Magnetic structures and anisotropic excitations
in Tb2Ti2O7 spin liquid
I.Mirebeau, S.Petit , A. Gukasov, J.Robert,
thesis S.Guitteny,
Laboratoire Léon Brillouin, CEA-Saclay
P.Bonville
DSM/IRAMIS/SPEC, CEA-Saclay
C.Decorse
ICMMO, Université Paris XI
H.Mutka, J.Ollivier, M.Boehm, P.Steffens
Institut Laue Langevin, Grenoble
A.Sazonov
LLB, Aachen University
Tb2Ti2O7: a hot topic
7 Posters at HFM’14
Kermarrec
Malkin
Fennel
Hallas
Kao
Sazonov
Yin
Why is Tb2Ti2O7 (or TTO) so interesting ?
Tb2Ti2O7: a hot topic
Spin
liquid
Antiferromagnetic
spin ice
quantum
spin ice
magnetoelastic
liquid
TTO
Spin
Glass
because nobody fully understands it!
Tb2Ti2O7: a hot topic
In the last 3 years
More and more sophisticated experiments
• Searching for a magnetization plateau : H //111
• Probing dispersive excitations
Influence of tiny defects
• ½ ½ ½ structure
• Competing SRO structures : Spin glass like vs. mesoscopic order
Coupling with the lattice
• magneto-elastic mode
• Dynamic Jahn-Teller transition and/or interactions between quadrupolar moments
Towards a more realistic description ?
Dipolar Spin ices: The Ising case
Mc. Clarthy- Gingras Rev Modern Phys. (
R2Ti2O7 pyrochlores R=Dy, Ho
Effective interaction Jeff = J+Ddip > 0
Tb nearby the threshold
Quantum fluctuations at
play: « quantum spin ice »
Molavian, Gingras, Canals, PRL (2008)
Molavian , Clarthy, Gingras arxiv0912.2957
Mc. Clarthy- Gingras Rev Progress Physics
77 056501(2014)
Tb
Dy
AF
FeF3
4in-4out
What about the Crystal
field ?
Ho
Dipolar spin ice
Spin ice
Den Hertog et al Phys. Rev. Lett. (1999)
Bramwell et al Phys. Rev. Lett (2000)
The crystal field
Tb3+ is a non-Kramers ion
Δ = 200 – 300K
Ho, Dy
spin ices
Δ ~ 1.5 meV
Gingras, PRB (2000)
Bonville, IM, PRB( 2007)
Bertin,Chapuis, JPCM(2012)
Zhang, Fritsch, PRB (2014)
Klekovina- Malkin J Opt. Phys. (2014)
Δ = 10-20K (Tb)
𝜓1 𝑗 𝜓2 = 𝜓2 𝑗 𝜓1 = 0
𝜓1 𝐽𝑧 𝜓1 = −4𝑎 2 + 5𝑏2
𝜓2 𝐽𝑧 𝜓2 = - 𝜓1 𝐽𝑧 𝜓1
•
Cao et al PRL(2009)
Strong but finite <111> anisotropy
•
No exchange fluctuations allowed within
the GS doublet
No intensity scattered by neutrons
Splitting of the Ground state doublet
In molecular field approach
Quantum mixing in the GS.
1st order perturbation
0th order perturbation
Δ ~ 1.5 meV
Δ ~ 1.5 meV
ℎ
|𝜓′1 = |𝜓1 + Δ |𝜓1 𝑒
h
|𝜓′2 = |𝜓2 +
ℎ
| 𝜓2 𝑒
Δ
d
|𝜓′1 =
|𝜓1
− |𝜓2
2
+ |𝜓2
|𝜓′2 =
Simplest case: entangled
wave2 functions
h: molecular field
|𝜓1
D: quantum mixing
I α | 𝜓 ′1 𝐽 𝜓 ′ 2 |2 . 𝛿 𝜔 − 𝐸1 − 𝐸2
ℎ
I α (∆)2
(gjµBh/)2
1
2
( 𝜓1 𝐽𝑧 𝜓1
𝜓′1 𝐽𝑧 𝜓′2 =
- 𝜓2 𝐽𝑧 𝜓2 )= 𝜓1 𝐽𝑧 𝜓1 ≠ 0
But 𝜓′1 𝐽𝑧 𝜓′1 =0
(0.75/15)22.10-3
gJµB/kB= 1 for Tb !
Splitting of the Ground state doublet
In molecular field approach
Quantum mixing in the GS.
1st order perturbation
0th order perturbation
Δ ~ 1.5 meV
Δ ~ 1.5 meV
ℎ
|𝜓′1 = |𝜓1 + Δ |𝜓1 𝑒
h
|𝜓′2 = |𝜓2 +
ℎ
| 𝜓2 𝑒
Δ
h: molecular field
d
•
Very small intensity associated with
GS fluctuations (with resp. to CF )
Spin ice anisotropy: magnetization
plateau
Molavian, Gingras, Canals PRL(2007)
Molavian, McClarthy, Gingras arxiv(2009)
− |𝜓2
2
+ |𝜓2
|𝜓′2 =
Simplest case: entangled
wave2 functions
Virtual crystal field model
•
|𝜓′1 =
|𝜓1
|𝜓1
Two singlet ground state
•
•
each singlet is non magnetic : no static signal
the transition has a large spectral weight
• Jahn-Teller distortion?
Bonville et al PRB(2011), PRB (2014)
Searching for a magnetization plateau
Using Magnetization, susceptibility, MuSR : a controversial situation
low field anomalies of the susceptibility:
No plateau in the isothermal magnetization
MuSR
Baker PRB (2012)
Yin et al PRL(2013)
Lhotel et al PRB-RC (2012)
cross over regime in the dynamics
Fritsch , PRB(2014)
TF~200-400 mK
Spin glass-like freezing ?
Legl et al PRL (2012)
Searching for a magnetization plateau
Using neutrons :
magnetic structure for H//111
• Exclude all-in all out structure
• Gradual reorientation of the Tb moments in the
Kagome plane (keeping 1in- 3 out) without Kagome
ice structure
See poster A. Sazonov
Searching for a magnetization plateau
D=0 no mixing
A. Sazonov et al PRB(2013)
Field Irreversibilities
•
No evidence for the 1/3 plateau at ~2µB expected
at very small fields (down to 80mK)
•
quantitative agreement with MF model assuming
a dynamical JT distortion:
•
•
4 moment values and angles
M(H) for H//100, 111, 110
Spin glass like freezing?
• see poster A. Sazonov
Spin fluctuations at very low temperature
Using unpolarized neutrons
2 components in the neutron cross section
• elastic (dominant)
• inelastic (low energy)
inelastic
See also:
Takatsu et al.
JPCM (2011)
Fritsch et al
PRB(2013)
elastic
D=0.25K
•
•
•
•
Pinch points
diffuse maxima at ½ ½ ½ positions
becomes structured at low T
well accounted for by 2 singlet model + anisotropic
exchange
Static character not reproduced by the 2 singlet model
diffuse scattering
The main features of the diffuse scattering are reproduced
b = -0.13T/µB ; DQ=0.25K
3d-map
Experiment
Phase diagram
6T2 ( LLB)
Energy integrated
intensity
Simulation
P. Bonville et al Phys. Rev. B (2011)
Simulation with
•
•
•
•
•
anisotropic exchange
dipolar interactions
CF
JT distortion along equivalent 100,
010, 001 cubic axes.( preserves the overall
cubic symmetry)
Dynamical JT (average Structure factors
and not intensities)
Q dependence of the elastic scattering
• Pinch points in both compounds: Coulomb phase
𝑇𝑏2 𝑇𝑖2 𝑂7 - 50 mK
no spectral weight at Q=0
½ ½ ½ maxima : AF correlations
S.Petit & al, PRB 86 (2012)
𝐻𝑜2 𝑇𝑖2 𝑂7 - 50 mK
strong spectral weight at Q=0
T.Fennell & al, Science 326 (2009)
Analysis of the pinch points
Strongly anisotropic correlations of algebric nature
conservation law in TTO spin liquid analogous to the ice rules
S.Guitteny & al, PRL 111 (2013)
What are the spin component involved?
T. Fennell et al
PRL(2012)
Polarization analysis
Longitudinal polarimetry separates spin components
neutron polarization P// Z
Neutron cross section
•
•
•
•
Non spin flip: N+ <MZ.Mz>
Spin Flip
<My.My>
Correlations along Q (or x)
between spin components M┴Q
z
Mz
Ho2Ti2O7
Z //110
Fennell Science (2009) : Ho2Ti2O7
PRL (2013) Tb2Ti2O7
1
3
x// Q
4
2
My
1’
y
x
Q
2’
NSF: correlations « up-down » 1-1’ or 2-2’:
Weak
(2 Spins, between T)
SF: correlations « 2in-2 out » 1-2-3-4:
Strong
(4 spins, in a T)
Polarization +energy analysis
Longitudinal polarimetry separates spin components
neutron polarization P// Z
Neutron cross section
•
•
•
•
Non spin flip: N+ <MZ.Mz>
Spin Flip
<My.My>
Tb2Ti2O7
Z //110
T=50 mK
1
3
Look at the dispersion
z
Mz
My
1’
x// Q
4
2
Correlations along Q (or x)
between spin components M┴Q
Fennell Science (2009) : Ho2Ti2O7
PRL (2013) Tb2Ti2O7
2’
Mz: « up-down » correlations: relaxing (Quasi-E)
My: « 2 in-2out » correlations : dispersing (Inel.)
y
x
Q
Low energy excitations
First observation of a dispersive excitation in fluctuating disordered medium
Mz
• In all directions
• Quasi-élastic
• Strong fluctuations
My
• Along (h,h,h)
• quasi-élastic
• along (h,h,2-h) et (h,h,0)
• propagating excitation
• no gap (Δres = 0,07meV)
• Disperses up to 0,3 meV
• intensity varies like 1/ω
S. Guitteny et al PRL(2013)
18
Nature of the static SRO? the ½ ½ ½ order
Short range vs. mesoscopic order
In single crystals
½ ½ ½ diffuse maxima
Fennel PRL (2012)
Fristch PRB(2012)
• Short range ~8-10 A
Petit PRB (2012)
• below ~0.4K
• Vanish in a small field ( ~200G)
In powders
½ ½ ½ Mesoscopic structure
• Over 30-50A
• Associated with Cp anomaly
• tuned by minute defects in Tb content
Taniguchi PRB RC(2011)
See also poster E. Kermarrec
powder samples Tb2+xTi2-xO7+y
T=50mK
Mesoscopic structure for x=0 and x=0.01
Difference pattern: I(50 mk)- I(1K)
½½½
½ ½ 3/2
X=0
Neutron counts
N
½ ½ 5/2
3/2 3/2 1/2
X=0
exp: P. Dalmas de Réotier
2 q (deg)
Symmetry analysis
space group Fd-3M, K= ½ ½ ½
2 orbits with no common IR
N
1
2
3
4
site
000
¾¼½
¼½¾
½¾ ¼
Champion, PRB (2001)
Stewart, Wills JPCM(2004)
Gd2Ti2O7
K // local <111> axis
no intensity at ½ ½ ½
site 1
Sites 2-4
No way to build a strong ½ ½ ½ peak for Ising spins!
Needs to break either Ising anisotropy or cubic symmetry
Systematic search of magnetic structures
•
•
•
1T
cfc translations (cubic cell : a)
K= ½ ½ ½ (magnetic unit cell: 2a)
•
No vectors of the IR
along the local <111> axes
• Contributions to ½ ½ ½
cancel by symmetry
The best structures (x=0)
moments remain close to local <111>axes (3-10 deg)
« Monopole layered structure »
« AF -Ordered spin ice »
X=0
X=0
M=1.9(4) µB/Tb; Lc =60 A (Y=1.4)
Correlation length ~30 -50 A
The best structures (x=0)
moments remain close to local <111>axes (<10 degs)
« Monopole layered structure »
Ferrimagnetic piling of SI Tetrahedra
« AF -Ordered spin ice »
Fritsch PRB (2012)
AF packed OSI cubic cells,
Z//001
MZ
S. Guitteny (thesis)
derived from Tb2Sn2O7 I. M et al PRL (2005)
The best structures (x=0)
moments remain close to local <111>axes (<10 degs)
« Monopole layered structure »
Ferrimagnetic piling of SI Tetrahedra separated by
monopole layers
« AF -Ordered spin ice »
Fritsch PRB (2012)
AF packed OSI cubic cells, separated by SI tetrahedra
with M 
Z//001
MZ
Full of monopoles, but compatible with a distortion
No monopoles, but symmetry breaking at each cubic cell
no possible LRO?
Calculated diffuse scattering
In a single crystal, correlation length reduced to 2 cubic cells
« AF -Ordered spin ice »
« Monopole layered structure »
4
3
3
0, 0, l
0, 0, l
4
2
2
1
1
1
2
h, h, 0
Experiments
Petit PRB (2013)
Fennel PRL (2013)
Fritsch PRB(2013)
3
4
1
2
h, h, 0
3
4
The ½ ½ ½ order: summary
• ½ ½ ½ order cannot propagate without breaking the cubic symmetry
• different structures and/or K orientations may compete (in space, time) yielding:
• SRO (single crystal)
• mesoscopic orders (powders, tuned by x)
• Spin glass like irreversibilities : Yin (2013), Fritsch PRB (2014) , Lhotel (2013)
• 2 physical mechanisms at play for the magnetic excitations
• Relaxation (quasielastic)
• Dispersive excitations
• Analog to the double dynamics in SP particles or quantum molecular magnets
Quasielastic or slow relaxations
(thermally activated ,QT)
Magneto-elastic modes as a switching
mechanism?
Inelastic
modes
Probing the magneto-elastic coupling
Interaction between 1st excited CF doublet and acoustic phonon branch
Guitteny PRL(2013)
see also:
Fennel PRL(2013)
this conf.
M. Ruminy : next talk
Other probes
• pressure induced magnetic order
IM et al Nature 2002, PRL(2004)
•Elastic constants
Klekovina-Malkin
J. Phys. 2011, J. Opt. Phys. 2014
•Thermal conductivity
Li et al PRB(2013)
Summary: what is new in TTO?
• Quantum mixing in the GS doublet due to quadrupolar order: a necessary ingredient
•
MF
•
JT distortion
« exchange » int. between quadrupolar
moments
Magnetoelastic coupling
•
Non-Kramers character is crucial
Gehring-Gehring (1985)
Savary-Balents PRL(2012)
Lee-Onoda-Balents PRB(2012)
• First observation of dispersive anisotropic excitations
in a fluctuating disordered medium
Two types of dynamics : relaxation, excitations
• Competing SI correlations with K=½ ½ ½
•
•
•
•
Not compatible with cubic symmetry
Tuned by off-stoechiometries
With different time and length scales
Associated with glassy behaviour
poster Malkin
x=0.01
coexistence of LRO and mesoscopic orders
• Mesoscopic: M= 1.3µB/Tb
• LRO: M=0.3 µB/Tb
Pressure induced structures
Under pressure : a phase with larger unit cell is also stabilized
I.M et al Nature
(2002)