Neutron Scattering from Geometrically Frustrated Antiferromagnets Spins on corner-sharing tetrahedra Paramagnetic phase Long Range Ordered phase (ZnCr.
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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
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
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