Simulation of ultrafast electron diffraction intensity under coherent acoustic phonons

The generation and propagation of coherent acoustic phonons can produce periodic modulation of the crystal structure, which exists as collective atomic motion, and induce phase transition.^1–4 Upon femtosecond-laser excitation, coherent longitudinal acoustic (LA) waves usually appear in the material due to thermoelastic effects or deformation potential generated by the carriers in the sample.^5,6 The excitation of the coherent transverse acoustic (TA) waves is noted when the instability of the crystal structure or disoriented lattice is considered.^7–10 In addition, several works have investigated the mechanism of piezoelectrically induced coherent phonons.^11,12

Laser-induced coherent phonons typically contain multiple frequencies. In thick films, where the optical penetration depth is considerably smaller than the sample thickness, a spatially non-uniform stress distribution in longitudinal direction is excited. Both even and odd high harmonics can be observed in the coherent oscillation, corresponding to the propagating strain waves bouncing back and forth in the film.^13,14 When the optical penetration depth is much larger than the thickness, uniform stress is generated in the film, resulting in a standing wave with odd high harmonics only.^13,15 The modulation effect of coherent acoustic phonons on the crystal structure makes them easily detectable by ultrafast electron diffraction (UED). However, common samples used in UED measurements have an optical penetration depth comparable to the film thickness,¹⁶ where the traveling and standing waves can coexist in the system, thereby complicating the analysis. In particular, LA and TA phonons can be simultaneously excited in a given system, generating complex multifrequency oscillation signals.^8,17 Structural fluctuations induced by coherent phonons can be manifested in the diffraction intensity. A crucial yet rarely discussed issue is whether the experimentally observed oscillation frequencies in the diffraction can directly reflect the intrinsic phonon frequencies.^8,9,18 Recently, the complex behavior of electron diffraction under high harmonics and lattice distortion in graphite nanofilms has been discussed in detail, raising the possibility of the selective excitation of higher harmonics.¹⁹ However, several issues are yet to be discussed in detail, such as the effect of higher harmonics in TA phonons on the diffraction and mechanisms of in-phase and out-of-phase oscillation of the Friedel pairs.²⁰ 

In previous several UED works^8,9,17,20 reporting diffraction intensity oscillation in the tens of GHz range, multiple-frequency is usually attributed to the coexistence of TA and LA phonons, and in-phase oscillation is considered to be caused by initial deviation parameter. Here, we propose that the inhomogeneous of laser excitation has a strong influence on the diffraction intensity (both oscillation phase and frequency). In this work, we theoretically analyzed the complex correspondence between the frequencies of laser-induced coherent acoustic phonons, including the TA and LA modes, and electron diffraction intensity in two hypothetical 30-nm films with laser penetration depths of 10 and 600 nm. In addition, several excitation profiles with the rise time of 0–60 ps were used to investigate their influence on the atomic motion to give a comprehensive understanding of diffraction signal, such as the phase shift of oscillations observed in previous works.^21–24 Our results could give a new interpretation of the high harmonic diffraction signals, in-phase and out-of-phase oscillations, and provide further understanding of the uniformity of the laser distribution.

We specified that the laser is first irradiated on the upper surface of the sample, and the film thickness was set to 30 nm. Considering the region of interest (center of the Gauss laser) and timescale (<10 ps), a uniform excitation profile in the lateral direction is assumed, and the diffusion of the carrier and heat along the lateral direction are ignored. However, the longitudinal excitation profile is distinct due to the difference in the optical penetration depth. The lattice evolution under homogeneous and inhomogeneous excitation is obtained by assuming different optical absorption coefficients $α 1$ and $α 2$⁠. As shown in Fig. 1(a), the distribution of the laser energy in the film is approximately uniform when the penetration depth $δ 1$ (⁠ $1/ α 1$⁠) is 600 nm. Meanwhile, when $δ 2$ (⁠ $1/ α 2$⁠) is 10 nm, a significant inhomogeneity is created with the laser confined to a limited film depth. The carrier and heat diffusion along the longitudinal direction will smooth the inhomogeneity of laser deposited energy in different timescale. The carrier diffusion mainly occurs before the rapid relaxation through electron-phonon coupling (∼ps), then the heat diffusion dominates the thermalization within the sample, which may take several nanoseconds for a thickness of 30 nm, depending on thermal diffusivity.^25,26 To simplify, we ignore the carrier and thermal diffusion along the longitudinal direction.

Dynamic simulation of depth-dependent atomic displacement in the two films with optical penetration depth of 10 and 600 nm is shown in Figs. 2(a) and 2(b), respectively (The rise time τ = 0 ps). The homogeneous absorption of laser leads to the excitation of a typical standing wave. However, for inhomogeneous absorption, in addition to the standing wave, a traveling wave propagating along the surface normal becomes evident. Figure 2(c) plots the calculated shear displacements at the top layer of the film (z = 0) with the optical penetration depth $δ 1=$ 600 nm, showing an oscillation period T = 30 ps. The position of the first peak and amplitude of the oscillations under different rise times exhibits distinct features. The longer rise time τ could attenuate the amplitude and push oscillation toward a positive direction. We used Δt to denote the shift of the first peak position relative to the position, where delay = T/2 = 15 ps. The phase shift $2πΔt/T$ will increase with the rise time τ, but eventually converge to π/2 which can be analyzed by Green function. The corresponding information on the film with the optical penetration depth $δ 2=$ 10 nm is shown in Fig. 2(d), in which the phase shift and amplitude attenuation exhibit similar behaviors as in the homogeneous condition.

To quantify the effect of the rise time τ on the oscillation amplitude with the specific frequency, fast Fourier transform (FFT) was conducted on the oscillation profile [black curves in Figs. 2(c) and 2(d)] with the rise time τ = 0 ps, as shown in Fig. 3(a) and 3(c). By performing FFT of atomic motions at the top surface calculated with different rise time τ, the amplitude of each frequency against τ/T can be extracted as shown in Fig. 3(b) and 3(d), in which the positions where the amplitude drops by one order of magnitude are marked with the vertical black lines. The results shown in Fig. 3 indicate the suppressed oscillation in both homogeneous and inhomogeneous excitation as the rise time increases. In addition, the higher harmonics are more sensitive to the rise time. It is worth noting that there is only odd harmonics under homogeneous excitation, where the longitudinal excitation profile (and also the boundary condition) has central inversion symmetry about the center of the sample; however, the even harmonics appears under inhomogeneous excitation where the inversion symmetry is broken.

To investigate the effect of the coherent phonon modes on the diffraction, kinematical diffraction simulations were performed in the proposed crystal (⁠ $a$ = 3.52 Å, $b$ = 3.52 Å, $c$ = 7 Å, $α$ = 90°, $β$ = 90.3°, $γ$ = 90°). The sample surface is parallel to the plane where lattice vectors $a$ and $b$ lie in, similar to the layered films obtained by mechanical exfoliation. In the calculation of the shear mode, the direction of the atomic displacement on the top surface is set along the $a$ direction [Fig. 4(a)]. For the breathing mode, the displacement is along the $c *$ direction. For the diffraction simulation and analyses in this and next section, the rise time $τ$ in stress $σ ext$ and damping factor λ are set as zero for simplicity. Subsequently, Eqs. (1) and (2) can be totally determined, and the atomic displacements $μ z , t$ can be numerically solved, which are used for the simulation of the diffraction in the following text.

Under inhomogeneous excitation, the simulated temporal intensity evolutions of the Bragg peak (200) and $( 2 ¯00)$ with the same deviation parameter $s z=0.01$Å⁻¹ are indicated by the red and blue curves in Fig. 4(d), respectively. The corresponding intensity decomposition is shown by black dashed line (⁠ $ΔI Δ s z$⁠) and green solid line (⁠ $ΔA$⁠). $ΔI Δ s z$ mainly shows fundamental mode features (fundamental with j = 1 is dominant, with the oscillation period T = 30 ps), whereas high harmonic features are dominant in $ΔA$ (second harmonic with j = 2 is dominant, with period of T/2). Another key point is that the oscillation of $ΔI Δ s z$ in $(200)$ and $( 2 ¯00)$ exhibits an out-of-phase feature [ $phase ( 200 )=π$⁠, $phase ( 2 ¯ 00 )=0$] whereas $ΔA$ shows an in-phase oscillation (both phases are $π$⁠) in the Friedel pairs. Typically, the Friedel pairs have the same diffraction intensity at the same initial deviation parameter (⁠ $s 0 , G= s 0 , − G$⁠), which is determined by the central symmetry of the structural factors. However, the $ΔI Δ s z$ and $ΔA$ would cause the unequal change of the diffraction in Friedel pairs G and –G. Oscillations caused by $ΔA$ within a Friedel pairs are always in-phase, while whether oscillations caused by $ΔI Δ s z$ within a Friedel pairs are out-of-phase or in-phase sensitively depends on the initial deviation parameter, which is related to the tilt angle. Corresponding information of the film with the optical penetration depth $δ=$ 600 nm is shown in Fig. 4(e). In homogeneous excitation, intensity change $ΔI Δ s z$ is dominant. In addition, $ΔI Δ A$ mainly shows the oscillation with period of T/4 and a very small amplitude.

We use the mean change of lattice constant $Δ c ′ ¯$ (⁠ $Δ c ′ ¯= ∑ n z ( c n z ′ − c ′ ¯ ) / N z$⁠) to denote the mean lattice expansion or shearing. Figure 5 plots the temporal evolution of $Δ c ′ ¯$ and $S$ with the modulation of shear mode in the crystal. It is clear that the oscillation of $Δ c ′ ¯$ shows an period of 30 ps, while $S$ show high harmonic features, which is consistent with the results shown in Figs. 4(d) and 4(e). These results also support the conclusion that the intensity change $ΔI Δ s z$ is mainly caused by the mean change of lattice, while $ΔI Δ A$ is stemming from the lattice disorder.

After decomposing the intensity, the frequency distribution of the diffraction intensity and its relationship to the coherent phonon mode can be clarified. The shear modes can induce the intensity change of the in-plane peaks, whereas the breathing modes exert influence on the out-of-plane peaks. Given that the strain wave propagates along the reciprocal vector $c *$⁠, the FFT of the diffraction intensity of the peak (200) and $(0 2 ¯2)$ is performed to analyze the influence of the TA phonons (shear mode) and LA phonons (breathing mode), respectively. The FFT results of peak (200) under shear mode induced by homogeneous (⁠ $δ 1=$ 600 nm) and inhomogeneous (⁠ $δ 2=$ 10 nm) excitation is shown in Fig. 6(a), where the upper and lower curves are obtained based on the temporal oscillation data shown in Figs. 4(f) and 4(d), respectively. Figure 6(b) shows the corresponding information of peak $(0 2 ¯2)$ under the breathing mode. It should be noted that, in order to view peak $(0 2 ¯2)$⁠, the crystal should be tilted to [011] zone axis. In the actual experiment, we used the sine function $y= A isin(2π f it+ φ i)$ to describe the oscillation with the specific frequency $f i$⁠. All the oscillations shown in Fig. 6 have a phase of 0 or $π$ due to the used step-function-like temporal excitation profile (⁠ $σ ext$ with $τ=0$⁠). We specify the amplitude of FFT as >0 when $φ i=π$⁠, while <0 when $φ i=0$⁠, resulting in the negative values shown in Fig. 6.

Figures 6(a) and 6(b) indicate that the homogeneous excitation mainly causes the shift of the reciprocal rod, which is manifested by the intensity oscillations $ΔI Δ s z$ with frequency $f 1$⁠. Meanwhile, the oscillation of $ΔA$ at the frequency of $4 f 1$ indicates slight changes in the amplitude of the reciprocal rod. In contrast, inhomogeneous excitation causes more drastic reductions in the intensity of the reciprocal rod, resulting in a strong oscillation of $ΔA$ with frequency of $2 f 1$⁠. These findings apply to both the breathing and shear modes; however, the behavior in the breathing mode is more complex. As shown in Fig. 6(b), the breathing mode causes additional oscillation of $ΔA$ with frequency $f 1$⁠, which is not related to the uniformity of the optical excitation.

The frequency of atomic displacements under coherent phonons contains multiple components, mainly includes the fundamental (⁠ $f 1$⁠), second harmonic (⁠ $f 2$⁠), and third harmonic (⁠ $f 3$⁠) components. To analyze the origin of the various frequencies of the diffraction in Figs. 6(a) and 6(b), atomic displacements under each frequency component based on the equilibrium position are introduced into the diffraction simulation. The results are shown in Figs. 6(c)–6(f), in which $μ homo$ and $μ inhomo$ denote the atomic equilibrium position after laser excitation. While, $μ f i$ represents the atomic displacements induced by the vibration with frequency $f i$⁠. Corresponding schematic can be seen in Fig. 1(c).

For the shear mode under homogeneous excitation [Fig. 6(c)], the fundamental mode (frequency: $f 1$⁠) generates the oscillation of diffraction intensity $ΔI Δ s z$ with the same frequency $f 1$⁠, which behaves as an out-of-phase oscillation in the Friedel pairs. The third harmonic mode produces the oscillation of $ΔA$ and $ΔI Δ s z$ with frequencies of $6 f 1$ and $3 f 1$⁠, respectively. Notably, the simultaneous presence of the fundamental and third harmonic modes exhibit a weak $4 f 1$ (⁠ $ΔA$⁠) signal in the diffraction, indicating that the influence of each mode is not completely independent. Thus, the oscillation of $ΔA$ exhibits $4 f 1$ signals for both the shear and breathing modes in the film with the penetration depth of 600 nm. These conclusions remain valid for the breathing mode under homogeneous excitation [Fig. 6(d)], except the influence of the fundamental mode with the atomic displacements of $μ homo+ μ f 1$⁠, which causes additional oscillations of $ΔA$ with frequency of $f 1$⁠.

The comparison of Figs. 6(c) and 6(e) [or Figs. 6(d) and 6(f)] indicates that, for the inhomogeneous excitation, the oscillation of $ΔA$ shows obvious frequency $i · f 1$ under the i-th order harmonic (i > 1), which is absent under the homogeneous condition. In addition, the amplitude of the oscillation with frequency $2 i · f 1$ exhibits obvious attenuation, which is attributed to the asymmetric distribution of the atomic equilibrium position with respect to the center of the film. Diffraction signals in inhomogeneous excitation exhibit a $2 f 1$ oscillation not only because of the displacement $μ f 2$ of atoms at the second harmonic, but also by the equilibrium position of the atoms $μ inhomo$⁠. The equilibrium position of atoms in the oscillation has great effect on diffraction, that is the reason why $μ homo$+ $μ f 3$ and $μ inhomo$+ $μ f 3$ show significant difference in diffraction [Figs. 6(c) and 6(e), or Figs. 6(d) and 6(f)], and why the second harmonics manifested in the diffraction intensity oscillation is much stronger than the true phonon oscillation amplitude [Figs. 3(c), 6(c), and 6(e)]. The asymmetric atomic distribution also depends on the initial position $μ z , 0$ of each lattice layer [Eq. (5)], indicating that $μ z , 0$ can significantly influence the oscillation of $ΔA$ with frequency $i · f 1$ under the ith order harmonics (i > 1). The change in the initial position induced by the lattice distortion was discussed in previous works.^34,35 However, this was not considered in this work.

The relationship between the frequencies of the diffraction intensity and the coherent phonons now can be clarified. In the breathing or shear modes generated by the homogeneous excitation, the oscillation of the diffraction intensity in $(0 2 ¯2)$ or (200) with frequency $f 1$ is attributed to the fundamental mode, whereas the frequency $4 f 1$ stems from the coupling of the fundamental and third harmonic modes. For the inhomogeneous excitation, additional $2 f 1$ signal is ascribed to the second harmonic mode, whereas the weak $3 f 1$ signal is stemming from the third harmonic mode. It should be mentioned that the finite electron–phonon coupling constant in the actual material would result in a non-zero value of $τ$⁠. According to the results in Fig. 3(b), the high-frequency signals will be significantly attenuated, and therefore, it is difficult to experimentally observe the $4 f 1$ signal in homogeneous excitation.

It should be noted that the sample is assumed to be absolutely smooth and there are no wrinkles in the film. The LA and TA phonons are simply described by one dimensional wave equation and exist independently in the sample. However, the real material may consist of many wrinkles or micro-patches separated by defects, and each patch may be different in size and orientation, resulting in its own oscillation direction. In addition, TA and LA phonons could be excited simultaneously in the sample, leading to more complicated behavior in oscillation frequency and phase.

In this study, the phase shift and amplitude suppression in oscillation due to the rise time of the stress were analyzed. The results of this work are useful for understanding the generation of coherent phonons under different mechanisms. For example, long electron–phonon coupling time or small thickness can lead to a large relative rise time τ/T. This significantly weakens the amplitude of thermoelasticity induced coherent phonons, especially the higher harmonics. In addition, we demonstrated the complex behaviors of the diffraction intensity under structural modulation of the coherent LA and TA phonons generated by homogeneous and inhomogeneous distribution. The correspondence between the frequencies of the diffraction intensity oscillation and high harmonics of the strain waves was analyzed by decomposing the diffraction intensity. Thus, the phonon-induced in-phase and out-of-phase intensity oscillation in the Friedel pairs and unequal change were elaborated. These analyses deepened the understanding of the structural behaviors under coherent acoustic phonons and their corresponding influence on electron diffraction, which facilitated further structural modulations by the coherent phonons in the UED device.

This work was supported by the National Key Research and Development Program of China (Grant No. 2021YFA1301502), the National Natural Science Foundation of China (Grant Nos. U22A6005 and 12074408), the Guangdong Major Scientific Research Project (No. 2018KZDXM061), the Youth Innovation Promotion Association of CAS (No. 2021009), the Scientific Instrument Developing Project of the Chinese Academy of Sciences (Grant Nos. YJKYYQ20200055, ZDKYYQ2017000, and 22017BA10), the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant Nos. XDB25000000 and XDB33010100), Beijing Municipal Science and Technology Major Project No. Z201100001820006, the IOP Hundred Talents Program (No. Y9K5051), the Postdoctoral Support Program of China (No. 2020M670501), and the Synergetic Extreme Condition User Facility (SECUF).

The authors have no conflicts to disclose.

Yongzhao Zhang: Conceptualization (equal); Data curation (lead); Investigation (lead); Methodology (equal); Visualization (lead); Writing – original draft (lead). Jun Li: Funding acquisition (equal); Methodology (equal). Wentao Wang: Writing – review & editing (equal). Huanfang Tian: Writing – review & editing (equal). Wenli Gao: Writing – review & editing (equal). Jianqi Li: Funding acquisition (equal); Writing – review & editing (equal). Shuaishuai Sun: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Writing – review & editing (equal). Huaixin Yang: Conceptualization (equal); Funding acquisition (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.