Frustration Driven Lattice Distortion in Y 2 Mo 2 O 7 Eva Sagi and - PowerPoint PPT Presentation

Frustration Driven Lattice Distortion in Y 2 Mo 2 O 7 Eva Sagi and Amit Keren Physics department, Technion

Outline  What is frustration? Why is it interesting?  Why Y 2 Mo 2 O 7 ?  Experimental results.  Computer simulations-no temperature.  Computer simulations- crystal “melting”.  Conclusions.

Geometrical Frustration  AF Hamiltonian and triangular geometry- not all near- neighbor spin interactions can be satisfied: FRUSTRATION.    H J S S ij i j i , j

The Heisenberg Hamiltonian 2   J J   .           2 H S S S S J ij i j i i   2 2     i , j i i    The only requirement for minimum of energy: S 0 . i   i  The frustration is “shared” among bonds.

Heisenberg Hamiltonian on the Pyrochlore Lattice  Infinite set of mean field ground states with zero net spin on all tetrahedra.  Each tetrahedron has an independent degree of freedom in the ground state!  No barriers between mean field ground states.  Infinite degeneracy, no single ground state can be selected by Heisenberg Hamiltonian- lower-order terms become significant.

Is Exchange C onstant ?    H J S S ij i j ij  J ij is controlled by higher energy physics that we like to consider irrelevant at low energies. • Atomic spacing • Orbital overlap • Orbital occupancy • Localized or itinerant electronic states  These degrees of freedom can become relevant if H produces “degenerate” state.  The lattice might distort, changing the value of the exchange, if the cost in elastic energy is smaller than the gain in magnetic energy.

Example- the kagome lattice J J ? J J a J a J b

Suggestion for Relief of Degeneracy- Magnetoelastic Distortion       k         2  H J J ' r S S r ij i j ij   2 i , j Effective Exchange Elastic Term  models the electrostatic k potential near its minimum. J '  is the change in the exchange integral with change in interatomic distance.

Theoretical Ground State, T=0       k         2  H J J ' r S S r ij i j ij   2 i , j  Find minimal value of normal vibrational coordinates in the presence of magnetoelastic term   J ' r S S . ij i j  Arrange distorted tetrahedrons on pyrochlore lattice.  Net zero spin on each tetrahedron. Tchernyshyov et al., PRB 66 (2002)

The q=0 State  The minimum energy state for a single tetrahedron can be arranged on the pyrochlore lattice in one of two q=0 configurations.  The q=0 distortion: tetrahedrons with identical orientation distort the same way. Tchernyshyov et al., PRB 66 (2002)

The q=0 State- Characteristics  2/3 strong (shortened) bonds,  1/3 weak (lengthened) bonds,  collinear spins  2/3 bonds with antiparallel spins , 1/3 bonds with parallel spins.

Searching for Frustration Driven Distortion

How will the system behave at T → 0?  CW Material spin type spin T c Low T phase Ref. value (K) (K) MgV 2 O 4 isotrop. 1 -750 45 LRO Baltzer et al '66 ZnV 2 O 4 isotrop. 1 -600 40 LRO Ueda et al '97 CdCr 2 O 4 isotrop. 3/2 -83 9 LRO Baltzer et al '66 MgCr 2 O 4 isotrop. 3/2 -350 15 LRO Blasse and Fast '63 ZnCr 2 O 4 isotrop. 3/2 -392 12.5 LRO S.-H. Lee et al '99 FeF 3 isotrop. 5/2 -230 20 LRO Ferey et al. '86 Y 2 Mo 2 O 7 isotrop. 1 -200 22.5 spin glass Gingras et al. '97 Y 2 Mn 2 O 7 isotrop. 3/2 17 spin glass Reimers et al '91 Tb 2 Mo 2 O 7 anisotr. 6 and 1 25 spin glass Greedan et al '91 Gd 2 Ti 2 O 7 isotrop. 7/2 -10 1 LRO Radu et al '99 Er 2 Ti 2 O 7 anisotr. -25 1.25 LRO Ramirez et al '99 Tb 2 Ti 2 O 7 anisotr. -19 spin liquid? Gardner et al '99 Yb 2 Ti 2 O 7 anisotr. 0 0.21 LRO Ramirez et al '99 Dy 2 Ti 2 O 7 Ising 7.5 1/2 0.5 1.2 spin ice Ramirez et al '99 Ho 2 Ti 2 O 7 Ising 8 1/2 1.9 spin ice Harris et al ''97  We chose Y 2 Mo 2 O 7 as a candidate to look for frustration-driven distortion, since it is a spin glass, and we want to understand the origin of the disorder in this material.

Y 2 Mo 2 O 7 Characteristics  Cubic pyrochlore A 2 B 2 O 7  Magnetic ion Mo 4+ , spin 1  AF interaction, θ CW =200K, J= θ CW /z~33K.  Spin-Glass transition at 22.5K

Experimental Motivation: Y 2 Mo 2 O 7  Booth et al.,XAFS: the Mo tetrahedra are in fact disordered from their ideal structure, with a relatively large amount of pair distance disorder, in the Mo-Mo pairs and perpendicular to the Y- 2 -3 -1 1 -2 3 Mo pairs (2000). -4 4 5 1 -5 0 2  Keren & Gardner, NMR: -1 6  (  f) (a.u.) T=92.4K -6 3 7 many nonequivalent 89 Y sites, -2 8 -7 9 possibly stemming from a T=200K 89  ( H 0 - H ext )/2  lattice distortion (2001). -50 0 50 100 150 200  f (KHz)

Experimental Data  DC magnetization.   SR.  High resolution neutron diffraction.

DC magnetization  Measure sample magnetization with moving sample magnetometer.  Observe phase transition to spin-glass. Phase transition

What is  SR?  100% spin polarized p muons.  Muon life time :  n  2.2 μ sec. e +   Positron emitted preferentially in the muon spin direction.  Collect positrons, obtain distribution of muon spin orientations.

 SR N B (t) N F (t) t A(t) t         t /   1   N t Bg N e A P t 0 0

Muon Relaxation Mechanisms  Relaxation caused by dynamical field fluctuations, consists of both longitudinal relaxation caused by fluctuations in the xy plane, and dynamical transverse relaxation caused by fluctuations in the z direction.  Static relaxation,which is reversible. It is caused by field inhomogeneities in the sample ∆B which are responsible for dephasing in the xy plane.

The μ SR Experiment  TF μ SR: measure both static and dynamic relaxation.  LF μ SR: measure dynamic relaxation.  Simultaneous TF and LF measurements, H=6000G, 20 0 K<T<240 0 K.  Subtract LF relaxation from TF relaxation- obtain relaxation from static fields only → compare to magnetization.

μ SR Data               1 / 2 exp cos A t A R t t Bg 0 TF   Bg        1 / 2 A t A 0 exp R t (a) 0.2 LF H=6kG 0.0 0.15 -0.2 T=45.9K Asymmetry 0.10 (b) 0.2 Asymmetry 0.0 T(K)= 0.05 23.2 -0.2 30.2 T=30.2K 45.9 0.00 (c) 0.2 0 2 4 6 8 Time (  sec)  Relaxation increases as 0.0 temperature is decreased. -0.2 T=23.2  TF data displayed in rotating- 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 reference-frame, H=5600G. Time (  sec)

μ SR Data R TF -transverse relaxation rate R LF -longitudinal relaxation rate          1 / 2  t P t P e cos t Temperature (K) static 0   24.5 55.0 42.4 35.4 29.6 2    1 / 2 1 / 2 R R TF LF    H 10  TF 10 -1 ] -1 ]  ,R TF [  sec 1 R LF [  sec 0.1  ∆ increases 1 Rlf exponentially Rtf fast with  0.01 0.008 0.009 0.010 0.011 0.012 0.013 0.014 increasing χ .  [emu/mol]

What Does it Mean?        The muon’s Hamiltonian: Η I H H  TF int   S  H A r  Mean field: int  M   H S A - magnetic coupling I - muon spin  Relaxation function S - electronic spin measured by  SR:     dA           TF  P t P cos 1 A H A 0 Evolution of polarization Averaging over of a single muon different muons

 We want the relation between what we measure in μ SR and what happens in matter:            1 / 2   t   1 A   P t P e cos t   0 A f          A A     1 / 2    t e cos A H t A dA  TF    A  H  TF  represents the width of the distribution. A   As the temperature is lowered, the ratio and therefore A , ,  grows, and the distribution widens.

Conclusions from Magnetic Measurements:  The change in the muon environment indicates that atoms shift!  However…

High Resolution Neutron Diffraction  No evidence for periodic  Neutron scattering data for rearrangement of the atoms, Y 2 Mo 2 O 7 show uniform from  SR or neutrons . shrinking of the unit cell with decreasing temperature.

Is something wrong with theory?  Valid only for T=0 ; we’re not there yet…  Only first order distortional terms were taken into account.  Assumption of zero net spin on each tetrahedron ; not necessarily true in the presence of a magnetoelastic distortion.  q=0 is guessed to be the ground state; the guess might be wrong…