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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
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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
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Spin Hall effect of electrons
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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)
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• 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
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• 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
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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.)
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Spin Hall effect of light
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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)
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)||()( 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)
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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
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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)
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Hall effect of magnons
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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)
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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
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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
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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
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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
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– 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)
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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
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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?
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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
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Thermal Hall effect of magnon
• semiclassical theory
• If ∇T≠0 there appear net current + ≠0
Q Ej j j : heat current
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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
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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
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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
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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
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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
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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
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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
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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).
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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
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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)
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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
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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)
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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
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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
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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
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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).
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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