Magnetic Frustration at TripleAxis S H Lee S

Magnetic Frustration at Triple-Axis S. -H. Lee, S. Park, P. M. Gehring NCNR q Magnetism, Neutron Scattering, Geometrical Frustration q Zn. Cr 2 O 4: The Most Frustrated Magnet How are the fluctuating spins in the Spin Liquid phase correlated with each other? How does nature respond to the ground state degeneracy? q Summary

Zn. Cr 2 O 4 Space group Fd 3 m Lattice of B sites : Corner-sharing tetrahedra O A Cr 3+ (3 d 3) B Edge-sharing octahedra eg H = -J S Si. Sj nn 3 d t 2 g Free Ion dxy, dyz, dzx Cubic Field

Magnetic Neutron Scattering Neutron: Ø Wavelength comparable with interatomic spacings Ø Penetrating bulk properties are measured Ø has spin s = ½ so interacts with atomic moments Scattering by atomic magnetic moments: I = (0. 54)2 S (S+1) Magnetic scattering intensities can be comparable to nuclear scattering !!

Neutron Scattering Sample ki , Ei , s Incident neutrons ? Scattered neutrons kf , Ef , s measures scattering cross section as a function of Q and d 2 s (Q, ) d. Wd

Magnetic Neutron Scattering Cross Section Spin-Spin Correlation Function Fourier Transform d 2 s d. Wd < SR(t) SR (0) > The probability of finding a spin in a spin state SR at positionmoment R at time t when another spin at R’ at time t=0 Ordered is in a spin state SR/ Fluctuating moment G ~ h/t 0 0 Time Long range order Short range order Q G : relaxation time k : intrinsic linewidth k ~ 1/x R-R t : lifetime x : correlation length

Phase Diagram for an Ordinary Magnet F = Hmag – TS = - S J Si Sj - TS T (K) Paramagnet TS > | Hmag | AFM or FM | Hmag | > TS QCW= Jz. S(S+1)/3 k. B Ground State : Long range (anti)ferromagnetically ordered state. Low Energy Excitations : linear spin waves around the ordered state.

Paramagnetic Neutron Scattering Neel phase: Function Whatkindof ofmagneticscatteringsignalin in. QQCorrelation and spacewould youexpect? What and you Cross Section (1) Any Elastic 2 signal? Fourier Transform ds (2) Any Inelastic signal? d. Wd < Sin. R(t) (0) Q-space? > If any, what kind of shape the S -R and Ordered moment Fluctuating moment G ~ h/t 0 0 Time Long range order Short range order Q G : relaxation time k : intrinsic linewidth k ~ 1/x R-R t : lifetime x : correlation length

Geometrical Frustration AF M A simplest example: a Triangle of three antiferromagnetic Ising spins AFM H = -J S Si. Sj All exchange interactions can not be satisfied. ? E=-|J| A tetrahedron with four isotropic spins S Si=0 Zero energy modes in the ground state manifold Geometrical frustration leads to a large degeneracy in the ground state

Why Zn. Cr 2 O 4 ? Space group Fd 3 m Lattice of B sites : Corner-sharing tetrahedra O A Cr 3+ (3 d 3) B Edge-sharing octahedra eg Interaction between nearest neighbor Cr 3+ ions through direct overlap of t 2 g orbitals is dominant. 3 d t 2 g Free Ion Geometrically frustrated dxy, dyz, dzx Cubic Field Best candidate for a system with H = -J S Si. Sj nn

Phase Diagram for Geometrical Frustration F = Hmag – TS = - S J Si Sj - TS T (K) Paramagnet QCW= Jz. S(S+1)/3 k. B ? Geometrically frustrated What is the nature of the low temperature phase ?

Theory of spins with AFM interactions on corner-sharing tetrahedra H = -J S Si. Sj SPIN TYPE SPIN LOW T METHOD Value PHASE Isotropic S=1/2 Spin Liquid Exact Diag. Isotropic S= Spin Liquid MC sim. REFERENCE Canals and Lacroix PRL ’ 98 Reimers PRB ’ 92 Moessner, Chalker PRL ’ 98

Issues q Nature of the Spin Liquid State How are the fluctuating spins in the SL phase correlated with each other? q Novel Phase Transition? How does nature respond to the ground state degeneracy? You will be able to answer the questions after the demonstration session at SPINS!

A Phase Transition in Zn. Cr 2 O 4 A. P. Ramirez, unpublished QCW = -390 K TN = 12. 5 K Phase transition What is the nature of the phase transition?

Neutron Scattering from Zn. Cr 2 O 4 SHL et al. , PRL 84, 3718 (2000) Paramagnetic fluctuations in frustrated AFM Local spin resonance in magneto-elastic LRO phase

Keep in mind !!! Neutron scattering is the most powerful technique to study magnetism!! Fourier Transform d 2 s d. Wd < SR(t) SR (0) > Ordered moment Fluctuating moment G ~ h/t 0 0 Time Long range order Short range order Q G : relaxation time k : intrinsic linewidth k ~ 1/x R-R t : lifetime x : correlation length

SPINS : Cold Neutron Triple-Axis Spectrometer