Topological Phononic Crystals with One-Way Elastic Edge Waves

Pai Wang,1 Ling Lu,2 and Katia Bertoldi1, 3, ∗

1School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA2Department of Physics, MIT, Cambridge, Massachusetts 02139, USA

3Kavli Institute, Harvard University, Cambridge, Massachusetts 02138, USA(Dated: 8th April 2015)

We report a new type of phononic crystals with topologically non-trivial bandgaps for both longit-udinal and transverse polarizations, resulting in protected one-way elastic edge waves. In our design,gyroscopic inertial effects are used to break the time-reversal symmetry and realize the phononicanalogue of the electronic quantum Hall effect. We investigate the response of both hexagonal andsquare gyroscopic lattices and observe bulk Chern number of 1 and 2, indicating that these struc-tures support single and multi-mode edge elastic waves immune to back-scattering. These robustone-way phononic waveguides could potentially lead to the design of a novel class of surface wavedevices that are widely used in electronics, telecommunication and acoustic imaging.

Topological states in electronic materials, includingthe quantum Hall effect [1] and topological insulat-ors [2, 3], have inspired a number of recent develop-ments in photonics [4, 5], phononics [6–10] and mechan-ical metamaterials [11–14]. In particular, in analogy tothe quantum anomalous Hall effect [15], one-way wave-guides in two-dimensional photonic systems have beenrealized by breaking time-reversal symmetry [16–18].Moreover, very recently unidirectional edge channelshave also been proposed for scalar acoustic waves byrotating fluids [19, 20] and for elastic waves by Coriolisforce in a non-inertial reference frame [21]. However,the latter system is very difficult to implement in solidstate devices.

Here, we introduce gyroscopic phononic crystals,where each lattice site is coupled with a spinning gyro-scope that breaks time-reversal symmetry in a well-controlled manner. In both hexagonal and square lat-tices, gyroscopic coupling opens bandgaps character-ized by Chern numbers of 1 and 2. As a result, atthe edge of these lattices both single-mode and multi-mode one-way elastic waves are observed to propagatearound arbitrary defects without backscattering.

To start, we consider a hexagonal phononic crys-tal with equal masses (m2 = m1) connected by lin-ear springs (green and black rods in Figs. 1a, 1band 1c). The resulting unit cell has four degrees offreedom specified by the displacements of m1 and m2

(U = [um1x , um1

y , um2x , um2

y ]). Consequently, there area total of four bands in the band structure (Fig. 1e).Note that this is the minimal number of bands requiredto open a complete bandgap, since the first two elasticdispersions are pinned at zero frequency. The phononicband structures are calculated by solving the dispersionequation [22] [

K(µ)− ω2M]U = 0 (1)

for wave vectors µ within the first Brillioun zone. Here,ω denotes the angular frequency of the propagatingwave and M = diag{m1, m1, m2, m2} is the massmatrix. Moreover, K is the 4×4 stiffness matrix as a

function of the Bloch wave vector µ. The band struc-ture of this simple lattice is shown in Fig. 1e and hasa quadratic degeneracy between the third and fourthbands at the center of the Brillouin zone and a com-plete gap between the second and third bands due tothe lack of inversion symmetry. However, this gap istopologically trivial, since time-reversal symmetry isnot broken and the Chern numbers of the bands areall zero.

In order to break time-reversal symmetry, we intro-duce gyroscopic coupling [23–25] and attach each massin the lattice to the tip of the rotational axis of a gyro-scope, as shown in Fig. 1d. Note that the other tipof the gyroscope is pinned to the ground to preventany translational motion, while allowing for free rota-tions. Because of the small-amplitude in-plane wavespropagating in the phononic lattice, the magnitude ofthe tip displacement of the each gyroscope is given by

Utip = h sin θ ≈ hθ = hΘeiωt for |Θ| � 1, (2)

where h and θ denote the height and nutation angleof the gyroscope (Fig. 1b) and Θ is the amplitude ofthe harmonic change in θ. Interestingly, the couplingbetween the masses in the lattices and the gyroscopesinduces an in-plane gyroscopic inertial force perpendic-ular to the direction of Utip [24, 26, 27]:

Fg = ±iω2αUtip, (3)

where α is the spinner constant that characterizes thestrength of the rotational coupling between two inde-pendent inertias in the 2D plane. As a result, the massmatrix in Eq. (1) becomes,

M = M +

0 iα1 0 0−iα1 0 0 0

0 0 0 iα2

0 0 −iα2 0

, (4)

where α1 and α2 denote the spinner constants of thegyroscopes attached to m1 and m2, respectively. We

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Figure 1: Ordinary and Gyroscopic Phononic Crys-tals: (a) Schematic of the hexagonal lattice. The purpleand grey sphere represent concentrated masses m1 and m2,respectively. The green and black straight rods representmass-less linear springs with stiffness k1 and k2 = k1/20,respectively. The dashed cell is the primitive cell of the lat-tice. (b) Schematic of a gyroscope with the top tip pinnedto a mass in the lattice. (c) Unit cell for the ordinary(non-gyroscopic) phononic crystal. (d) Unit cell for thegyroscopic phononic crystal. (e) Band structure of the or-dinary (non-gyroscopic) phononic crystal (α1 = α2 = 0).The inset is the Brillouin zone. (f) Band structure of thegyroscopic phononic crystal (α1 = α2 = 0.3m1) with theChern numbers labeled on the bulk bands. The frequenciesare normalized by ω0 =

√k1/m1.

note that the imaginary nature of the gyroscopic iner-tial effect indicates directional phase shifts with respectto the tip displacements, which breaks time-reversalsymmetry.

We now consider a gyroscopic hexagonal lattice withα2 = α1 = 0.3m1 and show its band structure in Fig.1f. Comparing this to the original band structure re-ported in Fig. 1e, we observe that the original quad-ratic degeneracy, between the third and fourth bands,is opened into a full band gap. Moreover, we alsofind that the original bandgap between the second andthird bands first closes and then reopens as we gradu-ally increase the magnitude of α1 and α2. In partic-ular, for α1 = α2 = 0.07m1 the gap is closed and apair of Dirac cones at K points emerges (see Fig. S1

Figure 2: Edge modes in Gyroscopic Phononic Crys-tal: (a) 1D band structure showing bulk bands (black dots)and edge bands (colored lines). Red solid lines representedge modes bound to the top boundary, while blue dashedlines represent edge modes bound to the bottom bound-ary. (b) Modal displacement fields of top edge states withnegative group velocities. (c) Modal displacement fields ofbottom edge states with positive group velocities

for details). Importantly, for α2 = α1 = 0.3m1 (Fig.1f) both bandgaps are topologically-nontrivial as high-lighted by the non-zero Chern numbers labeled on thebands (the calculations conducted to compute these to-pological invariants are detailed in the SupplementaryMaterials [27]). Therefore, in the frequency ranges ofthese nontrivial bandgaps, we expect gapless one-wayedge states, whose number is dictated by the sum ofChern numbers below the bandgap.

To verify the existence of such one-way edge states,we perform one-dimensional (1D) Bloch wave analyseson a supercell comprising 20 × 1 unit cells, assumingfree boundary conditions for the top and bottom edges.In full agreement with the bulk Chern numbers, theband structure of the supercell shows one one-wayedge mode on each edge in both bandgaps. For modesbound to the top edge (Fig. 2b), the propagationcan only assume negative group velocities (red solidlines with negative slope in Fig. 2a). On the otherhand, the modes bound to the bottom edge (Fig. 2c)possess positive group velocities (blue dashed lineswith positive slope in Fig. 2a). Since these edge modesare in the gap frequency range where no bulk modesmay exist, they cannot scatter into the bulk of thephononic crystal. Furthermore, their uni-directionalgroup velocities guarantee the absence of any backscattering and result in the topologically protectedone-way propagation of vibration energy.

To show the robustness of these edge states, we con-duct transient analysis on a finite sample comprising20×20 unit cells with a line defect on the right bound-ary created by removing twelve masses and the springsconnected to them (Fig. 3a). A harmonic force excit-ation, F0e−iωt, is prescribed at a mass site on the bot-tom boundary (red arrow in Fig. 3a) with frequencywithin the bulk bandgap between the second and third

3

t = 2T0

t = 12T0

t = 22T0

t = 32T0

Figure 3: Transient Response of a gyroscopic phononiccrystal consisting of 20 × 20 unit cells with a line defect onthe right boundary: Snapshots of the displacement field at(a) t = 2T0, (b) t = 12T0, (c) t = 22T0 and (d) t = 32T0,

where T0 =√m1/k1 is the characteristic time scale of the

system. Starting from t = 0, a time-harmonic excitationforce F(t) = [Fx(t), Fy(t)] = [1, 1]F0e

−iωt is prescribed atthe site indicated by the red arrow.

bands (ω/ω0 = 1.3). In Fig. 3 we plot snapshots ofthe velocity field at different time instances, t/T0 = 2,12, 22 and 32, where T0 =

√m1/k1 is the character-

istic time scale of the system. Remarkably, because oftheir topological protection, the edge modes circum-vent both the sharp corner and the line defect withoutany reflection.

Next, we investigate the effect of the lattice geometryand start with an ordinary square phononic crystalwith masses m1 connected by springs with elastic con-stant k1. To make the lattice statically stable, we addan additional mass m2 = m1 at the center of each unitcell and connect it to its four adjacent m1 masses bysprings with elastic constant k2 = 2k1 (See Figs. 4aand 4b). The band structure for this lattice (shown inFig. 4d) contains a pair of three-fold linear degeneracyamong the first, second and third bands at theX pointsof the Brillouin zone. Note that this type of degener-acy, consisting of a locally flat band and a Dirac point,is known as the “accidental Dirac point” [28]. Interest-ingly, while previously this degeneracy has been foundto be very sensitive to the system parameters [28], inour lattice it robustly appears at the X points whenm1 = m2. Upon the introduction of gyroscopic inertialeffects (α2 = α1 = 0.3m1), these three-fold degener-ate points are lifted and a gap is created between thesecond and third bands. The Chern numbers of thetwo bulk bands below the gap is two, predicting theexistence of two topological edge states. In Figs. 4fand 4g, we plot the band structure of the correspond-ing 20×1 supercell, highlighting the two one-way edge

Figure 4: Square Lattice results: (a) Schematic of thesquare lattice. The purple and grey sphere represent con-centrated masses m1 and m2, respectively. The black andgreen straight rods represent mass-less linear springs withstiffness k1 and k2 = 2k1, respectively. (b) Unit cellfor the ordinary (non-gyroscopic) phononic crystal. (c)Unit cell for the gyroscopic phononic crystal. (d) Bandstructure of the ordinary (non-gyroscopic) phononic crys-tal (α1 = α2 = 0). The insets is the Brillouin zone. (e)Band structure of the gyroscopic phononic crystal (α1 =α2 = 0.3m1) with the Chern numbers labeled on the bulk

bands. Frequencies are normalized by ω0 =√k1/m1. (f)

1D band structure showing bulk bands (black dots) and to-pological edge bands (red solid lines for edge modes boundto the top boundary. Note that the edge bands that arebound to the bottom boundary are not shown here. (g)Modal displacement fields of the edge states shown in (f).

modes and their modal displacement fields.

To summarize, we demonstrated that gyroscopicphononic crystals can support topologically non-trivialgaps, within which the edge states are unidirectionaland immune to back-scattering. Moreover, transientanalysis confirmed that the propagation of such topo-logical edge waves is robust against large defect andsharp corners. We note that phononic crystals [29, 30]

4

and acoustic metamaterials [31–35] that enable manip-ulation and control of elastic waves have received signi-ficant interest in recent years [22, 36], not only becauseof their rich physics, but also for their broad rangeof applications [37–49]. Interestingly, the edge wavemodes in phononic crystals are important in manyscenarios [50–52], including vibration control [53] andacoustic imaging [45]. However, most of reported stud-ies have focused on topologically trivial surface wavesthat can be easily scattered or localized by defects [50].Therefore, the work reported here could open new av-enues for the design of phononic devices with specialproperties and functionalities on edges, surfaces andinterfaces.

This work has been supported by NSF throughgrants CMMI-1120724, CMMI-1149456 (CAREER)and the Materials Research Science and EngineeringCenter (DMR-1420570). K.B. acknowledges start-upfunds from the Harvard School of Engineering and Ap-plied Sciences and the support of the Kavli Instituteand Wyss Institute at Harvard University. L.L. wassupported in part by U.S.A.R.O. through the ISN un-der Contract No. W911NF-13-D-0001, in part by theMRSEC Program of the NSF under Award No. DMR-1419807 and in part by the MIT S3TEC EFRC ofDOE under Grant No. de-sc0001299. The authors arealso grateful to Scott Skirlo, Timothy H. Hsieh andDr. Filippo Casadei for ispirational discussions and toDr. Farhad Javid and Xin You for their support withgraphics.

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