Introduction

Intense laser-matter interaction plays an important role in many applications including inertial confinement fusion1, laser micromachining2, and material synthesis3. Ultrafast laser excitation can drive matter into extremely nonequilibrium states, in which the hot electron and cold lattice coexist. The subsequent atomistic dynamics is therefore a long-standing challenge because it is governed by the interplay between excited-electron-modulated potential energy surface (PES)4, electron-ion coupling5, and geometric characteristics of irradiated samples6.

Tremendous efforts based on time-resolved probing techniques and simulations have provided valuable insights into the nonthermal behaviors7,8,9,10,11, kinetics of laser-driven melting12,13,14,15, and electron-phonon coupling16,17,18. The related processes from cold solid to hot liquid and plasma are the typical multiscale dynamics due to the cascade of interrelated processes triggered by the laser excitation, both in timescale and size scale. Therefore, it is of great difficulty and importance to construct a well-coordinated picture between experimental and theoretical efforts. For example, the dynamics of laser-excited Au is still under debate4,7,19, regarding the phonon behaviors driven by laser excitation. In these cases, different priori assumptions on material response were usually made20,21.

The above obscure stems from the technical limitations that the present methods cannot capture both the nonthermal nature and intrinsic scale of the laser-induced process at the same time. For ab initio methods such as time-dependent density functional theory (DFT), the sizes are limited to 101~103 atoms and 102~104 fs, unable to access a realistic representation of structural transformations of irradiated samples. While for classical molecular dynamics simulations coupled with two-temperature model (TTM-MD)22,23, the implementation of empirical potential like embedded-atom-method (EAM) is limited in prior knowledge and model complexity, thus can hardly capture the high dimensional dependence of PES on both atomic local environments and electron occupations for a wide range of temperature and density19,24, leading to the inadequate description of nonthermal nature of laser-driven processes. Therefore, bringing the advantage of ab initio and large-scale molecular dynamics including nonthermal effect becomes the route one must take.

In this paper, we developed an ab initio atomistic-continuum model by combining two-temperature-model (TTM) with an extended deep potential molecular dynamics (DPMD), as illustrated in Fig. 1. When the ultrafast laser interacts with solids, the electrons quickly thermalized at timescales of femtoseconds, producing highly nonequilibrium states (electron temperature Te ≫ ion temperature Ti). The hot Te will result in the redistributed charge density first and then modify the PES of ions. To capture this physics, we introduced the laser-excited PES by constructing electron-temperature-dependent deep neural network potential and coupled the PES into an additional electron continuum subsystem via the TTM-MD framework. In this way, we can directly simulate the whole electron-ion coupled dynamics during the laser-driven processes with large-scale simulations within ab initio accuracy. We take tungsten as an example and systematically validate the accuracy of our model in describing lattice dynamics, thermophysical properties, and laser heating process in both equilibrium and laser-excited states, by comparing with the related experimental results recently13.

Fig. 1: Schematic diagram of workflow for efficient and accurate simulation of laser-driven atomistic dynamics.
figure 1

a ETD-DP model. Te is the electron temperature, regarding the electron occupation distribution. The free energy A, force F, virial Ξ, electronic entropy Se, and electronic heat capacity Ce can be inferred through the backpropagation algorithm. b Iterative concurrent learning scheme is used to efficiently sample atomic configurations for a wide range of equilibrium and nonequilibrium conditions. c Hybrid atomistic-continuum approach. The evolution of the electron subsystem allows atomistic system transits between different PES, and the Langevin thermostat is introduced to mimic non-adiabatic energy exchange between electron and lattice.

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Results

Construction of laser-excited PES

Recent efforts have demonstrated the success of the machine learning model toward large-scale simulations of ab initio quality at extreme conditions25,26,27,28,29,30,31,32, but most of the studies focus on the equilibrium-state and ground-state applications. Here the electron-temperature-dependent deep potential (ETD-DP) model is implemented in the framework of deep potential method25,33,34 to model the laser-driven dynamics.

To avoid constructing hand-crafted features or kernels for different types of bulk systems, a general end-to-end symmetry-preserving scheme is adopted35. As illustrated in Fig. 1a, the ETD-DP model consists of an embedding network and a fitting network. The embedding network is designed to transform the coordinate matrices \({{{\mathcal{R}}}}\) to symmetry preserving features, encoded in the descriptor \({{{\mathcal{D}}}}\). The fitting network is a standard fully connected feedforward neural network, mapping the descriptor to the atomic contribution of total energies.

The newly introduced parameter, electron temperature Te, is used to characterize the laser modulation on PES, in which electron occupation distribution is far away from electron-ion equilibrium states. This ETD-DP is defined as

$$\begin{array}{r}A=A({{{\mathcal{R}}}},{T}_{{{{\rm{e}}}}})=\mathop{\sum}\limits_{i}{{{{\mathcal{N}}}}}_{{\alpha }_{i}}({{{{\mathcal{D}}}}}_{{\alpha }_{i}}({{{{\bf{r}}}}}_{i},{\{{{{{\bf{r}}}}}_{{{{\bf{j}}}}}\}}_{j\in n(i)}),{T}_{{{{\rm{e}}}}})\end{array}$$
(1)

where \(A({{{\mathcal{R}}}},{T}_{{{{\rm{e}}}}})\) is the potential energy depends on the local atomic environment (\({{{\mathcal{R}}}}\)) and \({T}_{{{{\rm{e}}}}},{{{{\mathcal{N}}}}}_{{\alpha }_{i}}\) denotes the neural network of specified chemical species of αi of atom i, and the descriptors \({{{{\mathcal{D}}}}}_{{\alpha }_{i}}\) describes the symmetry preserved local environment of atom i with its neighbor list n(i) = {j∣rji < rcut}.

To generate an ETD-DP model, the new degree of freedom, Te, will dramatically expand the sampling space in the data labeling process, introducing expensive computational costs. Therefore, an iterative concurrent learning scheme36 is highly required to efficiently sample atomic configurations under both equilibrium (Te = Ti) and nonequilibrium conditions (Te ≠ Ti). As shown in Fig. 1b, to explore the density-temperature space with different electron occupations (ρ, Ti, Te), a variety of crystal structures are used as the initial configurations to run multiple DPMD simulations. An ensemble of ETD-DP is trained with the same dataset but with different parameter initializations. The model deviation, denoted as the maximum standard deviation of the predicted atomic forces by the ensemble of ETD-DP, is used to evaluate whether the explored atomic configurations should be sent to generate referenced ab initio energies, forces, and virial tensors (see Supplementary Fig. 1).

Two-temperature model coupled DPMD (TTM-DPMD)

To model the whole ultrafast laser-driven processes from cold solid to plasma, we should couple the quantum electron subsystem and strongly coupled ionic subsystem. Here, we implemented our laser-excited PES into the TTM-MD framework22,23,37, going beyond traditional ground-state EAM and neural-network-driven PES descriptions. As shown in Fig. 1c, the heat conduction equation of the electron continuum characterizes the temporal evolution of electron occupations, thus governing the transition of the ionic system between different Te-dependent PES. Langevin dynamics is incorporated to mimic the dynamic electron-ion collisions23,37,38,39. The TTM-DPMD is defined as follows,

$${C}_{e}({T}_{{{{\rm{e}}}}})\frac{\partial {T}_{{{{\rm{e}}}}}}{\partial t}\,=\,\nabla \cdot ({\kappa }_{e}\nabla {T}_{{{{\rm{e}}}}})-{g}_{ei}({T}_{{{{\rm{e}}}}})({T}_{{{{\rm{e}}}}}-{T}_{{{{\rm{i}}}}})+S({{{\bf{r}}}},t)$$
(2)
$${m}_{i}\frac{{d}^{2}{{{{\bf{r}}}}}_{i}}{d{t}^{2}}\,=\,-{\nabla }_{i}A({T}_{{{{\rm{e}}}}})-{\gamma }_{i}{{{{\bf{v}}}}}_{i}+\tilde{{{{{\bf{F}}}}}_{{{{\bf{i}}}}}}(t)$$
(3)

where Ce is the electronic heat capacity, κe the electronic thermal conductivity, gei the electron-phonon coupling constant, S(r, t) the laser source. The ions evolve on the Te-dependent PES \(A({{{\mathcal{R}}}},{T}_{{{{\rm{e}}}}})\), and suffer fluctuation-dissipation forces \(-{\gamma }_{i}{{{{\bf{v}}}}}_{i}+\tilde{{{{{\bf{F}}}}}_{{{{\bf{i}}}}}}(t)\) from electron sea. Here γi is the friction parameter that characterizes the electron-ion equilibration rate, relating to the electron-phonon coupling constant through γ = geimi/2nikB, where ni is the ion number density, kB the Boltzmann constant, mi the atomic mass. The \(\tilde{{{{{\bf{F}}}}}_{{{{\bf{i}}}}}}(t)\) term is a stochastic force term with a Gaussian distribution, whose mean and variance are given by \(\langle \tilde{{{{{\bf{F}}}}}_{{{{\rm{i}}}}(t)}}\rangle =0\) and \(\langle \tilde{{{{{\bf{F}}}}}_{{{{\bf{i}}}}}}(t)\cdot \tilde{{{{{\bf{F}}}}}_{{{{\bf{i}}}}}}({t}^{{\prime} })\rangle =2{\gamma }_{i}{k}_{{{{\rm{B}}}}}{T}_{{{{\rm{e}}}}}\delta (t-{t}^{{\prime} })\).

In TTM-DPMD, by practically choosing the electron temperature or ionic temperature in the meshgrid as the additional parameter in the ETD-DP model, the ions can evolve under laser-excited PES (Te ≫ Ti) or ground-state PES (Te = Ti), so that we can separate the nonthermal effects defined by the electronic excitation from thermally driven atomic dynamics and phase transformation.

Validating neural network model for laser-excited tungsten

To validate the effectiveness of the extended DP model, we chose tungsten as our target system. Tungsten is a typical transition metal, with half-filled d bands that are sensitive to Te. Upon laser excitation, tungsten is expected to go through a complicated dynamic process including possible nonthermal solid-solid phase transition8,20,21,24, attracting much attention but remains ambiguous. Here we generate a Te-dependent deep-neural-network tungsten model by learning from DFT data calculated with the generalized gradient approximation (GGA) of the exchange-correlation functional40 using VASP package41. The atomic configurations used in the training set are collected from a wide range of (ρ, Ti, Te) conditions, covering the phase space of the body-centered-cubic (BCC), close-packed structure, uniaxially distorted crystalline, and the liquid structures. More details about DP training can be found in Supplementary Information.

Here we pay special attention to the thermodynamic properties of equilibrium tungsten that are closely related to the laser heating process. The melting temperature predicted by ground-state DPMD (3550 K; see Supplementary Information) is in consistence with the previous DFT-MD (3450 ± 100 K42) and Gaussian approximation potentials simulations (3540 K43), which confirms that the present PES can reproduce melting with DFT accuracy. Furthermore, the dependence of DPMD-predicted enthalpy on temperature along isobaric heating conditions is shown in Fig. 2a, and the experimental data agree very well with our DPMD predictions, especially in the liquid regime44. The estimated enthalpy of fusion at the melting point (ΔHm = 237 ± 20 kJ kg−1) is also close to the DFT-MD values (250 kJ kg−1)45 and other experimental values (see Supplementary Table 1 and Supplementary Fig. 3 in Supplementary Information).

Fig. 2: Validating the accuracy of ETD-DP model.
figure 2

a Temperature dependence of enthalpy under isobaric heating (p = 1 bar) with the reference temperature of 300 K. The blue line, the black cross, and gray triangle denotes the DPMD results, previous DFT-MD prediction45, and isobaric expansion experimental data44 respectively. b Phonon dispersion of laser-excited tungsten (ρ0 = 19.15 g cm−3). The black cross and white squares represent the individual KS-DFT calculation and experimental measurements63.

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Based on calculated thermophysical properties, we can determine the complete melting threshold ϵm, which is the laser energy that is sufficient to drive the complete melt of the samples. We found ϵm = 0.92 ± 0.04 MJ kg−1, corresponding absorbed pump fluence is 53.0 ± 2.2 mJ cm−2 for 30-nm-thick tungsten film (more details in Supplementary Information). Such values are in agreement with the estimated values from experimental results13,44,46,47,48,49, in which energy density is approximately 0.94 MJ kg−1 (pump fluence of 53.8 mJ cm−2). The density decrease at elevated temperature as shown in Supplementary Fig. 4, is also consistent with the experimental measurements48,49,50.

The lattice dynamics that requires high-order derivatives of PES were further investigated. As shown in Fig. 2b, the phonon dispersion curves of BCC tungsten under both equilibrium and nonequilibrium states are well-reproduced compared with the DFT results. In particular, compared with the phonon dispersion at Te = 300 K, the directional phonon softening is observed along the H − N and N − Γ path in the first Brillouin zone at elevated electron temperature (Te = 10000 K), which can be attributed to the delocalization half-filled d bands8,24. The depopulation of such a strong directional component in electronic bonding weakens the directional forces and may drive the crystalline structures toward close-packed forms. These results indicate that the neural network PES can provide faithful prediction-related properties in consistency with experiments or ab initio method.

Direct ab initio simulations of laser-driven dynamics

It is stressed that our explicit electron-temperature-dependence PES can well capture the nonthermal nature of laser-excited metals. When implemented in the TTM-DPMD framework, it allows us to establish a comprehensive understanding of laser-induced nonequilibrium states within ab initio accuracy. Recent time-resolved ultrafast electron and X-ray diffraction experiments collect direct quantitative structural information of laser-driven processes13, providing a benchmark for the validation of the present newly developed methods. Here, we apply TTM-DPMD to directly simulate the dynamic response of tungsten nanofilm under different absorbed laser energy densities.

In TTM-DPMD simulations, full-scale ab initio description in one-dimension of polycrystalline (PC) 30-nm-thick tungsten nanofilm is considered, according to relevant UED experiment13. For PC systems, large size including 752,650 atoms is used to describe crystal grains with random shapes, orientations, and different types of boundaries. The size of each grain ranges from ~5 nm to ~7 nm and each grain contains more than 104 atoms (totally reaching millions of atoms), which cannot be achieved by the traditional time-dependent DFT simulations. Moreover, an extra 30 nm vacuum space perpendicular to the laser incident direction is set to allow free surface response to the internal stress relaxation, and extra spring forces are introduced for atoms in the bottom regime relating to their initial lattice site, to present bonding to the substrate (see Supplementary Information).

Considering the ballistic transportation of excited electron in tungsten (the mean free path ~33 nm), we assumed the uniform deposition of laser energy with a relatively low energy density of 0.08 MJ kg−1 (corresponding to absorbed laser fluence of 4.8 mJ cm−2). In this case, a moderate two-temperature state is created at the initial stage, where maximum electron temperature can reach 4400 K. Through electron-ion energy exchange, the system quickly reaches thermal equilibrium (Te = Ti ~ 920 K) at t = 5 ps.

The structure factor is calculated to extract the decay dynamics of the Laue diffraction peak (LDP)51, which is an important quantity to diagnose the structural dynamics in experiments13. As shown in Fig. 3a, based on TTM-DPMD simulations with the inclusion of laser-driven excited states, the temporal evolution of the normalized intensity of (211) LDP agrees well with UED measurements. Conversely, the results from simulations by ground-state PES deviate from experimentally measured values significantly. It is interesting to say that the thermal process (Te = Ti) exhibits remarkably slower decay dynamics than the process with excited states (Te≥Ti) upon such laser fluence. By further investigating the lattice vibration of bulk tungsten, we note that even under moderate nonequilibrium state (Te = 5000 K), a relative increase of over 10% in mean square displacement (MSD) can be observed under isobaric heating condition, as shown in Fig. 3b. The enhancement of lattice vibration can be attributed to the hot-electron-induced phonon softening (Fig. 3c). Such nonequilibrium and nonthermal effects therefore modify the dynamics of diffraction signals according to Debye-Waller formula, in which the decay of LDP is relating to the temporal evolution of lattice temperature and temperature dependence of MSD. The quantitative consistency between our simulations and experiments validates our model further, and then provides a chance to further elucidate laser excitation effects.

Fig. 3: Capturing nonthermal effect with TTM-DPMD approach.
figure 3

Comparison of a temporal evolution of (211) diffraction peak intensity in structure factor under absorbed laser energy density of 0.08 MJ kg−1, compared with experimental data13. b Temperature dependence of mean square displacement with isobaric constraints and c phonon density of states (PDOS), obtained under equilibrium condition (blue) and nonequilibrium condition (orange).

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By increasing laser energy density up to 0.80 MJ kg−1, the irradiated tungsten nanofilm starts with more severe nonequilibrium states (Te = 11200 K, Ti = 300 K). As presented in Fig. 4a, b, the evolution of tungsten nanofilm predicted by ground-state PES (Te = Ti) is a purely thermal process governed by electron-ion coupling. With increased lattice temperature, the system first evolves along the equilibrium isochore in the first 4 ps, where the ionic kinetic pressure accumulates to ~10 GPa. Then the thermal pressure is gradually released due to the existence of a free surface. Although the thermal expansion process leads to temperature and density decrease, the gradient in the thermodynamic profile is slight and the whole system can be considered homogeneous.

Fig. 4: Hot electron modifies the thermodynamic pathway.
figure 4

Comparison of a, c thermodynamic pathway and b, d temporal evolution of the thermodynamic profile of nanofilm, predicted by ground-state PES and laser-excited PES, respectively. In (a, c), the red arrows indicate the evolution path of the selected regime (z = 14.0 nm) in the tungsten nanofilm, and the thermodynamic state is highlighted by colored stars every 1 ps. In (d), the black dashed lines are used to highlight the propagation of stress waves, whose slope represents a constant propagation speed of ~4.3 km s−1.

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When laser-induced changes in the PES are included (Te≥Ti), the thermodynamic pathway and thermodynamic profile are totally different. As shown in Fig. 4c, d and Supplementary Fig. 7, the ultrafast excitation of electrons results in the buildup of extra pressure on a sub-picosecond timescale. Such hot-electron-contributed pressure increases monotonically with increased laser energy density, from ~1 GPa with Te,0 = 4400 K (ϵ = 0.08 MJ kg−1) to ~ 17 GPa with Te,0 = 11200 K (ϵ = 0.80 MJ kg−1). The tungsten nanofilm then quickly responds to this nonthermal internal stress, triggering anisotropic volume relaxation dynamics. As a result, a significant inhomogeneity is demonstrated in the thermodynamic profiles. In Fig. 4d, the propagation and reflection of stress waves can be identified with a velocity of ~4.3 km s−1, accompanied by a density decrease of ~1 g cm−3. With the existence of a free surface, the buildup and uniaxial relaxation of nonthermal stress can strongly influence the thermodynamic pathway, especially under high laser fluence, which cannot simply be assumed to be isochoric or isotropically isobaric (details see Supplementary Figs. 8 and 9). We highlight that such real-time material response captured by TTM-DPMD simulation provides microscopic insights into previous controversial issues on the nonthermal behavior of laser-excited matter19,20.

Discussion

In this work, we developed the deep learning model to perform large-scale ab initio simulations on the laser-induced atomistic dynamics, with quantum accuracy on the nonthermal effects. To validate the accuracy, special attention is paid to recent experiments. We successfully reproduce the experimental data with our model. It is therefore verified that the laser-excited states have profound effects on the thermodynamic evolution and structural transformation dynamics. More importantly, the combination of deep learning techniques with a hybrid continuum-atomistic approach bridges the theoretical method and experimental observations, providing a path to establish an accurate and complete understanding of the atomistic dynamics under ultrafast laser interactions.

Methods

DP training

The ETD-DP models for tungsten are generated with DeePMD-kit packages52,53 by considering Te as atomic parameter. Deep Potential Generator (DP-GEN)36, has been adopted to sample the most compact and adequate dataset that guarantees the uniform accuracy of ETD-DP in the explored configuration space. We consider BCC structure (54 atoms) and liquid structure (54 atoms) as the initial configurations and run DPMD under NVT and NPT ensemble (both isotropic and uniaxial constraints are considered), where temperatures range from 100 K to 6000 K, pressure ranges from −15 to 60 GPa, and corresponding electronic temperature ranges from 100 K to 25,000 K. The training sets consist of 6366 configurations under equilibrium condition (Te = Ti) and 6820 configurations sampled under a two-temperature state (Te > Ti).

For DP training, the embedding network is composed of three layers (25, 50, and 100 nodes) while the fitting network has three hidden layers with 240 nodes in each layer. The total number of training steps is set to 1,000,000. The radius cutoff rcut is chosen to be 6.0 Å. The weight parameters in loss function for energies pE, forces pf, and virials pV are set to (0.02, 1000, 0.02) at the beginning of training and gradually change to (1.0, 1.0, 1.0). The accuracy of generated DP model is tested in the whole sampling configuration. The prediction errors are shown in Supplementary Fig. 2. For the equilibrium dataset (Te = Ti), the root mean square error (RMSE) of energy σE is 5.191 meV per atom, the RMSE of forces σf is 0.363 eV Å−1, and the RMSE of pressure σp is 0.157 GPa. For the nonequilibrium dataset (Te ≠ Ti), the RMSE of energies, forces, and pressures is 2.390 meV per atom, 0.093 eV Å−1, and 0.089 GPa respectively.

The self-consistency calculations are all performed with the VASP package54. The Perdew-Burke-Erzerhof (PBE) exchange-correlation functional is used55, and the pseudopotential takes the projector augmented-wave (PAW) formalism56,57. The sampling of the Brillouin zone is chosen as 0.2 Å−1 under ambient conditions (Te ≤ 300 K), and 0.5 Å−1 for high temperature.

TTM-DPMD simulation setting

We perform TTM-DPMD simulations with LAMMPS package58 through modified EXTRA-FIX packages23. The electronic heat capacity is calculated by individual DFT calculations \({C}_{{{{\rm{e}}}}}={T}_{{{{\rm{e}}}}}\frac{\partial {S}_{{{{\rm{e}}}}}}{\partial {T}_{{{{\rm{e}}}}}}\), which is consistent with previous calculations59. The electron-phonon coupling factor is set to constant (G0 = 2.0 × 1017 W m−2 K−1) according to relevant ultrafast electron diffraction experiments13. The electron thermal conductivity is described by the Drude model relationship, \({\kappa }_{{{{\rm{e}}}}}({T}_{{{{\rm{e}}}}},{T}_{{{{\rm{i}}}}})=\frac{1}{3}{v}_{{{{\rm{F}}}}}^{2}{C}_{{{{\rm{e}}}}}({T}_{{{{\rm{e}}}}}){\tau }_{{{{\rm{e}}}}}({T}_{{{{\rm{e}}}}},{T}_{{{{\rm{i}}}}})\), where vF is Fermi velocity and τe(Te, Ti) is the total electron scattering time defined by the electron-electron and electron-phonon scattering rates, \(1/{\tau }_{{{{\rm{e}}}}}=1/{\tau }_{{{{\rm{e}}}}-{{{\rm{e}}}}}+1/{\tau }_{{{{\rm{e}}}}-{{{\rm{ph}}}}}=A{T}_{{{{\rm{e}}}}}^{2}+B{T}_{{{{\rm{i}}}}}\). The coefficients A = 2.11 × 10−4 K−2 ps−1, B = 8.4 × 10−2 K−1 ps−1, vF = 9710 Å ps−1 are adopted60. The duration of laser pulse is set to 130 fs. Since the mean free path of laser-excited electrons is ~33 nm in tungsten61, the electrons are heated uniformly due to the ballistic transport. Therefore, optical penetration of laser energy can be neglected for simplicity.

For the atomic system, the simulation size of polycrystalline sample is set to 30 × 20 × 20 nm, containing 752,650 atoms, with an extra 30-nm vacuum space along the x direction to minic the free boundary condition. Extra spring forces are introduced for atoms in the bottom 5 Å relating to their initial lattice site to present bonding to the substrate.

Phonon spectra calculation

To validate the accuracy of the ETP-DP model, we investigate the lattice dynamics that need high-order derivatives of PES. We use the finite displacement method to calculate the phonon dispersion with ALAMODE package62 as a postprocessing code. The forces are calculated in 5 × 5 × 5 supercell with cell lattice parameter a0 = 3.17104 Å. The atomic displacement is set to 0.01 Å, and the interatomic force constants are extracted from KS-DFT and DPMD calculation respectively. The dynamical matrices are derived from these force displacement data to obtain phonon dispersion spectra.

Ultrafast electron diffraction pattern

To extract the decay of Laue diffraction peak (LDP) intensities as in the UED experiments, we performed the ultrafast electron diffraction simulations with DIFFRACTION package51 to obtain the structure factor S(Qx, Qy) defined as follows,

$$S=\frac{{F}^{* }F}{N}$$
(4)
$$F({{{\bf{Q}}}})=\mathop{\sum}\limits_{i}{f}_{i}({{{\bf{Q}}}};\lambda ){e}^{i2\pi {{{\bf{Q}}}}\cdot {{{{\bf{r}}}}}_{i}}$$
(5)

where Q = (Qx, Qy, Qz) the wave vector, fi the atomic scattering factor, λ the wavelength of incident electron, ri the coordinates of atom i. Here, simulated 3.2 MeV electron radiation (λ ~ 0.34 pm) is used to create selected area electron diffraction (SAED) patterns according to relevant experiments13, and the SAED patterns aligned on the [100] axis (Qz = 0) are constructed by selecting reciprocal lattice points intersecting a 0.01 Å−1 thick Ewald sphere slice. A detailed discussion can be found in Supplementary Figs. 5 and 6 in Supplementary Information.