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,

I.Mirebeau, S.Petit , A. Gukasov, J.Robert,thesis S.Guitteny, Laboratoire Léon Brillouin, CEA-Saclay

P.BonvilleDSM/IRAMIS/SPEC, CEA-Saclay

C.DecorseICMMO, Université Paris XI

H.Mutka, J.Ollivier, M.Boehm, P.SteffensInstitut Laue Langevin, Grenoble

A.Sazonov LLB, Aachen University

Magnetic structures and anisotropic excitationsin Tb2Ti2O7 spin liquid

Tb2Ti2O7: a hot topic

Why is Tb2Ti2O7 (or TTO) so interesting ?

7 Posters at HFM’14Kermarrec Malkin Fennel Hallas KaoSazonovYin

Tb2Ti2O7: a hot topic

because nobody fully understands it!

TTO

quantum spin iceSpin

liquid

Antiferro-magnetic spin ice

magneto-elastic liquid

Spin Glass

Tb2Ti2O7: a hot topic

More and more sophisticated experiments

Influence of tiny defects

Coupling with the lattice

In the last 3 years

• Searching for a magnetization plateau : H //111• Probing dispersive excitations

• ½ ½ ½ structure• Competing SRO structures : Spin glass like vs. mesoscopic order

• magneto-elastic mode• Dynamic Jahn-Teller transition and/or interactions between quadrupolar moments

Towards a more realistic description ?

Mc. Clarthy- Gingras Rev Modern Phys. ( Dipolar Spin ices: The Ising case

R2Ti2O7 pyrochlores R=Dy, Ho Effective interaction Jeff = J+Ddip > 0

Dipolar spin iceAF

FeF3

4in-4out

Spin ice Den Hertog et al Phys. Rev. Lett. (1999)

Bramwell et al Phys. Rev. Lett (2000)

Tb

DyHo

Tb nearby the thresholdQuantum fluctuations at play: « quantum spin ice » Molavian, Gingras, Canals, PRL (2008)Molavian , Clarthy, Gingras arxiv0912.2957Mc. Clarthy- Gingras Rev Progress Physics 77 056501(2014)

What about the Crystal field ?

The crystal field

Δ = 200 – 300K Ho, Dy spin ices

Δ = 10-20K (Tb)

Tb3+ is a non-Kramers ion

Strong but finite <111> anisotropy

Δ ~ 1.5 meV

=

=

= -

• No exchange fluctuations allowed within the GS doublet

• No intensity scattered by neutrons

Gingras, PRB (2000)Bonville, IM, PRB( 2007) Bertin,Chapuis, JPCM(2012) Zhang, Fritsch, PRB (2014) Klekovina- Malkin J Opt. Phys. (2014)

Cao et al PRL(2009)

Δ ~ 1.5 meV

= + = +

I α I α

h: molecular field

Splitting of the Ground state doublet

In molecular field approach

Δ ~ 1.5 meV

= =

- =

But =0

dh

Quantum mixing in the GS.

1st order perturbation 0th order perturbation

Simplest case: entangled wave functions

(gjµBh/)2 (0.75/15)22.10-3

D: quantum mixing

gJµB/kB= 1 for Tb !

Δ ~ 1.5 meV

= + = +

h: molecular field

Splitting of the Ground state doublet

In molecular field approach

Δ ~ 1.5 meV

= =

dh

Quantum mixing in the GS.

1st order perturbation 0th order perturbation

Simplest case: entangled wave functions

Virtual crystal field model

• Very small intensity associated with GS fluctuations (with resp. to CF )

• Spin ice anisotropy: magnetization plateau

Two singlet ground state

• each singlet is non magnetic : no static signal• the transition has a large spectral weight • Jahn-Teller distortion?

Molavian, Gingras, Canals PRL(2007)Molavian, McClarthy, Gingras arxiv(2009)

Bonville et al PRB(2011), PRB (2014)

Searching for a magnetization plateauUsing Magnetization, susceptibility, MuSR : a controversial situation

low field anomalies of the susceptibility:

MuSR Baker PRB (2012)

Legl et al PRL (2012)

No plateau in the isothermal magnetization

cross over regime in the dynamics

Yin et al PRL(2013)

Lhotel et al PRB-RC (2012)

Spin glass-like freezing ? TF~200-400 mKFritsch , PRB(2014)

Searching for a magnetization plateauUsing 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

• 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

Field Irreversibilities

Spin glass like freezing?

A. Sazonov et al PRB(2013)

D=0 no mixing

• see poster A. Sazonov

Spin fluctuations at very low temperatureUsing unpolarized neutrons

2 components in the neutron cross section• elastic (dominant) • inelastic (low energy)

elastic

• Pinch points• diffuse maxima at ½ ½ ½ positions

inelastic

• becomes structured at low T• well accounted for by 2 singlet model + anisotropic

exchange

D=0.25K

See also:

Takatsu et al. JPCM (2011)

Fritsch et al PRB(2013)

Static character not reproduced by the 2 singlet model

diffuse scatteringb = -0.13T/µB ; DQ=0.25K

Phase diagram

P. Bonville et al Phys. Rev. B (2011)

3d-map Experiment

Simulation6T2 ( LLB)

The main features of the diffuse scattering are reproduced

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)

Energy integrated intensity

- 50 mK - 50 mK

S.Petit & al, PRB 86 (2012) T.Fennell & al, Science 326 (2009)

Q dependence of the elastic scattering • Pinch points in both compounds: Coulomb phase

strong spectral weight at Q=0 no spectral weight at Q=0 ½ ½ ½ maxima : AF correlations

Analysis of the pinch points Strongly anisotropic correlations of algebric nature

conservation law in TTO spin liquid analogous to the ice rules

What are the spin component involved?

S.Guitteny & al, PRL 111 (2013)

T. Fennell et al PRL(2012)

Polarization analysisFennell Science (2009) : Ho2Ti2O7

PRL (2013) Tb2Ti2O7

Longitudinal polarimetry separates spin components

xZ //110

x// Q

1

2

3 4

1’

2’

Neutron cross section

• Correlations along Q (or x)• between spin components M┴Q

Ho2Ti2O7

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)

Q

z

yMz

My

neutron polarization P// Z

• Non spin flip: N+ <MZ.Mz>• Spin Flip <My.My>

Polarization +energy analysisFennell Science (2009) : Ho2Ti2O7

PRL (2013) Tb2Ti2O7

Q

z

yMz

Myx

Z //110

x// Q

1

2

3 4

1’

2’

Tb2Ti2O7 Look at the dispersion

Mz: « up-down » correlations: relaxing (Quasi-E)My: « 2 in-2out » correlations : dispersing (Inel.)

T=50 mK

Longitudinal polarimetry separates spin components

Neutron cross section

• Correlations along Q (or x)• between spin components M┴Q

neutron polarization P// Z

• Non spin flip: N+ <MZ.Mz>• Spin Flip <My.My>

18

Low energy excitations

• 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

First observation of a dispersive excitation in fluctuating disordered medium

Mz

S. Guitteny et al PRL(2013)

Nature of the static SRO? the ½ ½ ½ order

½ ½ ½ diffuse maxima• Short range ~8-10 A• below ~0.4K• Vanish in a small field ( ~200G)

Fennel PRL (2012)Fristch PRB(2012)Petit PRB (2012)

In single crystals

In powders½ ½ ½ Mesoscopic structure• Over 30-50A• Associated with Cp anomaly• tuned by minute defects in Tb content

Taniguchi PRB RC(2011)

Short range vs. mesoscopic order

See also poster E. Kermarrec

powder samples Tb2+xTi2-xO7+y

½ ½ ½ ½ ½ 3/2

½ ½ 5/2

3/2 3/2 1/2

X=0

Mesoscopic structure for x=0 and x=0.01

Difference pattern: I(50 mk)- I(1K)

T=50mK

N

X=0

exp: P. Dalmas de Réotier 2 q (deg)

Neu

tron

cou

nts

Symmetry analysis 2 orbits with no common IR

site 1

Sites 2-4

N site1 0 0 0 2 ¾ ¼ ½ 3 ¼ ½ ¾4 ½ ¾ ¼

space group Fd-3M, K= ½ ½ ½

Champion, PRB (2001) Stewart, Wills JPCM(2004) Gd2Ti2O7

No way to build a strong ½ ½ ½ peak for Ising spins!

Needs to break either Ising anisotropy or cubic symmetry

K // local <111> axis no intensity at ½ ½ ½

• No vectors of the IR along the local <111> axes• Contributions to ½ ½ ½ cancel by symmetry

Systematic search of magnetic structures • 1T • cfc translations (cubic cell : a)• K= ½ ½ ½ (magnetic unit cell: 2a)

The best structures (x=0)moments remain close to local <111>axes (3-10 deg)

M=1.9(4) µB/Tb; Lc =60 A (Y=1.4)

X=0X=0

Correlation length ~30 -50 A

« Monopole layered structure » « AF -Ordered spin ice »

Ferrimagnetic piling of SI Tetrahedra

moments remain close to local <111>axes (<10 degs)

Fritsch PRB (2012)

The best structures (x=0)

« AF -Ordered spin ice » « Monopole layered structure »

AF packed OSI cubic cells,

MZ

Z//001

S. Guitteny (thesis) derived from Tb2Sn2O7 I. M et al PRL (2005)

Ferrimagnetic piling of SI Tetrahedra separated by monopole layers

moments remain close to local <111>axes (<10 degs)

Fritsch PRB (2012)

The best structures (x=0)

« AF -Ordered spin ice » « Monopole layered structure »

AF packed OSI cubic cells, separated by SI tetrahedra with M

Full of monopoles, but compatible with a distortion No monopoles, but symmetry breaking at each cubic cell no possible LRO?

MZMZ

Z//001

Calculated diffuse scatteringIn a single crystal, correlation length reduced to 2 cubic cells

h, h, 0

0, 0

, l

0, 0

, l

h, h, 0

« Monopole layered structure » « AF -Ordered spin ice »

Experiments

Petit PRB (2013)Fennel PRL (2013)Fritsch PRB(2013)

1

2

3

4

1 2 3 41 2 3 4

1

2

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

Magneto-elastic modes as a switching mechanism?

Quasielastic or slow relaxations (thermally activated ,QT)

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 orderIM et al Nature 2002, PRL(2004)

•Elastic constantsKlekovina-Malkin J. Phys. 2011, J. Opt. Phys. 2014

•Thermal conductivityLi et al PRB(2013)

Summary: what is new in TTO?• Quantum mixing in the GS doublet due to quadrupolar order: a necessary ingredient

• JT distortion « exchange » int. between quadrupolar moments

• Magnetoelastic coupling

• Non-Kramers character is crucial

• 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

Gehring-Gehring (1985) Savary-Balents PRL(2012) Lee-Onoda-Balents PRB(2012)

MF

poster Malkin

x=0.01

coexistence of LRO and mesoscopic orders

• Mesoscopic: M= 1.3µB/Tb

• LRO: M=0.3 µB/Tb

I.M et al Nature (2002)

Under pressure : a phase with larger unit cell is also stabilized

Pressure induced structures