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Topological chiral magnonic edge mode in a magnonic crystal

Ryuichi Shindou, Ryo Matsumoto, and Shuichi Murakami

Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, Japan

(Dated: April 17, 2012)

Topological phases have been explored in various fields in physics such as spintron-

ics, photonics, liquid helium, correlated electron system and cold-atomic system. This

leads to the recent foundation of emerging materials such as topological band insula-

tors, topological photonic crystals and topological superconductors/superfluid. Here,

we propose topological magnonic crystal which provides protected chiral edge modes

for magnetostatic spin waves. Based on a generic linearized Landau-Lifshitz equation,

we show that a magnonic crystal with the dipolar interaction usually acquires spin

wave band with non-zero Chern integer. We argue that such magnonic systems are

accompanied by the same integer numbers of chiral magnonic edge modes, along which

the spin wave can propagate in a unidirectional manner without being backward scat-

tered. Being a robust ‘one-way spin-wave guide’, the proposed chiral magnonic edge

mode makes it possible the magnonic Fabry-Perot type interferometer, which can be

utilized as the spin-wave logic gate.

Topological phases in condensed matters have been at-

tracting much attention because of their fascinating phys-

ical properties. Discoveries of topological band insula-

tors1–7open up emerging research paradigm on spin-orbit

interaction physics. Relativistic spin-orbit interaction in

the topological band insulator endows its Bloch electron

bands with non-trivial global structures, which lead to

novel surface metallic states5and topological magneto-

electric effect.8Superconductor analogues of topological

insulators have exotic edge modes9or bound states,10

which are composed only of ‘real-valued’ fermion field

dubbed as Majorana fermion.

bound states are experimentally confirmed,11,12foster-

ing much prospect of the realization of quantum comput-

ers.13Photonics analogue of topological phase with chiral

edge modes are proposed theoretically14,15and are subse-

quently designed in actual photonics crystals.16Unidirec-

tional propagations of electromagnetic wave along these

chiral edge modes were experimentally observed, which

provides these metamaterials with unique photonic func-

tionality.

Some aspects of these

Here, we theoretically propose a magnonics analogue

of topological phases, which has topologically-protected

chiral edge mode for spin waves.

lective propagation of precessional motions of magnetic

moments in magnets. Depending on its wavelength, spin

waves are classified into two categories. One is exchange

spin-wave with the shorter wavelength, whose motion is

driven by the quantum-mechanical exchange interactions

(‘exchange-dominated’ region). The other is magneto-

static spin wave with the longer wavelength,17,18whose

propagation is caused by the long-range dipolar interac-

tion (‘dipolar’ region). Magnonics research investigates

how these spin waves propagate in the sub-micrometer

length scale and sub-nanosecond time scale.19–25Like in

other solid-state technologies such as photonics, phonon-

ics and plasmonics, the main application direction is to

explore ability of the spin wave to carry and process

information. Especially, the propagation of spin waves

Spin wave is a col-

in periodically modulated magnetic materials dubbed as

magnonic crystals21–23,26,27are of one of its central con-

cern. Owing to the periodic structuring, the spin wave

spectrum in magnonic crystal acquires allowed frequency

bands of spin wave states and forbidden-frequency bands

(band gap).

We introduce a topological Chern number in these

spin-wave bands with a band gap. Based on a simple

magnonic crystal with wide varieties of material parame-

ters, we predict that the first Chern number for the low-

est spin-wave band usually takes non-zero integer values,

when the wavelength is in the dipolar region. The rele-

vant length scale turns out to be from sub-µm to hun-

dreds of µm, well within the range of nanosize fabrica-

tion. The nonzero Chern integer in magnonic crystals

results in the same integer numbers of topological chiral

edge modes, along which the spin wave can propagate in a

unidirectional way. Thanks to the topological protection,

this propagation is generically free from any static back-

ward scatterings. The unidirectional motion can be ex-

perimentally measurable especially in yttrium iron garnet

(YIG), where the coherence length of magnons is on the

order of centimeters.20We argue that these edge modes

can be easily channelized, twisted, split and manipulated,

which enables to construct novel magnonic devices such

as spin-wave logic gate.

Chern number in boson systems

To introduce topological Chern number for the magne-

tostatic spin wave, let us consider a general quadratic

Bogoliubov-de Gennes (BdG) Hamiltonian for boson

fields in the two-dimensional momentum space (k-space);

H =1

2

?

k

?

β†

kβ−k

?· Hk·

?

βk

β†

−k

?

. (1)

with β†

bosonic bands, and the 2N by 2N Hermite matrix Hk

is composed of the N by N normal parts (particle-hole

k≡ [β1,k,··· ,βN,k]. N denotes the number of

arXiv:1204.3349v1 [cond-mat.mes-hall] 16 Apr 2012

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2

FIG. 1: Magnonic crystal with chiral edge modes. Pe-

riodic array of holes is introduced into YIG, forming a square

lattice. Iron (Fe) is filled inside every hole.

channel) and N by N anomalous parts (particle-particle

channel).

This bosonic BdG Hamiltonian is diagonalized in terms

of a para-unitary matrix Tkinstead of a unitary matrix,28

?Ek

with [γ†

k,β−k], and Ekis a diagonal ma-

trix. From the commutation relation for the γ-field, the

orthogonality and completeness of the new basis are de-

rived as

T†

kHkTk=

E−k

?

, (2)

k,γ−k]T†

k= [β†

T†

kσ3Tk= σ3, Tkσ3T†

k= σ3

(3)

respectively, where the diagonal matrix σ3 takes ±1 in

the particle/hole space, i.e. [σ3]jm= δjmσj with σj =

+1 for j = 1,··· ,N and σj= −1 for j = N +1,··· ,2N.

In terms of the para-unitary matrix, the field strength

(Berry’s curvature) Bjand the gauge connection (gauge

field) (Aj,x,Aj,y) are introduced in the momentum space

for j-th magnonic band as;

Bj≡ ∂kxAj,y− ∂kyAj,x,

Aj,ν≡ iTr[Γjσ3T†

(4)

kσ3(∂kνTk)], (5)

respectively with j = 1,··· ,2N. Γjis a 2N by 2N diag-

onal matrix taking +1 for the j-th diagonal component

and zero otherwise. The Chern number (the first Chern

number) associated with the j-th band is given by the

momentum integral of the respective Berry’s curvature

over the two-dimensional first Brillouin zone (BZ),

?

This quantity can be shown to be an integer Cj =

n (see Supplementary Information for details), which

corresponds to a number of topological chiral edge

modes. Note that, due to the commutation relation

of the boson fields, the Chern integer defined in equa-

tions (4,5,6) generally takes a form different from those

Cj=

1

2π

BZ

d2kBj. (6)

Chern integers previously defined in electronic29and pho-

tonic14,15,30systems. In the following, we show that a

two-dimensional magnonic crystal (MC) with the dipolar

interaction usually supports spin-wave bands with non-

zero Chern integers.

2-d magnonic crystals and dipolar interaction

The MC considered is a ferromagnetic system with its

magnetization and exchange interaction modulated pe-

riodically in the 2-dimensional (x-y) direction. For sim-

plicity, we assume that the system is translationally sym-

metric along the z-direction, whereas the subsequent re-

sults are expected to be similar when the thickness of the

system in the z-direction becomes finite. The system is

composed of two kinds of ferromagnets; iron and YIG.

The unit cell of the MC is an ax× ay rectangle, inside

which iron is embedded into a circular region, while the

remaining region is filled with YIG (Fig. 1). The uniform

magnetic field H0is applied along the z direction, such

that the static ferromagnetic moment Msin both regions

is fully polarized in the z direction. Propagation of the

transverse moments (mx,my) is described by a linearized

Landau-Lifshitz equation26,27,31,32;

1

|γ|µ0

dm±

dt

= ± 2iMs

?∇ · Q∇?m±∓ 2im±

∓ iH0m±± ih±Ms

?∇ · Q∇?Ms

(7)

with ∇ ≡ (∂x,∂y), m±= mx± imyand h±= hx± ihy.

(hx,hy) stands for the transverse component of long-

ranged magnetic dipolar field h, which is related to

the ferromagnetic moment m ≡ (mx,my,Ms) via the

Maxwell equation, i.e. ∇×h = c−1∂tezand ∇·(h+m) =

0. The former two terms in the right hand side of equa-

tion (7) comes from short-ranged exchange interaction,

where Q denotes the square of the exchange interac-

tion length. Ms and Q take the values of iron inside

the circular region, while taking the values of YIG oth-

erwise. The filling fraction of the circular region with

respect to the total area of the unit cell is parameter-

ized by f. We restrict ourselves to the wavelength much

longer than atomic lattice constants of YIG and iron,

so that the use of this Landau-Lifshitz equation is jus-

tified. Since the velocities of relevant spin-wave modes

are much smaller than the speed of light, we can further

employ the magneto-static approximation, replacing the

Maxwell equations by ∇ × h = 0 and ∇ · (h + m) = 0;

hν= −∂νΨ, ∆Ψ = ∂xmx+ ∂ymy,

with ν = x,y. This in combination with equation (7)

gives a closed equation of motion (EOM) for the trans-

verse moments. In terms of normalized transverse fields;

(8)

β(r) ≡

m+(r)

?2Ms(r), β†(r) ≡

m−(r)

?2Ms(r). (9)

with a proper commutator [β(r),β†(r?)] = δ(r −r?), the

coupled EOM reduces to an equivalent generalized eigen-

value problem with a Hermitian matrix Hk(see Methods

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3

for details);

id

dt

?

βk

β†

−k

?

= σ3Hk

?

βk

β†

−k

?

, (10)

where β†

of β†(r) with respect to the reciprocal vector G; β†

[··· ,β†

EOM is diagonalized in terms of a para-unitary matrix

Tkintroduced in equation (2) as

?

kdenotes a vector composed of the Fourier series

k≡

k(G),···] and β(r) =?

k,Gβk(G)e−i(k+G)r. The

id

dt

γk

γ†

−k

?

=

?Ek

−E−k

??

γk

γ†

−k

?

.

[Ek]j gives the dispersion of the j-th magnonic band

(j = 1,···), while [Tk]a,j (a = ··· ,G,···) stands for

the periodic part of the j-th magnonic Bloch wavefunc-

tion. Correspondingly, the Chern integer is defined for

each magnonic band via equations (6), (4) and (5).

The topological Chern integer thus calculated always

reduces to zero, whenever the system considered is ei-

ther time-reversal symmetric; H∗

symmetric with the mirror plane being perpendicular to

the xy plane, e.g. H(kx,ky)= H(kx,−ky). Owing to the

dipolar interaction, however, the anomalous part of Hk

acquires a complex-valued phase factor which depends

on the momentum in the chiral way (see Methods for de-

tails). As a result, these two symmetries are generically

absent in the 2-d MCs and the bosonic Chern integer can

take a non-zero integer value.

This situation is quite analogous to what the rela-

tivistic spin-orbit interaction does in ferromagnetic met-

als33,34and topological band insulators.1,2,4,5Moreover,

contrary to the relativistic spin-orbit interaction, the

strength of the dipolar interaction is an experimentally

tunable parameter in the MCs.21Namely, the dimen-

sional analysis on equation (7) indicates that, when the

characteristic length scale of the MC (the unit cell size

λ ≡√axay) becomes larger than the typical exchange

length√Q, the dipolar interaction prevails over the ex-

change interaction.

−k= Hk, or mirror-

Chiral magnonic bands in MC

We found that the Chern integer of the lowest magnonic

band, C1, is always quantized to be 2 for the longer

λ, while the integer reduces to zero for the shorter λ

(Fig. 2a). The respective quantization is protected by

a finite direct band gap between the lowest band and

the second lowest band (Fig. 2b). In the intermediate

regime of λ, these two bands get closer to each other.

With the four-fold rotational symmetry (r ≡ ay/ax= 1),

the gap closes at the two X-points at a critical value

of λ (∼ 0.28µm), where the two bands form gapless

Dirac spectra (Fig. 2c). Without the four-fold symmetry

(r ?= 1), the band touching at one of the two X points

and that of the other occur at different values of λ. These

band-touchings are P1(π,0,λc,1) and P2(0,π,λc,2) in the

FIG. 2:

magnonic band and band dispersions of several lowest

magnonic bands. a, Chern-integer phase diagram for the

lowest magnonic band as a function of the linear dimension

of the unit cell size (λ ≡√axay) and the aspect ratio of the

unit cell (r ≡ ay/ax) with f = π×10−2. The phases are char-

acterized by the Chern integer of the lowest magnonic band,

C1. b, Band dispersions of the lowest three magnonic bands

with r = 1, λ = 0.35µm and f = π ×10−2. A magnonic band

gap appears between the first and the second lowest band,

whose size is on the order of 10−3THz. c, Band dispersions

of the lowest and second lowest magnonic band at the critical

point which intervenes the phase II (C1 = 2) and the phase

IV (C1 = 0). Dirac cones with same chiralities are formed at

the two inequivalent X points.

Chern-integer phase diagram for the lowest

3-dimensional parameter space subtended by two crystal

momenta kxand kyand the unit cell size λ (Fig. 3).

As in the fermionic case,35,36a band-touching point in

the 3-d parameter space generally plays a role of the dual

magnetic monopole (charge). The dual magnetic field

considered is a vector field defined in the parameter space

(kx,ky,λ), generalized from equation (4) as the ‘rotation’

of three-component gauge field Aj≡ (Aj,x,Aj,y,Aj,λ);

Bj≡ ∇ × Aj, (11)

with ∇ ≡ (∂kx,∂ky,∂λ). Here the third component of

the gauge field Aj,λis newly introduced in the same way

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FIG. 3:

monopoles in the 3-dimensional parameter space sub-

tended by the crystal momenta (kx,ky) and the lin-

ear dimension of the unit cell size (λ ≡√axay) for

r ≡ ay/ax > 1. Small green spheres denote the magnetic

monopoles, which emit the dual magnetic field (blue arrows).

We only show the first quadrant of the Brillouin zone. Three

phases found in Fig. 2a with r > 1 are separated by the

λ = λc,2 plane and the λ = λc,1 plane. A magnetic monopole

located on each plane means that the ‘phase transition’ is

accompanied by the change of the Chern integer.

Schematic configuration of dual magnetic

as previously done in (5);

Aj,λ≡ iTr[Γjσ3T†

kσ3(∂λTk)]. (12)

j specifies either one of the two magnonic bands which

form the band-touching. At the band-touching point, the

dual magnetic field for the respective bands has a dual

magnetic charge, whose strength is quantized to be 2π

times integer (see Supplementary Information).

We found that dual magnetic charges for the lowest

band at the band touching points at P1and P2are both

+2π (Fig. 3); ∇ · B1 = 2πδ(λ − λc,1)δ(kx− π)δ(ky) +

2πδ(λ−λc,2)δ(kx)δ(ky−π), where λc,1< λc,2for ay> ax

(r > 1) and λc,1> λc,2for ay< ax(r < 1). The Gauss

theorem suggests that, when the unit cell size λ is varied

across one of these band-touching points (λc,1 or λc,2),

the Chern integer for the lowest magnonic band changes

by unity, e.g.

C1|λ>λc,1− C1|λc,1>λ=

1

2π

?

S1

dn · B1= 1,

where S1is a small sphere enclosing the magnetic charge

at P1in the 3-d parameter space. This leads to the phase

diagram in Fig. 2a, which describes the Chern integer of

the lowest magnonic band as a function of the unit cell

size λ and the aspect ratio r. There are four phases,

divided by the curves λ = λc,1and λ = λc,2.

A dual magnetic charge is a quantized object, so that

it can neither disappear by itself nor change gradually.

Upon any small change of material parameters, two mag-

netic charges can only move around in the 3-d parameter

space. Moreover, in the presence of the spatial inver-

sion symmetry Hk = H−k, their locations are always

restricted at the X points. As a result, the global struc-

ture of the phase diagram depicted in Fig. 2a widely

holds true for other combinations of material parame-

ters. In fact, for r = 1 and f = π × 10−2, we found

λc= 0.370µm for iron (circular region) and YIG (host),

and λc= 0.372µm for cobalt (circular region) and YIG

(host). When varying the filling fraction for iron (cir-

cular region) and YIG (host) with r = 1, we found

λc = 0.274µm for f = 4π × 10−2, and λc = 0.348µm

for f = 9π × 10−2. From these observations, we expect

that, even if materials in the circular region is replaced

simply by the vacuum, the MC systems with larger unit

cell size still belongs to topological non-trivial phases,

having nonzero Chern integers.

Chiral magnonic edge mode in MC

The chiral phases with non-zero Chern integers have chi-

ral magnonic edge modes, which are localized at the

boundary with the phase with zero Chern integer (phase

IV) or the vacuum. These chiral modes carry spin wave

in a unidirectional manner, whose dispersions go across

the band gap between the lowest and the second lowest

band. As an illustrative example, we consider a bound-

ary (y axis) between the MC in phase III and MC in the

phase IV, whose Chern integers for the lowest magnonic

bands differ by unity (Fig. 4a). The existence of a chiral

magnonic edge mode between these two is shown from a 2

by 2 Dirac Hamiltonian derived near their phase bound-

ary (λ = λc,1). The Hamiltonian generally takes a fol-

lowing form (see Supplementary Information),

Heff= ω0τ0+ κ(x)τ3− ia∂xτ1− ib∂yτ2.

τj denotes the Pauli matrices subtended by the two-

fold degenerate eigenstates with their eigen-frequency ω0,

which are formed by the lowest magnonic band and sec-

ond lowest one at P1. a and b are constant material pa-

rameters which are positive-valued. The difference of the

Chern integers (C1) for the two phases is represented as a

change of the sign of the Dirac mass term κ(x); we hence

suppose that positive κ(x) for x > 0 is for the phase III

and and negative κ(x) for x < 0 is for the phase IV (see

Fig. 4a); limx→±∞κ(x) = ±κ∞. The Hamiltonian has a

following eigenstate;37,38

?xκ(x?)dx??1

(13)

ψk(r) ∝ eikye−1

a

i

?

.

which is localized at the boundary (x = 0). In terms

of the surface crystal momentum k along the y axis, the

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FIG. 4: Chiral magnonic edge modes. a, geometry of

the system. bc, dispersions for Dirac Hamiltonian with the

P¨ oschl-Teller potential15κ(x) = κ∞tanh(x/d) for κ∞d =

0.9a (b) and κ∞d = 2.9a (c).

(‘bulk’) have a gap, and one chiral edge state runs across the

gap. In c there are some nonchiral edge states, whereas in b

there is not. Nonetheless, the number of chiral edge modes is

one, which is determined solely from the difference between

the Chern integers for the two phases.

The bulk magnonic bands

corresponding eigen-frequency is given by E = ω0+ bk.

This connects the lowest magnonic band lying at E ≤

ω0−κ∞and the second lowest band at E ≥ ω0+κ∞(see

Fig. 4b,c). The mode is chiral, since it propagates along

the boundary in a unidirectional way, vk≡ ∂kE = b. The

argument so far can be generalized into other situations.

The generalization suggests that the phase with C1= 1

at r > 1 (Phase III) or r < 1 (Phase I) has a chiral edge

mode at its boundary with vacuum, whose dispersion

crosses the direct band gap at the (π,0) or (0,π)-point

respectively, while the phase with C1= 2 (Phase II) has

both at its boundary with vacuum.

The chiral magnonic edge modes can be easily twisted

or split by the change of the size (λ) and shape (r) of

the unit cell, which we demonstrate in Fig. 5a,b.

Fig. 5a, the MC in the phase II (r = 1) is connected

with the other MC in the phase III (r > 1), whose Chern

integer for the lowest band differ by unity. The bound-

ary between these two MC systems supports the chiral

In

magnonic edge mode which runs across the direct band

gap at (0,π) point. The existence of such an edge mode

can be shown from the other 2 by 2 Dirac Hamiltonian

derived near λ = λc,2. This means that the two chiral

magnonic edge modes propagating along the boundary

of the the MC in the phase II (r = 1) are spatially di-

vided into two, where one mode goes along the boundary

of another MC in the phase III (r > 1), while the other

goes along the boundary between these two MCs. This

configuration realizes a ‘spin-wave current splitter’, an

alternative to those proposed in other geometries.40

Discussion

By calculating a newly-introduced bosonic Chern inte-

ger for spin wave bands, we argue that two-dimensional

normally-magnetized magnonic crystal acquires chiral

edge modes for magnetostatic wave in the dipolar regime.

Each mode is localized at the boundary of the sys-

tem, carrying magnetic energies in a unidirectional way.

Thanks to the topological protection, spin-wave propa-

gations along these chiral edge modes are generally ro-

bust against imperfections of the lattice periodicity and

boundary roughness; they are free from any types of

elastic backward scatterings with moderate strength.39

This robustness makes it possible a magnonic analogue

of Fabry-Perot interferometer as discussed below.

The interferometer is made up of a coupled of chiral

magnonic edge modes encompassing a single topological

MC (see Fig. 5c). Parts of the MC are spatially con-

stricted by the hole inside, so as to play a role of the

‘point contact’ between these edge modes.41A unidirec-

tional spin-wave propagation is induced in a chiral mode

via an antenna attached to the boundary. The wave is

divided into two chiral edge modes at a point contact.

Two chiral propagations merge into a single chiral propa-

gation at the other point contact. Depending on a phase

difference between these two, the superposed wave ex-

hibits either a destructive interference or a constructive

one, which is detected as an electric signal from the other

antenna. Note that a local application of magnetic field

changes the velocity of either one of the two chiral edge

modes locally as vk= b → vk= b + ∆b(r). This mod-

ification results in a phase shift of the wave, which thus

changes the interference pattern observed in the output

signal. With the use of this local magnetic field as an

external control,19,42the interferometer can serve as a

solid-state based magnonic logic gate.

Though chiral edge modes are robust against static

perturbations, magnetic energies excited in the edge

mode decay into either phonon states or other magnon

states in the bulk via inelastic scatterings. The associated

decay time or coherence length depends on specific ma-

terials, and spin wave propagation along the chiral edge

mode survives only within this coherence length. Due to

the absence of conduction electrons, however, spin waves

in magnetic insulators are generally known to have long

coherence lengths. For example, the coherence length in

YIG in the magnetostatic regime becomes on the order of