Frustration and fluctuations in diamond antiferromagnetic spinels Leon

Frustration and fluctuations in diamond antiferromagnetic spinels Leon Balents Doron Bergman Jason Alicea Simon Trebst Emanuel Gull Lucile Savary Sungbin Lee

Degeneracy and Frustration p p p Classical frustrated models often exhibit “accidental” degeneracy The degree of (classical) degeneracy varies widely, and is often viewed as a measure of frustration E. g. Frustrated Heisenberg models in 3 d have spiral ground states with a wavevector q that can vary n n n FCC lattice: q forms lines Pyrochlore lattice: q can be arbitrary Diamond lattice J 2>|J 1|/8: q forms surface

Accidental Degeneracy is Fragile p Diverse effects can lift the degeneracy n n n Thermal fluctuations F=E-TS Quantum fluctuations E=Ecl+Esw+… Perturbations: p p p Further exchange Spin-orbit (DM) interaction Spin-lattice coupling Impurities Questions: n n What states result? Can one have a “spin liquid”? What are the important physical mechanisms in a given class of materials? Does the frustration lead to any simplicity or just complication? Perhaps something useful?

Spinel Magnets p Normal spinel structure: AB 2 X 4. B cubic Fd 3 m A X p Consider chalcogenide X 2 -=O, S, Se n p Valence: QA+2 QB = 8 A, B or both can be magnetic.

Deconstructing the spinel p A atoms: diamond lattice n p Bipartite: not geometrically frustrated B atoms: pyrochlore lattice n Two ways to make it: A B Decorate bonds Decorate plaquettes

Frustrated diamond spinels

Road map to A-site spinels p Many materials! Co. Rh 2 O 4 1 s=2 Co 3 O 4 5 s = 5/2 Mn. Sc 2 S 4 10 Mn. Al 2 O 4 20 Co. Al 2 O 4 Very limited theoretical understanding… V. Fritsch et al. (2004); N. Tristan et al. (2005); T. Suzuki et al. (2007) p Naïvely unfrustrated Fe. Sc 2 S 4 900 s = 3/2 Orbital degeneracy

Major experimental features p Significant diffuse scattering which is temperature dependent for TÀTN =2. 3 K n Correlations developing in spin liquid regime

Major Experimental Features p Correlations visible in NMR Loidl group, unpublished

Major Experimental Features p Long range order in Mn. Sc 2 S 4: n n n TN=2. 3 K Spiral q=(q, q, 0) Spins in (100) plane Lock-in to q=3¼/2 for T<1. 9 K Reduced moment (80%) at T=1. 5 K q

Major experimental features p Anomalous low temperature specific heat

Major Experimental Features p “Liquid” structure factor at low temperature in Co. Al 2 O 4: n No long range order

Frustration p Roth, 1964: 2 nd and 3 rd neighbor interactions not necessarily small n p Exchange paths A-X-B-X-A Minimal theory: n Classical J 1 -J 2 model J 1 J 2 p Néel state unstable for J 2>|J 1|/8

Ground state evolution p Coplanar spirals Neel Evolving “spiral surface” q 0 p 1/8 Spiral surfaces:

Effects of Degeneracy: Questions p Does it order? n n p Pyrochlore: no order (k arbitrary) FCC: order by (thermal) disorder (k on lines) If it orders, how? n And at what temperature? Is f large? Is there a spin liquid regime, and if so, what are its properties? p Does this lead to enhanced quantum fluctuations? p

Low Temperature: Stabilization p There is a branch of normal modes with zero frequency for any wavevector on the surface (i. e. vanishing stiffness) n p Naïve equipartion gives infinite fluctuations Fluctuations and anharmonic effects induce a finite stiffness at T>0 n Fluctuations small but À T: n Leads to non-analyticities

Low Temperature: Selection p Which state is stabilized? n “Conventional” order-by-disorder p p 1/8 Normal mode contribution Need free energy on entire surface F(q)=E-T S(q) Results: complex evolution! 1/4 ~1/2 ~2/3 Green = Free energy minima, red = low, blue = high

Tc: Monte Carlo p Parallel Tempering Scheme (Trebst, Gull) Co. Al 2 O 4 Mn. Sc 2 S 4 Tc rapidly diminishes in Neel phase “Order-by-disorder”, with sharply reduced Tc Reentrant Neel

Spin Liquid: Structure Factor p Intensity S(q, t=0) images spiral surface Analytic free energy Numerical structure factor Mn. Sc 2 S 4 p Spiral spin liquid: 1. 3 Tc<T<3 Tc Order by disorder 0 “hot spots” visible Spiral spin liquid Physics dominated by spiral ground states

Capturing Correlations p Spherical model n Predicts data collapse Peaked near surface Mn. Sc 2 S 4 Structure factor for one FCC sublattice Nontrivial experimental test, but need single crystals… Quantitative agreement! (except very near Tc)

Comparison to Mn. Sc 2 S 4 p Structure factor reveals intensity shift from full surface to ordering wavevector Experiment Theory J 3 = |J 1|/20 A. Krimmel et al. PRB 73, 014413 (2006); M. Mucksch et al. (2007)

Degeneracy Breaking p Additional interactions (e. g. J 3) break degeneracy at low T Order by disorder 0 J 3 Spiral spin liquid paramagnet Mn. Sc 2 S 4 Two ordered states! Spin liquid only

Comparison to Mn. Sc 2 S 4 p Ordered state q=2 (3/4, 0) explained by FM J 1 and weak AF J 3 “Spin liquid” with Qdiff 2 diffuse scattering ordered 0 1. 9 K High-T paramagnet 2. 3 K =25 K qq 0 A. Krimmel et al. (2006); M. Mucksch et al. (2007)

Magnetic anisotropy p Details of Mn. Sc 2 S 4 cannot be described by Heisenberg model n Spins in <100> plane p n Not parallel to wavevector q=(q, q, 0): ferroelectric polarization? Wavevector “locks” to commensurate q=3¼/2

Landau theory Order parameter p Coplanar state p Spin plane p

Order of energy scales Spiral surface formed Specific q selected p ? Spin spiral plane chosen ? Lock-in Require symmetry under subgroup of space group preserving q =(q, q, 0)

Landau Theory p Free energy (q=(q, q, 0)) p Phase diagram n Direction of n Observed spin order in Mn. Sc 2 S 4

Mechanisms? p Dipolar interactions n n Effect favors n=(110) Very robust to covalency corrections and fluctuations p p Dzyaloshinskii-Moriya interactions n p Quantum fluctuations reduce moment by 20% but do not change dipole favored order Ineffective due to inversion center Exchange anisotropy n Depending upon significance of first and second neighbor contributions, this can stabilize n=(100) order

Predictions related to anisotropy Lock-in occurs as observed p Spin flop observable in magnetic field not along (100) axis p n p Observed at B=1 T field (Loidl group, private communication) Order accompanied by electric polarization, tunable by field

Impurity Effects p Common feature in spinels n n p “inversion”: exchange of A and B atoms Believed to occur with fraction x ~ 5% in most of these materials Related to “glassy” structure factor seen in Co. Al 2 O 4? n But: why not in Mn. Al 2 O 4, Co. Rh 2 O 4, Mn. Sc 2 S 4?

Impurity Effects: theory p A hint: recall phase diagram Co. Al 2 O 4 Mn. Sc 2 S 4

Sensitivity to impurities Seems likely that Co. Al 2 O 4 is more sensitive to impurities because it lies near “Lifshitz point” p What about spiral degeneracy for J 2>J 1/8? p Competing effects: p n n p Impurities break “accidental” spiral degeneracy: favors order Different impurities prefer different wavevectors: favors disorder Subtle problem in disordered “elastic media”

Swiss Cheese Picture p A single impurity effects spin state only out to some characteristic distance » & ¸ n Stiffness energy » Constant q here

Swiss Cheese Picture p A single impurity effects spin state only out to some characteristic distance » & ¸ n Stiffness energy » p local patches of different q

Comparison to Co. Al 2 O 4 p Close to J 2/J 1=1/8 n p Co. Al 2 O 4 |q|! 0: ¸ ! 1 : large » “Theory”: Experiment T. Suzuki et al, 2007 “Theory”: average over spherical surface Mn. Sc 2 S 4

Outlook p Combine understanding of A+B site spinels to those with both n Many interesting materials of this sort exhibiting ferrimagnetism, multiferroic behavior… Take the next step and study materials like Fe. Sc 2 S 4 with spin and orbital frustration p Identification of systems with important quantum fluctuations? p