1. 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 Broholm Johns Hopkins University and NIST Center for Neutron Research Supported by the NSF through DMR-9453362 and DMR-0074571 SrCr 9p Ga 12-9p O 19 KCr 3 (OD) 6 (SO 4 ) 2 ZnCr 2 O 4

2. HFM2000 6/11/00 Collaborators S.-H. LeeNIST and University of MD M. AdamsISIS Facility, RAL G. AeppliNEC Research Institute E. BucherUniversity of Konstanz C. J. CarlileISIS Facility, RAL S.-W. CheongBell Labs and Rutgers Univ. M. F. CollinsMcMaster University R. W. ErwinNIST L. HellerMcMaster University B. HessenBell Labs/Shell T. J. L. JonesISIS Facility, RAL T. H. KimRutgers University C. KlocBell Labs N. LacevicJohns Hopkins University T. G. PerringISIS Facility, RAL A. P. RamirezBell Labs W. Ratcliff IIIRutgers University A. TaylorISIS Facility, RAL

3. HFM2000 6/11/00 A simple frustrated magnet: La 4 Cu 3 MoO 12 Masaki Azuma et al. (2000)

4. HFM2000 6/11/00 Magnetization and order of composite trimer spins

5. HFM2000 6/11/00 Theory of spins with AFM interactions on corner-sharing tetrahedra 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

6. HFM2000 6/11/00 SrCr 9p Ga 12-9p O 19 : Kagome’ sandwich Isolated spin dimer Kagome’-Triangular-Kagome’ sandwich (111) slab of pyrochlore/spinel AFM A. P. Ramirez et al. PRL (1990)

7. HFM2000 6/11/00 Fe & Cr Jarosite: coupled Kagome’ layers A. S. Wills et al. (2000) AM 3 (OH) 6 (SO 4 ) 2 Townsend et al (1986) Lee et al. (1997) KCr 3 (OH) 6 (CrO 4 ) 2

8. HFM2000 6/11/00 ZnCr 2 O 4 : Corner sharing tetrahedra

9. HFM2000 6/11/00 Magnetic Neutron Scattering The scattering cross section is proportional to the Fourier transformed dynamic spin correlation function Fluctuation dissipation theorem:

10. HFM2000 6/11/00 NIST Center for Neutron Research

11. HFM2000 6/11/00

12. Detection system on SPINS neutron spectrometer

13. HFM2000 6/11/00 Spatial correlation length saturates for T 0 SrCr 9p Ga 12-9p O 19 for T<  CW

14. HFM2000 6/11/00 Relaxation Rate decreases for T 0 Paramagnetic state: Fluctuating AFM spin clusters that largely satisfy exchange interactions dimer Kagome’ sandwich SrCr 9p Ga 12-9p O 19

15. HFM2000 6/11/00 Spin glass in concentrated frustrated AFM  Does quenched disorder play a role?  What is structure of SG phases?  What are excitations in SG phases?

16. HFM2000 6/11/00 Spin glass transition in SCGO(p=0.92)

17. HFM2000 6/11/00 The role of disorder at SG transition Higher Cr concentration Higher T f 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)

18. HFM2000 6/11/00 Structure of frozen phase Short range in plane order Kagome’ tri-layer correlations

19. HFM2000 6/11/00 Excitations in a frustrated spin glass Gapless 2D Halperin-Saslow “Spin Waves” SCGO(p=0.92)

20. HFM2000 6/11/00 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?

21. HFM2000 6/11/00 Phase transition to LRO in Cr-Jarosite >90% Cr 75-90% Cr

22. HFM2000 6/11/00 Weak order / strong fluctuations

23. HFM2000 6/11/00 Impurity enhanced LRO in (DO 3 )Fe 3-x Al y (SO 4 ) 2 (OD) 6 Wills et al (1998) Wills et al. (2000)  Intensity (arb) 100% Fe 89% Fe Only reported Jarosite w/o LRO Diamagnetic impurities yield LRO

24. HFM2000 6/11/00 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?

25. HFM2000 6/11/00 First order phase transition in ZnCr 2 O 4 Dynamics: Low energy paramag. Fluctuations form a resonance at 4.5 meV Statics: Staggered magnetization tetragonal lattice distortion

26. HFM2000 6/11/00 Local spin resonance in ordered phase Paramagnetic fluctuations in frustrated AFM Paramagnetic fluctuations in frustrated AFM Local spin resonance in magneto-elastic LRO phase Local spin resonance in magneto-elastic LRO phase

27. HFM2000 6/11/00 Low T excitations in ZnCr 2 O 4 : Magnetic DOS Q-dep. of E-integ. intensity C ABBC A A: Bragg peaks B: Spin waves C: Resonance D: Upper band D

28. HFM2000 6/11/00 Spectra at specific Q Spin wavesResonance

29. HFM2000 6/11/00 Dispersion relation for resonance ZnCr 2 O 4 single crystals T=1.5 K

30. HFM2000 6/11/00 Structure factor for resonance Extended sharp structures in reciprocal space Extended sharp structures in reciprocal space Fluctuations satisfy local constraints Fluctuations satisfy local constraints ZnCr 2 O 4 T=1.4 K

31. HFM2000 6/11/00 Comparing resonance to PM fluctuations 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

32. HFM2000 6/11/00 Why does tetragonal strain encourage Neel order? Edge sharing n-n exchange in ZnCr 2 O 4 depends strongly on Cr-Cr distance, r : Cr 3+ O 2- From series of Cr-compounds: r The effect for a single tetrahedron is to make 4 bonds more AFM and two bonds are less AFM. This relieves frustration! Tetragonal dist.

33. HFM2000 6/11/00 Magnetic order in ZnCr 2 O 4 - Viewed along tetragonal c-axis tetrahedra have zero net moment => this is a mean field ground state for cubic ZnCr 2 O 4 Tetragonal distortion lowers energy of this state compared to other mean field ground states: In a strongly correlated magnet this shift may yield

34. HFM2000 6/11/00 Analysis of magneto-elastic transition in ZnCr 2 O 4 Free energy of the two phases are identical at T C From this we derive reduction of internal energy of spin system T F tet, F cub TCTC Tetrag. AFM Cubic paramagnet

35. HFM2000 6/11/00 Direct measurement of confirms validity of analysis From first moment sum-rule for the dynamic spin correlation function we find When a single Heisenberg exchange interaction dominates. Inserting magnetic scattering data acquired at 15 K and 1.7 K we get where S(Q,  ) changes LRO develops from a strongly correlated state

36. HFM2000 6/11/00 Analogies with Spin Peirls transition? There are similarities as well as important distinctions!

37. HFM2000 6/11/00  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 k B T/ 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 ZnCr 2 O 4 Y 2 Mo 2 O 7