As a fundamental degree of freedom, phonon chirality is expected to promote the development of quantum information technology just like electron spin. Currently, central to this area is the realization of efficient transmission and control of chiral information. In this paper, we propose an approach by integrating topological theory, leveraging topologically invariant Chern numbers, to encode hexagonal lattice systems. Our investigation reveals the presence of topologically protected chiral interface states within the shared band gaps of both trivial and non-trivial system units. By precisely modulating the magnetic field distribution within the encoding system, we can effectively manipulate the topological pathways. Building upon this framework, we design and implement a chiral phonon three-port device. Through dynamic calculations, we demonstrate the transmission process of chiral information, showcasing the chiral phonon switching effect and logical OR operation. Our findings not only establish a fundamental mechanism for the manipulation and control of phonon chiral information but also provide a promising direction for research in harnessing chirality degrees of freedom in practical applications.

Phonons are ubiquitous quasiparticles in condensed matter physics. Phononics, developed based on the study of phonon manipulation,1–8 has now become a potential physical dimension for information processing after electronics and photonics. In recent years, phonons have been found to possess chirality9 and have been experimentally verified,10–12 which has brought an additional controllable degrees of freedom to the field of fundamental physics.13–17 Considering the important application of electron spin in spintronic devices,18–20 the appearance of analogous phonon chirality related to angular momentum and polarization has caused researchers to think about phonon-based devices. Current studies have explored the selective coupling of chiral phonons with electrons and circularly polarized light for spin-related information transfer,9,10,21 but how to directly control the chirality degree of freedom and transmit the information remains to be further investigated.

We try to combine phonon chirality and topological theory22–24 to explore potential possibilities. The topologically protected phonon mode is not affected by impurities and defects in the crystal lattice, thus can achieve one-way dissipationless transport, which provides the possibility for robust transmission of phonon chiral information. Meanwhile, the edge modes that can be excited at fixed frequencies often exist in topological systems,25–27 which can facilitate the excitation and transmission of single phonon modes with predetermined chirality. In addition, topological states also have a wide range of applications due to their great flexibility in the control of robust edge modes. Expanded to the fields of acoustic and elastic waves that possess spin angular momentum,28–31 related research is also developing,32–39 such as acoustic antennas,32 acoustic delay lines,33 elastic router,34 etc. The manipulation of transport particles in these studies can provide some approaches for designing chiral phonon devices. Furthermore, in order to achieve automatic control and reconfigurability of chiral phonon propagation paths, we draw on the encoding methods used in metamaterials. Coding metamaterials were initially used to control electromagnetic waves and were later introduced into the field of acoustics to manipulate sound waves.40–42 Here, we utilize the means of coding to construct the propagation path of chiral phonons.

In this work, by resorting to topological interface states, we realize the transmission and control of phonon chiral information in coded systems. Using the Chern number to label the system units, we first investigate the influence of controllable external magnetic fields on the hexagonal system to identify the common bandgap between topologically trivial and non-trivial units. Next, the properties of supercells containing zigzag or armchair interfaces are calculated, focusing on the chirality and group velocity of the topological interface states within the bandgap, which serves as the foundation for designing topological paths. Subsequently, we employ magnetic fields to control the transmission of chiral information and design a three-port chiral phonon device.

Figure 1(a) shows a schematic of the coding system featuring elements 0 and 1. By adopting an appropriate encoding sequence, the fixed transport paths can be constructed, enabling steady travel of chiral phonons along the designated paths as indicated by the arrows. Figure 1(b) shows the coding unit of the system, aiming to realize three types of coding units, namely, the 0 state without a magnetic field, the 1 state with a magnetic field along the z direction, and the −1 state with a magnetic field along the –z direction. These states correspond to the value of Chern numbers, which are caused by different system characteristics. Therefore, at the interface where the topologically trivial unit 0 and the topologically non-trivial unit ±1 are spliced together, we expect that topological interface states can be achieved, and the chiral information carried by the interface state will be stably transmitted along the shared edge.

FIG. 1.

(a) Schematic diagram of a coding system. The red arrows indicate the transport paths of the chiral phonons. (b) Schematic diagram of coding unit. 0, 1, and −1 are the coded digital elements of the unit, corresponding to the Chern numbers. Red and blue spheres represent the atoms with mass m A and m B, respectively. The lattice constant of the hexagonal system: a = 1 Å, the longitudinal force constant: K T = 0.144 eV / ( u Å 2 ), and the transverse one: K T = K L / 4. (c) Phase diagrams of the Chern number obtained from different system conditions. (d) Phonon dispersion relationship. Left panel: the system type encoded as 0. m A / m B = 2.0, λ = 0. Right panel: the system type encoded as 1. m A / m B = 2.0 , λ = 10 THz. The shaded area represents the shared bandgap of the two types of systems: 0.0276 0.0312 eV.

FIG. 1.

(a) Schematic diagram of a coding system. The red arrows indicate the transport paths of the chiral phonons. (b) Schematic diagram of coding unit. 0, 1, and −1 are the coded digital elements of the unit, corresponding to the Chern numbers. Red and blue spheres represent the atoms with mass m A and m B, respectively. The lattice constant of the hexagonal system: a = 1 Å, the longitudinal force constant: K T = 0.144 eV / ( u Å 2 ), and the transverse one: K T = K L / 4. (c) Phase diagrams of the Chern number obtained from different system conditions. (d) Phonon dispersion relationship. Left panel: the system type encoded as 0. m A / m B = 2.0, λ = 0. Right panel: the system type encoded as 1. m A / m B = 2.0 , λ = 10 THz. The shaded area represents the shared bandgap of the two types of systems: 0.0276 0.0312 eV.

Close modal
Here, we only consider the two-dimensional motion of atoms in the xy plane. For a general lattice, the Hamiltonian of the system under the harmonic approximation can be written as
(1)
where u and p are the displacement and momentum terms, respectively, and K is the force constant matrix of the whole system. When the system is in an external magnetic field, the introduction of spin–phonon interactions causes the magnetic field to be able to act indirectly on electrically neutral phonons. The form of the Hamiltonian is43 
(2)
where A is an antisymmetric real matrix with the block diagonal elements Λ α = ( 0 λ λ 0 ) for the αth atom site. λ has a frequency dimension, and its magnitude is proportional to the magnetic field. For Eq. (2), it can be further written as H = p T p + 1 2 u T ( K A 2 ) u + u T A p, where u T A p can be interpreted as spin–phonon interaction term. The introduction of spin–phonon interaction can break the time reversal symmetry of the system, thereby bringing about topological phenomena.

Figure 1(c) shows the sum of the Chern number for the energy band below the frequency gap as a function of the magnetic field and the atomic mass ratio. It can be seen that the Chern number jumps from 0 to (±)1 in a relatively strong magnetic field, and the system turns to topologically non-trivial, which means the appearance of topologically protected edge states. Furthermore, if we want to realize a common edge state at the interface between trivial and non-trivial systems, their phonon dispersions are required to have a common bandgap, so the inversion symmetry of the two types of systems must be broken. For the system with only inversion symmetry broken, the symmetry of the system at zero magnetic field determines that it has a trivial bandgap structure. When the magnetic field is added and continuously strengthened, in the first stage, as the energy bands gradually turn to degenerate in the K valley, the bandgap is closed. The Chern number jumps from zero to non-zero, transforming into a non-trivial system. The second stage of the increase in the magnetic field will bring about the reopening of degeneracy accompanied by a widening of the non-trivial bandgap. When the magnetic field becomes too large, the energy bands gradually tend to degenerate at the Γ point, causing the bandgap to close. Based on the understanding of the changing characteristics of the phonon dispersion, we can find suitable parameters to make the systems have a wide common frequency gap. Figure 1(d) shows the dispersion relation of two types of systems with strong inversion symmetry breaking. They have a clear shared bandgap that is topologically non-trivial, and topological interface energy bands across the gap can exist.

For the supercell of upper-lower type as shown in Fig. 2(a), there is a pair of edge bands in the frequency gap [see Fig. 2(c)]. Furthermore, we focus on phonon chirality information, which can be characterized by calculating phonon polarization,9 
(3)
where ϕ is the eigenvector of the phonon mode, the polarization operator,
(4)
| R α and | L α are defined as a set of bases vectors of right-handed and left-handed circular polarization, respectively. s p h z > 0 indicates that the phonon is right-handed, and s p h z < 0 means that it is left-handed. Similarly, the polarization of sublattice in the primitive cell can also be calculated,
(5)
Figure 2(e) shows the polarization distribution of the sublattice for the selected modes. For one of them, the polarization corresponding to the obvious chirality vibration is distributed at the lower edge of unit 0 and the upper edge of unit 1, which means that the mode is the topological interface state. Its group velocity is along the x direction, and our calculation shows that the phonon polarization value is 0.22, which indicates that it is right-handed. For the other mode, it can be seen that the atoms at the lower edge of the supercell have obvious chirality vibration. It is the topological edge mode of the supercell and is not the focus of our attention.
FIG. 2.

Chiral interface modes of the supercells composed of digital units. (a) and (b) are the schematic diagram for 0/1 supercell structure of upper-lower type with the zigzag middle interface along the x direction and the left-right type with the armchair middle interface along the y direction, respectively. (c) and (d) are the phonon dispersion of 0/1 supercells with zigzag and armchair interfaces, respectively. Red and blue asterisks indicate topologically protected interface and boundary states in the bandgap, and the black dashed box indicates the common frequency gap range. (e) and (f) Phonon polarization distributions of atoms in the primitive cell corresponding to the topological interface states (the red line) and edge states (the blue line). (e) 80 atoms along the y direction as the primitive cell. (f) 96 atoms along the x direction as the primitive cells. Atoms 1–48 are in the first row, and atoms 49–96 are in the second row. (g) The phonon dispersion relationships of 1/0 (top), 0/−1 (middle), and −1/0 (bottom) supercells containing zigzag interfaces. The modes marked in the figure correspond to phonon energy 0.0292 eV. The gray arrow indicates the direction of the group velocity of the interface state, and ± indicates that the mode is right/left-handed.

FIG. 2.

Chiral interface modes of the supercells composed of digital units. (a) and (b) are the schematic diagram for 0/1 supercell structure of upper-lower type with the zigzag middle interface along the x direction and the left-right type with the armchair middle interface along the y direction, respectively. (c) and (d) are the phonon dispersion of 0/1 supercells with zigzag and armchair interfaces, respectively. Red and blue asterisks indicate topologically protected interface and boundary states in the bandgap, and the black dashed box indicates the common frequency gap range. (e) and (f) Phonon polarization distributions of atoms in the primitive cell corresponding to the topological interface states (the red line) and edge states (the blue line). (e) 80 atoms along the y direction as the primitive cell. (f) 96 atoms along the x direction as the primitive cells. Atoms 1–48 are in the first row, and atoms 49–96 are in the second row. (g) The phonon dispersion relationships of 1/0 (top), 0/−1 (middle), and −1/0 (bottom) supercells containing zigzag interfaces. The modes marked in the figure correspond to phonon energy 0.0292 eV. The gray arrow indicates the direction of the group velocity of the interface state, and ± indicates that the mode is right/left-handed.

Close modal

For the 0/1 structure of left–right type containing an armchair interface [see Fig. 2(b)], there are also two energy bands across the frequency gap, as shown in Fig. 2(d). One of them is localized at the supercell interface, and the other is localized at the right edge, as shown in Fig. 2(f). For the topological interface state, the adjacent atoms along the y direction in the primitive cell do opposite chiral vibrations, resulting in the mode polarization s p h z of almost 0. The phonons have no obvious chiral characteristic, which is caused by the nature of the interface itself. Once the interface mode is excited, it can only be transmitted one-way in the y direction. It is worth noting that since the modes at the zigzag and armchair interfaces are topologically protected, they can effectively convert each other. Therefore, in the whole encoding system, chiral phonons can propagate stably along the path constructed by the interface. For other supercell units in the coding system, we are more concerned with the upper-lower structures in which chiral modes exist. The phonon properties that can be transmitted are shown in Fig. 2(g). For the supercells of left-right type, although the interface states do not have chirality, the change of group velocity direction is consistent with the upper-lower type.

Next, we try to manipulate the transmission path of the mode and demonstrate the transmission process of chiral information. We design a lattice system composed of 4 × 4 coding units and then divided it into four regions. By regulating the magnetic field distribution in the middle two parts, the output of the port can be controlled, as shown in Figs. 3(a)–3(c).

FIG. 3.

Three-port chiral phonon device. (a)–(c) Control diagram of input (I) and output (O) by three distributions of magnetic field along the z direction. (g)–(i) The case when the magnetic field is reversed. (d)–(f) and (j)–(l) are the snapshots of the corresponding angular momentum propagation along the topological path, respectively. The excited phonon energy is 0.0292 eV, the time is 125 T , and T = 2 π / ω is the period of atomic vibration.

FIG. 3.

Three-port chiral phonon device. (a)–(c) Control diagram of input (I) and output (O) by three distributions of magnetic field along the z direction. (g)–(i) The case when the magnetic field is reversed. (d)–(f) and (j)–(l) are the snapshots of the corresponding angular momentum propagation along the topological path, respectively. The excited phonon energy is 0.0292 eV, the time is 125 T , and T = 2 π / ω is the period of atomic vibration.

Close modal

Using the method of dynamic simulation, we specifically demonstrate the transport process in Figs. 3(d)–3(f), and the digits marked on the system are the codes of each unit. We input the eigen information of interface mode of 0/1 supercell into the I port, that is, the eigenfrequency is used as the excitation frequency, and the position and initial phase information contained in the eigenvector are used as the initial value of the rotational motion of the atoms. With time, the periodic vibrations of the atoms at the input port are transmitted to the nearby atoms. Since the group velocity of the input mode is along the x direction, right-handed phonon can transport to the right along the interface of the 0/1 unit of upper-lower type, and the topological protection characteristic prevent the mode from spreading into the bulk during the transmission process. It can be seen that the phonon can bypass the corners on the transmission path with almost no backscattering, which is caused by the efficient conversion of topological edge states at different interfaces. What we show here is the transport of phonon angular momentum in real space, and the angular momentum can characterize the chiral vibration of the atoms of the phonon mode. For the transport of topologically protected interface states, the angular momentum distribution pattern of atoms near the interface of output port at the complete cycle moment is almost consistent with the polarization distribution of the input signal.

When the magnetic field in the middle parts of the system becomes zero, the transmission path of the phonon mode changes, as shown in Fig. 3(e). At the first corner, the originally upward transmission turns downward, which is because the longitudinal interface of the 0/1 unit only allows the mode with the group velocity along the y direction to pass, while the 1/0 type is just the opposite, allowing phonons with the −y direction to transport. In this case, the output port O1 is closed, and the right-handed phonon mode is output at the port O2.

Figure 3(f) shows the case where both transmission paths are open at the same time. The atom vibration at the node can be regarded as an excitation source, with signals input at the ports of the two longitudinal interfaces, resulting in simultaneously transmission of the phonons in opposite directions. For this system, the right-handed phonon signal input at the I port can be output at the O1 and O2 ports at the same time. However, the energy is shunt during the transmission process, so that the signal strength output by O1 and O2 is not as strong as that of the input port. Using p1 and p0 to, respectively, represent the presence and absence of chiral signals at each port, we can label the system as {p1, (p1, p1)}. Figure 4(a) further shows the average vibration intensity of the interface atoms for each port in this case. It can be seen that the O1 port is weaker than O2, and the corresponding O2 port can show a stronger output signal. Here, we have designed a three-port device that functions as a single-pole double-throw switch, and the corresponding chiral signals are listed in the left part of Fig. 4(b). For this three-port system, it has the characteristic of one-way conduction from left to right, and the reverse transmission is suppressed.

FIG. 4.

(a) Input and output signals of a three-port device. λ > 0 represents the signal transmission in the x direction when the magnetic field is along the z direction. λ < 0 represents signal transmission in the −x direction when the magnetic field is reversed. p1 (p0) indicates that there is (no) input or output signal at the corresponding port. ± indicates that the measured port signal is right/left-handed. (b) Vibration intensity of each input and output port of {p1,(p1,p1)} system and {(p1,p1),p1} system.

FIG. 4.

(a) Input and output signals of a three-port device. λ > 0 represents the signal transmission in the x direction when the magnetic field is along the z direction. λ < 0 represents signal transmission in the −x direction when the magnetic field is reversed. p1 (p0) indicates that there is (no) input or output signal at the corresponding port. ± indicates that the measured port signal is right/left-handed. (b) Vibration intensity of each input and output port of {p1,(p1,p1)} system and {(p1,p1),p1} system.

Close modal

The right panel of Fig. 3 shows the situation when the magnetic field is reversed. For this situation, the system becomes a device with two input ports and one output port, and having the characteristic of one-way conduction along the −x direction. We can control the opening and closing of the input port by changing the distribution of the magnetic field. It is worth noting that if both input ports are turned on at the same time and the interface mode of the 0/−1 supercell unit are excited at the I1 and I2 ports, the left-handed phonon is transmitted along the −x direction, the angular momentum output at the O port is doubled, and the chiral signal is strengthened. We label the system as {(p1, p1), p1)}, and an enhanced atomic vibration at the interface for the output port can be observed [see Fig. 4(a)]. The right part of Fig. 4(b) shows the chiral signals transmitted by the three-port device when the magnetic field is reversed. We can see that as long as one port has a left-handed signal input, there must be a left-handed signal output, and the logical OR operation is realized.

In this encoding system, its operating frequency range is theoretically consistent with the width of the bandgap, that is, 6.68 7.55 THz ( 0.0276 0.0312 eV), and the operating frequency bandwidth is 0.87 THz. Within this bandwidth range, the three-port system can exhibit the properties we have demonstrated, and the input and output signals will not be changed. Note that in order for system performance not to be affected by boundary effects, the size of the coding unit cannot be too small.

In summary, we have established a controllable encoding system using magnetic field and designed a three-port chiral phonon device. With eigen excitation at the input port, chiral information can be stably transmitted along topological interfaces and efficiently transformed at the corners of zigzag and armchair interfaces. By precisely controlling the distribution and direction of the magnetic field within the encoding system to manipulate the transmission path of the mode, we achieved the chiral phonon switching effect and the logical OR operation. This work enables the manipulation and application of phonon chirality, which is expected to introduce new mechanisms for the encoding, transmission, and operation of information.

This work was supported by the National Key Research and Development Program of China (No. 2023YFA1407001) and by the Department of Science and Technology of Jiangsu Province (No. BK20220032). X.L. acknowledges support from the Postgraduate Research and Practice Innovation Program of Jiangsu Province under Grant No. KYCX22_1544.

The authors have no conflicts to disclose.

Xiaozhe Li: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Resources (lead); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Yang Long: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Tingting Wang: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Yan Zhou: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Validation (equal); Writing – review & editing (equal). Lifa Zhang: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (lead); Resources (equal); Software (equal); Supervision (lead); Writing – review & editing (lead).

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

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