Dynamics and Thermodynamics of Artificial Spin Ices - and the Role of Monopoles

Transcription

1 Dynamics and Thermodynamics of Artificial Spin Ices - and the Role of Monopoles Gunnar Möller Cavendish Laboratory University of Cambridge Roderich Moessner Max Planck Institute for the Physics of Complex Systems Dresden MMM 2011, Phoenix, Arizona Nov 3rd, 2011

2 Overview Introduction: Spin Ice Spin-Ice: an attractive target model engineering magnetic degrees of freedom Square Ice: a two-dimensional representation of Spin-Ice Soft-Spin model frustrated magnetism and long-range dipolar interactions Thermodynamics & a model of dynamics in Artificial Spin-Ice Kagomé-Ice: a different frustrated magnetic state Conclusions Cascade of low-temperature ordered phases: Kagomé Ice I & II Mapping to the dimer model on the hexagonal lattice

3 Target-model for this talk Spin-ice: a frustrated magnetic state equivalent to ice Mapping from ice I h to spins on the pyrochlore lattice Ground-states satisfy the ice rules Ice rule: There are precisely two hydrogen atoms near each oxygen atom (H 2 O survives) In spin language: each tetrahedron satisfies the rule two in - two out O H S 6 out of 16 possible spin configurations on a single tetrahedron satisfy the ice-rule for a lattice of tetrahedra, a macroscopic degeneracy arises [observed e.g. in Dy 2 Ti 2 O 7, Ramirez et al., Nature (1999)]

4 Spin Ice: Neither ordered nor disordered No order as in ferromagnet extensive degeneracy Not disordered like a paramagnet ice rules conservation law Consider magnetic moments µ i as (lattice) flux vector field Ice rules µ = 0 µ = A Local constr. emergent gauge struct. algebraic spin correlations structure factor

5 Neutron scattering: evidence for bow-ties Ice rules µ = 0 µ = A Local constr. emergent gauge structure spin correlations (3 cos 2 θ 1)/r 3 Fennell et al.2009 NIST 2009 (kagome ice) Kadowaki et al.2009

6 Elementary excitations: Magnetic monopoles magnetic Coulomb interaction E(r) = µ 0 qm 2 4π r deconfined monopoles charge q m q D /8000 [monopoles in H, not B] Castelnovo, et al, Nature 451, 42 (2008)

7 Elementary excitations: Magnetic monopoles magnetic Coulomb interaction E(r) = µ 0 qm 2 4π r deconfined monopoles charge q m q D /8000 [monopoles in H, not B] Castelnovo, et al, Nature 451, 42 (2008)

8 Elementary excitations: Magnetic monopoles magnetic Coulomb interaction E(r) = µ 0 qm 2 4π r deconfined monopoles charge q m q D /8000 [monopoles in H, not B] Castelnovo, et al, Nature 451, 42 (2008)

9 Artificial Spin Ice Two challenges: engineer spins, design their interactions strength ulations 1800 ith peri- System ith open d equa- Particle position as the fundamental degree of freedom L REVIEW LETTERS week ending 1 DECEMBER of freedom2006 QI, BRINTLINGER, AND CUMINGS Colloids in bimodal optical traps tunable potential (a) well: threshold (1) for reversal charged colloids 3D Coulomb unit of -colloid interactions b form, Libál et al., PRL (2006) =r ij FIG. 2. Color online a MFM data presented by Tanaka et al. Local properties of site as degree Arrays of nano-magnets J 1 (b) J 2 J 3 PHYSICAL REVIEW B 77, magnetization of permalloy islands as local Ising degree of freedom long-range dipolar interactions flexibility of geometries, manufactured by lithography Wang et al., Nature (2006), Qi et FIG. 3. Color online a An in-focus TEM image of our fabricated kagome structure scale bar: 1 m. Inset: A design image of the entire lattice scale bar: 10 m; the individual elements cannot be seen at this scale. b A TEM image of the same kagome structure with Lorentz contrast. c A Lorentz TEM simulation using a contrast transfer function reveals the single-domain magnetic moment direction based on the dark-bright edge contrast; using this, six spins in b are labeled with their directions. The two circles in b indicate clockwise and counterclockwise closed loops. al.(2008) & others

10 Spin-Ice in Two Dimensions Think of square-ice as the projection of spin-ice onto the plane same ice-rules apply to vertices two in - two out enforced by short-range Hamiltonian H = J ij σ i σ j, J ij J <i,j>, i,j model is integrable [ Lieb, 1967 ] µ = 0 gauge theories macroscopic ground state degeneracy S 0 = 3 4 ln 4 3 No magnetic materials are known which realize square-ice. How could this state be realized in nature?

11 Artificial Square Ice Artificially manufacture degrees of freedom, using the tool-box of lithography & nano-technology J 1 J 2 J 3 Wang et.al., Nature (2006) effective Ising degrees of freedom µ = σ µ ˆd, σ = ±1 Magnetic moments have long-range dipolar interactions [resolved by projective equivalence in 3D, Isakov et al.(2005)]

12 Artificial Square Ice Artificially manufacture degrees of freedom, using the tool-box of lithography & nano-technology J 1 J 2 J 3 Wang et.al., Nature (2006) effective Ising degrees of freedom µ = σ µ ˆd, σ = ±1 Magnetic moments have long-range dipolar interactions [resolved by projective equivalence in 3D, Isakov et al.(2005)]

13 Modelling Artificial Square Ice Q: Can we find a two-dimensional system that, as in 3D, implements spin ice in presence of long-range interactions? Write Hamiltonian for Ising-degrees of freedom σ i,α H = D iα,jβ σ iα σ jβ (iα),(jβ) l cell index i, sublattice index α model dipoles as linearly extended entities D iα,jβ = ρ 2 d r µ d riα d r iα d r jβ 3[d r iα ˆr ij ][d r jβ ˆr ij ] jβ rij 3 linear density of magnetic dipole moment ρ µ d µ = ρ µ d l

14 Dumbbell picture: Dipoles as pairs of Magnetic Monopoles Have already assumed linear distribution of dipole moments Infinitesimal dipoles equal a pair of opposite charges d µ = 2Qd l For any linear distribution C, only the charge density and location of the end-points matters r U( r) = d l dl E dipole (l) C rl [ 1 = Q eff r r f 1 ] r r i + + Needle dipoles produce same field as dumbbell of two opposite magnetic monopoles charges placed at each of its ends

15 Soft-Spin (Mean Field) Model disregard discreteness of Ising variables σ H diagonal in momentum q with eigenvalues V n (q) Spin-Ice J 1 = J 2 = J Dipolar interaction cut-off at r max = 10a F-Model Only J 1, J 2 0 π q x π π q y π π q x π π q y π π q x π π q y π flat lower band indicates macroscopic degeneracy of the groundstate lower band with significant dispersion transition to an ordered state at T c 1.7J 1 J 1 = 3 2 J 2 lower band most similar to full dipole interaction

16 Idea: Fix inequality of J NN and J NNN Equality of nearest- and next-nearest neighbor interactions can be established by shifting one of the sublattices into the third dimension!

17 Idea: Fix inequality of J NN and J NNN h Equality of nearest- and next-nearest neighbor interactions can be established by shifting one of the sublattices into the third dimension!

18 Idea: Fix inequality of J NN and J NNN two parameters for the new geometry: h/a, l/a a h/a Ratio of J 2 /J h l l/a 0.8 Equality of nearest- and next-nearest neighbor interactions can be established by shifting one of the sublattices into the third dimension!

19 Idea: Fix inequality of J NN and J NNN two parameters for the new geometry: h/a, l/a Ratio of J 2 /J h/a l/a 0.8 Equality of nearest- and next-nearest neighbor interactions can be established by shifting one of the sublattices into the third dimension! l a: J 1 = J 2 if endpoints form tetrahedron

20 Self-screening of the dipolar interactions π q x π π q y π J(q) [J 1 ] F-model -0.2 K r = 1 r = 2 r = 3 r = 4 r = r = 6 X r = 8 r = 10 r = 20 BZ r =100 r = X π/2 0 π/sqrt(2) K q Parameters shown: l/a = 0.7, h/a = dispersion of lower band vanishes precisely, as l/a 1 analysis of energy eigenvalues along high symmetry axes for various cut-off r max long-range part of interaction is approximately self-screening

21 Thermodynamics of tweaked Square Ice % Type I Type II Type III Type IV T [J] ice regime S C Ising-Ice l/a=0.7, h/a=0.207 l/a=0.7, h/a= T [J] [type III defects = Monopoles, from: GM and R.Moessner PRL (2006)] Clearly visible temperature interval of ice regime with S = S Lieb = 3 4 ln different ordered states (ferro- or antiferro-magnetic) possible, depending on parameters l, h

22 Dynamics of Artificial Square-Ice Experimental results of Wang et al. do not match the predictions for a thermal ensemble of spins New Model In thermodynamic simulations, defects (type III and type IV vertices) are being annealed out at rather high temperatures Try to understand the limitations of the dynamics 1 Greedy Dynamics Simulate an inhibited zero temperature Monte-Carlo dynamics Accept only moves which gain at least a threshold energy θ Average over independent downhill runs starting from random initial conditions Characterize final configurations obtained in this procedure with the experimental observables G. Möller and R. Moessner. Phys. Rev. Lett. 96, (2006)

23 Greedy Dynamics: Results Vertex distribution and local correlations for the downhill algorithm 40 Type I Type II Type III 20 Type IV θ [J 1 ] NN θ [J 1 ] [dashed lines: l/a = 0.7, solid lines: l/a = 0.95] Without a threshold, most non ice-rule defects are still being annealed out (recombining oppositely charged monopoles) At intermediate θ (plateau 3), results are close to experimental values L T

24 A phenomenological model 2 Model activation of spins by a rotating magnetic field to model the experimental situation of Wang et.al. keep energy threshold θ add coupling to a magnetic field with arbitrary orientation (rare spin-flips) linear decay of B-field with rate κ fix (θ, κ) to fit all experimental data good fit to experimental data % Type I Type II Type III Type IV a [nm] dotted: needle-hamiltonian dashed: modified ratio of J 1 /J 2, close to interactions calculated from finite element simulation by Wang et.al. [recent works: Nisoli et al.(2007), Budrikis et al.(2010) - steady state]

25 Intermediate Summary Geometry planar Square ice has an ordered groundstate (2 deg.) full frustration is obtained in a sublattice shifted geometry, by restoring symmetry between all NN interactions Thermodynamics modified square ice has a wide temperature window realizing spin ice configurations Dynamics for the understanding of the experimental systems, require: threshold field for switching spins driving force of external B-field rate limited by recombination of oppositely charged monopoles G. Möller and R. Moessner. Phys. Rev. Lett. 96, (2006)

26 Kagomé-Ice: a different frustrated magnetic state three spins per vertex modified ice-rule: two in - one out and vice versa 6 out of 8 possible vertex configurations in set of groundstate ice -states high zero-point entropy: S 0 = 1 n [2 ln n ( ) 2n ] = 2 3 ln thermodynamic regime of Kagomé ice naturally robust to long-range interactions due to perfect symmetry between all spins in a vertex long-range dipolar interactions cause an ordering transition at low temperature in the limit of l/a 1, Kagomé ice is obtained exactly experiments: Cummings group (2008), Schiffer group (2010)

27 Reminder: Dumbbell picture + + Apply this picture to the understanding of Kagomé ice

28 States of Kagomé-Ice in the monopole picture Consider long dipoles l/a 1 Ice-rules total charge on each vertex Q = ±1 At low enough temperature, obtain ordered state of opposite charges ±1 on neighboring vertices

29 States of Kagomé-Ice in the monopole picture Consider long dipoles l/a 1 Ice-rules total charge on each vertex Q = ±1 At low enough temperature, obtain ordered state of opposite charges ±1 on neighboring vertices

30 States of Kagomé-Ice in the monopole picture Consider long dipoles l/a 1 Ice-rules total charge on each vertex Q = ±1 At low enough temperature, obtain ordered state of opposite charges ±1 on neighboring vertices

31 States of Kagomé-Ice in the monopole picture Consider long dipoles l/a 1 Ice-rules total charge on each vertex Q = ±1 At low enough temperature, obtain ordered state of opposite charges ±1 on neighboring vertices

32 States of Kagomé-Ice in the monopole picture Consider long dipoles l/a 1 Ice-rules total charge on each vertex Q = ±1 Q= 1 Q=+1 Proliferation of monopoles in Kagomé-Ice, as opposed to monopoles being elementary excitations in Square-Ice

33 States of Kagomé-Ice in the monopole picture Consider long dipoles l/a 1 Ice-rules total charge on each vertex Q = ±1 Q= 1 Q=+1 At low enough temperature, obtain ordered state of opposite charges ±1 on neighboring vertices

34 Mapping from magnetic dipoles to dimers on the hexagonal lattice Vertex charges not fundamental degrees of freedom Exponential number of states possible for ordered charge state Can map ground-states of the ordered charged state to dimer coverings of the hexagonal lattice

35 Mapping from magnetic dipoles to dimers on the hexagonal lattice Vertex charges not fundamental degrees of freedom Exponential number of states possible for ordered charge state Can map ground-states of the ordered charged state to dimer coverings of the hexagonal lattice

36 Counting states in the set of groundstates Use mapping of dimers on the hexagonal lattice to the antiferromagnet on the dual triangular lattice Read off entropy of effective dimer model from triangular AFM: S kagomé ice II = 1 3 S triangular 0.108

37 Counting states in the set of groundstates Use mapping of dimers on the hexagonal lattice to the antiferromagnet on the dual triangular lattice Read off entropy of effective dimer model from triangular AFM: S kagomé ice II = 1 3 S triangular 0.108

38 Counting states in the set of groundstates Use mapping of dimers on the hexagonal lattice to the antiferromagnet on the dual triangular lattice Read off entropy of effective dimer model from triangular AFM: S kagomé ice II = 1 3 S triangular 0.108

39 Thermodynamics of dipolar Kagome-Ice Monte-Carlo simulations confirm the existence of an effective dimer model between the ordering temperature T d and the transition temperature to the ice regime T k. T ( J 1 ) e-04 1e-05 a) paramagnet Ice I T [ Q 2 /a ] 1e Ice II T=0.40 Q 2 /a T=0.84 ε 2 Q 2 /a ε = 1 l/a ordered ε 0.1 b) ε = 0.05 C S 0 1e T (Q 2 / a) G. Möller and R. Moessner, Phys. Rev. B 80, (R) (2009) C ( 10-2 /k B )

40 Low temperature order: beyond local interactions Need to resolve substructure of vortex-charges: multipole expansion in ɛ = 1 l/a E = q2 a = { 2 3ɛ + ( 3 2 α 2 ) ɛ + (γ + δ )ɛ2 }. b) for ordered groundstate (as shown left) monopole-monopole: α (Madelung energy) dipole-dipole term: γ monopole-quadrupole: δ (courtesy of Ravi Chandra)

41 Thermodynamics of dipolar Kagome-Ice Monte-Carlo simulations confirm the existence of an effective dimer model between the ordering temperature T d and the transition temperature to the ice regime T k. T ( J 1 ) e-04 1e-05 a) paramagnet Ice I T [ Q 2 /a ] 1e Ice II T=0.40 Q 2 /a T=0.84 ε 2 Q 2 /a ε = 1 l/a ordered ε 0.1 b) ε = 0.05 C S 0 1e T (Q 2 / a) G. Möller and R. Moessner, Phys. Rev. B 80, (R) (2009) C ( 10-2 /k B )

42 Conclusions Exciting possibilities to manufacture frustrated compounds Square Ice Proposed 2D square ice geometry with robust spin ice-regime Characterized out of equilibrium dynamics for artificial ice in a phenomenological model G. Möller and R. Moessner, Phys. Rev. Lett. 96, (2006) Kagomé Ice Kagomé-Ice is realized in dipolar nano-magnetic arrays Dumbbell picture of monopoles yields intuitive understanding Two distinct Kagomé-Ice phases, including a partially ordered phase mapping to dimer coverings of the hexagonal lattice G. Möller and R. Moessner, Phys. Rev. B 80, (R) (2009)