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

Overview Introduction Square Ice Kagome-Ice & more Conclusions

Dynamics and Thermodynamics of ArtificialSpin Ices - and the Role of Monopoles

Gunnar MollerCavendish Laboratory

University of Cambridge

Roderich MoessnerMax Planck Institute for the Physics of Complex Systems

Dresden

MMM 2011, Phoenix, ArizonaNov 3rd, 2011


Overview

Introduction: Spin Ice

Spin-Ice: an attractive target modelengineering ‘magnetic’ degrees of freedom

Square Ice: a two-dimensional representation of Spin-Ice

Soft-Spin modelfrustrated magnetism and long-range dipolar interactionsThermodynamics & a model of dynamics in Artificial Spin-Ice

Kagome-Ice: a different frustrated magnetic state

Cascade of low-temperature ordered phases: Kagome Ice I & IIMapping to the dimer model on the hexagonal lattice

Conclusions


Target-model for this talkSpin-ice: a frustrated magnetic state equivalent to ice

Mapping from ice Ih to spins on the pyrochlore lattice

Ground-states satisfy the ice rules

Ice rule: There are precisely twohydrogen atoms near each oxygenatom (H2O survives)

In spin language: each tetrahedronsatisfies the rule ‘two in - two out’

O

H

S

6 out of 16 possible spin configurations on a singletetrahedron satisfy the ice-rule

for a lattice of tetrahedra, a macroscopic degeneracy arises[observed e.g. in Dy2Ti2O7, Ramirez et al., Nature (1999)]


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 ~µias (lattice) ‘flux’ vector field

Ice rules ⇔ ∇ · ~µ = 0 ⇒ ~µ = ∇× ~ALocal constr. ⇒ emergent gauge struct.

→ algebraic spincorrelations

→ structure factor


Neutron scattering: evidence for bow-ties

Ice rules ⇔ ∇ · ~µ = 0 ⇒ ~µ = ∇× ~ALocal constr. ⇒ emergent gauge structure

→ spin correlations ∼ (3 cos2 θ − 1)/r3

Fennell et al.2009

NIST 2009 (kagome ice)Kadowaki et al.2009


Elementary excitations: Magnetic monopoles

magnetic Coulombinteraction

E (r) = −µ0

4π

q2m

r

� deconfined monopoles� charge qm ≈ qD/8000

[monopoles in H , not B]

Castelnovo, et al, Nature 451, 42 (2008)


Artificial Spin IceTwo challenges: engineer spins, design their interactions

Particle position as thefundamental degree of freedom

Colloids in bimodal optical trapscurs at constant temperature when the trap barrier strengthis varied.

We perform 2D Brownian dynamics (BD) simulationsfor systems of two sizes. System A contains N ! 1800interacting colloids and N ! 1800 optical traps with peri-odic boundary conditions in the x and y directions. SystemB has N ! 24 colloids and N ! 24 optical traps with openboundary conditions. In each case the overdamped equa-tion of motion for colloid i is:

!dRi

dt! Fcc

i " FTi " Fext

i " Fsi ; (1)

where the damping constant ! ! 1:0. We define the unit ofdistance in the simulation to be a0. The colloid-colloidinteraction force has a Yukawa or screened Coulomb form,Fcci ! #F0q2

PNi!jriV$rij% with V$rij% ! $1=rij%&

exp$#"rij%. Here, rij ! jri # rjj, rij ! $ri # rj%=rij, ri$j%is the position of particle i$j%; F0 ! Z'2=$4#$$0%, Z' is theunit of charge; $ is the solvent dielectric constant; q is thedimensionless colloid charge; and 1=" is the screeninglength, where " ! 4=a0 unless otherwise mentioned. Weneglect hydrodynamic interactions between colloids,which is a reasonable assumption for charged particles inthe low volume fraction limit. The thermal force FT ismodeled as random Langevin kicks with the propertieshFT

i i ! 0 and hFT$t%FT$t0%i ! 2!kBT%$t# t0%. Unless oth-erwise mentioned, FT ! jFT j ! 0. Fext

i represents an ex-ternally applied drive which is set to zero except for thebiased system, where Fext

i ! Fdc$x" y%.The substrate force Fs

i arises from elongated traps,shown schematically in Fig. 1(a), arranged in square struc-tures with lattice constant d, as in Fig. 1(b). Each trap iscomposed of two half-parabolic wells of strength fp andradius rp separated by an elongated region of length 2lwhich confines the colloid perpendicular to the trap axisand has a small repulsive potential or barrier of strength frparallel to the axis which pushes the colloid out of themiddle of the trap into one of the ends: Fs

ik !$fp=rp%r(ik!$rp # r(ik%r(ik " $fp=rp%r?ik!$rp # r?ik%r?ik "$fr=l%$1 # rkik%!$l # rkik%rkik. Here r(ik ! jri # rpk ( lpk

kj,r?;kik ! j$ri # rpk % ) pk

?;kj, ri (rpk ) is the position of colloid i(trap k), and pk

k (pk?) is a unit vector parallel (perpendicu-

lar) to the axis of trap k. We take 2l ! 2a0, rp ! 0:4a0, andd ! 3a0 unless otherwise noted. Elongated traps of thisform have been created in previous experimental work[11,12]. Our dimensionless units can be converted to physi-cal units for a particular system. For example, when a0 !2 &m, $ ! 2, and Z' ! 300e, such as in Ref. [14], F0 !2:5 pN and the trap ends are 0:2 &m apart at d ! 3. Wefind the ground state of each configuration using simulatedannealing.

The vertices are categorized into six types, listed inTable I, and we identify the percentage occupancy Ni=Nand energy Ei of each type. Type III and type IV verticeseach obey the ice rule of a two-in two-out configuration,

represented here by two colloids close to the vertex andtwo far from the vertex. Locally, the system would prefertype I vertices, but such vertices must be compensated byhighly unfavorable type VI vertices. The colloidal spin icerealization differs from the magnetic system, where north-north and south-south magnetic interactions at a vertexhave equal energy. For the colloids, interactions betweentwo filled trap ends raise the vertex energy Ei, whereas twoadjacent empty trap ends decrease Ei. Since particle num-ber must be conserved, creating empty trap ends at onevertex increases the particle load at neighboring vertices.As a result, the ice rules still apply to our system, but theyarise due to collective effects rather than from a localenergy minimization.

In Fig. 1(b) we illustrate a small part of system A withnoninteracting colloids at charge q ! 0. The distribution ofNi=N is consistent with a random arrangement. When weincrease q to q ! 1:3 so that the colloids are strongly

(b)

(c) (d)

(a)

FIG. 1. (a) Schematic of the basic unit cell with four doublewell traps each capturing one colloid. (b)–(d) Images of a smallportion of system A with N ! 1800. Dark circles: colloids;ellipses: traps. (b) Random vertex distribution at q ! 0.(c) Long-range ordered square ice ground state at q ! 1:3.(d) Biased system at q ! 0:4 with Fdc ! 0:02.

TABLE I. Electrostatic energy Ei=EIII for each vertex type. Anexample configuration for each vertex is listed; 1 (0) indicates acolloid close to (far from) the vertex.

Type Configuration Ei=EIII Type Configuration Ei=EIII

I 0000 0.001 IV 1001 7.02II 0001 0.0214 V 1101 14.977III 0101 1.0 IV 1111 29.913

PRL 97, 228302 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending1 DECEMBER 2006

228302-2

• tunable potential well: threshold

for reversal

• charged colloids → 3D Coulomb

interactions

Libal et al., PRL (2006)

Local properties of site as degreeof freedom

Arrays of nano-magnets

3JJ21J

ity of ice-rule-violating vertices among random ensembles,as Wang et al. have done. We present here an artificial spinice approach to the magnetic honeycomb structure that ad-dresses both of these experimental deficiencies.

In Ref. 16, magnetic force microscopy !MFM" is used toimage the magnetic structure of the kagome lattice. MFMworks by detecting escaped flux from the material, and inthese structures, it can therefore only yield information aboutthe excess flux at a given interaction vertex. The kagomelattice possesses three magnetic elements per Bravais latticepoint, with each element having a two-level degree of free-dom. However, MFM can only capture two-level informationat the interaction vertices, which number two per Bravaislattice point. For a lattice with n Bravais lattice sites, MFMresults may express up to 22n unique states, whereas the lat-tice can exhibit on the order of 23n !both with small correc-tions for the ice rule".

To demonstrate this deficiency, we use the data and modelpresented in Fig. 2 of Ref. 16. This figure contains MFMdata in Fig. 2!a" and a model of the magnetic moment ori-entations in Fig. 2!b", and both are reproduced here as Figs.2!a" and 2!b", respectively. To explicitly demonstrate the un-derdefined nature of moment configurations constructedfrom MFM on interacting vertices, one can construct anothermodel of magnetic orientations by selecting from a givenmoment map any head-to-tail chain of elements and thenreversing the entire chain. Two possible examples are shownhere in Figs. 2!c" and 2!d", and we estimate that there are onthe order of 212 other such configurations, only one of whichreflects the actual unknown configuration of the system. Thisuncertainty also makes second- and third-nearest-neighborcorrelations impossible to deduce. What is needed to adapt

the honeycomb network into a full-fledged physical model ofkagome spin ice is an imaging technique that directly andunambiguously records the internal magnetic flux of the wireelements. This can be achieved by the Lorentz-mode trans-mission electron microscopy, as we demonstrate below.

Our realization of the kagome structure is fabricated fromPermalloy !Ni80Fe20" using conventional electron-beam li-thography, followed by metal deposition and lift-off. Figure3!a" shows a transmission electron microscope !TEM" imageof our structure. The lines of the honeycomb are 500 nmlong, 110 nm wide, and 23 nm thick. At this scale, micro-magnetic simulations22 indicate that the connecting elementsare magnetized along their axis and act as macroscopic Isingspins with energy differences among the different configura-tions that support the ice-rule assumption.23 With stronganalogies to real spin ice, these simulations show that 85% ofthis nearest neighbor energy difference comes from a dipolarfield, with the remaining 15% coming from exchange energydue to the domain walls at the vertices. The total number ofelements in our realization is 12 864, large enough for en-semble results that are comparable with Monte Carlosimulations.17

To determine the directions of the single-domain ele-ments, we employ a TEM operating in Lorentz imagingmode, which is traditionally used to detect domain structuresof magnetic materials.24,25 To simulate the contrast of single-domain needle-shaped elements, we use a standard contrasttransfer function.26 Figure 3!c" shows that the images of thespin elements have overfocus Lorentz contrast featuring adark edge and a bright edge, depending on the magnetizationdirection. Simply, this can be explained by Lorentz-force de-flection when the electron beam passes through a magneticelement. Figure 3!b" shows a Lorentz-mode image corre-sponding to Fig. 3!a", and we can see that the elements havevaried contrast because of their varied magnetization direc-tions. Using a right-hand rule, we uniquely specify a direc-tion for each element, as shown by the colored arrows. Weverify the magnetic origin of the contrast both by through-

FIG. 2. !Color online" !a" MFM data presented by Tanaka et al.!Ref. 16". !b" A spin configuration proposed therein to describe thedata. #!c" and !d"$ Valid alternative configurations obtained by re-versing chains of elements, shown in different colors. In the generalcase, these chains include closed loops which can be reversedclockwise or counterclockwise.

FIG. 3. !Color online" !a" An in-focus TEM image of our fab-ricated kagome structure !scale bar: 1 !m". Inset: A design imageof the entire lattice !scale bar: 10 !m; the individual elements can-not be seen at this scale". !b" A TEM image of the same kagomestructure with Lorentz contrast. !c" A Lorentz TEM simulation us-ing a contrast transfer function reveals the single-domain magneticmoment 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.

QI, BRINTLINGER, AND CUMINGS PHYSICAL REVIEW B 77, 094418 !2008"

094418-2

• magnetization of permalloy islandsas local Ising degree of freedom• long-range dipolar interactions• flexibility of geometries,manufactured by lithographyWang et al., Nature (2006), Qi et

al.(2008) & others


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 =∑

<i ,j>,�i ,j�Jijσiσj , |Jij | ≡ J

model is integrable [ Lieb, 1967 ]

∇ · ~µ = 0 ⇒ gauge theories

macroscopic ground state degeneracyS0 = 3

4 ln 43

No magnetic materials are known which realize square-ice.How could this state be realized in nature?


Artificial Square Ice

Artificially manufacture degrees of freedom, using thetool-box of lithography & nano-technology

3JJ21J

Wang et.al., Nature(2006)

effective Ising degrees of freedom

~µ = σ|µ|d , σ = ±1

Magnetic moments have long-range dipolar interactions[resolved by ‘projective equivalence’ in 3D, Isakov et al.(2005)]


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 σi ,α

H =∑

(iα),(jβ)

Diα,jβσiασjβ

cell index i , sublattice index α

model dipoles as linearly extended entities

Diα,jβ = ρ2µ

sd~riαd~rjβ

d~riα·d~rjβ−3[d~riα·rij ][d~rjβ ·rij ]r3ij

linear density of magnetic dipole moment ρµ

l

d~µ = ρµd ~


Dumbbell picture: Dipoles as pairs of Magnetic Monopoles

Have already assumed lineardistribution of dipole moments

Infinitesimal dipoles equal a pair ofopposite charges d~µ = 2Qd ~

For any linear distribution C, onlythe charge density and location ofthe end-points matters

U(~r) =

∫ ~r

∞d ~′ ·

∫Cd` ~Edipole(`)

∣∣∣~r`′

= Qeff

[1

|~r −~rf |− 1

|~r −~ri |

]+−

+−

Needle dipoles produce same field as dumbbell of two oppositemagnetic monopoles charges placed at each of its ends


Soft-Spin (Mean Field) Model

disregard discreteness of Ising variables σ ⇒ H diagonal inmomentum q with eigenvalues Vn(q)

Spin-IceJ1 = J2 = J

xq

qy

π

π

−π

−π

flat lower bandindicatesmacroscopicdegeneracy of thegroundstate

Dipolar interactioncut-off at rmax = 10a

xq

qy

π

π

−π

−π

lower band withsignificantdispersiontransition to anordered state atTc ≈ 1.7J1

F-ModelOnly J1, J2 6= 0

xq

qy

π

π

−π

−π

J1 = 3√2J2

lower band mostsimilar to fulldipole interaction


Idea: Fix inequality of JNN and JNNN

Equality of nearest- and next-nearest neighbor interactionscan be established by shifting one of the sublattices into thethird dimension!



h




two parameters for the new geometry:h/a, l/a

h

a

l

Ratio of J2/J1

0.95

0.8

0.6

0.3

0.2

0.1

0.0

0.0 0.2 0.4 0.6 0.8

h/a

l/a

1.05

1.2

1.4

1.6




two parameters for the new geometry:h/a, l/a

Ratio of J2/J1

0.95

0.8

0.6

0.3

0.2

0.1

0.0

0.0 0.2 0.4 0.6 0.8

h/a

l/a

1.05

1.2

1.4

1.6

Equality of nearest- and next-nearest neighbor interactionscan be established by shifting one of the sublattices into thethird dimension! l → a: J1 = J2 if endpoints form tetrahedron


Self-screening of the dipolar interactions

xq

qy

π

π

−π

−π

Parameters shown:l/a = 0.7, h/a = 0.207

dispersion of lower bandvanishes precisely, asl/a→ 1

X π/2 0 π/sqrt(2) K|q|

-0.4

-0.3

-0.2

-0.1

0

J(q)

[J 1]

F-modelr = 1r = 2r = 3r = 4r = 5r = 6r = 8r = 10r = 20r =100r =500

X

K

0

BZ

analysis of energy eigenvaluesalong high symmetry axes forvarious cut-off rmax

long-range part of interaction isapproximately self-screening


Thermodynamics of tweaked Square Ice

0 1 2T [J]

-40

-20

0

20

40

60

80

%

Type I Type II Type III Type IV

0

0.2

0.4

0.6 S

Ising-Icel/a=0.7, h/a=0.207l/a=0.7, h/a=0.205

0 1 2T [J]

0

0.2

0.4

0.6 C

-��

ice regime

[type III defects = Monopoles, from: GM and R.Moessner PRL (2006)]

Clearly visible temperature interval of ice regime withS = SLieb = 3

4 ln 43 ≈ 0.212

different ordered states (ferro- or antiferro-magnetic) possible,depending on parameters l , h


Dynamics of Artificial Square-Ice

Experimental results of Wang et al. do not match thepredictions for a thermal ensemble of spins ⇒ New Model

In thermodynamic simulations, defects (type III and type IVvertices) are being annealed out at rather high temperatures

Try to understand the limitations of the dynamics

1 Greedy DynamicsSimulate an inhibited zero temperature Monte-Carlo dynamicsAccept only moves which gain at least a threshold energy θAverage over independent ‘downhill’ runs starting from randominitial conditionsCharacterize final configurations obtained in this procedurewith the experimental observables

G. Moller and R. Moessner. Phys. Rev. Lett. 96, 237202 (2006)


Greedy Dynamics: Results

Vertex distribution and local correlations for the downhillalgorithm

0 2 4 6 8 10 12θ [J

1]

-40

-20

0

20

40 Type IType IIType IIIType IV

0 2 4 6 8 10 12θ [J

1]

-0.2

0

0.2

0.4

NN

L

T

1

23

4

12

3

4

[dashed lines: l/a = 0.7, solid lines: l/a = 0.95]

Without a threshold, most non ice-rule defects are still beingannealed out (recombining oppositely charged monopoles)

At intermediate θ (plateau 3), results are close toexperimental values


A phenomenological model

2 Model activation of spins bya rotating magnetic field tomodel the experimentalsituation of Wang et.al.

keep energy threshold θ

add coupling to a magneticfield with arbitraryorientation (rare spin-flips)

linear decay of B-field withrate κ

fix (θ, κ) to fit allexperimental data

⇒ good fit to experimental data

300 400 500 600 700 800 900a [nm]

-30

-20

-10

0

10

20

30

%

Type IType IIType IIIType IV

dotted: ‘needle-Hamiltonian’

dashed: modified ratio ofJ1/J2, close to interactionscalculated from finite elementsimulation by Wang et.al.

[recent works: Nisoli et al.(2007), Budrikis et al.(2010) - steady state]


Intermediate Summary

Geometry

planar ‘Square ice’ has an ordered groundstate (2× deg.)

full frustration is obtained in a sublattice shifted geometry, byrestoring symmetry between all NN interactions

Thermodynamics

modified square ice has a wide temperature window realizingspin ice configurations

Dynamics

for the understanding of the experimental systems, require:

threshold field for switching spinsdriving force of external B-field

rate limited by recombination of oppositely charged monopoles

G. Moller and R. Moessner. Phys. Rev. Lett. 96, 237202 (2006)


Kagome-Ice: a different frustrated magnetic state

three spins per vertex ⇒ modifiedice-rule: “two in - one out” and vice versa

6 out of 8 possible vertex configurationsin set of groundstate ‘ice’-states

high zero-point entropy:

S0 = 1n ln

[2n(

68

) 2n3

]= 2

3 ln 3√2≈ 0.501

thermodynamic regime of Kagome ice naturally robust tolong-range interactions due to perfect symmetry between allspins in a vertex

long-range dipolar interactions cause an ordering transition atlow temperature

in the limit of l/a→ 1, Kagome ice is obtained exactly

experiments: Cummings group (2008), Schiffer group (2010)


Reminder: Dumbbell picture

+−

+−

⇒ Apply this picture to the understanding of Kagome ice


States of Kagome-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 oppositecharges ±1 on neighboring vertices





Q=−1

Q=+1

Proliferation of monopoles in Kagome-Ice, as opposed tomonopoles being elementary excitations in Square-Ice






Q=−1

Q=+1



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 dimercoverings of the hexagonal lattice


Counting states in the set of groundstates

Use mapping of dimers on the hexagonal lattice to theantiferromagnet on the dual triangular lattice

Read off entropy of effective dimer model from triangularAFM: Skagome ice II = 1

3Striangular ≈ 0.108


Thermodynamics of dipolar Kagome-Ice

Monte-Carlo simulations confirm the existence of an effectivedimer model between the ordering temperature Td and thetransition temperature to the ice regime Tk .

✫

✫

✫

✫

✫

✫ ✫

0 0.2 0.4 0.6 0.8 1

ε = 1 − l/a

1e-05

1e-04

0.001

0.01

0.1

1

10

T (

J1 )

1e-04 0.01 1 100

T (Q2/ a)

0

0.1

0.2

0.3

0.4

0.5

0.6CS

✫

✫

✫✫

✫

✫

0.001 0.01 0.1ε

1e-0

40.

011

T [

Q2 /a

] T=0.40 Q2/a

T=0.84 ε2 Q

2/a

1e-04 0.01 1 100

T (Q2/ a)

0

1

2

3

C (

10-2

/kB )

b) ε = 0.05a) paramagnet

Ice I

Ice IIordered

G. Moller and R. Moessner, Phys. Rev. B 80, 140409(R) (2009)


Low temperature order: beyond local interactions

Need to resolve substructure of vortex-charges: multipoleexpansion in ε = 1− l/a

+ ...= ++

⇒ E =q2

a

{−2√

3ε+ (

3

2− α

2) +

3

2ε+ (γ +

δ

4+

3

2)ε2

}.

Q=+1

Q=−1

b)a) for ordered groundstate (as shown left)

monopole-monopole: α ≈ 1.5422197(Madelung energy)

dipole-dipole term: γ ≈ −2.226947

monopole-quadrupole: δ ≈ −0.5829489

(courtesy of Ravi Chandra)


Thermodynamics of dipolar Kagome-Ice

Monte-Carlo simulations confirm the existence of an effectivedimer model between the ordering temperature Td and thetransition temperature to the ice regime Tk .

✫

✫

✫

✫

✫

✫ ✫

0 0.2 0.4 0.6 0.8 1

ε = 1 − l/a

1e-05

1e-04

0.001

0.01

0.1

1

10

T (

J1 )

1e-04 0.01 1 100

T (Q2/ a)

0

0.1

0.2

0.3

0.4

0.5

0.6CS

✫

✫

✫✫

✫

✫

0.001 0.01 0.1ε

1e-0

40.

011

T [

Q2 /a

] T=0.40 Q2/a

T=0.84 ε2 Q

2/a

1e-04 0.01 1 100

T (Q2/ a)

0

1

2

3

C (

10-2

/kB )

b) ε = 0.05a) paramagnet

Ice I

Ice IIordered



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 aphenomenological model

G. Moller and R. Moessner, Phys. Rev. Lett. 96, 237202 (2006)

Kagome Ice

Kagome-Ice is realized in dipolar nano-magnetic arrays

Dumbbell picture of monopoles yields intuitive understanding

Two distinct Kagome-Ice phases, including a partially orderedphase mapping to dimer coverings of the hexagonal lattice


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

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