Neutron Scattering of Frustrated Antiferromagnets

Satisfaction without LRO Paramagnetic phase Low Temperature phase

Spin glass phase Long range order Spin Peierls like phase

Conclusions

Collin BroholmJohns Hopkins University and NIST Center for Neutron Research

Supported by the NSF through DMR-9453362 and DMR-0074571

SrCr9pGa12-9pO19

KCr3(OD)6(SO4)2 ZnCr2O4

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CollaboratorsS.-H. Lee NIST and University of MDM. Adams ISIS Facility, RALG. Aeppli NEC Research InstituteE. Bucher University of KonstanzC. J. Carlile ISIS Facility, RALS.-W. Cheong Bell Labs and Rutgers Univ.M. F. Collins McMaster UniversityR. W. Erwin NISTL. Heller McMaster UniversityB. Hessen Bell Labs/Shell T. J. L. Jones ISIS Facility, RAL T. H. Kim Rutgers UniversityC. Kloc Bell LabsN. Lacevic Johns Hopkins UniversityT. G. Perring ISIS Facility, RAL A. P. Ramirez Bell LabsW. Ratcliff III Rutgers University

A. Taylor ISIS Facility, RAL

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A simple frustrated magnet: La4Cu3MoO12

Masaki Azuma et al. (2000)

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Magnetization and order of composite trimer spins

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Theory of spins with AFM interactions on corner-sharing tetrahedra

SPIN TYPE SPINVALUE

LOW TPHASE

METHOD REFERENCE

Isotropic S=1/2 Spin Liquid Exact Diag. Canals and LacroixPRL'98

Isotropic S= Spin Liquid MC sim. Reimers PRB'92Moessner, ChalkerPRL'98

Anisotropic S= Neel order MC sim. Bramwell, Gingras,ReimersJ. Appl. Phys. '94

What is special about this lattice and this spin system?• Low coordination number• Triangular motif• Infinite set of mean field ground states with zero net spin on all tetrahedra• No barriers between mean field ground states• Q-space degeneracy for spin waves

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SrCr9pGa12-9pO19 : Kagome’ sandwich

Isolated spin dimer

Kagome’-Triangular-Kagome’ sandwich(111) slab of pyrochlore/spinel AFM A. P. Ramirez et al. PRL (1990)

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Fe & Cr Jarosite: coupled Kagome’ layers

A. S. Wills et al. (2000)

AM3(OH)6(SO4)2

Townsend et al (1986)

Lee et al. (1997)

Material Cr/Fe concentr. CW (K) Tf (K)

CW/Tf

(D3O)Fe3(SO4)2(OD)6 0.97 -700(5) 13.8 51NaFe3(SO4)2(OD)6 0.95 -667(5) 42, 62 16, 11

(ND4)Fe3(SO4)2(OD)6 0.91 -640(5) 46,62 14, 10AgFe3(SO4)2(OD)6 0.89 -677(4) 51 13RbFe3(SO4)2(OD)6 0.87 -688(5) 47 15

(D3O)Fe3-xAly(SO4)2(OD)6 0.89 -720(5) 25.5 28KFe3(SO4)2(OD)6 ? -600 46 13KFe3(CrO4)2(OD)6 ? -600 65 9KCr3(SO4)2(OD)6 >0.9 -70(5) 1.8 39KCr3(SO4)2(OD)6 0.8(1) -54(2) 1.55 34

KCr3(OH)6(CrO4)2

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ZnCr2O4 : Corner sharing tetrahedra

Material CW TN

CW/TN

MgV2O4 -750 45 17ZnV2O4 -600 40 15CdCr2O4 -83 9 9MgCr2O4 -350 15 23ZnCr2O4 -392 12.5 31

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Magnetic Neutron Scattering

fi kkQ

fi EE

The scattering cross section is proportional to the Fourier transformed dynamic spin correlation function

ik fk

Q

2

''R

)'( )0(S)(S1

2

1),(

RRR

RRQiti teN

edtQ

S

Fluctuation dissipation theorem:

,1," 2 QegQ B S

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NIST Center for Neutron Research

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Detection system on SPINS neutron spectrometer

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Spatial correlation length saturates for T 0

SrCr9pGa12-9pO19

for T<CW

a

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Relaxation Rate decreases for T 0

Paramagnetic state:Fluctuating AFM spin clusters that largely satisfyexchange interactions

dimer

Kagome’sandwich

SrCr9pGa12-9pO19 SrCr9pGa12-9pO19

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Spin glass in concentrated frustrated AFM

Does quenched disorder play a role?

What is structure of SG phases?

What are excitations in SG phases?

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Spin glass transition in SCGO(p=0.92)

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The role of disorder at SG transition

Higher Cr concentrationHigher Tf

Sharper features in S(Q)Two scenarios remain viable : A) Strong sensitivity to low levels of disorder B) Intrinsic disordered frozen phase

Martinez et al. (1992)

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Structure of frozen phase

Short range in plane order 33 Kagome’ tri-layer correlations

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Excitations in a frustrated spin glass

"

TTTC

Gapless 2D Halperin-Saslow “Spin Waves”

SCGO(p=0.92)

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Long range order in frustrated AFM

Clarify role of symmetry breaking interactions, quenched disorder and “order by disorder” effects in stabilizing LRO.

Are there anomalous critical properties at phase transitions to LRO?

What are the excitations in the ordered phase of a highly frustrated magnet?

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Phase transition to LRO in Cr-Jarosite

>90% Cr

75-90% Cr

40

N

CW

T

33

0Q

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Weak order / strong fluctuations

Sg

M

B

Bq

3

1

)3(1.1

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Impurity enhanced LRO in (DO3)Fe3-

xAly(SO4)2(OD)6

Wills et al (1998)

Wills et al. (2000)

Int

ensi

ty (

arb)

100% Fe100% Fe

89% Fe89% Fe

Only reportedJarosite w/oLRO

Diamagneticimpurities yield LRO

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Magneto-elastic effects in frustrated AFM

Can magneto-elastic coupling relieve frustration?

What is magnetic and lattice strain configuration in ordered phase?

Determine energetics of a spin-Peierls like transition for frustrated magnets.

What are excitations from the ordered phase?

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First order phase transition in ZnCr2O4

Dynamics:• Low energy paramag. Fluctuations form a resonance at 4.5 meV

Statics:• Staggered magnetization• tetragonal lattice distortion

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Local spin resonance in ordered phase

Paramagneticfluctuations infrustrated AFM

Paramagneticfluctuations infrustrated AFM

Local spin resonancein magneto-elasticLRO phase

Local spin resonancein magneto-elasticLRO phase

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Low T excitations in ZnCr2O4:

Magnetic DOS Q-dep. of E-integ. intensity

C

A

B

B

C

A

A: Bragg peaksB: Spin wavesC: ResonanceD: Upper band

D

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Spectra at specific Q

Spin waves

Resonance

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Dispersion relation for resonance

ZnCr2O4 single crystals T=1.5 K

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Structure factor for resonance

Extended sharpstructures in reciprocal space

Extended sharpstructures in reciprocal space

Fluctuations satisfy local constraintsFluctuations satisfy local constraints

ZnCr2O4 T=1.4 K meV 6meV 3

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Comparing resonance to PM fluctuations

meV 6meV 3 meV 7.0

• Paramagnetic fluctuations and resonance satisfy same local constraints. • Transition pushes low energy fluctuations into a resonance w/o changing spatial correlations

15 K1.4 K

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Why does tetragonal strain encourage Neel order?

meV06.0d

d

meV04.0d

d

2

0

0||

r

JrJ

r

JrJ

a

ca

Edge sharing n-n exchange in ZnCr2O4 depends strongly on Cr-Cr distance, r :

Cr3+

O2-

AmeV40

d

d /r

JFrom series of Cr-compounds:

r

The effect for a single tetrahedron is to make 4 bonds more AFM and two bondsare less AFM. This relieves frustration!

Tetragonal dist.

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Magnetic order in ZnCr2O4

-Viewed along tetragonal c-axis

•tetrahedra have zero net moment => this is a mean field ground state for cubic ZnCr2O4

•Tetragonal distortion lowers energy of this state compared to other mean field ground states:

meV07.052

1|| JJH

MFS

•In a strongly correlated magnet this shift may yield

MFStNB HTk

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Analysis of magneto-elastic transition in ZnCr2O4

Free energy of the two phases are identical at TC

lHScTsH

ScTl

HsHF

0

From this we derive reduction of internal energy of spin system

meV/Cr21.0

meV/Cr04.02221116

3meV/Cr17.0

sH

acCal

H

ScT

T

F tet, F

cub

TCTetrag. AFM

Cubic paramagnet

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Direct measurement of confirms validity of

analysis sH

From first moment sum-rule for the dynamic spin correlation function we find

0

0

0

2

sin1

,1

2

3

QrQr

QSe

H s

When a single Heisenberg exchange interaction dominates. Inserting magneticscattering data acquired at 15 K and 1.7 K we get

meV)5(35.0 sH

S

S

cBS

H

JH

TkH

where S(Q,) changes

LRO develops froma strongly correlated state

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Analogies with Spin Peirls transition?

There are similarities as well as important distinctions!

Spin-Peirls

ZnCr2O4

Quantum critical above TC yes yesOrder suppressed to | T/CW| <<1 due to low D frustra-

tionChange of lattice symmetry at TC enableslower energy spin state

yes yes

Low energy magnetic spectral weight ispushed into resonance

yes yes

Order of phase transition second firstLow T phase isolated

singletNeelLRO

TC S is significant energy scale no yes

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Low connectivity and triangular motif yields cooperative paramagnet for|T/CW|<<1. The paramagnet consists of small spin clusters with

no net moment, which fluctuate at a rate of order kBT/ h.

Spinels can have entropy driven magneto-elastic transition to Neel order with spin-Peirls analogies.

The ordered phase has a spin-resonance, as expected for under-constrained and weakly connected systems.

Pyrochlore’s can have a soft mode transition to a spin-glass even when there is little or no quenched disorder.

Variations of sub-leading interactions in pyrochlore’s give different types of SRO in different compounds.

Lattice distortions may be a common route to relieving frustration and lowering the free energy of geometrically frustrated magnets.

Conclusions

Tetragonal

ZnCr2O4

Y2Mo2O7