Neutron Scattering from Geometrically Frustrated Antiferromagnets

1. Neutron Scattering from Geometrically Frustrated Antiferromagnets Collin Broholm Johns Hopkins University and NIST Center for Neutron Research Spins on corner-sharing tetrahedra Paramagnetic phase Long Range Ordered phase (ZnCr2O4) Spin-glass phase (Y2Mo2O7) Concluding phase Supported by the NSF through DMR-9453362

2. Collaborators S.-H. Lee NIST and University of MD S.-W. Cheong Bell Labs and Rutgers Univ. T. H. Kim Rutgers University W. Ratcliff III Rutgers University J. Gardner Chalk River Nuclear Lab B. D. Gaulin McMaster University N. P. Raju McMaster University J. E. Greedan McMaster University Experiments performed at NIST center for Neutron Research

3. 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

4. Some non-disordered cubic insulators with spins on corner sharing tetrahedra B-spinel Pyrochlore Subjects of this talk

5. Magnetic Neutron Scattering The scattering cross section is proportional to the Fourier transformed dynamic spin correlation function Fluctuation dissipation theorem:

7. AFM correlations in Y2Mo2O7for T<|QCW|=200 K

8. ZnCr2O4: short range dynamic correlations for |T/QCW|<<1 0 0.5 1.0 1.5 2 2.5 Q (A-1) • Points of interest: • 2p/Qr0=1.4 • => nn. AFM correlations • No scattering at low Q • => satisfied tetrahedra • Relaxation rate of order kBT • => quantum critical hw (meV)

9. Spin Fluctuations in Paramagnetic phase of ZnCr2O4 Lorentzian relaxation spectrum: Near Quantum Critical spin system: No indication of finite T cross over or phase transition in cubic phase

10. Spin resonance for T<TC hw (meV) T=TC+: kBT is the energy scale T<TC : Spin resonance at

11. Low T excitations in ZnCr2O4: B C D A B Magnetic DOS Q-dep. of E-integ. intensity C A: Bragg peaks B: Spin waves C: Resonance D: Upper band A

12. First order phase transition in ZnCr2O4 • Dynamics: • Low energy paramag. • Fluctuations form a • resonance at 4.5 meV • Statics: • Staggered magnetization • tetragonal lattice distortion

13. Why does tetragonal strain encourage Neel order? Cr3+ O2- Tetragonal dist. Edge sharing n-n exchange in ZnCr2O4 depends strongly on Cr-Cr distance,r: 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!

14. 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: • In a strongly correlated magnet • this shift may yield

15. Analysis of magneto-elastic transition in ZnCr2O4 Ftet, Fcub TC T Cubic paramagnet Tetrag. AFM Free energy of the two phases are identical at TC From this we derive reduction of internal energy of spin system

16. Direct measurement of confirms validity of analysis where S(Q,w) changes 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 LRO develops from a strongly correlated state

17. Analogies with Spin Peirls transition? There are similarities as well as important distinctions!

18. 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

19. 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: • G(T) decreases linearly with • T and extrapolates to • Tg=23 K derived from • AC-susceptibility

20. Y2Mo2O7 : Q-dep. of elastic magnetic scatteringin spin glass phase • 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 that differ between members of pyrochlore family control G.S. selection. • 2p/Q0r0=4.4 • x/d =1.5

21. Conclusions ZnCr2O4 • Low connectivity and triangular motif yields cooperative paramagnet for|T/QCW|<<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. Y2Mo2O7 Tetragonal