Artificial spin ice: Frustration by design

Peter Schiffer Pennsylvania State University

Joe Snyder, Ben Ueland, Rafael Freitas, Ari Mizel Princeton: Bob Cava, Joanna Slusky, Garret Lau

Ruifang Wang, Cristiano Nisoli, Jie Li, Bill McConville, BJ Cooley, Nitin Samarth, Vin Crespi Minnesota: Mike Lund and Chris Leighton Funding: ARO, NSF OUTLINE

Introduction to frustration and spin ice 1. How does spin ice freeze? 2. New class of materials: Stuffed Spin Ice 3. New way to be frustrated: Artificial Spin Ice Frustration

AFM AFM The common case of disordered magnets: ?

Random FM and AFM FM interactions AFM

Generic definition of Frustration: A system's inability to simultaneously minimize all of the interaction energies between its components resulting in multiple ground states

Disorder and frustration cause spins to freeze in random → “Spin Glass” configuration at low T Spin Glasses: A few characteristics

Behavior both history Broad range of length and time dependent scales and relaxation times Cooled in 1.0 M(T) Finite Field 0.8 '' 0.6 Cooled in PEAK

Zero Field χ Single τ

'' / '' 0.4 T χ χac(T) Low Freq. 0.2 0.0 1E-3 0.01 0.1 1 10 100 High Freq. f / f PEAK

T Geometrical magnetic frustration: not based on disorder

Competition of local interactions between spins on a regular lattice → high degeneracy of states from geometry of lattice AFM

AF M M F A

? What’s interesting about geometrically frustrated magnets? Continuum of energetically equivalent states

→ spins do not order at T ~ |ΘW|

150 2.0 NaTiO SrCr Ga O 2 2 400 9p 12-9p 19 5.45 atom/nm

Cd Mn Se 1.5 0.55 0.45 300 100 Cd Mn Se χ 0.65 0.35 1.0 200 2 Θ 1/ p = 0.89 5.2 atom/nm 0.61 50 CsMnFeF W 6 3 0.39 Monolayer of He on graphite 100 0.5 preplated with HD

Siqueira et al., PRL (1993) 0 0 0.0 0 100 200 300 400 0 100 200 300 01234567 Temperature (K) Temperature (K) Temperature (mK) New ground states theoretically expected Anderson, Villain, Chandra, Shender, Berlinsky, Moessner, Chalker, LaCroix, Henley, Gingras, Lhuillier… New ground states seen experimentally Spin liquids, Spin glasses without disorder, Spin ice… The “Spin Ice” Materials

Pyrochlore Lattice: corner-sharing tetrahedra of spins

Spin ices: Ho2Ti2O7, Dy2Ti2O7 ,Ho2Sn2O7….. Insulators with big rare-earth moments M. J. Harris et al. (1997) What makes a spin ice: “Two-in/Two out” • Crystal fields cause the rare-earth spins to be uniaxial along <111> directions (200 K energy scale)

• FM and dipole interactions cause spins to align two-in/two-out on each tetrahedron (2 K energy scale)

High degeneracy of possible states on pyrochlore lattice Why call it a spin ice?

In frozen water (H2O), each oxygen is surrounded by four hydrogens. Two are close to it, and two are closer to another oxygen ion.

Large degeneracy of states (Pauling 1945) invoked to explain the observed “ground state entropy” in ice “Spin Ice” residual entropy seen in heat capacity experiments 6 Ramirez et al. (1999) Rln2 Dy Ti O R(ln2 - 1/2ln3/2) 2 2 7 Ground State has predicted 4 residual entropy in zero field, same as ordinary ice

2 c (T ) S = P dT ∫ T 0 Integrated Entropy (J/moleK) Integrated Entropy 024681012 T (K) Possibly spins are in metastable state (not in equilibrium) -- glassy ground state with onset T ~ 3K Part 1: How does spin ice freeze?

Measure a.c. and d.c. magnetic susceptibility

look for spin-glass-like freezing in Dy2Ti2O7

Snyder et al. Nature 2001 similar work done by Matsuhira et al. J. Phys. Cond. Mat. 2001

Snyder et al. Phys. Rev. Lett. 2003 Snyder et al. Phys. Rev. B 2004 (two papers) Magnetization data: No sign of freezing above 1.8 K

No difference between field-cooled and zero-field-cooled data 2

1 M / M

Δ

3 0 10

-1 0 10203040 Temperature [K] AC susceptibility – much more interesting

χ 8 Dy2Ti2O7 dc 10 Hz 6 100 Hz 1,000 Hz

10,000 Hz 4

[emu/G mole] 2 ' χ

0 0 10203040

T(K) Spin freezing at T ~ 16 K and second feature at T ~ 3K Freezing is strongly frequency dependent (Arrhenius law) Lower temp. feature in χ(T) is another freezing

] 12 7

O 0.5 Hz 2 Ti

2 5 Hz

Dy 9 1 Hz 10 Hz

6 ] 7

O 2 6 500 Oe Ti

50 Hz 2 Dy 3 100 Hz 200 Hz

' [emu/Oe mol 0 500 Hz 4 χ emu/mol 3

6 ] 7 5 1 Hz 5 Hz 50 Hz O FC

2 0.5 Hz 10 Hz 2 Ti 2 4

Dy 100 Hz ZFC 3

0 2 200 Hz [10 Magnetization 0.0 0.2 0.4 0.6 0.8 1.0 1.2

1 500 Hz Temperature (K) 0 '' [emu/Oe mol

χ -1 0.6 0.8 1.0 1.2 1.4 Temperature [K] Snyder et al. PRB 2004 Spin ice freezing involves very narrow range of relaxation times

0.82 K 1.0 6 0.90 K 1.01 K

] 1.27 K 7 5 O 0.8 2 1.50 K Ti 2

'' 1.81 K 6 K Dy 4 5.36 K 0.6 12 K PEAK

16 K χ

single 3

'' / 0.4

χ 2 '' [emu/Oe mol '' [emu/Oe 0.2 H = 0 χ 1

0.0 0 1E-3 0.01 0.1 1 10 100 0.1 1 10 100 1000 Frequency [Hz] f / f PEAK No broad range of time-scales typical of spin glasses: This freezing is something very different! Understand double-freezing in Dy2Ti2O7 from spin relaxation time

Maximum in χ"(freq) gives the relaxation time, τ

T = 7.5 K T = 12 K T = 16 K 0.5 f = 1/τ ]

Dy 0.4 FWHM single τ 0 Oe 0.3

2.5 kOe 0.2 5 kOe 8 kOe 0.1 10 kOe '' [emu/Oemol χ 0.0 10 100 1k 10k 10 100 1k 10k 10 100 1k 10k Frequency [Hz] Crossover from thermal to quantum spin relaxation at T ~ 13 K

Snyder et al., PRL 2003 10 Dy2Ti2O7

1 0 Oe [ms] τ 4 kOe 10 kOe 0.1 51015 Temperature [K] Two types of relaxation clearly seen in a field

0.18 Dy Y Ti O 1.6 0.4 2 7

] 0.16 Dy 0.14

0.12

9 K 0.10 10 K 0.08 11 K 12 K 0.06 H = 5 kOe '' [emu/Oe mol 13 K χ 0.04 10 100 1k 10k Frequency [Hz] Thermal relaxation re-emerges at low temp.

0 10 Thermal Spin Relaxation

10-1

-2 10 [s] τ 10-3 Quantum Spin Relaxation 10-4 0 5 10 15 Temperature [K] Low temperature spin-correlations are apparently suppressing the quantum relaxation Re-entrant thermal spin relaxation explains

double freezing in Dy2Ti2O7

100 8 χ dc -1 10 Hz 10 6 100 Hz 1,000 Hz

-2 10 4 10,000 Hz [s] τ [emu/G mole]

-3 '

10 χ 2

-4 10 0 051015 0 10203040 Temperature [K] T(K) • Thermal fluctuations freeze out (single ion process, Ehlers et al.) • Quantum fluctuations until spins get correlated • Correlated spins relax very slowly Part 2: Stuffed Spin Ice What happens if we add more magnetic ions to the lattice? Ho lattice only

Start with Ho2Ti2O7 and replace Ti with Ho ions get Ho2(Ti2-xHox)O7-x/2

Lau et al. (Nature Physics, in press) Stuffing the lattice: Ho2(Ti2-xHox)O7-x/2 Randomly replace some Ti4+ with Ho3+ Stuffing the lattice

Ho Ti Stuffing changes the connectivity of spins

+ Ho -Ti

x = 0 x = 0.67 corner sharing edge sharing Magnetic entropy of the stuffed spin ice 6 Rln(2) 5 Spin Ice 4 cP (T ) K)] S = dT Ho x = 0 ∫ 3 T 0.3 2 0.67

S [J/(mol 1 Ho (Ti Ho )O 2 2-x x 7-x/2 0 0 4 8 12 16 20 24 T (K) Entropy recovery is similar for all x!! Ground state entropy also for stuffed case Total Entropy vs. Stuffing

K)] 5.5 R ln 2 Ho H = 1 T H = 0 5.0 Entropy is unchanged within 4.5 measurement 4.0 Spin Ice uncertainty

3.5 Total Entropy S [J/(mol S Entropy Total 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x Open Questions: • What macroscopic state exists as approach fcc lattice? • How common is the “ground state” entropy? Part 3: Artificial Spin Ice Wang et al., Nature 2006 A major limitation in the study of magnetic frustration is that we are limited by the materials • Cannot easily change lattice spacing/geometry

• Cannot control defects locally

• Cannot image individual spins

What if we use modern nanometer-scale lithography to create an artificial frustrated system? Artificial frustrated systems with superconductors Superconducting rings or Josephson junction loops with flux trapped Davidovic et al. 1996, 1997 Hilgenkamp, Kirtley, Tsuei et al. 2003, 2005 What if we make “artificial spins” using ferromagnetic material?

Etch islands out of a ferromagnetic film If islands are small enough, they will be single-domain (like big spins) Shape anisotropy results in Ising-like spins with direction given by island shape Islands interact through local magnetic fields Do ferromagnetic islands really behave like big spins? Yes! Choose shape such that islands are single-domain

AFM

MFM

Imre et al. 2006 Are island moments controlled by interaction? Yes!

5 μm

AFM MFM Making frustrated 2-D network out of ferromagnetic islands

1 1 2 2 0 0

1 2 0

1 1 2 2 0 0

1 1 2 2 0 0

Advantage of lithographically defined network: • Fine control of island shape, size, and spacing • Direct investigation of individual islands by MFM • Flexibility in introducing defects Dipolar interactions in one vertex of the perpendicular square lattice

Favorable pair configuration

Unfavorable pair configuration

The six pairs of dipole interactions cannot be minimized simultaneously: FRUSTRATION! Magnetization configurations on single vertex Expect certain distribution if orientations are random Lowest energy states Type I (12.5%) are two-in/ two-out: Type II (25%) LIKE SPIN ICE!!

Type III (50%)

Type IV (12.5%) Sample details

AFM image

• Patterned with e-beam lithography and lift-off technique. • Composition: Permalloy (80% Ni + 20% Fe) • Island size: 80nm * 220nm, 25nm thick • Lattice parameter of 320 – 880 nm (center to center) Important energy scales of islands • Island size: 80nm * 220nm, 25nm thick • Island moment: ~ 3×107 Bohr magnetons • Zeeman energy of single island: ~ 2 * 104 K in external field of 10 Oe • Shape anisotropy energy of each island: ~ 4 * 106 K • Dipole-Dipole energy of two nearest neighbors: ~ 104 K depending on separation Sample Preparation: How prepare magnetic state Cannot use thermal energy to randomize magnetic moments (would need > 104 K)

Instead rotate sample in a magnetic field which is stepped down in magnitude with switching polarity H H

time

Sample Use MFM to scan the arrangement of island moments Can clearly see individual islands and single domain moments When islands are closely spaced, can clearly see disordered state Use MFM to determine the arrangement of island moments Can clearly see local moments and vertex types on the lattice

Figure 2. Wang et al.

1 µ m Distribution of vertices depends on island spacing Count percentages of different vertex types

80 Type I & II (Ice rule) Type I Type II Type I (12.5%) 60 Type III Type IV

40 Type II (25%)

20 Type III (50%) Percentage of vertices Percentage

0 300 450 600 750 900 Lattice spacing (nm) Type IV (12.5%) More ice-like vertices seen for closely spaced islands As spacing increases, the percentages of vertex types converge to those expected for random moments

40 Type I & II (Ice rule) Type I Type I (12.5%) Type II Type III 20 Type IV

Type II (25%) 0

Type III (50%) DIfference from random random from DIfference

distribution of vertices (%) of vertices distribution -20

300 400 500 600 700 800 900 Type IV (12.5%) lattice spacing (nm) Pair correlations decrease with increasing lattice spacing

T Define correlation as: 0.3 NN NN NN +1 if pair minimizes L L dipole energy 0.2 NN NN

T T -1 if pair maximizes 0.1 dipole energy L

Average Correlation Value Correlation Average 0.0 Essentially no longer range correlations 300 400 500 600 700 800 900 Lattice Spacing (nm) JUST LIKE PYROCHLORE ICE Why correlations are stronger with transverse neighbors

0.3 NN T NN NN 0.2 L L T 0.1 L NN NN

Correlation Value 0.0 T 300 400 500 600 700 800 900

Lattice Spacing (nm)

“L” neighbors are frustrated “T” neighbors are not by NN interactions frustrated by NN interactions Technology from interacting arrays: recent work suggests device possibilities

Notre Dame group Imre et al. Science, 2006 Artificial frustrated systems: the fun is just beginning… New control of frustrated lattices is possible • make different lattices • control strength of interactions • put in defects New measurements • look at dynamics in magnetic field • push to superparamagnetic limit • measure entropy to compare to “real” spin ice

Perhaps relate to recording on patterned media….