted electron wavefront,^40,59 resulting in a CBED profile well described by

in which Δθ is the change in diffraction angle (relative to the Bragg angle θ_B), G_hkl the corresponding reciprocal lattice vector, u(z) the atomic displacement field, and $F$ the Fourier transformation along the graphite z-axis. The corresponding deformation gradient tensor F is given (for small deformations, as relevant here) by the gradient of the displacement field, i.e., F = Ι₃ + ∇u. Notably, the line profiles depend on the projection G_hkl·u(z) [Eq. (4)], so that the cross sections for individual Bragg conditions are sensitive to different components of the displacement vector field and thereby to the polarization of the involved phonon modes.

In Fig. 5, we exemplarily compare the experimental time-dependent (422) line profiles at the membrane edge and in the continuous film with predicted profiles according to Eq. (4), utilizing the numerically simulated displacement fields. For the continuous part of the graphite film, a periodic change of the Bragg line width is observed (with a period T_expansion/2), which is well reproduced within the numerical strain model [Figs. 5(a) and 5(c), left panel]. Approximately at delay times of maximum film expansion and compression, sharp Bragg lines are obtained due to the intermediate nearly homogeneous ε_zz strain distribution within the film, as, for example, visible in Fig. 4(c) at Δt = 30 ps. The slight time lag between Bragg line shift and line broadening as well as their relative amplitude sensitively depends on the optical excitation depth and the resulting transient stress profile. In particular, the experimental width of the Bragg line profiles cannot be reproduced if one considers the optical penetration depth in graphite of δ_p = 36 nm (Ref. 99) alone. Instead, a good agreement is obtained for an excitation depth spatially spread to about 90 nm (see supplementary material, SI 5), which may be caused by fast interlayer electron or ballistic phonon transport.^100,101 Furthermore, the asymmetry at the crests of the oscillatory Bragg line movement is reproduced well in the simulations by adopting an 8-ps coupling time of the initial excitation to the experimentally detected coherent out-of-plane motion, similar to the time constant observed for the increase of the in-plane atomic mean square displacement.^78–80 

For the strain dynamics induced at the membrane edge, the more complex behavior of Bragg line profiles [Figs. 5(b) and 5(c), right panel] is a result of the superposition of expansion and shear deformation, resulting in different projections of the displacement field u(z) onto reciprocal lattice vectors G_hkl [cf. Eq. (4)]. The main features of the experimental line shapes are regained in the numerical strain simulation, including the decreasing intensity maximum after Δt = 0 with a pronounced line sub-structure between 23 and 60 ps [Fig. 5(b)]. In addition, also the general experimental trend of partial line re-focusing between 60 and 90 ps and increased broadening between 90 and 140 ps is found in the simulation. Microscopically, the line shapes sensitively depend on the relative amplitudes and phases of the expansional and shear wave modes, allowing for a sensitive mapping of nanophononic strain fields. The remaining difference between the experimental and simulated line profiles may indicate the break-down of classical continuum mechanics at the length and time scales considered here. Further developments are required to properly account for the impact of the initial non-thermal phonon distribution and mode specific phonon-phonon interactions on ultrafast transport processes and the transient local lattice stress, particularly relevant for the nanoscale geometries considered here.

We demonstrated the quantitative mapping of a time-dependent structural distortion in a nanoscale geometry, utilizing ultrafast convergent beam diffraction with a raster-scanned ultrashort electron probe. Our technique is applicable to a wide variety of locally structured thin-film sample systems. In particular, we believe that U-CBED opens a new avenue for achieving a quantitative description of ultrafast processes relevant in nanophononic devices, potentially allowing for a precise tailoring of nanostructure and function. With the temporal resolution demonstrated here, U-CBED is also capable of imaging phonon modes up to the terahertz regime, which will enable us to address the flow of thermal energy on its intrinsic time and length scales. Such capabilities may help to unravel the influence of local dissipation channels in complex materials, transport processes across designed interfaces, and nonlinear phononic interactions.

See supplementary material for details on the experimental setup (SI 1), sample preparation (SI 2), data collection and analysis (SI 3), Debye Waller analysis (SI 4), numerical simulations (SI 5), and near field characterization (SI 6) (PDF). Movie showing delay-dependent change in CBED intensity (difference pattern) probed in a continuous part of the membrane (M 1) and close to its edge (M 2) (AVI).

We gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft (No. DFG-SPP-1840 “Quantum Dynamics in Tailored Intense Fields” and No. DFG-SFB-1073 “Atomic Scale Control of Energy Conversion,” project A05), support by the Lower Saxony Ministry of Science and Culture, and funding of the instrumentation by the DFG and VolkswagenStiftung. W.L. would like to acknowledge support by the Director Fund of WNLO (Grant No. WNLOZZYJ1501) and the National Natural Science Foundation of China (Grant No. 11574094).