1. Neutron Scattering from Geometrically Frustrated Antiferromagnets Spins on corner-sharing tetrahedra Paramagnetic phase Long Range Ordered phase (ZnCr 2 O 4 ) Spin-glass phase (Y 2 Mo 2 O 7 ) Concluding phase Collin Broholm Johns Hopkins University and NIST Center for Neutron Research Supported by the NSF through DMR-9453362

2. Collaborators S.-H. LeeNIST and University of MD S.-W. CheongBell Labs and Rutgers Univ. T. H. KimRutgers University W. Ratcliff IIIRutgers University J. GardnerChalk River Nuclear Lab B. D. GaulinMcMaster University N. P. RajuMcMaster University J. E. GreedanMcMaster 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 Y 2 Mo 2 O 7 for T<|  CW |=200 K

8. ZnCr 2 O 4 : 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  /Qr 0 =1.4 => nn. AFM correlations No scattering at low Q => satisfied tetrahedra Relaxation rate of order k B T => quantum critical

9. Spin Fluctuations in Paramagnetic phase of ZnCr 2 O 4 Lorentzian relaxation spectrum: Near Quantum Critical spin system: No indication of finite T cross over or phase transition in cubic phase

10. h  (meV) Spin resonance for T<T C T=T C+ : k B T is the energy scale T<T C : Spin resonance at

11. 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

12. 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

13. 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.

14. 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

15. 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

16. 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

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 T g is approached from above Decrease of inelastic scattering below T g No change in spectrum for T<T g

19. Statics and dynamics of spinglass transition in Y 2 Mo 2 O 7 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 T g =23 K derived from AC-susceptibility

20. Y 2 Mo 2 O 7 : Q-dep. of elastic magnetic scattering in spin glass phase 2  /Q 0 r 0 =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 that differ between members of pyrochlore family control G.S. selection.

21.  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