Berry curvature and Hall Effect of Magnons in Insulating Ferromagnets

Shuichi Murakami (Tokyo Institute of Technology)

Collaborators: Ryo Matsumoto (Tokyo Inst. Tech.)

Ryuichi Shindou (Tokyo Inst. Tech.)

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

• R. Matsumoto, S. Murakami, arXiv:1106.1987 (to appear in Phys. Rev. B)

KITP, UCSB

Oct.26, 2011

outline

1. Introduction spin Hall effect of electrons

spin Hall effect of photons

2. Semiclassical theory: rotational motion, Hall effect

3. Linear response theory

4. Application (Lu2V2O7, YIG)

5. Summary

self-rotation edge current

Thermal Hall effect

Spin Hall effect of electrons

Spin Hall effect

• p-type semiconductors

(SM, Nagaosa, Zhang, Science (2003))

Electric field Transverse pure spin current

( )S O

H U k

intrinsic spin Hall effect= spin-orbit coupling + Berry phase in k-space

• n-type semiconductors with Rashba coupling

(Sinova, Culcer, Niu, Sinitsyn, Jungwirth, and MacDonald, Phys. Rev. Lett. (2004))

Magnetic field

Magnetnot needed

Electric manipulation of spins

(without magnetic field)

• Determined by Bloch wf.

various Hall effects

spin Hall effect

Semiclassical theory Adams, Blount; Sundaram,Niu, …

( )1( )

n

n

E kx k k

k

k e E

n( : band index)

: periodic part of the Bloch wf.knu xki

knknexux

)()(

• Enhanced near band crossing

: Berry curvature = “magnetic field in k-space”

Boltzmann

transport

( )n n

n

u uk i

k k

Energy band Berry

curvature

• 3D n-type semicond., Kerr rotation

• Kato, Myers, Gossard, Awschalom, Science (2004)

• Sih et al. , Nature Phys. (2005) n-GaAs, n- (In,Ga)As

• Sih et al., PRL (2006)

• Stern et al., PRL(2006) n-ZnSe

• 2D p-type semicond., spin LED

• J. Wunderlich et al., PRL(2005)

p-GaAs

E

• Metal (Pt, Al,Au…) -- Inverse SHE

• Saitoh, Ueda, Miyajima, Tatara, APL (2006) Pt

• Valenzuela and Tinkham, Nature(2006) Al

• Seki et al., Nature Materials(2008) Au

• Metal (Pt) -- SHE & Inverse SHE

• Kimura, Otani, Sato,

Takahashi, Maekawa, PRL (2007) Pt

Experiments on spin Hall effect

RT

RT

RT

RT

G. Y. Guo, SM, T. W.Chen, N. Nagaosa, Phys. Rev. Lett. (2008)

Large spin Hall effect in Pt: first-principle calc.

Band crossing

near Fermi energy

( d-orbitals)

11240

cm

S

11360

cm

S

at RT

Cf. Experiment

(Kimura et al.)

Spin Hall effect of light

Spin Hall effect of light

-- Analogous to electrons --

Onoda, SM, Nagaosa, Phys. Rev. Lett. (2004)

Onoda, SM, Nagaosa, Phys. Rev. E (2006)

also in Dooghin et al. Phys. Rev. A (1992)

Libermann et al., Phys. Rev. A (1992)

)()(

rn

crv

Semiclassical eq. of motion

: slowly varying

)( k

: Berry curvature

)( k

: gauge field

Shift of a trajectory of light beam

“Spin Hall effect of light”

Polarization change

z : polarization

1

1)(

3k

kk

In the vacuum

Left circular pol.

right

Chiao,Wu(’86) : theory

Tomita,Chiao(’86) : experiment

zkkiz

rvkk

zkzkkrvr

)(

)(

)(ˆ)(

Geometrical optics

“Fermat’s principle”

spin-orbit coupling of light

(transverse only)

)||()( zzkk

krvr

k

k

k

Anomalous velocity due to Berry phase

Anomalous velocity =

2n

1n

)(21

nn

= transverse

shiftAnomalous velocity =

)( rvkk

1

1)(

3k

kk

Isotropic medium

Left circular pol.

Right circular pol.

Transverse shift at the interface

Theory: Fedorov (1955)

Experiment: Imbert(1972)

Imbert shift

Onoda, SM, Nagaosa, Phys. Rev. Lett. (2004)

Onoda, SM, Nagaosa, Phys. Rev. E (2006)

spin-orbit coupling of light

(transverse only)

Hosten, Kwiat, Science 319,787 (2008)

Magnitude of the shift

Width of the beam is much larger not easy to observe.

Experiments on imbert shift : Shift of light beam in reflection/refraction

28 total reflections shift is enhanced

Imbert, Phys. Rev. D5, 787 (1972)

total reflection refraction

Simulation

Dielectric constant Photonic band structure

Onoda, SM, Nagaosa, Phys. Rev. Lett.93, 083901 (2004)

Enhanced Berry

curvature near

band crossing

Spin Hall effect of light enhanced in photonic crystals

Large shift

enhancement

(2 orders of magnitude)

Hall effect of magnons

Introduction：magnons

magnon (spin wave)

• low energy excitation in magnets

(fluctuation from the ground state)

• Bose distribution function

• spin current (magnon transport)e.g.) Kajiwara et al., Nature 464, 262 (2010)

Introduction

• magnon transport in magnetic insulator

Kajiwara et al., Nature 464, 262 (2010)

(insulator !)

SHE

ISHE

• long distance (~1[mm])

• no Joule heating by electrons

• room temperature

electric current

spin current

spin wave

Motivation

electron system magnon system

• charge –e

• fermion

• no charge

• boson

wave nature

band structure

Berry phase in k-space

Berry curvature effect can be expected in magnon systems!

self-rotation edge currentThermal Hall effect

Magnon thermal Hall effect by Berry curvature

– previous works --

Theory:

• S. Fujimoto, Phys. Rev. Lett. 103, 047203 (2009).

• H. Katsura, N. Nagaosa, and P. A. Lee, Phys. Rev. Lett.104, 066403 (2010).

Experiment & theory:

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

Lu2V2O7 : Ferromagnet

Dyaloshinskii-Moriya interaction Berry phase

2

,

2Im

2

x y n k n k n k

n k

n k x y

u H u

V k k

Rotational motions of electrons in a magnetic field

due to Berry curvature

1. self-rotation ( cyclotron motion)

Yafet (1963)

Chang and Niu, PRB 53, 7010 (1996)

L is expressed by Berry curvature

2. edge current

: wave packet

: Berry curvature

Orbital

magnetic

moment

Gradient of confinement potential Hall current

– Electrons: charge e, fermion

– Magnons: no charge, boson

Wave nature = described by Bloch wf.

V(r)： confinement potential

From electrons to magnons (spin waves)

( )1( )

n

n

E kx k k

k

k V

electrons

( )n n

n

u uk i

k k

: Berry curvature

( )1( )

n

n

E kx k k

k

k e E x B

magnons

•Exchange magnons (quantum-mechanical)

e.g. Lu2V2O7

•Magnetostatic spin waves (classical)

e.g. YIG (yttrium iron garnet)

Orbital motions of magnon wavepacket

・self-rotation motion

・edge current of magnon

: Berry curvature

where

l r v : (reduced) angular momentum

Along the edge

: confining potential

Thermal Hall effect of magnon

• The gradient of temperature isnot dynamical force but statistical force.

deflection does

not occur

• How about in the semiclassical picture?

Thermal Hall effect of magnon

• semiclassical theory

• If ∇T＝0 there is no net current + = 0

• If ∇T≠0 there appear net current + ≠0

Along the edge

Edge current

Thermal Hall effect of magnon

• semiclassical theory

• If ∇T≠0 there appear net current + ≠0

Q Ej j j : heat current

Magnon Thermal Hall conductivity (Righi-Leduc effect)

where

Note:

It is different from previous results by linear response theory

(Katsura et al. PRL (2010), Onose et al., Science (2010)).

2

20

2

2

2

1lo g

1(1 ) lo g (lo g ) 2 L i ( )

tc d t

t

Semiclassical theory

Some terms missing !

modified linear response theory = identical result with semiclassical theory

Q xy yx

j T

Linear response theory

J: magnon current

JE: energy current

: gravitational field

Lij: transport coefficient

L. Smrčka and P. Středa, J. Phys. C, 10, 2153 (1977)

H. Oji, P. Středa, PRB 31, 7291 (1985)

cf) for electrons

J. M. Luttinger, Phys. Rev. 135, A1505 (1964)

• L11: Hall effect

• L12=L21: Nernst effect

• L22: thermal Hall effect

1 1 1 2

1J L U T L T

T T

1 2 2 2

1

EJ L U T L T

T T

MijSij0

Linear response theory• Linear response theory (to external field)

equilibriumdeviation

from equilibrium

• Mij: new correction term

•Theory： H. Katsura, N. Nagaosa, and P. A. Lee,

PRL 104, 066403 (2010)

•Experiment： Y. Onose, et al., Science 329, 297 (2010)

• Sij: calculated by Kubo formula

equilibrium deviation from equilibrium

Density matrix Current

0 0 1 0 0 1T r ( ) T r ( ) + T r ( ) + T r ( )J j r j r j r j r

0 0 1 0 0 1T r ( ) T r ( ) + T r ( ) + T r ( )

E E E E EJ j r j r j r j r

Linear response theory

Correction terms of currents

due to external fields

Equilibrium currents

Currents in the presence of fields

Magnon current

Energy current

0

1,

2j j

j

j r v r r

0 0

1,

2E

j r H j r

0 1 0 0

1,

2j

j

j r j r j r j r j r r

0 1

0 0 0 0

1 1, , , , ,

2 4

E E E

E j j j

j j

j r j r j r

j r j r U r j r r H H r j r

1

' ,2

j j

j j

H U r H r

0 0 1 0 0 1T r ( ) T r ( ) + T r ( ) + T r ( )J j r j r j r j r

0 0 1 0 0 1T r ( ) T r ( ) + T r ( ) + T r ( )

E E E E EJ j r j r j r j r

Linear response theory

1 1 1 2

1J L U T L T

T T

1 2 2 2

1

EJ L U T L T

T T

i j i j i jL S M

( )T r ( ) ( )i j i j i j

d G d GS i j j H j H j d

d d

1 2

1( )T r ( )

2M H r v r v d

1 10M

2 2( )T r ( ) ( )T r ( ) ,

4

iM H r v r v d H v v d

Magnon current

Energy current

Correction terms of currents

due to external fields

Linear response theoryFormula in terms of Berry curvature

2

2

2 2

1(1 ) lo g (lo g ) 2 L i ( )c

i j i j i jL S M

1(1 ) lo g 1 lo gc

Q EJ J J

Thermal Hall conductivity

Heat current

Correction

terms

Linear response theory

• New correction term

• Thermal Hall conductivity

the same result as the semicalssical theory !

These arises from

the orbital motion of magnon.

ex)

22

2

2 20

1 1lo g (1 ) lo g (lo g ) 2 L i ( )

tc d t

t

Example 1: Thermal Hall effect of magnons in Lu2V2O7

H. Katsura et al., PRL 104, 066403 (2010)

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

Lu2V2O7 : Ferromagnet

Dyaloshinskii-Moriya interaction Berry phase

Our work:

2

,

2Im

2

x y n k n k n k

n k

n k x y

u H u

V k k

2

2

,

2Im

xy n k n kB

n k

n k x y

u uk Tc

V k k

2

20

1lo g

tc d t

t

Matsumoto,Murakami, PRL106,197202 (2011)

arXiv: 1106.1987 (2011).

magnon magnetostatic spin wave

wave length Short (nm) Long (μm)

dominant interaction exchange

interaction

demagnetizing field

(dipole interaction)

mechanism quantum

mechanical

classical

electromagnetism

coherence length (typical value) [nm]~[μm] ~[mm] in some magnets

We similarly apply this theory to the magnetostatic spin wave!

Magnons: classical vs. quantum

Magnetostatic mode in YIG film

A. Kreisel, F. Sauli, L. Bartosch, and P. Kopietz, Eur. Phys. J. B 71, 59 (2009)

E.g.：Magnetostatic backward volume mode (MSBVM)

Magnetostatic mode

( demagnetizing field)

Magnon

( exchange)

Example 2: Magnetostatic modes in ferromagnetic films (YIG)

2

M

k

(a) MSSW (magnetostatic surface mode)

(b) MSBVW (magnetostatic backward volume mode)

(c) MSFVW (magnetostatic forward volume mode)

• Multiband system

wavefunction :

z

( )( )

( )

x

y

m zz

m z

m

Wavelength =

exchange interaction negligible

film demagnetization field (i.e dipolar interacition) =dominant

m - m m

Exchange interaction = isotropic : 2E Jk

Generalized eigenvalue eq.

• Landau-Lifshitz (LL) equation

• Maxwell equation

• Boundary conditions

M: magnetization, g: gyromagnetic ratio, H: external magnetic field

00 HB , (magnetostatic

limit)

//2//121HHBB

,

/ 2

/ 2

ˆ( ) ( , ) ( ) ( )L

H M yL

z d z G z z z z

m m m

wavespin the offrequency function, sGreen' the ofmatrix 22

fieldmagnetic static ion,magnetizat saturation

film, , film, the of thickness , y

::ˆ

::

0

0:,

00

00

gg

G

HM

zi

iLMH

MH

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

)( HMM

gdt

d

Example 2: Magnetostatic modes in ferromagnetic films (YIG)

Kalinikos, Slavin, J. Phys. C: Solid State Phys. 19, 7013 (1986).

2

M

k

2 2 2 1

2

21

2

( ) s in s in 2 c o s c o s c o s s in s in c o s s in 2ˆ ( , )

s in s in c o s s in 2 s in

P Q P Q P

Q P P

G z z iG G iG GG z z

iG G G

( , ) ex p ' , ( , ) ( , ) sg n ( )2

P Q P

kG z z k z z G z z G z z z z

Berry curvature

Generalized eigenvalue eq.

( )( )

( )

x

y

m zz

m z

m

z

/ 2

/ 2

ˆ( ) ( , ) ( ) ( )L

H M yL

z d z G z z z z

m m m

1n y n m m

( )n n

n yk i

k k

m m

( )n n

n

u uk i

k k

cf: for electrons: Schrödinger eq.

Example 2: Magnetostatic modes in ferromagnetic films (YIG)

Cf. Bogoliubov Hamiltonian

generalized eigenvalue problem.

We introduce the Berry curvature for the magnetostatic spin wave:

g

g

kk

n

y

n

n

kkmm

k,,

)( Im

(a) MagnetoStatic Surface Wave (MSSW)

(b) MagnetoStatic Backward Volume Wave (MSBVW)

(c) MagnetoStatic Forward Volume Wave (MSFVW)

0)( kz

n because of the symmetry

(2-fold in-plane rotation + time reversal )

We can expect the Berry curvature to be nonzero !

Example 2: Magnetostatic modes in ferromagnetic films (YIG)

Zero Berry curvature

Berry curvature for the MSFVW mode

2

2

12

1)(

n

Hn

H

z

n

kk

k

2

( ) 1c o s

( )

x x yn

y x y

m z i k kp k z

m y k i kN

2 2 2 2

: n o rm a liz a tio n c o n s ta n t

, M H M

H H

N

n : band index of the MSFVW mode

R. W. Damon and H. van de Vaart, J. Appl. Phys. 36, 3453 (1965)

Magnetostatic forward volume waves (MSFVW)

in ferromagnetic films (YIG)

2tan 1, 1

np p kz p

: Eigenmode eq. (n=0,1,2,…)

Wavefunctions

Numerical results:

Dispersion rof the MSFVW mode (H0/M0=1.0),

for n=0～5.

Berry curvature of MSFVW mode.

• R. Matsumoto, S. Murakami, Phys. Rev. Lett. 106,197202 (2011), arXiv: 1106.1987 (2011).

SummaryWavepacket of spin wave (magnon)

• Berry curvature

• 2 types of rotational motions

self rotation

edge current

• Magnon thermal Hall effect (Righi-Leduc effect)

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

• R. Matsumoto, S. Murakami, arXiv:1106.1987

l r v