Three-Dimensional Topological States of Phonons with ......2019/06/08 · Research 10 10 0 0 −10...

Research ArticleThree-Dimensional Topological States of Phonons with TunablePseudospin Physics

Yizhou Liu123 Yong Xu 123 and Wenhui Duan124

1State Key Laboratory of Low Dimensional Quantum Physics Department of Physics Tsinghua University Beijing 100084 China2Collaborative Innovation Center of Quantum Matter Beijing 100084 China3RIKEN Center for Emergent Matter Science (CEMS) Wako Saitama 351-0198 Japan4Institute for Advanced Study Tsinghua University Beijing 100084 China

Correspondence should be addressed to Yong Xu yongxumailtsinghuaeducn

Received 8 June 2019 Accepted 5 July 2019 Published 31 July 2019

Copyright copy 2019 Yizhou Liu et al Exclusive Licensee Science and Technology Review Publishing House Distributed under aCreative Commons Attribution License (CC BY 40)

Efficient control of phonons is crucial to energy-information technology but limited by the lacking of tunable degrees of freedomlike charge or spin Here we suggest to utilize crystalline symmetry-protected pseudospins as new quantum degrees of freedom tomanipulate phonons Remarkably we reveal a duality between phonon pseudospins and electron spins by presenting Kramers-like degeneracy and pseudospin counterparts of spin-orbit coupling which lays the foundation for ldquopseudospin phononicsrdquoFurthermore we report two types of three-dimensional phononic topological insulators which give topologically protected gaplesssurface states with linear and quadratic band degeneracies respectively These topological surface states display unconventionalphonon transport behaviors attributed to the unique pseudospin-momentum locking which are useful for phononic circuitstransistors antennas etc The emerging pseudospin physics offers new opportunities to develop future phononics

Recently intensive research effort has been devoted to findingnovel topological states of phonons including the quan-tum anomalous Hall-like [1ndash15] and quantum spin Hall-likestates [16ndash21] These new quantum states of phonons arecharacterized by topologically protected gapless boundarymodes within the bulk gap which are useful for variousapplications like high-efficiency phononic circuitsdiodesand offer new paradigms for future phononics [12ndash14]However experimental realization of two-dimensional (2D)topological states is quite challenging for phonon systemsSpecifically the quantum anomalous Hall-like states requirebreaking time reversal symmetry of phonons which remainsexperimentally illusive The quantum spin Hall-like statesrely on the pseudospin degeneracy protected by crystallinesymmetries that typically get broken at the one-dimensional(1D) edges [21] In contrast crystalline symmetries of three-dimensional (3D) systems can preserve simultaneously inthe bulk and on the surface enabling the topological pro-tection Importantly most solid materials are crystalized in3D lattices Nevertheless despite a few preliminary workson phononic topological semimetals [22ndash25] 3D phononictopological insulators (TIs) have rarely been reported before

as far as we knowThis is possibly because the spin (Kramers)degeneracy and spin-orbit coupling (SOC) which are essen-tial to TIs are natively missing for phonons On the otherhand phonons are elementary excitations of lattice vibrationswith zero charge and spin The lacking of tunable degreesof freedom considerably limits their device applications Inthis context it is of critical importance to develop newquantum degrees of freedom for phonons In light of thegreat success of spintronics future research of phononicswould be greatly enriched if one could establish any cor-relations between phonon pseudospins and electron spins[26]

In this Article we provided a guiding principle todesign 3D phononic TIs as well as topological semimetalsby utilizing crystalline symmetry-protected pseudospinscharacterized by Kramers-like degeneracy quantizedpseudoangular momenta and nonzero Berry curvatureRemarkably we revealed pseudospin counterparts ofthe intrinsic and Rashba-Dresselhaus SOC namely thepseudo-SOC of phonons which builds a duality betweenphonon pseudospins and electron spins The duality featureenables exploring the physics and applications of phonon

AAASResearchVolume 2019 Article ID 5173580 8 pageshttpsdoiorg103413320195173580

2 Research

pseudospins by borrowing ideas from spintronics thusopening new opportunities for ldquopseudospin phononicsrdquo

1 Design Principle of Phononic TIs

An essential requirement of TIs is band degeneracies atno less than two high symmetry momenta (HSM) in theboundary Brillouin zone (BZ) [27ndash29] The requirement issatisfied for electrons with spin degeneracies protected bytime reversal symmetry However phonons do not have realspins invoking different strategies for building phononic TIsNaturally one could apply crystalline symmetries that areprevalent in solid materials to realize Kramers-like degen-eracies Such symmetries should also be preserved whenprojected onto the surface However for 2D spinless casesno such kind of crystalline symmetry has higher than 1Dirreducible representations at more than one HSM in the1D edge BZ implying that 2D phononic TIs protected bycrystalline symmetries are forbidden [21] The constraint isreleased for a variety of 3D lattices where multiple banddegenerate HSM can exist in the 2D surface BZ [28ndash31]Thusthe construction of 3D phononic TIs is feasible in principle

The Hamiltonian of phonons resembles a tight-bindingHamiltonian of spinless electrons with fixed 119901119909119910119911 orbitals[13] The symmetry representation of phonons is Γphonon =Γequiv otimes Γvector where Γequiv is the equivalence representationof atomic sites and Γvector is the representation of a 3D polarvector [32] Here we will not thoroughly discuss all possible3D crystalline symmetries but focus on 119862119899V (119899 = 3 4 6)symmetries that show interesting topological physics forelectrons [30 31] Take1198626V lattices as an example Γequiv is a119873dimensional representation where119873 is the number of atomicsites in a unit cell When all the atomic sites are inequivalentΓequiv is decomposed into 119873 1D irreducible representations1198601 119873 ge 2 is necessary to build a phononic TI that requiresat least four bands For simplicity we choose 119873 = 2 asdisplayed in Figure 1(a) which gives Γequiv = 21198601 In themomentum space the high-symmetry lines Γ-119860 and 119870-119867in the bulk BZ are projected onto the HSM Γ and 119870 in thesurface BZ [Figure 1(a)] which have1198626V and1198623V symmetriesrespectively Under 1198626(3)V symmetry Γvector = 1198601 oplus 119864 where1198601 and 119864 are 1D and 2D irreducible representations for basisfunctions of 119901119911 and 119901119909plusmn119894119901119910 respectivelyTherefore Γphonon =21198601 oplus 2119864 implying that 119901119909 plusmn 119894119901119910 (119901119911) are doubly degenerate(nondegenerate) along the high-symmetry lines Note thatfrequencies of in-plane vibrations (119901119909 plusmn 119894119901119910) are typicallyhigher than those of out-of-plane ones (119901119911) leading to weakhybridization between the doublet and singlet statesWe thusfocus on in-plane vibrations [33] for which aZ2 classificationof band topology is permitted [30]

2 Pseudospin- and Topology-Related Physics

Considering that the double degeneracy of 119901119909plusmn119894119901119910 resemblesthe Kramers degeneracy of spins we introduce a pseudospinindex to label the Kramers-like states There exist two typesof phonon modes including in-phase (I) and out-of-phase(O) vibrations between the two atomic sites A and B Their

pseudospin states [Figure 1(b)] are defined in coordinates of(119909119860 119910119860 119909119861 119910119861)119879 as10038161003816100381610038161003816Iuarr⟩ = (120576+ sin 120579k 120576+ cos 120579k)119879 10038161003816100381610038161003816Ouarr⟩ = (120576minus cos 120579k minus120576minus sin 120579k)119879 10038161003816100381610038161003816Idarr⟩ = 119894 (120576minus sin 120579k 120576minus cos 120579k)119879 10038161003816100381610038161003816Odarr⟩ = 119894 (minus120576+ cos 120579k 120576+ sin 120579k)119879 (1)

where 120576plusmn = (1 plusmn119894)radic2 and the vibrational magnitude ofeach atomic site is determined by 120579k isin [0 1205872] that is k-dependent They are orthonormal and form a complete basisof in-plane vibrations

The pseudospin states are featured by well-defined quan-tized pseudoangular momenta about 119911-axis for k along thehigh-symmetry lines due to the 1198626(3) rotational symmetryThe pseudoangularmomentumoperator is expressed as 119869ph =120590119911119904119911 [33] where 120590119911 and 119904119911 are Pauli matrices with 120590119911 = plusmn1and 119904119911 = plusmn1 refer to pseudospin up (down) and I (O)vibrational modes respectively The phonon pseudoangularmomentum 119895ph is composed of two parts including a localpart determined by the on-site orbital and a nonlocal partcontributed by the inter-site Bloch phase change [34] Hereinthe nonlocal part is zero and 119895ph is fully determined by thelocal part giving 119895ph = 1 for |Iuarr⟩ and |Odarr⟩ and 119895ph = minus1 for|Idarr⟩ and |Ouarr⟩ as depicted in Figure 1(b)

The symmetry-adapted 119896sdot119901methodwas applied to deriveeffective Hamiltonians of phonons near HSM by using thebasis set |Iuarr⟩ |Ouarr⟩ |Idarr⟩ |Odarr⟩ [33] Specifically the BlochHamiltonian matrix 119867(k) was first expanded by the lowestorders of k (referenced to HSM) Then by writing down thecorrespondingmatrices of symmetry operators and requiring119867(k) commute with them some matrix elements of 119867(k)are enforced to be zero The remaining nonzero part givesthe effective Hamiltonian The HSM are classified into type-I(1198701198701015840 and1198671198671015840) and type-II (Γ and119860) which have 1198623V and1198626V symmetries respectively andwill be discussed separatelyImportantly the conservation of pseudoangular momentumrequires that basis states of the same 119895ph (eg |Iuarr⟩ and |Odarr⟩)are coupled only by terms like 119896119911 1198962119909 + 1198962119910 and 1198962119911 whilebasis states of different 119895ph (eg |Iuarr⟩ and |Ouarr⟩) are coupledonly by terms like 119896plusmn and 1198962plusmn The effective Hamiltonian nearthe type-I HSM is written as 119867 = 1198670 + 119867119868 + 119867119877119863 where1198670 = diag(119872k minus119872k119872k minus119872k)

119867119868 = ( 0 Δ 1119896+ 0 Δ 2119896119911Δ 1119896minus 0 Δ 2119896119911 00 Δ 2119896119911 0 minusΔ 1119896minusΔ 2119896119911 0 minusΔ 1119896+ 0 )119867119877119863 = ( 0 0 minus1198941198621119896+ 00 0 0 1198941198622119896minus1198941198621119896minus 0 0 00 minus1198941198622119896+ 0 0 )

(2)

Research 3

Γ

Γ Γ

M K

(a)

A

KM

R

H

(b)

a

A

B

(c) (d)

I

O

(e)

c2

c1

Γ uarr⟩

）uarr

darr

）darr

jＪＢ = minus1

jＪＢ = minus1

jＪＢ = +1

jＪＢ = +1

8

6

4

2

0KM A HR R

66

66

63

60

69

64

62

Γ M ＋

＋

⟩ ⟩

⟩

Figure 1 (a) Atomic structure (left) and Brillouin zone (right) of a 3D triangular lattice with two sublattices A and B Bond lengths aredenoted by lattice parameters 119886 1198881 and 1198882 (b) Schematic pseudospin states of in-phase ldquoIrdquo (out-of-phase ldquoOrdquo) vibrational modes whosepseudoangular momentum 119895ph is labelled (c) Dispersion curves of a type-I phononic TI Blue (red) color is used to denote the contributionof I (O) vibrational modes (d) Local density of states (LDOS) of the (001) surface where higher (lower) LDOS are colored red (blue) (e)Schematic phonon dispersion and pseudospin textures in the 119896119911 = 0 plane near119870 The bottom panel displays pseudospin textures of O (left)and I (right) vibrational modes from the top view

119872k = 119872 minus 1198611(1198962119909 + 1198962119910) minus 11986121198962119911 119896plusmn = 119896119909 plusmn 119894119896119910 Thecurvature parameters 1198611 and 1198612 typically have the samesign

Remarkably this effective Hamiltonian resembles the 3DBernevig-Hughes-Zhang (BHZ) model with broken inver-sion symmetry for electrons 1198670 + 119867119868 is exactly the same asthe typical 3D BHZ model [35ndash37] where 119867119868 is attributedto the intrinsic SOC For electrons119867119877119863 arises in conditionsof broken inversion symmetry which includes plusmn1198941198622119896∓ and∓1198941198621119896plusmn terms corresponding to the Rashba (119896119910120590119909 minus 119896119909120590119910)and Dresselhaus (119896119910120590119909 +119896119909120590119910) SOC respectively In analogy119867119868 and119867119877119863 are called the intrinsic and Rashba-Dresselhauspseudo-SOC for phonon pseudospins In this sense phononsand electrons share essentially the sameHamiltonian thoughtheir underlying physics is distinctly different This impliesa ldquodualityrdquo between phonon pseudospins and electron spinsTherefore we can study the topological and quantum physicsof phonon pseudospins by borrowing ideas from spintronicsThis key result could lay the foundation for an emerging fieldof ldquopseudospin phononicsrdquo

Then we will discuss phononic topological properties bythe effectiveHamiltonianWhen excluding119867119877119863 and selecting119896119911 = 0 119867 reduces to the 2D BHZ model which givesquantized pseudospin-resolved Chern numbers Cuarr(darr) [38]

As type-I HSM exist in pairs Cuarr(darr) = plusmn2 when a bandinversion occurs (ie 1198721198611 gt 0) The sum of C119904 = (Cuarr minusCdarr)4 contributed by all HSM mod 2 gives a topologicalinvariant Z2 The 3D phononic TI phase is characterizedby Z2 = 1 The inclusion of 119867119877119863 introduces intrabandcoupling between opposite pseudospins which removes thepseudospin degeneracy except at the HSM Then Cuarr(darr) getsill defined but theZ2 topological classification remains valid[30]TheZ2 topological invariant will not be affected by119867119877119863as far as the bulk band gap keeps open when adiabaticallyturning on119867119877119863

To demonstrate the nontrivial topological states weexplicitly studied lattice vibrations in a 1198626V lattice [Fig-ure 1(a)] The interatomic interactions between the nearestand next-nearest neighbors were described by longitudinaland transverse force constants as done previously [34] Theout-of-plane vibrations typically have minor influence ontopological properties of in-plane vibrations [33] which arethus neglected for simplicity We systematically searched thewhole space of interatomic coupling parameters and foundthat the required band inversions can be obtained by a widerange of coupling parameters Details of calculation methodsand parameters related to the following discussions weredescribed in Supplemental Material [33]

4 Research

(b) (c)(a)

9

6

6

7

8

3

0

70

65

KM A HR R

Γ M Γ ＋－－

Figure 2 (a) Dispersion curves of a type-II phononic TI Blue (red) color is used to denote the contribution of I (O) vibrational modes (b)LDOS of the (001) surface where higher (lower) LDOS are colored red (blue) (c) Schematic phonon dispersion and pseudospin textures inthe 119896119911 = 0 plane near 119860 The right panel displays pseudospin textures of O (top) and I (bottom) vibrational modes from the top view

Figure 1(c) presents phonon dispersion curves with aband inversion between I and O vibrational modes at119870 Thisband inversion leads to a nontrivial band topology Z2 = 1as confirmed by our calculations of hybrid Wannier centers[39 40] that display partner switching between Kramers-likepairs [33] Moreover there is a frequency gap between thetwo kinds of bands The system is thus a 3D phononic TI Ahallmark of phononic TIs is the existence of gapless surfacestate within the bulk gap which is topologically protectedwhen the corresponding symmetry is preserved On the(001) surface where the 1198626V symmetry preserves we indeedobserved a single pair of gapless Dirac-cone-shaped surfacebands located near 119870 and 1198701015840 [Figure 1(d)] as warranted bythe bulk-boundary correspondence [41]

Figure 1(e) displays schematic pseudospin textures of bulkbands in the 119896119911 = 0 planeThere would be no net pseudospinpolarization if excluding the Rashba-Dresselhause pseudo-SOC interaction 119867119877119863 Interestingly when including 119867119877119863Rashba-like and Dresshause-like pseudospin textures evolvein the O and I bands respectively We further consideredthe topological surface states (TSSs) near the Dirac pointwhich are described by the effective Hamiltonian (referencedto the Dirac frequency) 119867surf = V119863(119896119909120590119909120591119911 + 119896119910120590119910) where120591119911 = plusmn1 refers to 119870 (1198701015840) valley V119863 is the group velocity atthe Dirac point Noticeably119867surf of each valley has the sameform as for TSSs of electrons [35] whose pseudospin texturesare schematically displayed in Figure 3(a) By adiabaticallyvarying k along the loop enclosing 119870 (1198701015840) the pseudospinvectors wind plusmn1 times giving quantized Berry phases of plusmn120587The similarity between phonon pseudospins and electronspins is thus well demonstrated for both bulk and surfacebands

3 Type-II Phononic TIs

Type-II phononic TIs are characterized by the existence ofband inversions at type-II HSM (Γ or 119860) Importantly dueto the 1198626 rotation symmetry all the linear terms of 119896plusmn areforbidden in the effective Hamiltonian near type-II HSMwhich is expressed as follows [33]

1198671015840 = 1198670 + 1198671015840119868 + 1198671015840119877119863= ( 119872k 12057511198962minus minus11989411988811198962minus 120575211989611991112057511198962+ minus119872k 1205752119896119911 11989411988821198962+11989411988811198962+ 1205752119896119911 119872k minus12057511198962+1205752119896119911 minus11989411988821198962minus minus12057511198962minus minus119872k

) (3)

Anewkindof intrinsic andRashba-Dresselhaus pseudo-SOC(1198671015840119868 and1198671015840119877119863) is thus introduced

Figure 2(a) presents dispersion curves of a type-IIphononic TI which is characterized by a band inversion at119860a finite frequency gap and a nontrivial band topologyZ2 = 1verified by the calculations of hybrid Wannier centers [33]We also calculated the surface states of the (001) termination[Figure 2(b)] which shows a quadratic band crossing at Γin contrast to a pair of linear band crossings at type-I HSMThe type-II TSSs are described by the effective Hamiltonian(referenced to the degenerate frequency) 1198671015840surf = 119863[(1198962119909 minus1198962119910)120590119909 minus 2119896119909119896119910120590119910] where 119863 is a coefficient Pseudospintextures of bulk bands [Figure 2(c)] are neither typicalRashba-like nor typical Dresselhaus-like but display newpseudospin-momentum locked features Pseudospin texturesof type-II TSSs are significantly different from those of type-I TSSs [Figure 3(a)] which are characterized by windingnumbers of plusmn2 and quantized Berry phases of plusmn21205874 Pseudospin-Momentum LockedPhonon Transport

One prominent feature caused by the pseudo-SOC 119867119877119863 or1198671015840119877119863 is that the pseudospin and momentum of phonons arelocked which plays an important role in determining trans-port properties Moreover type-I and type-II bulksurfacebands are featured by different kinds of pseudospin-momentum locking leading to distinct transport behaviorsTo demonstrate the concept we simulated phonon transportof TSSs in tunneling junctions [33] where a tunnelingbarrier is introduced into the gate region as schematicallydisplayed in Figure 3(a) The height of tunneling barrierin principle could be controlled by using a piezoelectric

Research 5

20 40 6000

0

05

10

source probegate

Transm

ission

(a) (c)(b)

Transm

ission

La

Type-I

Type-II

Normal

x

z y

1

0

1

60∘90∘

-90∘-60∘

-30∘

30∘

0∘

0Δ＄

Figure 3 (a) Schematic phonon tunneling junction where surface lattice vibrations of frequency 1205960 are excited (detected) by a point-likesource (probe) and a tunneling barrierΔ120596119863 is applied by a piezoelectric gate Schematic phonon dispersion and pseudospin textures of type-I(type-II) TSSs are displayed in the top (bottom) panel (b) Phonon transmission as a function of incident angle 120579 for type-I TSSs (red dash-dotted) type-II TSSs (purple solid) and normal surface states (black dashed) for which we used barrier widths of = 132119886 295119886 and 132119886respectively 119886 is the length parameter depicted in Figure 1(a) 120579 = 0 is defined along the 119909-axis (c) Phonon transmission as a function of 119871for 120579 = 0 1205960 = 01 and Δ120596119863 = 03 were usedgate that changes the on-site potential strain or interatomiccoupling of surface atoms [26] Figure 3(b) shows phonontransmissions as a function of incident angle 120579 for type-I and type-II TSSs Normal surface states with a linearband dispersion and no pseudospin-momentum lockingwere also studied for comparison Noticeably there existsresonant phonon transport (transmission equal to 1) alongsome particular directions which satisfy the constructiveinterference condition of Fabry-Perot resonances 119902119909119871 = 119899120587where 119902119909 is the 119909-component of wavevector in the gate region(Supplemental Figure S3) 119871 is the barrier width and 119899 is aninteger number Moreover phonon transmission of normalsurface states oscillates with varying 119871 which is also expectedby the Fabry-Perot physics

However substantially different results are found fortype-I and type-II TSSs Take 120579 = 0 for example 120579 = 0corresponds to transport along 119909-axis Transport of type-ITSSs keeps ballistic while phonon transmission of type-IITSSs decays exponentially with increasing 119871 [Figure 3(c)]These unconventional transport behaviors are insensitive tomaterial details indicating a topological origin Importantlythese physical effects can be well understood by pseudospinphysics Specifically for type-I (type-II) TSSs the forward-moving phononmodes have the same (opposite) pseudospinsbetween the upper and lower Dirac cones which correspondto transport channels of the sourceprobe and gate respec-tively [Figure 3(a)] Because of the perfectly matched (mis-matched) pseudospins phonons are able to transport acrossthe barrier with no (full) backscattering These quantumtransport phenomena are inherently related to the quantizedBerry phase 120587 (2120587) of type-I (type-II) TSSs which inducesdestructive (constructive) interferences between incident andbackscattered waves

Our findings suggest some potential applications of TSSsFor instance the excellent transport ability of type-I TSSs canbe utilized for low-dissipation phononic devicesThe stronglyangle-dependent transmission of type-II TSSs which areconfined to transport along some specific directions can beused to design directional phononic antennaMoreover type-II TSSs are promising for building efficient phononic tran-sistors because their phonon conduction in the tunnelingjunction can be switched off (on) by a finite (zero) barrier

5 Phononic Topological Semimetals

In addition to phononic TIs topological band inversions canintroduce other novel topological phases including topolog-ical nodal-ring semimetals and topological Weyl semimetalswhich are collectively called phononic topological semimet-als The essential physics is illustrated in Figure 4(a) Let usstart from 119867 = 1198670 and take two pairs of bands invertedat 119870 for example Generally the pseudospin degeneracy issplit by 119867119877119863 and a full band gap is induced by 119867119868 leadingto phononic TIs However phononic topological semimetalswould emerge if the opening of band gap was forbidden bysymmetryThis is possible here providing that band splittingsof 119867119877119863 have opposite signs between I and O vibrationalmodes (ie 11986211198622 gt 0) Then when in the presence of mirrorsymmetry 119872119911 (with the same atoms at A and B sites) thecrossing rings in the 119896119911 = 0 plane are protected to be gaplessintroducing topological nodal-ring semimetals In contrastunder broken 119872119911 the nodal rings are gapped out except forsome gapless points that are protected to exist in the Γ-119870line by mirror symmetry 119872119909 These gapless points are Weylpoints and the resulting phase is topologicalWeyl semimetal

6 Research

10

10

0

0minus10

minus10

(a) (b)

TSM TI

TSMTI

(c)

(00)

(d) (e)

C1C2lt0

C1C2gt0

H0+HRD H0+HRD+HI

H0

C1Δ1

C2Δ

1

()

(0)

8

6

68

68

64

64

60

60

2

4

0KM A AHR

Γ Γ

＋

＋

－

－

Γ

ΓΓ

Figure 4 (a) Schematic evolution of I (blue) and O (red) vibrational bands near 119870 under the influence of intrinsic pseudo-SOC 119867119868 andRashba-Dresselhause pseudo-SOC 119867119877119863 The 119867119877119863-induced pseudospin splittings of I and O vibrational bands have same (opposite) signswhen 11986211198622 lt 0 (11986211198622 gt 0) which can result in phononic TIs or topological semimetals (TSMs) as summarized in the topological phasediagram (b) (c) Dispersion curves of a phononic topological nodal-ring semimetal Blue (red) color is used to denote the contribution of I(O) vibrational modes (d) Zak phases along the (001) direction of the lowest two bands for the 2D surface BZ (e) LDOS of the (001) surfacewhere higher (lower) LDOS are colored red (blue)

A unified description of these topological phases is providedby the phase diagram presented in Figure 4(b)

Figure 4(c) presents dispersion curves of a phononictopological nodal-ring semimetal protected by119872119911 for whichequivalentA andB siteswere selected [33]The gapless featurein the 119896119911 = 0 plane is consistent with the symmetry analysisMoreover we calculated the Zak phase 120579Zak along the (001)direction for the 2D surface BZ where 120579Zak has quantizedvalues of 0 or 120587 [Figure 4(d)] The nodal rings appear atboundaries between regions of 120579Zak = 0 and 120579Zak = 120587implying their topological nature Furthermore phononicdrumhead surface states were observed by our surface-state calculations [Figure 4(e)] which is a hallmark featureof topological nodal-ring semimetals Their evolution withvarying phonon frequencies are displayed in SupplementalFigure S5

6 Guiding Principles for SearchingCandidate Materials

We would like to provide guiding principles for searchingcandidate materials by first-principles calculations (i) For afamily of materials having a specified crystalline symmetryuse group theory to determine band degeneracies at highsymmetry momenta (HSM) Topological phononic stateswould be allowed if having band degeneracies at more than

one HSM (ii) For a specific material calculate the phonondispersion and eigenstates at HSM Sort degenerate bandsat each HSM according to their phonon frequencies Thencheck whether the order of degenerate bands with differentirreducible representations varies between different HSMIf so a topological band inversion usually exists (iii) Forthe material with a topological band inversion calculate thephonon dispersion in the whole momentum space Thuswhether it is a topological insulator or semimetal can bedefined (iv) Determine the topological nature explicitly bycomputing hybrid Wannier centers and topological surfacestatesThe simple guiding principles could be applied for highthroughput discovery of phononic topological materials

7 Experimental Signatures of TopologicalPhononic Materials

Hallmarks of a topological phononic material include (i)the existence of topological band inversion for bulk phononmodes at HSM and (ii) the existence of topologically pro-tected gapless phononic states on the boundary One canuse infrared spectroscopy Raman techniques or inelasticneutron scattering to detect bulk phonon modes at HSMThe measured band degeneracy and band order at HSMcould be used to determine topological band inversionBesides one can use angle-resolved electron energy loss

Research 7

spectroscopy that is a surface-sensitive technique to measurephonon dispersion of surfaces Thus topological boundarystates of phonons can be directly probedThese experimentalsignatures in combination with first-principles calculationscould be used to determine topological nature of specificmaterials

8 Outlook

Our work sheds lights on future study of topological phonon-ics A few promising research directions are opened (i) Tosearch for new symmetry-protected phononic topologicalstates In addition to 119862119899V there are many other symmetrieslike (magnetic) space group symmetry time reversal sym-metry particle-hole symmetry etc and their combinationswhich could result in rich topological phases [28 42] (ii) Toexplore novel physical effects by breaking symmetry locallyor globally For instance the quantum anomalous Hall-likestates would emerge if time reversal symmetry breakingeffects were introduced to 2D TSSs of phononic TIs (iii)To investigate unconventional electron-phonon coupling andsuperconductivity caused by the pseudospin- and topology-related physics of phonons Many material systems couldsimultaneously host nontrivial topological states of electronsand phonons Thus TSSs of electrons and phonons are ableto coexist on the material surfaces Their interactions mightbe exotic for instance due to the (pseudo-)spin-momentumlocking (iv) To find realistic candidate materials for exper-iment and application Very few realistic materials of chiralphonons [21 34 43] and phononic topological semimetals[22ndash25] have been reported and plenty of unknown can-didate materials of various phononic topological phases areawaiting to be discovered It is helpful to develop machinelearningmethods for high throughput discovery of phononictopological materials

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This work was supported by the Basic Science Center Projectof NSFC (Grant No 51788104) the and Ministry of ScienceTechnology of China (Grants No 2016YFA0301001 No2018YFA0307100 and No 2018YFA0305603) the NationalNatural Science Foundation of China (Grants No 11874035No 11674188 and No 11334006) and the Beijing AdvancedInnovation Center for Future Chip (ICFC)

Supplementary Materials

(I) Derivation of effective Hamiltonians (II) topologicalphase diagram (III) interatomic coupling parameters (IV)hybrid Wannier centers (V) pseudospin textures (VI)phonon transport in tunneling junctions (VII) influence

of out-of-plane vibrations and (VIII) phononic topologicalsemimetal (Supplementary Materials)

References

[1] E Prodan andC Prodan ldquoTopological phononmodes and theirrole in dynamic instability of microtubulesrdquo Physical ReviewLetters vol 103 no 24 Article ID 248101 2009

[2] L Zhang J Ren J-S Wang and B Li ldquoTopological nature ofthe phonon hall effectrdquo Physical Review Letters vol 105 no 22Article ID 225901 2010

[3] Y-T Wang P-G Luan and S Zhang ldquoCoriolis force inducedtopological order for classical mechanical vibrationsrdquo NewJournal of Physics vol 17 Article ID 073031 2015

[4] P Wang L Lu and K Bertoldi ldquoopological phononic crystalswith one-way elastic edge wavesrdquo Physical Review Letters vol115 no 10 Article ID 104302 2015

[5] Z Yang F Gao X Shi et al ldquoTopological acousticsrdquo PhysicalReview Letters vol 114 no 11 Article ID 114301 2015

[6] A B Khanikaev R Fleury S H Mousavi and A Alu ldquoTopo-logically robust sound propagation in an angular-momentum-biased graphene-like resonator latticerdquo Nature Communica-tions vol 6 Article ID 8260 2015

[7] X Ni C He X-C Sun et al ldquoTopologically protected one-wayedge mode in networks of acoustic resonators with circulatingair flowrdquoNew Journal of Physics vol 17 Article ID 053016 2015

[8] V Peano C Brendel M Schmidt and F Marquardt ldquoTopolog-ical phases of sound and lightrdquo Physical Review X vol 5 no 3Article ID 031011 2015

[9] T Kariyado Y Hatsugai and Sci Rep ldquoManipulation of diraccones in mechanical graphenerdquo Scientific Reports vol 5 ArticleID 18107 2015

[10] R Fleury A B Khanikaev and A Alu ldquoFloquet topologicalinsulators for soundrdquoNature Communications vol 7 Article ID11744 2016

[11] R Susstrunk and S D Huber ldquoClassification of topologicalphonons in linearmechanicalmetamaterialsrdquo Proceedings of theNational Academy of Sciences of theUnited States of America vol113 no 33 pp E4767ndashE4775 2016

[12] S D Huber ldquoTopological mechanicsrdquo Nature Physics vol 12pp 621ndash623 2016

[13] Y Liu Y Xu S-C Zhang andW Duan ldquoModel for topologicalphononics and phonon dioderdquo Physical Review B CondensedMatter and Materials Physics vol 96 no 6 Article ID 0641062017

[14] Y Liu Y Xu andW Duan ldquoBerry phase and topological effectsof phononsrdquo National Science Review vol 5 p 314 2017

[15] H Abbaszadeh A Souslov J Paulose H Schomerus andV Vitelli ldquoSonic landau levels and synthetic gauge fields inmechanical metamaterialsrdquo Physical Review Letters vol 119 no19 Article ID 195502 1955

[16] R Susstrunk and S D Huber ldquoObservation of phononic helicaledge states in a mechanical topological insulatorrdquo Science vol349 no 6243 pp 47ndash50 2015

[17] S H Mousavi A B Khanikaev and Z Wang ldquoTopologicallyprotected elastic waves in phononic metamaterialsrdquo NatureCommunications vol 6 Article ID 8682 2015

[18] L-H Wu and X Hu ldquoScheme for achieving a topologicalphotonic crystal by using dielectric materialrdquo Physical ReviewLetters vol 114 no 22 Article ID 223901 2015

8 Research

[19] R K Pal M Schaeffer and M Ruzzene ldquoHelical edge statesand topological phase transitions in phononic systems using bi-layered latticesrdquo Journal of Applied Physics vol 119 no 8 ArticleID 084305 2016

[20] C He X Ni H Ge et al ldquoAcoustic topological insulator androbust one-way sound transportrdquo Nature Physics vol 12 pp1124ndash1129 2016

[21] Y Liu C-S Lian Y Li Y Xu and W Duan ldquoPseudospinsand topological effects of phonons in a kekule latticerdquo PhysicalReview Letters vol 119 no 25 Article ID 255901 2017

[22] T Zhang Z Song A Alexandradinata et al ldquoDouble-weylphonons in transition-metal monosilicidesrdquo Physical ReviewLetters vol 120 Article ID 016401 2018

[23] H Miao T T Zhang L Wang et al ldquoObservation of doubleweyl phonons in parity-breaking FeSirdquo Physical Review Lettersvol 121 no 3 Article ID 035302 2018

[24] J Li Q Xie S Ullah et al ldquoCoexistent three-component andtwo-componentWeyl phonons in TiS ZrSe andHfTerdquo PhysicalReview B Condensed Matter and Materials Physics vol 97 no5 Article ID 054305 2018

[25] Q Xie J Li M Liu et al ldquoPhononic weyl nodal straightlines in high-temperature superconductor MgB2rdquo 2018httpsarxivorgabs180104048

[26] N Li J Ren L Wang G Zhang P Hanggi and B Li ldquoCol-loquium Phononics Manipulating heat flow with electronicanalogs and beyondrdquo Reviews of Modern Physics vol 84 no 3p 1045 2012

[27] L Fu C L Kane and E J Mele ldquoTopological insulators in threedimensionsrdquo Physical Review Letters vol 98 no 10 Article ID106803 2007

[28] C-X Liu R-X Zhang and B K VanLeeuwen ldquoTopologicalnonsymmorphic crystalline insulatorsrdquo Physical Review B vol90 no 8 Article ID 085304 2014

[29] X-Y Dong and C-X Liu ldquoClassification of topological crys-talline insulators based on representation theoryrdquo PhysicalReview B vol 93 no 4 Article ID 045429 2016

[30] L Fu ldquoTopological crystalline insulatorsrdquo Physical ReviewLetters vol 106 no 10 Article ID 106802 2011

[31] A Alexandradinata C Fang M J Gilbert and B A BernevigldquoSpin-orbit-free topological insulators without time-reversalsymmetryrdquo Physical Review Letters vol 113 no 11 Article ID116403 2014

[32] M S Dresselhaus G Dresselhaus and A Jorio Group TheoryApplication to the Physics of CondensedMatter Springer BerlinGermany 2008

[33] See Supplemental Material for details of calculation methodsand parameters

[34] L Zhang andQ Niu ldquoChiral phonons at high-symmetry pointsin monolayer hexagonal latticesrdquo Physical Review Letters vol115 no 11 Article ID 115502 2015

[35] H Zhang C X Liu X L Qi X Dai Z Fang and S C ZhangldquoTopological insulators in Bi2Se3 Bi2Te3 and Sb2Te3 with asingle Dirac cone on the surfacerdquo Nature Physics vol 5 no 6pp 438ndash442 2009

[36] X-L Qi and S-C Zhang ldquoThe quantum spin Hall effect andtopological insulatorsrdquo Physics Today vol 63 no 1 p 33 2010

[37] X-L Qi and S-C Zhang ldquoTopological insulators and super-conductorsrdquo Reviews of Modern Physics vol 83 no 4 ArticleID 1057 2011

[38] C L Kane and E J Mele ldquoQuantum spin hall effect ingraphenerdquo Physical Review Letters vol 95 no 22 Article ID226801 2005

[39] M Taherinejad K F Garrity and D Vanderbilt ldquoWanniercenter sheets in topological insulatorsrdquo Physical Review B vol89 no 11 Article ID 115102 2014

[40] A Alexandradinata X Dai and B A Bernevig ldquoWilson-loop characterization of inversion-symmetric topological insu-latorsrdquo Physical Review B vol 89 no 15 Article ID 155114 2014

[41] M Z Hasan and C L Kane ldquoColloquium topological insula-torsrdquo Reviews of Modern Physics vol 82 no 4 Article ID 30452010

[42] C Fang and L Fu ldquoNew classes of three-dimensional topo-logical crystalline insulators Nonsymmorphic and magneticrdquoPhysical Review B vol 91 no 16 Article ID 161105 2015

[43] H Zhu J Yi M Y Li et al ldquoObservation of chiral phononsrdquoScience vol 359 no 6375 pp 579ndash582 2018

2 Research

pseudospins by borrowing ideas from spintronics thusopening new opportunities for ldquopseudospin phononicsrdquo

1 Design Principle of Phononic TIs

An essential requirement of TIs is band degeneracies atno less than two high symmetry momenta (HSM) in theboundary Brillouin zone (BZ) [27ndash29] The requirement issatisfied for electrons with spin degeneracies protected bytime reversal symmetry However phonons do not have realspins invoking different strategies for building phononic TIsNaturally one could apply crystalline symmetries that areprevalent in solid materials to realize Kramers-like degen-eracies Such symmetries should also be preserved whenprojected onto the surface However for 2D spinless casesno such kind of crystalline symmetry has higher than 1Dirreducible representations at more than one HSM in the1D edge BZ implying that 2D phononic TIs protected bycrystalline symmetries are forbidden [21] The constraint isreleased for a variety of 3D lattices where multiple banddegenerate HSM can exist in the 2D surface BZ [28ndash31]Thusthe construction of 3D phononic TIs is feasible in principle

The Hamiltonian of phonons resembles a tight-bindingHamiltonian of spinless electrons with fixed 119901119909119910119911 orbitals[13] The symmetry representation of phonons is Γphonon =Γequiv otimes Γvector where Γequiv is the equivalence representationof atomic sites and Γvector is the representation of a 3D polarvector [32] Here we will not thoroughly discuss all possible3D crystalline symmetries but focus on 119862119899V (119899 = 3 4 6)symmetries that show interesting topological physics forelectrons [30 31] Take1198626V lattices as an example Γequiv is a119873dimensional representation where119873 is the number of atomicsites in a unit cell When all the atomic sites are inequivalentΓequiv is decomposed into 119873 1D irreducible representations1198601 119873 ge 2 is necessary to build a phononic TI that requiresat least four bands For simplicity we choose 119873 = 2 asdisplayed in Figure 1(a) which gives Γequiv = 21198601 In themomentum space the high-symmetry lines Γ-119860 and 119870-119867in the bulk BZ are projected onto the HSM Γ and 119870 in thesurface BZ [Figure 1(a)] which have1198626V and1198623V symmetriesrespectively Under 1198626(3)V symmetry Γvector = 1198601 oplus 119864 where1198601 and 119864 are 1D and 2D irreducible representations for basisfunctions of 119901119911 and 119901119909plusmn119894119901119910 respectivelyTherefore Γphonon =21198601 oplus 2119864 implying that 119901119909 plusmn 119894119901119910 (119901119911) are doubly degenerate(nondegenerate) along the high-symmetry lines Note thatfrequencies of in-plane vibrations (119901119909 plusmn 119894119901119910) are typicallyhigher than those of out-of-plane ones (119901119911) leading to weakhybridization between the doublet and singlet statesWe thusfocus on in-plane vibrations [33] for which aZ2 classificationof band topology is permitted [30]

2 Pseudospin- and Topology-Related Physics

Considering that the double degeneracy of 119901119909plusmn119894119901119910 resemblesthe Kramers degeneracy of spins we introduce a pseudospinindex to label the Kramers-like states There exist two typesof phonon modes including in-phase (I) and out-of-phase(O) vibrations between the two atomic sites A and B Their

pseudospin states [Figure 1(b)] are defined in coordinates of(119909119860 119910119860 119909119861 119910119861)119879 as10038161003816100381610038161003816Iuarr⟩ = (120576+ sin 120579k 120576+ cos 120579k)119879 10038161003816100381610038161003816Ouarr⟩ = (120576minus cos 120579k minus120576minus sin 120579k)119879 10038161003816100381610038161003816Idarr⟩ = 119894 (120576minus sin 120579k 120576minus cos 120579k)119879 10038161003816100381610038161003816Odarr⟩ = 119894 (minus120576+ cos 120579k 120576+ sin 120579k)119879 (1)

where 120576plusmn = (1 plusmn119894)radic2 and the vibrational magnitude ofeach atomic site is determined by 120579k isin [0 1205872] that is k-dependent They are orthonormal and form a complete basisof in-plane vibrations

The pseudospin states are featured by well-defined quan-tized pseudoangular momenta about 119911-axis for k along thehigh-symmetry lines due to the 1198626(3) rotational symmetryThe pseudoangularmomentumoperator is expressed as 119869ph =120590119911119904119911 [33] where 120590119911 and 119904119911 are Pauli matrices with 120590119911 = plusmn1and 119904119911 = plusmn1 refer to pseudospin up (down) and I (O)vibrational modes respectively The phonon pseudoangularmomentum 119895ph is composed of two parts including a localpart determined by the on-site orbital and a nonlocal partcontributed by the inter-site Bloch phase change [34] Hereinthe nonlocal part is zero and 119895ph is fully determined by thelocal part giving 119895ph = 1 for |Iuarr⟩ and |Odarr⟩ and 119895ph = minus1 for|Idarr⟩ and |Ouarr⟩ as depicted in Figure 1(b)

The symmetry-adapted 119896sdot119901methodwas applied to deriveeffective Hamiltonians of phonons near HSM by using thebasis set |Iuarr⟩ |Ouarr⟩ |Idarr⟩ |Odarr⟩ [33] Specifically the BlochHamiltonian matrix 119867(k) was first expanded by the lowestorders of k (referenced to HSM) Then by writing down thecorrespondingmatrices of symmetry operators and requiring119867(k) commute with them some matrix elements of 119867(k)are enforced to be zero The remaining nonzero part givesthe effective Hamiltonian The HSM are classified into type-I(1198701198701015840 and1198671198671015840) and type-II (Γ and119860) which have 1198623V and1198626V symmetries respectively andwill be discussed separatelyImportantly the conservation of pseudoangular momentumrequires that basis states of the same 119895ph (eg |Iuarr⟩ and |Odarr⟩)are coupled only by terms like 119896119911 1198962119909 + 1198962119910 and 1198962119911 whilebasis states of different 119895ph (eg |Iuarr⟩ and |Ouarr⟩) are coupledonly by terms like 119896plusmn and 1198962plusmn The effective Hamiltonian nearthe type-I HSM is written as 119867 = 1198670 + 119867119868 + 119867119877119863 where1198670 = diag(119872k minus119872k119872k minus119872k)

119867119868 = ( 0 Δ 1119896+ 0 Δ 2119896119911Δ 1119896minus 0 Δ 2119896119911 00 Δ 2119896119911 0 minusΔ 1119896minusΔ 2119896119911 0 minusΔ 1119896+ 0 )119867119877119863 = ( 0 0 minus1198941198621119896+ 00 0 0 1198941198622119896minus1198941198621119896minus 0 0 00 minus1198941198622119896+ 0 0 )

(2)

Research 3

Γ

Γ Γ

M K

(a)

A

KM

R

H

(b)

a

A

B

(c) (d)

I

O

(e)

c2

c1

Γ uarr⟩

）uarr

darr

）darr

jＪＢ = minus1

jＪＢ = minus1

jＪＢ = +1

jＪＢ = +1

8

6

4

2

0KM A HR R

66

66

63

60

69

64

62

Γ M ＋

＋

⟩ ⟩

⟩







4 Research

(b) (c)(a)

9

6

6

7

8

3

0

70

65

KM A HR R

Γ M Γ ＋－－







) (3)




Research 5

20 40 6000

0

05

10

source probegate

Transm

ission

(a) (c)(b)

Transm

ission

La

Type-I

Type-II

Normal

x

z y

1

0

1

60∘90∘

-90∘-60∘

-30∘

30∘

0∘

0Δ＄






6 Research

10

10

0

0minus10

minus10

(a) (b)

TSM TI

TSMTI

(c)

(00)

(d) (e)

C1C2lt0

C1C2gt0

H0+HRD H0+HRD+HI

H0

C1Δ1

C2Δ

1

()

(0)

8

6

68

68

64

64

60

60

2

4

0KM A AHR

Γ Γ

＋

＋

－

－

Γ

ΓΓ









Research 7


8 Outlook




Acknowledgments





References



















8 Research


























Research 3

Γ

Γ Γ

M K

(a)

A

KM

R

H

(b)

a

A

B

(c) (d)

I

O

(e)

c2

c1

Γ uarr⟩

）uarr

darr

）darr

jＪＢ = minus1

jＪＢ = minus1

jＪＢ = +1

jＪＢ = +1

8

6

4

2

0KM A HR R

66

66

63

60

69

64

62

Γ M ＋

＋

⟩ ⟩

⟩







4 Research

(b) (c)(a)

9

6

6

7

8

3

0

70

65

KM A HR R

Γ M Γ ＋－－







) (3)




Research 5

20 40 6000

0

05

10

source probegate

Transm

ission

(a) (c)(b)

Transm

ission

La

Type-I

Type-II

Normal

x

z y

1

0

1

60∘90∘

-90∘-60∘

-30∘

30∘

0∘

0Δ＄






6 Research

10

10

0

0minus10

minus10

(a) (b)

TSM TI

TSMTI

(c)

(00)

(d) (e)

C1C2lt0

C1C2gt0

H0+HRD H0+HRD+HI

H0

C1Δ1

C2Δ

1

()

(0)

8

6

68

68

64

64

60

60

2

4

0KM A AHR

Γ Γ

＋

＋

－

－

Γ

ΓΓ









Research 7


8 Outlook




Acknowledgments





References



















8 Research


























4 Research

(b) (c)(a)

9

6

6

7

8

3

0

70

65

KM A HR R

Γ M Γ ＋－－







) (3)




Research 5

20 40 6000

0

05

10

source probegate

Transm

ission

(a) (c)(b)

Transm

ission

La

Type-I

Type-II

Normal

x

z y

1

0

1

60∘90∘

-90∘-60∘

-30∘

30∘

0∘

0Δ＄






6 Research

10

10

0

0minus10

minus10

(a) (b)

TSM TI

TSMTI

(c)

(00)

(d) (e)

C1C2lt0

C1C2gt0

H0+HRD H0+HRD+HI

H0

C1Δ1

C2Δ

1

()

(0)

8

6

68

68

64

64

60

60

2

4

0KM A AHR

Γ Γ

＋

＋

－

－

Γ

ΓΓ









Research 7


8 Outlook




Acknowledgments





References



















8 Research


























Research 5

20 40 6000

0

05

10

source probegate

Transm

ission

(a) (c)(b)

Transm

ission

La

Type-I

Type-II

Normal

x

z y

1

0

1

60∘90∘

-90∘-60∘

-30∘

30∘

0∘

0Δ＄






6 Research

10

10

0

0minus10

minus10

(a) (b)

TSM TI

TSMTI

(c)

(00)

(d) (e)

C1C2lt0

C1C2gt0

H0+HRD H0+HRD+HI

H0

C1Δ1

C2Δ

1

()

(0)

8

6

68

68

64

64

60

60

2

4

0KM A AHR

Γ Γ

＋

＋

－

－

Γ

ΓΓ









Research 7


8 Outlook




Acknowledgments





References



















8 Research


























6 Research

10

10

0

0minus10

minus10

(a) (b)

TSM TI

TSMTI

(c)

(00)

(d) (e)

C1C2lt0

C1C2gt0

H0+HRD H0+HRD+HI

H0

C1Δ1

C2Δ

1

()

(0)

8

6

68

68

64

64

60

60

2

4

0KM A AHR

Γ Γ

＋

＋

－

－

Γ

ΓΓ









Research 7


8 Outlook




Acknowledgments





References



















8 Research


























Research 7


8 Outlook




Acknowledgments





References



















8 Research


























8 Research


























Three-Dimensional Topological States of Phonons with ......2019/06/08 · Research 10 10 0 0 −10...

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Transcript of Three-Dimensional Topological States of Phonons with ......2019/06/08 · Research 10 10 0 0 −10...