Frustration and fluctuations in diamond antiferromagnetic spinels

1
Frustration and fluctuations in diamond
antiferromagnetic spinels

• Leon Balents
• Doron Bergman
• Jason Alicea
• Simon Trebst
• Emanuel Gull
• Lucile Savary
• Sungbin Lee

2
Degeneracy and Frustration

• 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 3d have
  spiral ground states with a wavevector q that can
  vary
• FCC lattice q forms lines
• Pyrochlore lattice q can be arbitrary
• Diamond lattice J2gtJ1/8 q forms surface

3
Accidental Degeneracy is Fragile

• Diverse effects can lift the degeneracy
• Thermal fluctuations FE-TS
• Quantum fluctuations EEclEsw
• Perturbations
• Further exchange
• Spin-orbit (DM) interaction
• Spin-lattice coupling
• Impurities
• Questions
• 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?

4
Spinel Magnets

• Normal spinel structure AB2X4 .

B
A
X

• Consider chalcogenide X2-O,S,Se
• Valence QA2QB 8
• A, B or both can be magnetic.

5
Deconstructing the spinel

• A atoms diamond lattice
• Bipartite not geometrically frustrated

• B atoms pyrochlore lattice
• Two ways to make it

A
B
Decorate bonds
Decorate plaquettes
6
Frustrated diamond spinels
7
Road map to A-site spinels
s 2

• Many materials!

Orbital degeneracy
FeSc2S4
s 5/2
CoRh2O4
MnSc2S4
Co3O4
1
900
10
20
5
CoAl2O4
MnAl2O4
s 3/2
Very limited theoretical understanding
V. Fritsch et al. (2004) N. Tristan et al.
(2005) T. Suzuki et al. (2007)

• Naïvely unfrustrated

8
Major experimental features

• Significant diffuse scattering which is
  temperature dependent for TÀTN 2.3K
• Correlations developing in spin liquid regime

9
Major Experimental Features

• Correlations visible in NMR

Loidl group, unpublished
10
Major Experimental Features

• Long range order in MnSc2S4
• TN2.3K
• Spiral q(q,q,0)
• Spins in (100) plane
• Lock-in to q3¼/2 for Tlt1.9K
• Reduced moment (80) at
• T1.5K

q
11
Major experimental features

• Anomalous low temperature specific heat

12
Major Experimental Features

• Liquid structure factor at low temperature in
  CoAl2O4
• No long range order

13
Frustration

• Roth, 1964 2nd and 3rd neighbor interactions not
  necessarily small
• Exchange paths A-X-B-X-A
• Minimal theory
• Classical J1-J2 model

J2
J1

• Néel state unstable for J2gtJ1/8

14
Ground state evolution

• Coplanar spirals

Evolving spiral surface
Neel
q
0
1/8

• Spiral surfaces

15
Effects of Degeneracy Questions

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

16
Low Temperature Stabilization

• There is a branch of normal modes with zero
  frequency for any wavevector on the surface (i.e.
  vanishing stiffness)
• Naïve equipartion gives infinite fluctuations
• Fluctuations and anharmonic effects induce a
  finite stiffness at Tgt0
• Fluctuations small but À T
• Leads to non-analyticities

17
Low Temperature Selection

• Which state is stabilized?
• Conventional order-by-disorder
• Need free energy on entire surface F(q)E-T S(q)
• Results complex evolution!

Normal mode contribution
1/8
1/4
1/2
2/3
Green Free energy minima, red low, blue high
18
Tc Monte Carlo

• Parallel Tempering Scheme (Trebst, Gull)

Tc rapidly diminishes in Neel phase
Order-by-disorder, with sharply reduced Tc
Reentrant Neel
19
Spin Liquid Structure Factor

• Intensity S(q,t0) images spiral surface

Numerical structure factor
Analytic free energy
MnSc2S4

• Spiral spin liquid 1.3TcltTlt3Tc

Spiral spin liquid
Physics dominated by spiral ground states
Order by disorder
0
hot spots visible
20
Capturing Correlations

• Spherical model
• Predicts data collapse

Peaked near surface
MnSc2S4
Structure factor for one FCC sublattice
Quantitative agreement! (except very near Tc)
Nontrivial experimental test, but need single
crystals
21
Comparison to MnSc2S4

• Structure factor reveals intensity shift from
  full surface to ordering wavevector

Experiment
Theory
J3 J1/20
A. Krimmel et al. PRB 73, 014413 (2006) M.
Mucksch et al. (2007)
22
Degeneracy Breaking

• Additional interactions (e.g. J3) break
  degeneracy at low T

Order by disorder
0
Two ordered states!
Spiral spin liquid
paramagnet
J3
Spin liquid only
23
Comparison to MnSc2S4

• Ordered state q2?(3/4,3/4,0) explained by FM J1
  and weak AF J3

High-T paramagnet
Spin liquid with Qdiff ? 2? diffuse scattering
ordered
0
1.9K
2.3K
25K
qq0
A. Krimmel et al. (2006) M. Mucksch et al. (2007)
24
Magnetic anisotropy

• Details of MnSc2S4 cannot be described by
  Heisenberg model
• Spins in lt100gt plane
• Not parallel to wavevector q(q,q,0)
  ferroelectric polarization?
• Wavevector locks to commensurate q3¼/2

25
Landau theory

• Order parameter
• Coplanar state
• Spin plane

26
Order of energy scales

• Require symmetry under subgroup of space group
  preserving q (q,q,0)

27
Landau Theory

• Free energy (q(q,q,0))
• Phase diagram
• Direction of n

Observed spin order in MnSc2S4
28
Mechanisms?

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

29
Predictions related to anisotropy

• Lock-in occurs as observed
• Spin flop observable in magnetic field not along
  (100) axis
• Observed at B1T field (Loidl group, private
  communication)
• Order accompanied by electric polarization,
  tunable by field

30
Impurity Effects

• Common feature in spinels
• 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
  CoAl2O4?
• But why not in
• MnAl2O4,
• CoRh2O4,
• MnSc2S4?

31
Impurity Effects theory

• A hint recall phase diagram

MnSc2S4
CoAl2O4
MnAl2O4
32
Sensitivity to impurities

• Seems likely that CoAl2O4 is more sensitive to
  impurities because it lies near Lifshitz point
• What about spiral degeneracy for J2gtJ1/8?
• Competing effects
• Impurities break accidental spiral degeneracy
  favors order
• Different impurities prefer different
  wavevectors favors disorder
• Subtle problem in disordered elastic media

33
Swiss Cheese Picture

• A single impurity effects spin state only out to
  some characteristic distance
• Stiffness energy

Constant q here
34
Swiss Cheese Picture

• A single impurity effects spin state only out to
  some characteristic distance
• Stiffness energy

• local patches of different q

35
Comparison to CoAl2O4

• Close to J2/J11/8
• q! 0 ! 1 large
• Theory

MnSc2S4
CoAl2O4
Experiment
T. Suzuki et al, 2007
Theory average over spherical surface
36
Outlook

• Combine understanding of AB site spinels to
  those with both
• Many interesting materials of this sort
  exhibiting ferrimagnetism, multiferroic behavior
• Take the next step and study materials like
  FeSc2S4 with spin and orbital frustration
• Identification of systems with important quantum
  fluctuations?