Neutron Scattering from Geometrically Frustrated

Antiferromagnets

Spins on corner-sharing tetrahedra

Paramagnetic phase Long Range Ordered phase (ZnCr2O4) Spin-glass phase (Y2Mo2O7) Concluding phase

Collin BroholmJohns Hopkins University and NIST Center for Neutron Research

Supported by the NSF through DMR-9453362

Collaborators

S.-H. Lee NIST and University of MDS.-W. Cheong Bell Labs and Rutgers Univ.T. H. Kim Rutgers UniversityW. Ratcliff III Rutgers UniversityJ. Gardner Chalk River Nuclear LabB. D. Gaulin McMaster UniversityN. P. Raju McMaster UniversityJ. E. Greedan McMaster University

Experiments performed at NIST center for Neutron Research

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

Some non-disordered cubic insulators

with spins on corner sharing tetrahedra

Material spintype

spinvalue

CW

(K)Tc

(K)Low T phase Ref.

MgV2O4 isotrop. 1 -750 45 LRO Baltzer et al '66ZnV2O4 isotrop. 1 -600 40 LRO Ueda et al '97CdCr2O4 isotrop. 3/2 -83 9 LRO Baltzer et al '66MgCr2O4 isotrop. 3/2 -350 15 LRO Blasse and Fast '63ZnCr2O4 isotrop. 3/2 -392 12.5 LRO S.-H. Lee et al '99FeF3 isotrop. 5/2 -230 20 LRO Ferey et al. '86Y2Mo2O7 isotrop. 1 -200 22.5 spin glass Gingras et al. '97Y2Mn2O7 isotrop. 3/2 17 spin glass Reimers et al '91Tb2Mo2O7 anisotr. 6 and 1 25 spin glass Greedan et al '91Gd2Ti2O7 isotrop. 7/2 -10 1 LRO Radu et al '99Er2Ti2O7 anisotr. -25 1.25 LRO Ramirez et al '99Tb2Ti2O7 anisotr. -19 spin liquid? Gardner et al '99Yb2Ti2O7 anisotr. 0 0.21 LRO Ramirez et al '99Dy2Ti2O7 Ising 7.5 1/2 0.5 1.2 spin ice Ramirez et al '99Ho2Ti2O7 Ising 8 1/2 1.9 spin ice Harris et al ''97

B-s

pin

el

Pyro

chlo

re

Subjects of this talk

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

AFM correlations in Y2Mo2O7 for T<|CW|=200 K

ZnCr2O4: short range dynamic correlations for

|T/CW|<<1

0 0.5 1.0 1.5 2 2.5 Q (A-1)

h

(meV

)

Points of interest:

• 2/Qr0=1.4 => nn. AFM correlations

• No scattering at low Q => satisfied tetrahedra

• Relaxation rate of order kBT => quantum critical

Spin Fluctuations in Paramagnetic phase of

ZnCr2O4

22),("

Q

QQQ

Lorentzian relaxation spectrum:

Near Quantum Criticalspin system:

TkT

TTkCT

B

QQ

BQ

1

1

3)(

)(

2

1

6.0

8.0

C

meV76.0Bk

No indication of finite T cross over or phase transition in cubic phase

h

(meV

)

Spin resonance for T<TC

T=TC+:

kBT is theenergy scale

T<TC :

Spin resonanceat

J

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

First order phase transition in ZnCr2O4

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

Statics:• Staggered magnetization• tetragonal lattice distortion

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.

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

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

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

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

Spin fluctuation spectrum versus T close to glass

transition

Points of interest:

• spectrum softens as Tg is approached from above

• Decrease of inelastic scattering below Tg

• No change in spectrum for T<Tg

Statics and dynamics of spinglass transition in Y2Mo2O7

Elastic scattering intensity:• Development of spin correlations static on the 50 ps time-scale of the experiment.

Inelastic scattering intensity:• Inelastic scattering decreases as spins cease to fluctuate.

Spin relaxation rate:• (T) decreases linearly with T and extrapolates to Tg=23 K derived from AC-susceptibility

Y2Mo2O7 : Q-dep. of elastic magnetic scattering

in spin glass phase

• 2/Q0r0=4.4• /d =1.5

Standard feaures:• short correlation length• Local cancellation of dipole moment

Unusual features:• period of spin structure is 4 n.n. spacings• No higher order peaks

Weak interactions thatdiffer between membersof pyrochlore family control G.S. selection.

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