Total-transmission and total-reflection of individual phonons in phononic crystal nanostructures

Similar to the periodic arrangement of atoms in crystals that gives rise to the electronic band structure, it is found that the artificial structures formed by periodically modulated physical properties can be used to control the transmission of various types of waves based on the band structure theory. For instance, the concept of photonic crystals was first proposed by Yablonovitch¹ and John² in the 1980s in which the photonic bandgaps can be formed via Bragg diffraction. Similarly, the phononic crystals formed by periodic structures with different elastic constants have also been developed for engineering the elastic bandgaps.^3–5 Although the phononic crystal was initially proposed to manipulate the transmission of acoustic waves,^3–9 there have been growing efforts^10–13 in applying the phononic crystal to the control of thermal waves as well, namely, the heat.

Since phonons are the energy carriers of heat, their particle nature has been widely studied in the literature to understand various scattering mechanisms for applications in thermal management^14–17 and thermoelectrics.^18,19 Thanks to the advancement in synthesis technology that ultra-thin superlattice structures with nanometer feature size can be synthesized in experiment,^20,21 the wave nature of phonons is attracting increasing interest.^20–28 For instance, Xie et al.²⁵ reviewed the recent advances in the study of the coherent phonon transport in periodic nanostructures. Hu et al.²⁷ reported the stopband formation mechanism in a multilayer array system. Zeng et al.²⁸ demonstrated via theoretical calculations that the twist angle can be used to control the phonon interference and the thermal conductance of molecular junctions. Recent studies^29–38 demonstrated that phononic crystal nanostructures have the potential to control thermal transport by engineering the phonon dispersion. The coherent interference of phonons gives rise to the peculiar behaviors that cannot be explained by the conventional particle scattering picture, such as the increase in thermal conductivity with the increase in interface density²¹ or interface number²⁰ and the phonon localization.^37,39 However, most of the existing studies^13,20,21,40 only indirectly reflect the wave nature of phonons via the macroscopic thermal transport coefficient, such as thermal conductivity. A systematic investigation on the phonon interference effect in the phononic crystal nanostructure at the microscopic phonon mode level is still lacking. In particular, the total-transmission and total-reflection phenomena for individual phonons have not yet been clearly demonstrated in the phononic crystal nanostructure.

In this work, we investigate directly the coherent nature of the individual phonon mode in the phononic crystal nanostructure based on the atomic level phonon wave-packet simulations. Taking monolayer graphene as a model system, we monitor the phonon propagation in one-dimensional graphene superlattice composed of isotopic carbon atoms. Both the constructive and destructive phonon interference phenomena have been clearly observed in the wave-packet simulations, which are characterized by the valleys and peaks in the phonon transmission coefficient, respectively. The physical conditions for achieving the phonon interference are discussed. More interestingly, we demonstrate in our simulations that both total-transmission and total-reflection of a particular phonon mode can be realized in the superlattice structure, providing a strong and direct proof for the wave nature of phonons at the individual model level.

The phonon wave-packet method⁴¹ is used to monitor the phonon transmission in the graphene superlattice structure. A phonon wave packet at wavevector q and branch index λ in the coordinate space is centered around R₀ with a spatial width η and can be described as⁴¹ 

where $uilαt$ represents the α component of displacement at time t for atom i in the unit cell labeled l, B represents the amplitude of the wave packet, ω is the phonon frequency, and ϵ_iλα is the phonon eigenvector for atom i. The parameter η is used to control the spatial width of a phonon wave packet, which is inversely proportional to the width of the phonon distribution function in the reciprocal space. This means that a larger spatial width η in the coordinate space corresponds to a better single-frequency phonon in the frequency space. Once q is specified, ϵ_iλα and ω are determined from the phonon dispersion relation by diagonalizing the dynamical matrix. The initial set of atom velocities can be computed from the time derivative of the atomic displacement as $vilαt=∂uilαt∂t$⁠. To analyze the phonon distribution in the reciprocal space, the mode amplitude in the reciprocal space for a given phonon mode (λ, q) can be computed by the Fourier transform of atomic displacements as

All wave-packet simulations in this work are performed by using the LAMMPS package⁴² with the optimized Tersoff potential⁴³ for graphene. The time step is set as 0.5 fs, and periodic boundary conditions are applied in all directions. Figure 1 shows the schematic graph of the wave-packet simulation. A thin layer of the graphene superlattice with length L is inserted into the center of a monolayer pristine graphene, acting as an interface layer. In optics, anti-reflection coatings are used to control the optical transmission efficiency between two different media, and the anti-reflection phenomenon is realized via the interference of reflected lights from the two interfaces.^44,45 More interestingly, is it possible to realize the total-transmission of thermal phonons by inserting an inhomogeneous layer in a homogeneous medium? Therefore, we construct such a sandwiched structure in our simulation. Here, the C–C bond length is set as $a=1.44Å$⁠, and we set $l0=3a$ and $q0=2πl0$ for convenience. The graphene superlattice structure is formed by carbon isotopes with atomic mass of 12 g/mol and 36 g/mol, and its period length L_P is set as L_P = n × l₀. Then, the length of the superlattice structure can be expressed as L = N_P × L_P, where N_P denotes the number of periods.

Since acoustic phonons make dominant contribution to thermal transport in most crystalline materials, we only consider the propagation of acoustic phonons in our simulation. In addition, we have verified that both longitudinal acoustic (LA) and transverse acoustic (TA) branches show similar coherent interference behaviors, so the TA phonon is used as an example to demonstrate the coherent effect in most part of this paper. Instead of phonon scatterings in the particle picture, as we focus on phonon coherence, which is a wave effect of phonons at a low temperature, a background temperature around 0 K is used in our simulations. This treatment has been widely used in the wave-packet simulations to understand the physical mechanisms in various studies.^41,46,47 A phonon wave packet with initial energy E_in is launched on the left pristine graphene segment and propagates toward the graphene superlattice along the zigzag (x) direction, with the initial phonon wavevector set as $qxG=kxq0,qyG=0,qzG=0$⁠. After transmitting into the graphene superlattice (interface layer), the wavevector is changed to $qxS$ in the superlattice due to the mode conversion at the interface. Finally, we measure in the pristine graphene segment on the right-hand side the amount of energy transmitted across the interface layer, E_out, and the transmission coefficient for each phonon wave packet is computed as

Here, Γ = 1 indicates total-transmission across the interface layer, while Γ = 0 indicates total-reflection by the interface layer.

We first examine the effect of superlattice periodicity on the phonon transport across the interface layer. Figure 2 shows the dependence of phonon transmission coefficient Γ on superlattice period length L_P for a fixed TA phonon wave packet with $qxG=0.025q0$ and η = 50a. For each superlattice period N_P, the phonon transmission coefficient Γ oscillates periodically with the increase in L_P, showing multiple peaks and valleys in the transmission function. The presence of these transmission peaks and valleys is clear evidence of the phonon interference effect.

Analogous to the wave interference in acoustics^4,48,49 and optics,^44,45 when the phonon wave packets reflected by superlattice interfaces are out of phase and interfere with each other, the resulting reflection is minimized,^11,50 giving rise to the maximized transmission. In this study, we refer to this phenomenon as destructive interference between the reflected phonon wave packets following the convention used in the literature.¹⁵ On the other hand, when the reflected wave packets are in phase, they can interfere constructively, leading to the minimized transmission. Therefore, the peaks and valleys observed in Fig. 2 are, in fact, caused by the destructive and constructive interference between the reflected phonon wave packets, respectively. Similar oscillation features of the phonon transmission function against the phonon frequency have also been reported in the confined nanoscopic films.^50,51

According to Bragg’s law, multiple scattering is required to enhance the phonon interference effect. This condition means that the phonon should travel sufficiently far in the superlattice to generate multiple reflected phonons.¹⁵ This aspect is clearly demonstrated with varying number of periods N_P in Fig. 2. For the same L_P, a convergent transmission value for peaks and valleys is gradually reached with increasing N_P. When the total length of the graphene superlattice is long enough (L = 240l₀), a convergent transmission function against L_P can be obtained. Therefore, a constant length of L = 240l₀ is used in the following discussion. Our simulation results clearly show that the phononic crystal structure should be long enough to ensure sufficient phonon interference and therefore the phononic bandgap.

Although the phonon interference effect has been demonstrated in Fig. 2, the total-transmission or total-reflection phenomenon is not observed for that particular phonon mode (⁠$qxG=0.025q0$⁠). Previous studies^50,51 have found that the transmission function also oscillates periodically with the increase in phonon frequency in a given structure. Therefore, to explore the effect of the phonon frequency on the transmission function, we carry out further simulations in a fixed superlattice structure (L = 240l₀ and L_P = 10l₀) for incident phonon wave packets with varying wavevector $qxG$ and spatial width η. As shown in Fig. 3, the transmission function exhibits an obvious periodic oscillation behavior with the variation of wavevector $qxG$ for η = 50a. The oscillation period between two adjacent transmission valleys is found to be $ΔqxG=0.0385q0$⁠.

More significantly, when increasing the width above η = 50a, the wavevectors corresponding to the peaks and valleys in the transmission function remain the same, but the value for the transmission peaks and valleys gradually approaches unity (total-transmission) and zero (total-reflection), respectively. When the wave packet is wide enough (η = 100a), both total-transmission and total-reflection can be realized at the particular wavevector (dotted line in Fig. 3). This is because unlike the plane wave with single-frequency in acoustics and optics, the phonon wave packet has a finite width η in the coordinate space, which gives rise to the finite spreading of the phonon distribution function in the reciprocal space [see Fig. 4(a)]. The increase in η narrows the phonon spreading in the reciprocal space, which makes the wave packet approach the plane wave limit, and therefore allows for the total-transmission or total-reflection at a particular wavevector. This interesting feature of total-transmission across the heterostructure interface might provide novel insight into the design of thermal interface materials⁴⁶ for heat dissipation applications. More importantly, our study demonstrates that both total-transmission and total-reflection can be realized in a phononic crystal structure via destructive and constructive phonon interference, which could be helpful for the design of phononic devices such as the phononic rectifier.⁵² 

Figure 4(a) shows an example of the phonon distribution function in the reciprocal space for the incident wave packet in pristine graphene (solid line) and the transmitted wave packet in the graphene superlattice (dashed line). The wavevector in pristine graphene $qxG=0.0932q0$ is shifted to $qxS=0.126q0$ in superlattice graphene due to the mode conversion at the interface. This is because the band folding in the phonon dispersion of superlattice graphene leads to the reduced group velocity of the acoustic branch, compared with that of pristine graphene. For the frequency-conserving elastic scattering process, the wavevector in superlattice graphene shifts to a larger value than the wavevector in pristine graphene due to the reduced phonon group velocity in superlattice graphene. The ratio between two wavevectors $ξ=qxSqxG$ is around 1.35. To examine whether this ratio depends on the incident wavevector, we repeat the wave-packet simulations with different incident wavevectors $qxG$ and record the transmitted wavevectors $qxS$⁠. As shown in Fig. 4(b), a linear dependence on $qxG$ is observed for $qxS$⁠. Furthermore, the linear fitting gives rise to a slope of 1.35, suggesting that the ratio ξ between two wavevectors is a constant in our study.

In optics and acoustics, wave interference takes place in a superlattice structure when the wave propagation distance is multiple times of half the wavelength. The propagation distance is twice the superlattice period when the wave travels back and forth across the unit cell of the superlattice. Therefore, the superlattice period must be equal to the integer times of the quarter wavelength, at which the destructive interference or constructive interference happens. This condition has been widely used in optics, such as the total-transmission films and the total-reflection films.⁴⁴ 

Correspondingly, to achieve total-transmission or total-reflection of phonons, the superlattice period L_P and the phonon wavelength l must satisfy the condition that $LP=m×l4$ (m is an integer). On the other hand, the phonon wavevector in the superlattice can be written as $qxS=2πl$⁠. Therefore, we can conclude a condition for the emergence of phonon interference in superlattice graphene as

Here, the integer m means a series of valleys and peaks in the transmission function, which correspond to the constructive and destructive phonon interference bands, respectively. For each constructive (or destructive) interference band, the value m is fixed, and thus, it is a constant on the right-hand side of Eq. (4), which means an inverse relationship between period length L_P and superlattice wavevector $qxS$⁠. In this study, we focus on the long wavelength phonons in superlattice graphene (small $qxS$ value) because the superlattice period L_P = n × l₀ (n is an integer) has a finite minimum value. Moreover, Δm = 2 represents two adjacent constructive (or destructive) interference bands. Therefore, the right-hand side of Eq. (4) predicts the difference between two adjacent constructive (or destructive) interference bands as $ΔLP×qxG2π=0.5$⁠. On the other hand, the separation between two adjacent valleys in Fig. 3 is found to be $ΔqxG=0.0385q0$ for a fixed superlattice with L_P = 10l₀. Based on the ratio ξ = 1.35 obtained from Fig. 4, we can get $ΔqxS=ξΔqxG=0.052q0$⁠. By plugging these values into the left-hand side of Eq. (4), the value of $ΔLP×qxS2π$ in Fig. 3 is equal to 0.52, which is in good agreement with the independent prediction of 0.5 from the theory. Such agreement highlights the validity of Eq. (4) for predicting the phonon interference.

To further verify the condition in Eq. (4), we simulate the propagation of phonon wave packets with varying incident wavevectors (⁠$qxG=0.01q0∼0.1q0$⁠) for both LA and TA phonons in the superlattice structures with a fixed total length (L = 240l₀) but varying period lengths (L_P = 2l₀ ∼ 40l₀). Figure 5 shows the contour plot against the superlattice wavevector $qxS$ and superlattice period L_P in which individual colors represent the transmission coefficients for each wave packet. The bands for maximum and minimum phonon transmission are clearly observed. The constructive interference between phonon wave packets results in minimum values of the transmission coefficient. These minimum values in Fig. 5 are connected together to form multiple bands depicted by the blue regions, which correspond to the phonon transmission bandgaps. In the same way, red regions show the locations of destructive interference bands with the maximum transmission coefficient. According to Eq. (4), $LP×qxS$ is constant for the same interference band. Therefore, we apply a hyperbolic fitting based on $LP×qxS2π=C$ for each constructive interference band. The black solid lines in Fig. 5 stand for the hyperbolic fitting curves. The obtained fitting constant C for the TA phonon is around 0.481, 1.02, 1.46, and 2.00 for each band [Fig. 5(a)], respectively. The difference between two adjacent bands is very small, and the averaged difference $ΔLP×qxS2π$ is about 0.51. This value is quantitatively in excellent agreement with the independent predictions of $ΔLP×qxS2π=0.52$ from Fig. 3. Similarly, the obtained constant C for the LA phonon is around 0.507, 1.02, 1.47, and 1.99 for each band [Fig. 5(b)], respectively, and the averaged value of $ΔLP×qxS2π$ is around 0.49, which is very close to the prediction of 0.5 from the theory. Quantitative agreement between predictions from different perspectives provides strong evidence that Eq. (4) is valid for determining the emergence of phonon interference.

In summary, we have studied the phonon interference in the graphene superlattice structure sandwiched between two pristine graphene segments via phonon wave-packet simulations. The constructive interference and destructive interference between the reflected phonons give rise to valleys and peaks in the transmission coefficient, respectively, leading to the periodic oscillation of the transmission function with the variation of the superlattice period length. Our simulation results reveal that the number of superlattice periods should be large enough to ensure sufficient interference between the multiple reflected phonons. More importantly, both total-transmission and total-reflection of individual phonons have been clearly demonstrated in this work when the spatial width of the wave packet is sufficiently large. The conditions for realizing the phonon interference have been proposed, which are quantitatively in good agreement with independent wave-packet simulations. Our study provides the direct evidence for the coherent interference effect at individual phonon mode levels, which might be helpful for the design of phononic devices based on the wave nature of phonons.

This project was supported, in part, by grants from the National Natural Science Foundation of China (Grant Nos. 12075168 and 11890703), the National Key Research and Development Program of China (Grant No. 2017YFB0406000), and the Science and Technology Commission of Shanghai Municipality (Grant Nos. 19ZR1478600 and 18JC1410900).

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