Transcript Spin-chirality induced anomalous Hall effect in pyrochlore ferromagnets

@ISSP
August 20th, 2003
Spin-chirality induced anomalous Hall effect
in pyrochlore ferromagnets
Y. Taguchi1, Y. Oohara2, H. Yoshizawa2, N. Nagaosa3,4,
T. Sasaki1, S. Awaji1, Y. Iwasa1, T. Tayama2, T. Sakakibara2,
T. Ito4, S. Iguchi3, K. Ohgushi3 and Y. Tokura3,4, 5
1Institute
for Materials Research, Tohoku University, Japan
2Institute for Solid State Physics, University of Tokyo, Japan
3Department of Applied Physics, University of Tokyo, Japan
4Correlated Electron Research Center, AIST, Japan
5Spin Super-Structure Project, ERATO, JST, Japan
Contents
1. Introduction of R2Mo2O7
2. Anomalous temperature dependence of sxy in
Nd2Mo2O7
3. Sign reversal of rH in Nd2Mo2O7
4. Sensitive dependence of sxy on the band-filling
in (Sm1-xCax) 2Mo2O7
5. Summary
1. Introduction
Pyrochlore-type structure : R2 Mo2 O7
Mo-sublattice, composed of
corner-sharing teterahedra.
(111)-plane is Kagome lattice.
Mo
R
Geometrical frustration in pyrochlore lattice
AF-coupled spin system
？
F-coupled spin system with
strong single-ion anisotropy
？
“Spin glass”
“Spin Ice”
M. J. Harris et al., Phys. Rev. Lett. 79, 2554 (1997)
A. P. Ramirez et al., Nature 399, 333 (1999)
Electron configuration
Mo4+ : 4d2
(111)
Mo ion is basically coordinated
octahedrally by O ions, but the
site-symmetry is D3d .
eg
Mo
4d
eg
eg
t2g
O2-
a1g
Oh
D3d
Phase diagram
Mo4+ : 4d 2 (constant filling)
100
Nd
Temperature (K)
80
Sm
60
Gd
40
Dy Tb
HoY
20 Er
Ferromagnetic
metal
Spin glass
insulator
0
1
small
1.05
3+
R Ionic Radius
W/U
1.1
large
Electronic structure
3
1
Y2Mo2O7
Eg～0.1 eV
R2Mo2O7 T = 10 K
1
Mo 4d
R = Sm
Mo 4d
R =Y
0
0
0.2 0.4
E (eV)
R = Sm
0
0
Sm2Mo2O7
2
-1
-1
σ(ω) (10 Ω cm )
2
3
-1
-1
Optical Conductivity (10 Ω cm )
・small Drude weight in Sm2Mo2O7
even in the ground state.
0.6
O 2p
O 2p
R=Y
1
2
3
4
Photon Energy (eV)
5
Mott-Hubbard type
(as opposed to CT type)
EF
2. Anomalous temperature dependence of
sxy in Nd2Mo2O7
Strong single-ion anisotropy
in Nd - moment
Anisotropy of Nd moments is
transmitted to Mo spins via the
f-d interaction.
“two-in, two-out”
Nd 4f
Nd
Jfd
Mo
Resistivity, magnetization, and neutron diffraction
TC
Below TC ～ 90 K
Nd2Mo2O7
30
Nd 4f
(localized moment)
Mo 4d
(conduction electron)
I(200)
I(111)
20
10
1.5
0
1
1
0.5
0
0
M(H= 0.5 T) ( m B/Mo)
Resistivity (mW cm) I ( m B2/2Nd2Mo2O7)
T*
0.5
*
100
Temperature (K)
T 50
0
150
Below T* ～40 K
T* is a crossover
temperature where Nd
moments begin to grow
rapidly, resulting in a
decrease of total M.
Magnetic Structure determined by the neutron diffraction experiment
“umbrella structure”
at 8 K
qN ～ 70～80°
mN ～ 2.2mB
qM< 10°
mM ～ 1.4mB
A magnetic unit cell contains
four Nd-moments and four
Mo-spins.
Nd 4f
at 40 K( = T*)
mN～0.2mB
mM～1.3mB
Mo 4d
anomalous Hall effect in magnetic metals
rH = RoH + 4 p RsM
Ey
jx
rH
H
4 p RsM
M
rH = Ey/jx = RoH + 4pRsM
RoH : ordinary term (proportional to H)
4pRsM : anomalous term (proportional to
M)
RoH
Magnetic Field
rH =
RoH
+
ordinary term
6
4pRsM
anomalous term
6
2K
5
Nd2Mo2O7
H = 0.5 T
H || (100)
r H (10-6 W cm)
10 K
4
TC = 89 K
20 K
30 K
40 K
3
50 K
60 K
2
r H (10-6 W cm)
H || (100)
4
TC
2
70 K
80 K
90 K
1
0
100 K
0
2 4 6 8 10
Magnetic Field (T)
0
0
50
100
Temperature (K)
low temperature data down to 0.5 K
6
Saturation of rH is observed
only below 2 or 1.5 K.
Nd2Mo2O7
H = 0.5 T
r H (10-6 W cm)
5
H || (100)
4
H || (110)
3
H || (111)
2
0
2
4
6
8
Temperature (K)
10
T*
-1
20
s
10
1
5
0
0
50
100
Temperature (K)
0
150
-1
H || (100)
H || (110) 2
H || (111)
3
0
15
Y. Taguchi, Y. Oohara, H. Yoshizawa,
N. Nagaosa, and Y. Tokura, Science
291, 2573 (2001)
s xx(H=0.5T) (10 W cm )
1
-1
-1
・sxy for every direction
continues to increase
down to 2 K.
2
xy(H=0.5 T) ( W cm ) MMo ( m B/Mo)
Anomalous T-dependence
of transverse conductivity
sxy =rH / (rxx2+rH2)
TC
Existing theories for anomalous Hall effect
Karplus and Luttinger, Phys. Rev. 95 (1954) 1154
L-S coupling of itinerant electron and imbalance of upand down-spin electron
R ∝ r2
s
J. Kondo, Prog. Theoret. Phys. 27 (1962) 772
interaction between conduction electron and localized
moment
rH ∝〈(m -〈m〉)3〉
Both theory predict
rH
0 when T
0.
Experimental results for
other ferromagnetic metals
Fe
La1-xCaxMnO3
P. Matl et al., Phys. Rev. B 57, 10248
(1998)
C. H. Chun, M. B. Salamon, Y.Tomioka, and
Y. Tokura, Phys. Rev. B 61, R9225 (2000)
Berry phase theory of anomalous Hall effect
J. Ye, Y. B. Kim, A. J. Millis, B. I. Shiraiman, P. Majumdar, and Z. Tesanovic,
Phys. Rev. Lett. 83, (1999) 3737
K. Ohgushi, S. Murakami, and N. Nagaosa, Phys. Rev. B 62, (2000) R6065
JH
Si
Sj
Carrier moving in a spin
background
with
strong
Hund's rule coupling JH ≫ t
tij = t0 cos(qij /2) exp(i aij)
A carrier moving in a topologically nontrivial spin background acquires a “Berry
phase” and feels fictitious magnetic field b.
anomalous Hall effect

  
b  S1  S2  S3
Theoretical calculation based
upon the Berry phase scenario
80
・Experimental result is
reproduced with Mo-spin
tilting angle of 4 - 5 degree.
-1
-1
sxy (W cm )
60
40
H || (100)
20
0
0
H || (111)
5
Tilting Angle (degree)
10
3. Sign reversal of rxy
Prediction by the Berry phase theory:
ρH
ρH
changes its sign for H || [111]
approaches zero without changing sign
for H || [100] and [110]
High-field measurements
Hall effect:
up to 27 T at 1.6 K
Magnetization: up to 23 T at 1.7 K, Vibrating-Sample Magnetometer
High magnetic-field was provided
by a hybrid magnet @ IMR, Tohoku University.
Low-T measurements
Magnetization: down to 50 mK for H < 12 T @ISSP, Univ. of Tokyo
Field-dependence of rH and M for H || [100] and [110]
Magnetization
M (mB/NdMoO3.5)
3
H || [100]
[100]
[110]
Mo : 1.4mB
2
H || [110]
1
T = 1.7K
T-independent below 1.7 K
70 mK for H || [100]
50 mK for H || [110]
gradual magnetization process
Mo
Nd : 111 - Ising spin, 2.3mB
rH (mW cm)
0
H || [100]
5
Nd 2Mo 2O7
T = 1.6 K
Hall effect
rH monotonously
approaches zero .
H || [110]
0
0
10
20
Magnetic Field (T)
30
c.f. T. Kageyama et al.JPSJ 70, 3006 (2001)
rH (mW cm)
M (mB/NdMoO3.5)
Field-dependence of rH and M for H || [111]
3
H || [111]
2
3-in, 1-out
2-in, 2-out
1
Mo
T = 1.7 K
65 mK
0
4
2
Nd2Mo2O7
H || [111]
T = 1.6 K
0
-2
0
Magnetization
10
20
Magnetic Field (T)
Mo : 1.4mB
Nd : 111 - Ising spin, 2.3mB
T-independent below 1.7 K
gradual magnetization process
c.f. Dy2Ti2O7
Hall effect
rH changes its sign at 7.5 T in
accord with the prediction.
30
Field dependence of Spin Chirality

   
bMo   S i  S j  S k nijk

H || [100]

i , j ,k
fictitious field that penetrates the Mo-tetrahedron
Low Field
High Field


bMo  H  0


bMo  H  0
Field dependence of Spin Chirality
H|| [111]
Low Field
High Field


bMo  H  0


bMo  H  0
4. Sensitive dependence of transverse conductivity
on the band-filling in (Sm1-xCax)2Mo2O7
anomalous Hall effect in (Sm0.9A0.1)2Mo2O7 : A = Ca2+, Y3+
In both cases, rxx and M show little variation.
In case of A = Ca, sxy shows large variation.
In case of A= Y, sxy shows little variation.
role of Ca-doping
・ to introduce scattering center
・ to partially remove f-d interaction
(Ca2+ is non-magnetic )
・ to change the band-filling
Important !
rxx and magnetization
Resistivity (m W cm)
1.5
R2Mo2O7
In general,
1
rH
0.5
R = Sm
Sm0.9Ca0.1
Sm0.9Y0.1
Magnetization ( m B/Mo)
0
rH
s xy  2
 2
2
r xx  r H r xx
r H  4pRs M
The variation of rxx and M is within 30 %.
1
We would expect little change of sxy.
0.5
0
However…...
Sm
Sm0.9Ca0.1
Sm0.9Y0.1
0
20 40 60 80
Temperature (K)
100
sxy is enhanced by as large as 800% !
Sm0.9Ca0.1
20
-1
-1
xy(H=0.5T) (W cm )
30
10
s
Sm
Sm0.9Y0.1
0
0
20 40 60 80
Temperature (K)
100
Explanation based upon the Berry phase mechanism
M. Onoda and N. Nagaosa,
J. Phys. Soc. Jpn. 71, 19 (2002).
s xy  e
2


n ,k



 
f ( n (k ))  k  An (k ) z
 
k  An (k )

f ( n (k )) : Fermi distribution func.


 
An (k )  i nk  k nk
 
 k  An (k ) : gauge flux density
sxy
Gauge flux density exhibits sharp
peaks at band crossing points. As a
result, sxy sensitively depends on
the position of chemical potential.
ky
kx
EF
Summary
The sxy in Nd2Mo2O7 continuously increases down to 2 K.
The rH in Nd2Mo2O7 changed its sign when the field was
applied along [111] direction while it monotonously
approached zero when applied along [100] and [110]
direction.
The sxy in (Sm1-xCax)2Mo2O7 shows large variation with the
change of band-filling in spite of little change of sxy and M.
These results suggest that the transverse conductivity in
these compounds are induced by the spin chirality.