Berry curvature and topological modes for magnons

Shuichi Murakami Department of Physics, Tokyo Institute of Technology

KITP, UCSB

Nov.14, 2013

Collaborators: R. Shindou (Tokyo Tech. Peking Univ.)

R. Matsumoto (Tokyo Tech.) J. Ohe (Toho Univ.) E. Saitoh (Tohoku Univ. )

Magnon thermal Hall effect

• Matsumoto, Murakami, Phys. Rev. Lett. 106, 197202 (2011)

• Matsumoto, Murakami, Phys. Rev. B 84, 184406 (2011)

Topological magnonic crystals

• Shindou, Matsumoto, Ohe, Murakami, Phys. Rev. B 87,174427 (2013)

• Shindou, Ohe, Matsumoto, Murakami, Saitoh, Phys. Rev. B87, 174402 (2013)

Phenomena due to Berry curvature in momentum space

• Hall effect Quantum Hall effect

chiral edge modes

• Spin Hall effect (of electrons) Topological insulators

helical edge/surface modes

•Spin Hall effect of light one-way waveguide in photonic crystal

• Magnon thermal Hall effect topological magnonic

crystal

Fermions

Bosons

Present work

Gapped Topological edge/surface

modes in gapped systems Gapless Various Hall effects

semiclassical eq. of motion for

wavepackets

( )1( )n

n

E kx k k

k

k eE

n( : band index)

: periodic part of the Bloch wf. knu

xki

knknexux

)()(

: Berry curvature ( ) n nn

u uk i

k k

- SM, Nagaosa, Zhang, Science (2003)

- Sinova et al., Phys. Rev. Lett. (2004)

Adams, Blount; Sundaram,Niu, …

Intrinsic spin Hall effect in metals& semiconductors

Spin-orbit coupling Berry curvature depends on spin

Force // electric field

2

2

,

( ) zBxy nk n

n k

k Tc k

V

Magnon Thermal Hall conductivity

22

2

2 20

1 1log (1 ) log (log ) 2Li ( )

tc dt

t

Q xy yxj T

T. Qin, Q. Niu and J. Shi,Phys. Rev. Lett. 107, 236601 (2011) R. Matsumoto, S. Murakami, Phys. Rev. Lett. 106, 197202 (2011)

Berry curvature

( )1( )n

n

E kx k k

k

k U

( ) n nn

u uk i

k k

: Berry curvature

equilibrium

Density matrix

Current

deviation by

external field

(1) Semiclassical theory ( 2) Linear response theory (Kubo formula)

Eq. of motion

: Bose distribution

Cf: different from previous works

Katsura, Nagaosa, and Lee, PRL.104, 066403 (2010).

Onose, et al., Science 329, 297 (2010);

Magnetic dipole interaction

• Dominant in long length scale (microns) • Similar to spin-orbit int. Berry curvature • Long-ranged nontrivial, controlled by shape

Magnetic domains

Generalized eigenvalue eq. : MSFVW mode

• Landau-Lifshitz (LL) equation

• Maxwell equation

• Boundary conditions

00 HB ,

//2//121 HHBB ,

0 0

0 0

1 0, , : thickness of the film, , m(z)

0 1

: saturation magnetization, : static magnetic field, film,

ˆ : 2 2 matrix of the Green's func

x y

H M z

x y

m imH M L

m im

M H z

G tion, : frequency of the spin wave

B. A. Kalinikos and A. N. Slavin, J. Phys. C 19, 7013 (1986)

)( HMM

dt

d

Magnetostatic modes

in ferromagnetic films (YIG)

/2

/2

ˆˆ ˆ( ) ( ) ( ) ( ) ( , ) ( )L

z H ML

H z z H z z dz G z z zm m m m m

H

Berry curvature

for magnetostatic forward volume wave mode

: Band structure for magnetostatic

forward volume-wave mode : Berry curvature

Berry curvature is zero for backward volume wave and surface wave

• R. Matsumoto, S. Murakami, PRL 106,197202 (2011), PRB84, 184406 (2011)

T: paraunitary matrix

Bosonic BdG eq. and Berry curvature

Generalized eigenvalue eq.

bosonic Bogoliubov-de Gennes Hamiltonian

Diagonalization

Berry curvature for n-th band

k k zH

k z k zT T

k z k zT T

z z

Linear response theory

Berry curvature

T: paraunitary matrix

Bosonic BdG eq. and Berry curvature

Generalized eigenvalue eq.

bosonic Bogoliubov-de Gennes Hamiltonian

Diagonalization

Berry curvature for n-th band

k k zH

k z k zT T

k z k zT T

z z

Linear response theory

Berry curvature

2.8MHz/Oe, 1750gauss, 300K

3000Oe, 17.2nm for YIG,

s

ex ex

M T

H l

85.8 10 W/Kmxy

(Example):

Topological chiral modes

in magnonic crystals

• R. Shindou, R. Matsumoto, J. Ohe, S. Murakami, Phys. Rev. B 87,174427

(2013)

• R. Shindou, J. Ohe, R. Matsumoto, S. Murakami, E. Saitoh,Phys. Rev. B87,

174402 (2013)

Phenomena due to Berry curvature in momentum space

• Hall effect Quantum Hall effect

chiral edge modes

• Spin Hall effect (of electrons) Topological insulators

helical edge/surface modes

•Spin Hall effect of light one-way waveguide in photonic crystal

• Magnon thermal Hall effect topological magnonic

crystal

Fermions

Bosons

Present work

Gapped Topological edge/surface

modes in gapped systems Gapless Various Hall effects

Chern number & topological chiral modes

Band gap Chern number for n-th band = integer

topological chiral edge modes

• Analogous to chiral edge states of quantum Hall effect.

• N>0 cw, N<0: ccw mode

bands below

Ch #(clockwise chiraledgestates in the gap at )n

n E

N E

2

Ch2

n nBZ

d kk

Berry curvature

k

bulk mode: Chern number= Ch1

Ch1 topological edge modes

bulk mode: Chern number= Ch2

bulk mode: Chern number= Ch3

(Ch1+Ch2) topological edge modes

( ) n nn

u uk i

k k

Chern number & topological chiral modes

Band gap Chern number for n-th band = integer

topological chiral edge modes

• Analogous to chiral edge states of quantum Hall effect.

• N>0 cw, N<0: ccw mode

bands below

Ch #(clockwise chiraledgestates in the gap at )n

n E

N E

2

Ch2

n nBZ

d kk

Berry curvature

k

bulk mode: Chern number= Ch1

Ch1 topological edge modes

bulk mode: Chern number= Ch2

bulk mode: Chern number= Ch3

(Ch1+Ch2) topological edge modes

( ) n nn

u uk i

k k

Why the Chern number is related

with the number of edge states within the gap?

Mathematics : Index theorem

Physics : Analogy to quantum Hall effect

(in electrons)

Quantum Hall effect of electrons:

-- Chern number & edge states --

Electric field //y

2

( )Fnk

xy nz

E E

eB k

( ) nk nkn

u uB k i

k k: Berry curvature

Hall conductivity for 2D systems (of electrons)

( )

eEy

E E

2 2( )

( )1 1 x

x x

ev eEyj f E j f E

L L E E

Current along x

2

22( )

1 y x

xy

v vie h f E

L E E

Hall conductivity (Kubo formula)

2

:filledband

Chxy n

n

e

h

2

Ch ( )2

n nzBZ

d kB k : Chern number = integer

integer QHE

Example: insulator

Hall conductivity is expressed as a sum of Chern numbers

at T=0

0

0at 0

at

t

t T

Laughlin gedanken experiment: Chern number & edge states

y

y

AL

0

2

0 0Ch Chx xy

y

y

y y y

e ej

TL h TL TLE

TL

ChQ e

Number of electron carried from left end to right end = Ch

Increase the flux charge transport along x

This charge transport is

between the edge states

on the left and right ends.

gapless edge modes exist.

E

j

Gradual change of vector pot. = change of wavenumber

Edge states at the right end

Integer quantum Hall effect

Ch=1

Ch=1

Ch=1 2 chiral

edge states

1 chiral

edge mode

0 edge mode

B

Topological photonic crystals Theory: Haldane, Raghu, PRL100, 013904 (2008) Experiment: Wang et al., Nature 461, 772 (2009)

Ch=0

Ch=1

1 Chiral

edge mode

Photonic gap

Photonic gap

Landau-Lifshitz equation

Maxwell equation (magnetostatic approx.)

Linearized EOM

exchange field (quantum mechanical short-range)

Dipolar field (classical, long range)

External field

• Saturation magnetization Ms • exchange interaction length Q

2D Magnonic Crystal : periodically modulated magnetic materials

YIG (host) Iron (substitute)

ax

ay

H// z modulated

bosonic Bogoliubov – de Gennes eq.

dipolar interaction

non-trivial Chern integer

topological chiral edge modes

(like in quantum Hall effect)

x y

y

x

a a

ar

a

: unit cell size

: aspect ratio of unit cell

YIG (host) Iron (substitute)

ax

ay

H// z `exchange’ regime

`dipolar’ regime

• Phase diagram

Magnonic gap between

1st and 2nd bands

2

1 1( )2BZ

d kC k

C1 (Chern number)

for 1st band

=Number of

topological chiral

modes within the gap

2nd Lowest band

Lowest magnon band

[10G

Hz]

chiral magnonic band in magnonic crystal

bulk

bulk f=4.2GHz

bulk

f=4.5GHz

bulk

f=4.4GHz

edge

Simulation (by Dr. Ohe) DC magnetic field : out-of-plane AC magnetic field : in-plane

Magnonic crystals with ferromagnetic dot array R. Shindou, J. Ohe, R. Matsumoto, S. Murakami, E. Saitoh, PRB (2013)

dot (=thin magnetic disc) cluster: forming “atomic orbitals”

convenient for (1) understanding how the topological phases appear

(2) designing topological phases

decorated square lattice decorated honeycomb lattice

Each island is assumed to behave as monodomain H//z

H//z

Equilibrium spin configuration

2.4, 2 1.2, 1.70, 1.0x y se e r V M

Magnetostatic energy

Magnonic crystals: decorated square lattice

Hext

Hext

Hext < Hc=1.71

Hext > Hc

Collinear // Hext

Tilted along Hext

• Spin-wave bands and Chern numbers

H=0

H=0.47Hc

H=0.82Hc

H=1.01Hc

H=1.1Hc

H=1.4Hc

Red: Ch=-1

Blue: Ch=+1

Time-reversal

symmetry

Topologically

trivial

•Small H<<Hc

•Large H>>Hc

Dipolar interaction is

weak

Nontrivial phases (i.e. nonzero Chern number )

in the intermediate magnetic field strength

topological chiral edge modes

Topologically nontrivial

= chiral edge modes

+1 chiral mode -1 chiral mode

Change of Chern number at gap closing event

parameter

Chi

Chj

Ch’i

Ch’j

( Chi +ChI =Ch’i +Ch’I: sum is conserved)

DChi = -DChj = ±1

: Dirac cone at gap closing

volume mode (=bulk)

gap closes

magnonic

(volume-mode)

bands

H=0.47Hc

H=0.76Hc

H=0.82Hc

Magnonic crystals: edge states and Chern numbers (1)

Red: Ch=-1

Blue: Ch=+1

+1 chiral mode

-1 chiral mode

-1 chiral mode

+1 chiral mode

Edge states

bulk Strip geometry

(bulk+edge)

“atomic orbitals” within a single cluster

Spin wave excitations: “atomic orbitals”

relative phase for precessions

H<Hc: noncollinear H>Hc:

collinear // z

nJ=1 and nJ=3 degenerate at H=0

nJ=0 softens

at H=Hc

nJ=2 is lowest at H=0: favorable for dipolar int.

nJ=0

(s-orbital)

nJ=+1

(px+ipy-orbital)

nJ=3

(px-ipy-orbital) nJ=2

(dx2-y2-orbital)

Equilibrium configuration

Energy levels of atomic orbitals

Magnonic crystals: tight-binding model with atomic orbitals

Gap closes at M

(example) :

H=0.47Hc H=0.82Hc

gap between 3rd and 4th bands

retain only nJ=0 and nJ=1 orbitals

tight binding model

complex phase for hopping

px+ipy orbitals

nJ=0

(s-orbital)

nJ=+1

(px+ipy-orbital)

+i

= Model for quantum anomalous Hall effect

(e.g. Bernevig et al., Science 314, 1757 (2006))

Design of topological magnonic crystals

• Metal or insulator ? either is OK

(for metal: damping might be a problem for observation of edge modes.)

• Symmetry restrictions:

• Usually, the magnonic crystals with in-plane magnetic field

is non-topological by symmetry reasons.

• Out-of-plane magnetic field is OK.

(cf.: Thin-film with out-of-plane magnetic field has nonzero

Berry curvature.)

• Magnonic gap is required

Strong contrast within unit cell is usually required.

1 ferromagnet and vacuum

2 ferromagnets with very different saturation magnetization

• Multiband & sublattice structure:

nonzero Berry curvature requires multiband structure

Sublattice structure is helpful for designing topological magnonic crystals.

Design of topological magnonic crystals

• Metal or insulator ? either is OK

(for metal: damping might be a problem for observation of edge modes.)

• Symmetry restrictions:

• Usually, the magnonic crystals with in-plane magnetic field

is non-topological by symmetry reasons.

• Out-of-plane magnetic field is OK.

(cf.: Thin-film with out-of-plane magnetic field has nonzero

Berry curvature.)

• Magnonic gap is required

Strong contrast within unit cell is usually required.

1 ferromagnet and vacuum

2 ferromagnets with very different saturation magnetization

• Multiband & sublattice structure:

nonzero Berry curvature requires multiband structure

Sublattice structure is helpful for designing topological magnonic crystals.

Summary • Magnon thermal Hall effect (Righi-Leduc effect)

• Topological chiral modes in magnonic crystals magnonic crystal with dipolar int.

Berry curvature & Chern number

Array of columnar magnet in other magnet phases with different Chern numbers by changing lattice constant

Array of disks non-zero Chern numbers

atomic orbitals tight-binding model reproduce spin-wave bands

• Matsumoto, Murakami, Phys. Rev. Lett. 106,197202 (2011)

• Matsumoto, Murakami, Phys. Rev. B 84, 184406 (2011)

• Shindou, Matsumoto, Ohe, Murakami, PRB87, 174427 (2013)

• Shindou, Ohe, Matsumoto, Murakami, Saitoh, PRB87, 174402 (2013)