Quantum Kagome Spin Liquids

Philippe MendelsLaboratoire de Physique des Solides, Université Paris-Sud, Orsay, France

Herbertsmithite ZnCu3(OH)6Cl2 Antiferromagnet

Quantum Kagome Spin Liquids: The case of Herbertsmithite

Philippe Mendels, Fabrice Bert, Areta Olariu, Andrej Zorko, Laboratoire de Physique des Solides, Université Paris-Sud, Orsay, France

F. Bert A. Olariu A. Zorko- J. C. Trombe, F. Duc, CEMES, Toulouse, P. Strobel, Grenoble , France

- M. de Vries, G. Nilsen, A. Harrison, Edinburgh, UK

- S. Nakamae, F. Ladieu, D. L’Hote, P. Bonville, CEA Saclay, France

- A. Mahajan’s group (IIT Mumbaï)

Quantum Kagome Spin Liquids

Part I

Novel states induced by frustration

Towards spin liquids

Néel versus Anderson

C

Querelles

0 ,. <−= ijjiij JSSJrr

HIntroduction

-200 0 200 400 600 8001/

χT

−θ0 200 400 600 800

χ

T

ΤΝ∼θ

Néel State:Antiferromagn.

+ Quantum fluctuations for S=1/2

Antiferromagnetism, any alternative?

?Geometrical Frustration of magnetic interactions

0rrrr

=++ CBA SSS

A

C

B

Class.

……=RVB

Antiferromagnetism, any alternative?

B

B

B

A

A

A

C

C

C

A

A

B

0rrrr

=++ CBA SSS

Edge sharing: triangular lattice

g = 2(XY spins)

Introduction

A BC

A

A

A

A

A

A

C

C

C

C

C

C

B

B

B

B

B

B

C

C

A

A

CA

B C

CA

BC

AC

B A

AC

A B

...

...

...

...

0rrrr

=++ CBA SSS

Corner sharing: classical kagomé lattice

Soft modesMacroscopic degeneracy

KagomeCorner sharing

geometry

Lecheminant, PRB 56, 2521 (1997) Waldtmann, EPJB 2, 501 (1998).

Exact diagonalizations

Fundamental ‘ spin liquid’ ≠ NéelRVB ? Mila, PRL 81, 2356 (2000)

Ener

gie

S=0 1 2

TriangularEdge sharing

geometry

S=0 1 2Fundamental Néel

<J/20

Antiferromagnetism, any alternative?

IDEAL Highly Frustrated Magnet- Heisenberg spins

- S=1/2 (quantum spins)- Corner sharing geometry:

Kagomé (2D) or pyrochlore (3D) lattice- No « perturbation » (anisotropy, dipolar interaction,

n.n. interactions, dilution)- For kagomé: stacking should keep the 2D kagomé planes

uncoupled

Introduction

0 1 2 3 4 5 6 7 8 90.0

0.5

1.0

1.5

2.0

λ (μ

s-1)

T (K)

Kagomé bilayers (Cr3+, S =3/2)

SrCr9pGa12-9pO19(SCGO)

Θ∼500 KTg ~ 4 K

~6.4 Å

Materials

• Exotic spin glass at only low T • High frustration ratio θ/Tg• Fluctuations at low T• Short correlation length• Field independant Cv• >3% spinless defects

T=70mK ~1/a

Spin glass

?

Jarosite FamilyAB3(SO4)2(OH)6

A= Na+, K+, Ag+, Rb+, H3O+, NH4+, ½ Pb2+, ½ Ba2+

B = Fe3+ (S=5/2), Cr3+ (S=3/2), V3+ (S=1)

Most of Jarosites order!(D.M. anisotropy)

Volborthite, Cu3V2O7(OH)2·2H2O

Z. Hiroi et al, J. Phys. Soc. Jpn 70 (2001) 3377

F. Bert et al, PRL, 95 (2005) 087203, Z. Hiroi, JPSJ

0 5 10 15 20 25 300,0

0,3

0,6

0,9

1,2

1,5

0

1

2

3

4

Volborthite

T g (K

)

spin vacancy (%at)

SCGO

S = 1/2High purity

But a transition

Another class of imperfections: J1 and J2

Quantum Kagome Spin Liquids

Part II

Novel states induced by frustration

Herbertsmithite: a true spin liquid?

Cu

ZnZn

ClCl

OHOH

Herbertsmithite:ZnCu3(OH)6Cl2Cu2+, S=1/2

Sciences, perspectives sept 2008

Herbertsmithite is the first example of a quantum kagomeantiferromagnet

perfect kagomé latticeno freezing at least down to J/4000

-> renewal ofthe search for scenarios for the kagomeground state

VBC : R.R.P. Singh and D.A. Huse, PRB (2007, 2008)Dirac spin liquid : Y. Ran et al, PRL (2007), PRB (2008) ....

1- ZnxCu4-x(OH)6Cl2 : paratacamite family x< 1

2- ZnCu3(OH)6Cl2: Herbertsmithite: ideal kagome ?

Gap?

Measurements of χlocal

Dynamical mesurements

Non-magnetic defects

Cu/Zn

« Perturb to reveal »

ClinoatacamiteCu2(OH)3Cl

HerbertsmithiteBraithwaite et al, 2004

Zn-paratacamiteZnxCu4-x(OH)6Cl2

Zn/Cu substitution rate

ZnxCu4-x(OH)6Cl2 atacamite family

Zn

Cu I Cu I

Cu I

Zn/Cu II

Cu I Cu I

Cu I

Cu III

Cu I Cu I

Cu II

Cu

ZnZnxxCuCu11--xx

x=0 x=0.33 x=1P21/n R-3m

Curie Weiss behavior for all x-> antiferromagnetic correlations

P. Shores et al, JACS (2005)

ZnxCu4-x(OH)6Cl2

J (AF)

J’ (AF,F?)

J : in-plane coupling AF ~ 175 KJ’: inter-plane coupling small, maybe Ferro

97°

119°

no transition for T<<J-> highly frustrated antiferromagnets

Ferromagnetic-like transition at TN~6K

Vanishes for x->1

0 50 100 150 200 250 300

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

χ (c

m3 /m

ol C

u)

temperature (K)

Herbersmithite x=1

-No sign of transition for T>2K-Low T Curie like upturn for T<50K

x=1

ZnxCu4-x(OH)6Cl2

µSR – Muon spin resonance, relaxation

μ+

S=1/2

Échantillon

μSR : ZnCu3(OH)6Cl2 , x=1

At 50mK, no sign of ordering relaxation arises fromsmall “static” nuclear fields.

P. Mendels et al, PRL 98, 077204 (2007)

0 5 100.0

0.2

0.4

0.6

0.8

1.0

Pol

aris

atio

n

time (μs)

50 mKH

Deuterated

HerbertsmithiteZero Field μSR

upper limit of a frozen moment for Cu2+, if any : 6x10-4 μB

No order or frozen disorder down to 50 mK despite J=175 K !

Also: ac-χ Helton et al, PRL 98 107204 (2007)μSR O. Ofer et al, cond-mat/0610540

μSR : ZnxCu4-x(OH)6Cl2

-x=0 : fully ordered below ~18KX.G. Zheng et al, PRL 95, 057201 (2005)

When x increases from 0 to 1 :-Oscillations are smeared out-A paramagnetic (x=1 type) component emerges at the expense of the frozen one

Large domain of stability 0.66<x<1 of a dynamical ground state-> surprisingly small influence of interlayer coupling

P. Mendels et al, PRL 98, 077204 (2007)

time (μs)

Pola

riza

tion

0.0 0.3 0.6

0.3

0.6

0.9

x = 0

x = 0.33

x =0.5x = 0.66

x = 1.0

x = 0.15

T=1.5K, Zero Field

M. Rigol and R. P. Singh, PRL 98, 207204 (2007)

-> need for additional terms to the KAF Hamiltonian to account for Herbertsmithite macroscopic susceptibility

Herbertsmithite macroscopic susceptibility

High temperature series expansion(G. Misguish, C. Lhuillier)

Magnetic defects : Zn/Cu intersite mixing

Zn in the kagome plane-> magnetic vacancy -> staggered magnetization-> effective paramagnetic defects (small moment?)

Cu on the Zn siteNearly free ½ spins

Cu

ZnZn defects

Neutron: structure refinement + HS.H. Lee et al, Nature Materials (2007)

M.A. de Vries et al, PRL 100, 157205 (2008)

dilution-> ~10%-> 9(2)%

P. Mendels et al., J. Phys.: Condens. Matter 19, 145224 (2007)

Magnetic defects : Zn/Cu intersite mixing

G. Misguich and P. Sindzingre, Eur. Phys. J. B 59, 305 (2007)

Susceptibility fit -> ~5% dilutionexact diagonalization+5% weakly interracting S=1/2 defects

Low T, High Field Magnetization -> ~7% dilution

0 50 100 150 200 2500.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

χ (c

m3 /m

ol C

u)

Mtot=Mdefect+ Hi

Mdefect~Brillouin(H/T)

χ

-~7% of interlayer Cu2+

- χi << χmacro at low T

F. Bert et al, PRB 76, 132411 (2007)

0 2 4 6 8 10 12 140

2

4

6

8

10

M/M

sat (

%)

H (T)

1.7 K

Schottky like anomaly in heat capacity

-> Kagome lattice dilution 6.5+/-1%

M.A. de Vries et al, PRL 100, 157205 (2008)

Dzyaloshinskii-Moriya interactions

M. Rigol and R. P. Singh, PRL 98, 207204 (2007)0.25 0.75 1.25 1.75T/J

Direct fit of susceptibility (no defect contribution):|Dz|=0.15J, |Dp|=0.3J

A. Zorko et al, PRL 101, 026405 (2008)

Broad room T ESR line <- magnetic anisotropy from DM|Dz|=0.08J, |Dp|~0.01J

Si Sj

DHDM=D.(Si∧Sj)

Dzyaloshinskii-Moriya interactions

O. Cepas et al, PRB 78, 140405 (R) (2008)

M. Elhajal et al, PRB 66, 014422 (2002)

For classical spins, DM stabilizes ordered phases (cf jarosites)

In the quantum case,a moment free phase survives up to D/J~0.1

-5 -4 -3 -2 -1 0 1 2 3 4 5

K ∝ χ

Δ χ

Line shift K :susceptibility χfrustr

Linewidth ΔH :spatially inhomogeneoussusceptibility (dilution)

NMR: principles

SIAHH Z

rr.+=

17O NMR, local susceptibility

OCu

Zn

Cl∆H ~ local susceptibility

17O: coupled to two Cu of the kagome plane

I = 5/2 → quadrupolar effects

ref

6.60 6.65 6.706.4 6.6 6.80.0

0.5

1.0

150 K 200 K 250 K 300 K

NM

R In

tens

ity

(nor

mal

ized

)

H (Tesla)

300 K

A. Olariu et al, PRL 100, 087202 (2008)

two magnetic site O next to a Zn defect in the kagome plane

CuCu

17OH

Cu

17O

Zn

~ 20% intensity -> ~ 5% Zn/Cu defects in kagome planes

6.4 6.5 6.6 6.7 6.8 6.90.0

0.5

1.0

6.5 6.6 6.7 6.8 6.9 7.0

In

tens

ité (n

orm

alis

ée)

175 K

H(Tesla)

β = 55°

Zn

Cu Cu

CuCu

O

-Susceptibility decreases below 50 K-> enhancement of short range correlations-> new energy scale

-No gap and Finite T->0 susceptibility

intrinsic or field effect, DM ?

0 100 200 3000

1

2

0

10

20

17O

lin

eshi

ft (%

)T (K)

χ SQU

ID (1

0-4 c

m3 /m

ol C

u)

Main line, susceptibility of the kagome planes

6.4 6.5 6.6 6.7 6.8 6.9 7.0

85 K

H (Tesla)

60 K

50 K

30 K

15 K

20 K

10 K

5 K

1.3 K

référence

0.47 K

40 K

D

M

175 K

0.1

1

10

0.1

1

0 15 30 45 60

17(1

/T1)

(ms-1

)

Temperature (K)

35Cl

63(1

/T1) (

ms-1

), 35

(1/T

1) (s-1

)

63Cu

17O

0 1 2

0.1

1

T 1-1 (m

s-1)

T-1 (K-1)

T1-1 ~ T 0.7+/-0.1

Low T Dynamics (NMR: T1)

63Cu and 35Cl NMR from T. Imai et al, PRL 100, 077203 (2008)

- No spin gap behavior-> in agreement with absence of a gap>0.1meV in INS : Helton et al, PRL 98 107204 (2007)

- original sub-linear T dependence

Low T Dynamics (neutrons: χ‘’)

Helton et al, PRL 98 107204 (2007)

χ’’(ω) ∼ ω−0.7(3)

• ∆<J/20 gap between singlet ground state and 1st triplet stateif any...

• no gap in the singlet sector.

Exact Diagonalization•Lecheminant, PRB 56, 2521 (1997) •Waldtmann et al., EPJB 2, 501 (1998).

No gap for Herbertsmithite!

- No gap (?) in exact diagonalizations (ED)- χ(T->0) well reproduced in ED

Could be an intrinsic property

- DM interaction mixes singlet and gappedtriplet and could restore a susceptibilityCould be an extrinsic property but one should

also explain dynamical properties

Two classes of models

-Valence Bond Crystals with fluctuations of quantum dimers but forming an ordered pattern: a small singlet – triplet gap

- RVB-type spin liquids: fractional excitations (spinons), e.g. algebraic spin liquids

. Algebraic decrease of correlations

. no gap

. Spinons are bound (singlet – spinon interact)

0 100 200 3000.0

0.5

1.0

1.5

Cu O

Zn

Cu O

Zn

OCu H

OCu H

Zn Cu

O

Zn Cu

O

Line

Shi

ft (%

)T (K)

S. Dommange et al, PRB 68 (2003) 224416

~30% oxygen sitesdimer localization arounda spin vacancy

~20% oxygen siteshalf shifted

-> ~5% spin vacancies in the kagome planes

Defect line

6.4 6.5 6.6 6.7 6.8 6.9 7.0

85 K

H (Tesla)

60 K

50 K

30 K

15 K

20 K

10 K

5 K

1.3 K

référence

0.47 K

40 K

D

M

175 K

I. Rousochatzakis et al, Phys.Rev.B 79, 214415 (2009)

ED: one impurity

Dommange et al., PRB 68, 224416 (2003); Lauchli et al. Phys. Rev. B 76, 144413 (2007)

-Dimers next to the impurity survive up to D~J

-DM interaction removes the singlet nature of the ground state

I. Rousochatzakis et al, Phys.Rev.B 79, 214415 (2009)

Conclusions on Herbertsmithite

- no order or frozen disorder down to 50mK despite J=175K and perturbations

- first S=1/2 antiferromagnet with perfect kagome lattice (3 fold symmetry)

Quantum spins on a perfect kagome lattice-> allows close comparison with theory-> renewed interest in understanding the GS

VBC : R.R.P. Singh and D.A. Huse, PRB (2007, 2008)Dirac spin liquid : Y. Ran et al, PRL (2007), PRB (2008) ....

- magnetic defects (rather complex, in plane and out of plane)which impact the low T thermodynamic measurements

-> local probe techniques-> second sample generation : controlled (or no) defects

- Local susceptiblity determined from 17O NMR:ground state appears to be gap-less with a finite susceptibilityintrinsic to Heisenberg kagome model or DM interaction closes the gap ?

(no field effect)

- sizeable DM interaction : D/J~0.1: probe criticality: on-going

PHFM 2010Perspectives in HighlyFrustrated Magnetism

Dresden , 19 – 23 April

Frustration and original ground states - Collaborations

PyrochloreTb2Sn2O7LLB, SaclayBert et al, PRL 2006

Volborthite, CEMES, ToulouseBert et al, PRL 2005

Kagome

Cr,

TriangularNaCrO2R. Cava, PrincetonOlariu et al, PRL 2006

Kagome+Spin anisotropyLangasitesInstitut Néel, GrenobleZorko et al, PRL (2008)

HerbertsmithiteF. Ladieux, S. Nakamae, P. Bonville

SPEC, CEA SaclayMA de Vries, A. Harrisson, EdinburghF. Duc, JC Trombe CEMES, Toulouse

P. Strobel, Institut Néel, Grenoble

Herbertsmithite:ZnCu3(OH)6Cl2Cu2+, S=1/2

Thank you!