Shuichi Murakami Department of Physics, Tokyo Institute of ...
Transcript of Shuichi Murakami Department of Physics, Tokyo Institute of ...
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)