Four-Dimensional Imaging of Lattice Dynamics using Inelastic Scattering

Time-resolved mapping of lattice dynamics in real- and momentum-space is essential to understand better several ubiquitous phenomena such as heat transport, displacive phase transition, thermal conductivity, and many more. In this regard, time-resolved diffraction and microscopy methods are employed to image the induced lattice dynamics within a pump-probe configuration. In this work, we demonstrate that inelastic scattering methods, with the aid of theoretical simulation, are competent to provide similar information as one could obtain from the time-resolved diffraction and imaging measurements. To illustrate the robustness of the proposed method, our simulated result of lattice dynamics in germanium is in excellent agreement with the time-resolved x-ray diffuse scattering measurement performed using x-ray free-electron laser. For a given inelastic scattering data in energy and momentum space, the proposed method is useful to image in-situ lattice dynamics under different environmental conditions of temperature, pressure, and magnetic field. Moreover, the technique will profoundly impact where time-resolved diffraction within the pump-probe setup is not feasible, for instance, in inelastic neutron scattering.


INTRODUCTION
Inelastic scattering of matter allows us to probe quasi-particles (QPs) such as phonons (quantized lattice vibrations), magnons (quantized spin excitations), and polarons [1][2][3][4] . These QPs have finite energy and lifetime that carries information about the intra-and inter-QPs interactions and their coupling strength, necessary to understand the materials' response to external stimuli. Thermal conductivity 5,6 , heat capacity [7][8][9] , and phase transitions 7,10,11 are among many of the material properties that are often described by directly invoking various QPs. Optical, x-ray, neutron, and electron scattering methods -for example, Raman scattering, inelastic x-ray scattering (IXS), inelastic neutron scattering (INS), and electron energy loss spectroscopy (EELS) -have been routinely employed to measure these QPs 1,3,4 .
Typically these measurements are performed in the momentum and energy domains (k-ω), and lack information on temporal dynamics, i.e., k-t and x-t imaging -time evolution of momentum or real space coordinates which ranges from femto-to several nano-seconds 12,13 .
To image k-t and x-t dynamics, a pump-probe setup having two ultrashort pulses, where the duration of the probe pulse must be shorter than the characteristic timescale of motion that is under probe, are required. Thanks to tremendous technological advancement, it has become possible to generate ultrashort x-ray and electron pulses, [14][15][16] and image the lattice dynamics in k-t and x-t domains. For example, the time-resolved x-ray and electron diffraction within a pump-probe configuration are used to image the lattice dynamics in the k-t domain [17][18][19][20][21][22][23][24] , and the ultrafast electron microscopy has recently been demonstrated for the imaging in the x-t domain at an unprecedented spatiotemporal resolution [25][26][27] . However, similar advances have not been taken for neutron sources to produce an ultra-short neutron pulse for imaging the lattice dynamics 28 . At this juncture, it is not straightforward whether one can employ neutron sources within a pump-probe setup with sufficient atomic-scale spatiotemporal resolution to image lattice or spin dynamics 28 .
In this work, we theoretically demonstrate that methods based on inelastic scattering are suitable to extract similar information as one could get from the time-resolved imaging of lattice dynamics in k-t or x-t domains. Our approach is general and equally applicable to IXS, INS, and EELS. In general, all these inelastic scattering based methods probe dynamical structure factor S(k, ω), apart from pre-factors, in experiments. The inelastic scattering methods provide k-ω resolved measurement of QPs and comprise a powerful way to investigate the correlated motion of atoms and electrons 1,3,4 . We should mention that in ideal conditions, irrespective of whether measurements are in k-ω, k-t, or x-t domains, they provide similar information after coordinate transformation(s). However, in practice, one measurement domain may have an advantage over the other. For example, under static environmental conditions of temperature, pressure, or magnetic field, the four-dimensional (4D) k-ω mapping of QPs is preferred because of its superior energy and momentum resolutions (∼0.1 meV and ∼0.5 nm −1 ) 2,29-31 , from which one can readily extract the QP energy, group velocity, and linewidth. On the other hand, the k-t domain is useful for tracking the temporal evolution of atomic motions upon photoexcitation-induced structural phase transitions 32 or the measurement of long-wavelength phonon lifetime (of the order of tens of picoseconds, which is not easily accessible in the k-ω domain). Moreover, mapping the acoustic phonon wavefronts or the nucleation of waves from defects and interfaces in nanostructures is better suited for the x-t domain 26,33 .
In the following, we show that S(k, ω) (obtained from experimental measurements or simulations) encodes all the essential information to image the lattice dynamics in the k-t and x-t domains after coordinate transformation without the causality violation. In particular, as we illustrate, our approach is well-suited to image the first-order states (i.e., emission or absorption of a single phonon at q 0 from inelastically scattering photons or disorderactivated continuum 34 ) and second-order 'squeezed' states 17 in the k-t domain (squeezed states are generated in the entire reciprocal lattice immediately after pumping the sample with a visible or near-infrared pump pulse) 17,35 . The temporal evolution and decay of the measured intensity from the change in phonon occupation at a given k point due to electronelectron, electron-phonon, and phonon-phonon scattering channels 17,36,37 are not explicitly included within the current framework. However, we consider the finite lifetime of first-or second-order states by including the phonon linewidths. Moreover, our methodology allows for x-t imaging of the coherent phonon dynamics from a point-like nucleation site or an extended defect. Our approach of imaging dynamics in the x-t domain can be directly compared with the electron microscopy data, as we demonstrate later by an example.

Computational approach
Silicon is used in the present work to demonstrate the proposed concept. The S(k, ω) is simulated in the (H, H, L) reciprocal plane following the same procedure as in our previous studies 38,39 . The range of energy transfer lies from 0 to 80 meV with a step size of 0.25 meV (i.e., energy resolution), whereas the momentum transfer range varies from (0, 0, 0) to (4,4,7) reciprocal lattice units (r.l.u.) with the step size of 0.025 r.l.u. (see Supplementary Fig. S3).
Here, a = 0.543 nm is used as the lattice parameter of silicon. After calculating S(k, ω) and using fluctuation-dissipation theorem, the imaginary part of the response function χ(k, ω) is In the following all results are shown at T = 300 K. It is known that the real part of the response function is related to its imaginary part by the Kramers-Kronig relation, 40 Here, P represents the principal value of the integral. Fourier transform is performed to that a (weak) optical pump pulse will also lead to the same χ(k, t) snapshots, as the pump pulse will generate first-and second-order states in the entire reciprocal space. The firstorder disorder-activated continuum is generated in the absence of perfect crystalline order 34 .
In contrast, the second-order squeezed states are generated by the coupling of a photon (momentum q 0) with the two phonon modes of near-equal and opposite momenta (i.e., at k and −k due to the conservation of momentum) 35 . Thus in the present study, the simulated snapshots can be considered to arise from either a point-like source in x or generation of first-and second-order states. As evident from Fig. 1, these generated phonons propagate

Response function in real-space
Not only time-resolved scattering methods in a pump-probe configuration provide the temporal evolution of correlation function, but also help us to visualize atomic motion (lattice dynamics) in the x-t domain 12,13 , for example, as in the time-resolved electron microscopy 7 experiments 25,26,33 . In the following, we demonstrate that momentum and energy-resolved inelastic scattering signal also provides the snapshots of the lattice dynamics in the x-t domain. For this purpose, we need to perform one more Fourier transform from momentum space to real space to obtain χ(x, t) from χ(k, t). Following Fourier relation, the spatial res-  lower panels, respectively. Before the disturbance at t < 0, the system is in equilibrium.
At t = 0, the system is struck with a negative disturbance. As a result of this disturbance, a positive recoil at the origin is generated, surrounded by a minimal positive build-up. As visible from the figure, the density-induced disturbance is propagating through the entire system as time evolves. At large time instances, the disturbance is still into the system, but the order is minimal due to the spread of energy into the system (no dissipation of energy from the system). Such time-resolved images can be captured nowadays using realspace femtosecond electron imaging, as recently demonstrated for the phonon nucleation and launch at a crystal step-edge in the WSe 2 flake 26 . At this point, it is important to mention that Abbamonte and co-workers have employed a similar reconstruction method of χ(x, t) from inelastic x-ray scattering data to visualize electron dynamics in various systems 41,47,48 . in germanium provides confidence and robustness of the proposed method. We believe that the current approach will be an alternative to time-resolved diffraction methods to image lattice dynamics and beneficial to the situations where time-resolved diffraction is not easy to perform, such as neutron scattering and in-situ measurement conditions.

Dynamical structure factor calculation
The dynamical structure factor S(k, ω) was calculated using the following expression: where f d (k) is the form factor for atom d (can be replaced by the neutron scattering length b d for inelastic neutron scattering), k = k' -k" is the wavevector or momentum transfer, and k" and k' are the final and incident wavevector of the scattered particle, respectively; q is the phonon wavevector, ω s is the eigenvalue and e ds is the eigenvector of the phonon corresponding to the branch index s, d is the atom index in the unit cell,

DATA AVAILABILITY
Data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

CODE AVAILABILITY
Code that support the findings of this study are available from the corresponding authors on reasonable request.    The disturbance is still in the system at large instances, but the ripples' height is low. There is no dissipation of energy from the system.