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Thermal fluctuations in artificial spin ice Vassilios Kapaklis1 *, Unnar B. Arnalds1, Alan Farhan2,3, Rajesh V. Chopdekar2,4, Ana Balan4, Andreas Scholl5, Laura J. Heyderman2,3 and Bjo¨rgvin Hjo¨rvarsson1 Artificial spin ice systems have been proposed as a playground for the study of monopole-like magnetic excitations1,2, similar to those observed in pyrochlore spin ice materials3. Currents of magnetic monopole excitations have been observed4, demonstrating the possibility for the realization of magneticcharge-based circuitry. Artificial spin ice systems that support thermal fluctuations can serve as an ideal setting for observing dynamical effects such as monopole propagation and as a potential medium for magnetricity investigations1,2. Here, we report on the transition from a frozen to a dynamic state in artificial spin ice with a square lattice. Magnetic imaging is used to determine the magnetic state of the islands in thermal equilibrium. The temperature-induced onset of magnetic fluctuations and excitation populations are shown to depend on the lattice spacing and related interaction strength between islands. The excitations are described by Boltzmann distributions with their factors in the frozen state relating to the blocking temperatures of the array. Our results provide insight into the design of thermal artificial spin ice arrays where the magnetic charge density and response to external fields can be studied in thermal equilibrium. Water ice is the archetype of geometric frustration, exhibiting disorder in the arrangement of oxygen–proton bonds, even at very low temperatures. Pauling5 argued that long-range ice rule order (two short and two long O–H distances for each O atom) cannot be obtained, resulting in a disordered ground state with macroscopic degeneracy. One year after Pauling’s publication, his postulation was experimentally confirmed by Giauque and Stout6. However, the concept of geometric frustration is not restricted to structural order. For example, spin frustration in rare-earth pyrochlores is found to mimic the geometric frustration in water ice through the formation of spin ice structures7,8. In spin ice, a spin pointing inward or outward from a tetrahedron corresponds to short or long O–H distances, respectively. Excitations in spin ice are therefore manifested as violations of the ice rule and are analogous to the existence of point charges or magnetic monopoles3,9,10. The monopoles carry an overall magnetic charge and the associated currents have been measured in materials such as Dy2Ti2O7 at low temperatures (,10 K)4. Two-dimensional nanopatterned arrays provide a new route for the investigation of geometric frustration in magnets1,2,11. Artificial spin ice structures, mimicking crystalline spin ice materials, can even be designed and imaged directly using real-space techniques11–15. These systems therefore provide an ideal platform for the investigation of ground state ordering and excitations. In artificial spin ice the arrangement of the moments arising from elongated single-domain nanopatterned magnetic islands can lead to excited states that possess magnetic charges16, analogous to the monopole excitations reported for rare-earth pyrochlores3,17. Until recently,
research on artificial spin ice has mainly been focused on static properties and frozen excitations12,16,18, as the thermal energy required to overcome the activation barriers for magnetization reversal (1 × 105 K) has not been accessible13,14,19. Field annealing artificial spin ice has provided some insight into ground state ordering11,20, but this approach suffers from limitations arising from quenched disorder and jamming21. Thermally driven dynamics in artificial spin ice can lead to extended ground state ordering12 and open up the prospect of exploring thermally induced magnetic charge excitations. The existence of thermal excitations has already been demonstrated for building blocks of spin ice structures using real-space magnetic imaging13,14. The available thermal energy (kBT, where kB is the Boltzmann constant and T is temperature) and the reversal energy barrier (Er) define the dynamic behaviour of the magnetic nanoislands. For elongated single-domain particles with uniaxial shape anisotropy, the magnetization reversal energy barrier is given by Er ¼ KV, where K is an effective uniaxial anisotropy constant and V their volume22. At thermal energies above the reversal energy, the particles act in many ways as super-paramagnetic entities19. In extended artificial spin ice structures, the role of excitations on the order–disorder transition has been demonstrated by magnetization measurements19, theoretical investigations23, and thermal relaxation in extended artificial square spin ice arrays has been observed by direct imaging15. Furthemore, annealing of artificial spin ice arrays has recently been shown to result in extended thermal ordering and crystallization of interacting magnetic charges24. To investigate a dynamic equilibrium of an extended spin ice structure requires the magnetic excitations to be observable under experimental conditions. To accomplish this, the magnetic properties of the islands and their interactions need to be adjusted to match the accessible temperature range. In the present work, this is achieved through the choice of the intrinsic ordering temperature of the island material and the distance between the islands. This allows us to determine the equilibrium states in the artificial spin ice structures and to show that these exhibit a strong temperature dependence relating to the interaction between the elements. Thermal excitations result in monopole-like magnetic charges, which, for the first time, can be observed in thermal equilibrium. Photoemission electron microscopy with X-ray magnetic circular dichroism (PEEM-XMCD) was used to determine the magnetization of the individual elements comprising the arrays, allowing for the determination of their vertex configurations and ordering. The islands are shaped into elongated structures, as shown in Fig. 1a, with a long axis of 470 nm and a width of 170 nm. The investigated arrays differ only in their periodicity (600 and 640 nm, respectively; Fig. 1b). A typical PEEM-XMCD image of the 600 nm array is shown in Fig. 1c. A clear magnetic contrast is visible for most
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Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120, Uppsala, Sweden, 2 Laboratory for Micro- and Nanotechnology, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland, 3 Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich, CH-8093 Zurich, Switzerland, 4 Swiss Light Source, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland, 5 Lawrence Berkeley National Laboratory (LBNL), 1 Cyclotron Road, Berkeley, California 94720, USA. * e-mail: [email protected] NATURE NANOTECHNOLOGY | ADVANCE ONLINE PUBLICATION | www.nature.com/naturenanotechnology
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Figure 1 | Geometry and magnetic imaging. a, The extended artificial square spin ice arrays are composed of 470 nm × 170 nm magnetic islands of 1.4 monolayers d-doped Pd(Fe). b, Two arrays of different periodicity (600 and 640 nm). c, A typical PEEM-XMCD image showing the 600 nm array, with a clear magnetic contrast for the d-doped Pd(Fe) islands at a temperature of 30 K. The array has been rotated 458 with respect to the X-ray sensitivity axis to fully resolve the magnetic configuration. In the PEEM-XMCD image, islands with magnetization pointing towards the right are visualized with a black contrast and islands pointing to the left with a white contrast. d, Determination of the magnetic configuration of part of the 600 nm square spin ice array at 30 K. The image illustrates the magnetic moment directions determined from the magnetic contrast of the PEEM-XMCD image on the left (region within the white frame), as shown by the black arrows. The black outlines of the nanomagnets indicate their position. An extended domain of ground state vertices can be seen, with the typical alternating chirality pattern depicted by blue and red circular arrows.
islands, enabling the determination of the magnetization direction. As a result, the state configuration for a vertex, where all neighbouring islands are assigned a magnetization vector, can be defined (Fig. 1d). A bias towards ground state ordering, with vertices following the ice rule (two in, two out) is seen, although excited vertex configurations, not obeying the ice rule, are also observed. In artificial square spin ice, each vertex can have one of 16 possible vertex configurations, which can be sorted in ascending energy (Supplementary Fig. 1). Types I and II comprise vertex states that obey the ice rule, with type II possessing a dipole magnetic moment. Types III and IV do not obey the ice rule. As the coupling strength increases, so does the energy difference between the 2
different vertex states. As a consequence, slow cooling to 30 K results in dissimilarities in the ordering of the two arrays, as seen in Fig. 2. Larger domains of type I vertices are present in the 600 nm array (Fig. 2a,b) than in the more weakly coupled 640 nm array (Fig. 2c,d). Furthermore, the populations of excited vertex states (types III and IV) are also significantly smaller in the strongly coupled array, which demonstrates the importance of the strength of the inter-island coupling for the obtained ordering25. The results are unambiguous, as both arrays were patterned on the same substrate and have undergone the same cooling protocol. The temperature dependence of the array states presented in Fig. 3 was determined from images at different temperatures
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Figure 2 | Determination of vertex states from magnetic imaging. a–d, PEEM-XMCD images and magnetic configurations of the 600 nm (a,b) and 640 nm (c,d) arrays at 30 K. The arrows in b and d depict the islands’ magnetization vectors. Green dots indicate ground state vertices of type I, blue dots indicate type II, and yellow and red cones types III and IV, respectively. The up and down cones depict the positive and negative charge of type III and IV vertices (Supplementary Fig. 1). Missing arrows in b and d indicate insufficient magnetic contrast to safely assign a magnetization direction. The 600 nm array has a higher number of ground state vertices than the 640 nm array, where the dipolar interaction between islands is weaker.
(Supplementary Fig. 2). In thermally active nanomagnetic systems the possibility of observing thermal fluctuations is determined by the timescale of thermal fluctuations and the timescale of imaging. The total acquisition time at each temperature was tm ¼ 500 s, consisting of fifty single frames of 10 s duration. As the temperature is raised, more islands will fluctuate between the two low-energy states, resulting in a lower number of islands that can be assigned a magnetization vector. As seen in Fig. 3, a difference in the temperature dependence is also obtained: the predominantly type I background breaks up and disappears at a substantially lower temperature for the weakly coupled 640 nm array than for the more strongly coupled 600 nm array. The temperature dependence of the number of identifiable island states for both arrays is shown in Fig. 4. A difference in the onset of thermal fluctuations is obtained, providing support for the effect of geometry on the energy landscape. The timescale t of the individual island fluctuations is given by t = t0 eEr /(kB T), where t0 is the inverse attempt frequency22, which dictates the temperature range over which PEEM measurements can be used to resolve the magnetic configuration of the islands. The probability of the magnetization not having switched after a time t is therefore given26 by P(t) ≈ e 2t/t. A Gaussian distribution of the island sizes is
assumed, which is used to capture the effect of imperfections in the arrays resulting in a distribution of the island moments and therefore the reversal energy barriers. The temperature dependence of the moment of the islands is included in the calculations (see Methods). A median value of the reversal energy barrier (E0) at 0 K is obtained from this analysis. The inverse attempt frequency t0 is assumed to be 1 × 10210 s for the islands in both arrays14,22, and the analysis yields E0 ¼ 9.7 × 10220 J (E0/kB ≈ 7.0 × 103 K) for the 600 nm array and 7 × 10220 J (E0/kB ≈ 5 × 103 K) for the 640 nm array with a standard deviation s equal to 10% of the median value of E0. As the temperature is lowered, P(t) does not reach unity because of a residual number of islands for which a magnetization vector cannot be unambiguously assigned, even at the lowest temperatures (Fig. 2). The very small magnetic moment from the buried 1.4 monolayers of Fe combined with the Pd top layer (see Methods) leads to a weak magnetic contrast. This results in an offset that is included in the fits presented in Fig. 4. The reversal energy barrier at 0 K is given by E0 ¼ Er(0 K) þ Ei (0 K), where Er is the intrinsic barrier and Ei arises from the inter-island interactions. All energies are temperature-dependent, because the magnetic moment decreases with increasing temperature. The intrinsic reversal barrier is determined using
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Figure 3 | Effect of inter-island interactions on the onset of vertex state fluctuations. a,b, Vertex states of the two arrays determined from PEEM-XMCD imaging, shown as a function of temperature for the 600 nm (a) and 640 nm (b) arrays. Green dots refer to ground state vertices of type I, blue dots to type II vertices, and yellow and red cones to types III and IV, respectively. The up and down cones depict the positive and negative charge of type III and IV vertices (Supplementary Fig. 1). Missing vertices result from the inability to unambiguously assign a magnetization direction for neighbouring islands on a timescale defined by the PEEM imaging. The weakly coupled array (640 nm) in b undergoes a transition to a fluctuating state at a lower temperature than the strongly coupled array (600 nm) in a.
magnetostatic analysis, applying a methodology proposed by Osborn27. The interaction energy was obtained from micromagnetic calculations assuming a collinear magnetization of the islands (see Methods). The contribution of the interaction energy to the reversal barrier can be estimated by considering the reversal of one element in a ground state configuration, resulting in the creation of two type III vertices. The energy cost for such a transition is Ei ¼ 2ET3–2ET1 , 1.0
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Figure 4 | Temperature dependence of magnetic fluctuations. Probability Pl of the island magnetization not having switched during the PEEM-XMCD acquisition time, determined by the number of islands that can be assigned a magnetization vector: 600 nm (blue squares) and 640 nm (red circles) arrays. Lines represent fits to a thermally activated magnetization reversal model, yielding median values of E0 ¼ 9.7 × 10220 J for the 600 nm array and E0 ¼ 7 × 10220 J for the 640 nm array and a standard deviation of s ¼ 10% for both E0 values. All error bars represent one standard deviation. 4
where ET3 and ET1 are the energies of type III and I vertices, respectively. For non-interacting islands, ln(tm /t0 )kB TB = 1/2m0 M 2 (TB ) DNV = Er (TB ) at the blocking temperature28 TB , where m0 is the vacuum magnetic permeability, M(TB) is the island magnetic moment at that temperature, DN represents the demagnetizing factors calculated using the Osborn methodology27, V is the island volume and tm is the measurement time. Using this approach, the blocking temperature of non-interacting was determined islands array to be 95 K. For interacting islands, ln tm /t0 kB TB = 1/2m0 M 2 array array (TB )DNV + Ei (TB ), where T array is the blocking temperature B of an island in the array. This results in an estimate of TBarray(640 nm) ¼ 108 K, which is identical to the experimentally obtained value of 108 K (Fig. 4). For the 600 nm array we obtain TBarray(600 nm) ¼ 116 K, which is lower than the experimental value of 130 K. The effective island reversal barrier is not only defined by the lattice spacing but also by the magnetic configuration of the vertices, altering the reversal energy barriers and therefore the fluctuation frequency for the different vertex states. Excited vertex states of types III and IV are therefore more short-lived than states of types I and II at any given temperature, as these have higher energy. The role of the energy landscape on the thermal behaviour of the arrays can therefore be explored by looking at the observed probabilities PT of the four vertex states. As shown in Fig. 5a the 600 nm array displays a high probability of type I vertex states over the examined temperature range. The onset of fluctuations is most prominently reflected in the decrease of type II and III probabilities above 110 K for the 600 nm array. The transition occurs at a lower temperature for the 640 nm array, enabling us to follow the full order-to-disorder transition. Here, type I vertices are revealed to have a reduced probability, almost equal to that for types II and III. Taking into account the degeneracy of the vertex states, this array still exhibits significant ground state ordering. The onset of vertex state specific transitions is revealed in Fig. 5b, with the higher-energy type III states beginning to fluctuate at lower
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Figure 5 | Thermal distributions, fluctuations and energy statistics of vertex states. a,b, Observed probabilities for all types of vertex state for the 600 nm (a) and 640 nm (b) arrays as a function of temperature, revealing the different decays of the vertex states. Dashed lines serve as guides to the eye, denoting the vertex specific onset observed for the 640 nm array. The distribution of vertex states unveils the boundary between a frozen and a thermally active state (boundary depicted by the vertical dashed lines) where investigations of thermal dynamics in artificial spin ice arrays are possible. c, Probability distributions for the vertex energy states. The energy values (ETn) for the strongly coupled (600 nm, blue open circles) and weakly coupled (640 nm, red open squares) arrays have been determined using micromagnetic calculations for the experimentally observed blocking temperatures. The number of vertices of each type are taken from images recorded at the lowest temperature used for measuring arrays. The probabilities are given by Boltzmann factors (exp(bEi)) and the related slopes of the fitted lines reflect the blocking temperature Tarray of each array. The coloured regions highlight the vertex types: I (green), II (blue), B III (yellow) and IV (red). All error bars represent one standard deviation.
temperatures than for types II and I. Furthermore, the rate at which vertex states decay while raising the temperature increases with increasing vertex energy. The results also illustrate the existence of a boundary between two distinct states: a frozen low-temperature state, where the population distribution of the vertex configurations is predominantly dictated by the coupling strength and cooling conditions, and a high-temperature state with dynamic and evolving vertex configurations reflecting the energy landscape of the array. In this dynamic state, excited vertices carrying a magnetic charge are mobile, hinting at new routes for achieving magnetic charge mobility, with implications for future device design. At this stage, it cannot be excluded that pattern defects or inhomogeneities can act as localization sites for such excitations. Intentional introduction of these in artificial spin ice structures could provide further information on their role and potential for magnetic charge manipulation (see Supplementary Information in ref. 16). Correcting the observed abundances of the different vertex states for their degeneracy reveals their probability with respect to energy. The observed probabilities are presented in Fig. 5c and can be described by a Boltzmann distribution. The presented vertex
populations are taken at the lowest temperatures for the 640 nm and 600 nm arrays. These correspond to frozen configurations of the arrays, well below their respective blocking temperatures, and the slopes relate to the state distributions at the array blocking temp. The slopes result in a ratio for the blocking temperaeratures Tarray B array array tures (TB (600 nm)/TB (640 nm)) of 1.6(2), which compares reasonably with the ratio of 1.2(1) extracted from the observations presented in Fig. 4 (numbers in parentheses denote standard deviations), demonstrating once again how the interaction strength modifies the thermal equilibrium state of artificial spin ice arrays. We therefore conclude that the energy landscape of two-dimensional nanomagnetic systems can be altered through the design of the lattice geometry and island material, thereby defining their thermal behaviour. The use of real-space imaging techniques allows unprecedented and detailed information on the magnetic states to be obtained, enabling observations of real thermal dynamics at mesoscopic length scales and over time13–15,19,29. In the case of artificial spin ice, this sets the scene for the design of systems where manipulation of magnetic charges is feasible. Control of the excitation populations and magnetic charges by
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means of the array geometry and temperature can serve as a platform for studies of charge mobility under externally applied magnetic fields or temperature gradients. In future investigations, this will provide insights into the utilization of magnetic charges in novel technologies such as magnetic cellular automata30,31. Finally, thermal artificial spin ice, with its intrinsic multiplicity of states, could be an ideal testbed for addressing the physics of emergence in magnetic systems.
Methods The extended artificial square spin arrays were patterned by a combination of electron-beam lithography and an argon-ion milling process13. The islands were made from d-doped Pd(Fe) thin films, consisting of 40 nm Pd, 1.4 monolayers of Fe and finally 2 nm Pd on top, all grown on a V seeding layer19,32. The remanent magnetization of the films vanishes at TC ¼ 250 K, while the in-field magnetization (at an applied field of m0H ¼ 50 mT) extends to T0 ¼ 290 K, as measured using magneto-optical Kerr effect magnetometry32. The temperature dependence of the in-field magnetization can be successfully described by the function M(T ) ¼ M0(1 2 T/T0)0.5. For comparison with the experimental data an island moment value of m0 ¼ 4 × 106mB at T ¼ 0 K was used. The magnetic moment of an individual island can also be determined from its area and the magnetization of the continuous film, measured using a superconducting quantum interference device (SQUID) magnetometer. The resulting island magnetic moment using this method at 0 K corresponds to 6 × 106mB. Micromagnetic calculations were carried out for individual islands and vertex configurations, assuming collinear magnetization and using the object oriented micromagnetic framework (OOMMF) for a thickness of the magnetic layer of 2 nm. The resulting energies are significantly higher than those obtained using simple point-dipole calculations. The obtained values were scaled for the temperature dependence of the magnetic moment M(T ) using the equation stated above, and the energies for all vertex states can thus be calculated for a temperature range from 0 K to T0. The orientation of the magnetization of the islands was determined using PEEM and XMCD33 at the Fe L3 edge. The experiments were performed at the PEEM3 station of the Advanced Light Source at the Lawrence Berkeley National Laboratory. The arrays were cooled in situ using a liquid-helium cryostat, and all measurements were performed in the absence of an externally applied magnetic field (Supplementary Fig. 2). As all the patterns were fabricated on the same substrate they also undergo the same cooling rates in the PEEM microscope stage.
Received 27 January 2014; accepted 2 May 2014; published online 8 June 2014
References 1. Nisoli, C., Moessner, R. & Schiffer, P. Colloquium. Artificial spin ice: designing and imaging magnetic frustration. Rev. Mod. Phys. 85, 1473–1490 (2013). 2. Heyderman, L. J. & Stamps, R. L. Artificial ferroic systems: novel functionality from structure, interactions and dynamics. J. Phys. 25, 363201 (2013). 3. Castelnovo, C., Moessner, R. & Sondhi, S. L. Magnetic monopoles in spin ice. Nature 451, 42–45 (2008). 4. Giblin, S. R., Bramwell, S. T., Holdsworth, P. C. W., Prabhakaran, D. & Terry, I. Creation and measurement of long-lived magnetic monopole currents in spin ice. Nature Phys. 7, 252–258 (2011). 5. Pauling, L. The structure and entropy of ice and of other crystals with some randomness of atomic arrangement. J. Am. Chem. Soc. 57, 2680–2684 (1935). 6. Giauque, W. & Stout, J. The entropy of water and the third law of thermodynamics. The heat capacity of ice from 15 to 273 8K. J. Am. Chem. Soc. 58, 1144–1150 (1936). 7. Harris, M. J., Bramwell, S. T., McMorrow, D. F., Zeiske, T. & Godfrey, K. W. Geometrical frustration in the ferromagnetic pyrochlore Ho2Ti2O7. Phys. Rev. Lett. 79, 2554–2557 (1997). 8. Bramwell, S. T. & Gingras, M. J. P. Spin ice state in frustrated magnetic pyrochlore materials. Science 294, 1495–1501 (2001). 9. Jaubert, L. D. C. & Holdsworth, P. C. W. Signature of magnetic monopole and Dirac string dynamics in spin ice. Nature Phys. 5, 258–261 (2009). 10. Bramwell, S. T. et al. Measurement of the charge and current of magnetic monopoles in spin ice. Nature 461, 956–959 (2009). 11. Wang, R. F. et al. Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands. Nature 439, 303–306 (2006). 12. Morgan, J. P., Stein, A., Langridge, S. & Marrows, C. H. Thermal ground-state ordering and elementary excitations in artificial magnetic square ice. Nature Phys. 7, 75–79 (2011).
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13. Arnalds, U. B. et al. Thermalized ground state of artificial kagome spin ice building blocks. Appl. Phys. Lett. 101, 112404 (2012). 14. Farhan, A. et al. Exploring hyper-cubic energy landscapes in thermally active finite artificial spin-ice systems. Nature Phys. 9, 375–382 (2013). 15. Farhan, A. et al. Direct observation of thermal relaxation in artificial spin ice. Phys. Rev. Lett. 111, 057204 (2013). 16. Mengotti, E. et al. Real-space observation of emergent magnetic monopoles and associated Dirac strings in artificial kagome spin ice. Nature Phys. 7, 68–74 (2011). 17. Bramwell, S. T. et al. Measurement of the charge and current of magnetic monopoles in spin ice. Nature 461, 956–959 (2009). 18. Ladak, S., Read, D. E., Perkins, G. K., Cohen, L. F. & Branford, W. R. Direct observation of magnetic monopole defects in an artificial spin-ice system. Nature Phys. 6, 359–363 (2010). 19. Kapaklis, V. et al. Melting artificial spin ice. New J. Phys. 14, 035009 (2012). 20. Ke, X., Li, J., Zhang, S., Nisoli, C., Crespi, V. H. & Schiffer, P. Tuning magnetic frustration of nanomagnets in triangular-lattice geometry. Appl. Phys. Lett. 93, 252504 (2008). 21. Budrikis, Z. et al. Disorder strength and field-driven ground state domain formation in artificial spin ice: experiment, simulation, and theory. Phys. Rev. Lett. 109, 037203 (2012). 22. Bedanta, S. & Kleemann, W. Supermagnetism. J. Phys. D 42, 013001 (2008). 23. Wysin, G. M., Moura-Melo, W. A., Mo´l, L. A. S. & Pereira, A. R. Dynamics and hysteresis in square lattice artificial spin ice. New J. Phys. 15, 045029 (2013). 24. Zhang, S. Crystallites of magnetic charges in artificial spin ice. Nature 500, 553–557 (2013). 25. Liba´l, A., Reichhardt, C. & Reichhardt, C. Realizing colloidal artificial ice on arrays of optical traps. Phys. Rev. Lett. 97, 228302 (2006). 26. Wernsdorfer, W. et al. Experimental evidence of the Ne´el–Brown model of magnetization reversal. Phys. Rev. Lett. 78, 1791–1794 (1997). 27. Osborn, J. A. Demagnetizing factors of the general ellipsoid. Phys. Rev. 67, 351–357 (1945). 28. Porro, J. M., Bedoya-Pinto, A., Berger, A. & Vavassori, P. Exploring thermally induced states in square artificial spin-ice arrays. New J. Phys. 15, 055012 (2013). 29. Heyderman, L. J. Artificial spin ice: Crystal-clear order. Nature Nanotech. 8, 705–706 (2013). 30. Cowburn, R. P. & Welland, M. Room temperature magnetic quantum cellular automata. Science 287, 1466–1468 (2000). 31. Imre, A. et al. Majority logic gate for magnetic quantum-dot cellular automata. Science 311, 205–208 (2006). 32. Papaioannou, E. Th., Kapaklis, V., Taroni, A., Marcellini, M. & Hjo¨rvarsson, B. Dimensionality and confinement effects in d-doped Pd(Fe) layers. J. Phys. 22, 236004 (2010). 33. Scholl, A. et al. Observation of antiferromagnetic domains in epitaxial thin films. Science 287, 1014–1016 (2000).
Acknowledgements The authors acknowledge support from the Knut and Alice Wallenberg Foundation, the Swedish Research Council, the Swedish Foundation for International Cooperation in Research and Higher Education and the Swiss National Science Foundation. The Advanced Light Source (ALS) is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the US Department of Energy (contract no. DE-AC02-05CH11231). The authors thank A. Young of the ALS for support during the PEEM experiments and A. Weber for development of the patterning processes and sample manufacture. The authors are also grateful to V. Guzenko for support with electron-beam lithography. V.K. would like to thank P.E. Jo¨nsson for discussions.
Author contributions B.H. and L.J.H. initiated the work. V.K., U.B.A., A.F., R.V.C., A.B. and A.S. performed the PEEM-XMCD imaging. V.K. and U.B.A. analysed the data. V.K., U.B.A. and B.H. wrote the paper. All authors discussed the results and commented on the manuscript.
Additional information Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to V.K.
Competing financial interests The authors declare no competing financial interests.
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Thermal fluctuations in artificial spin ice Vassilios Kapaklis1 *, Unnar B. Arnalds1, Alan Farhan2,3, Rajesh V. Chopdekar2,4, Ana Balan4, Andreas Scholl5, Laura J. Heyderman2,3 and Bjo¨rgvin Hjo¨rvarsson1 Artificial spin ice systems have been proposed as a playground for the study of monopole-like magnetic excitations1,2, similar to those observed in pyrochlore spin ice materials3. Currents of magnetic monopole excitations have been observed4, demonstrating the possibility for the realization of magneticcharge-based circuitry. Artificial spin ice systems that support thermal fluctuations can serve as an ideal setting for observing dynamical effects such as monopole propagation and as a potential medium for magnetricity investigations1,2. Here, we report on the transition from a frozen to a dynamic state in artificial spin ice with a square lattice. Magnetic imaging is used to determine the magnetic state of the islands in thermal equilibrium. The temperature-induced onset of magnetic fluctuations and excitation populations are shown to depend on the lattice spacing and related interaction strength between islands. The excitations are described by Boltzmann distributions with their factors in the frozen state relating to the blocking temperatures of the array. Our results provide insight into the design of thermal artificial spin ice arrays where the magnetic charge density and response to external fields can be studied in thermal equilibrium. Water ice is the archetype of geometric frustration, exhibiting disorder in the arrangement of oxygen–proton bonds, even at very low temperatures. Pauling5 argued that long-range ice rule order (two short and two long O–H distances for each O atom) cannot be obtained, resulting in a disordered ground state with macroscopic degeneracy. One year after Pauling’s publication, his postulation was experimentally confirmed by Giauque and Stout6. However, the concept of geometric frustration is not restricted to structural order. For example, spin frustration in rare-earth pyrochlores is found to mimic the geometric frustration in water ice through the formation of spin ice structures7,8. In spin ice, a spin pointing inward or outward from a tetrahedron corresponds to short or long O–H distances, respectively. Excitations in spin ice are therefore manifested as violations of the ice rule and are analogous to the existence of point charges or magnetic monopoles3,9,10. The monopoles carry an overall magnetic charge and the associated currents have been measured in materials such as Dy2Ti2O7 at low temperatures (,10 K)4. Two-dimensional nanopatterned arrays provide a new route for the investigation of geometric frustration in magnets1,2,11. Artificial spin ice structures, mimicking crystalline spin ice materials, can even be designed and imaged directly using real-space techniques11–15. These systems therefore provide an ideal platform for the investigation of ground state ordering and excitations. In artificial spin ice the arrangement of the moments arising from elongated single-domain nanopatterned magnetic islands can lead to excited states that possess magnetic charges16, analogous to the monopole excitations reported for rare-earth pyrochlores3,17. Until recently,
research on artificial spin ice has mainly been focused on static properties and frozen excitations12,16,18, as the thermal energy required to overcome the activation barriers for magnetization reversal (1 × 105 K) has not been accessible13,14,19. Field annealing artificial spin ice has provided some insight into ground state ordering11,20, but this approach suffers from limitations arising from quenched disorder and jamming21. Thermally driven dynamics in artificial spin ice can lead to extended ground state ordering12 and open up the prospect of exploring thermally induced magnetic charge excitations. The existence of thermal excitations has already been demonstrated for building blocks of spin ice structures using real-space magnetic imaging13,14. The available thermal energy (kBT, where kB is the Boltzmann constant and T is temperature) and the reversal energy barrier (Er) define the dynamic behaviour of the magnetic nanoislands. For elongated single-domain particles with uniaxial shape anisotropy, the magnetization reversal energy barrier is given by Er ¼ KV, where K is an effective uniaxial anisotropy constant and V their volume22. At thermal energies above the reversal energy, the particles act in many ways as super-paramagnetic entities19. In extended artificial spin ice structures, the role of excitations on the order–disorder transition has been demonstrated by magnetization measurements19, theoretical investigations23, and thermal relaxation in extended artificial square spin ice arrays has been observed by direct imaging15. Furthemore, annealing of artificial spin ice arrays has recently been shown to result in extended thermal ordering and crystallization of interacting magnetic charges24. To investigate a dynamic equilibrium of an extended spin ice structure requires the magnetic excitations to be observable under experimental conditions. To accomplish this, the magnetic properties of the islands and their interactions need to be adjusted to match the accessible temperature range. In the present work, this is achieved through the choice of the intrinsic ordering temperature of the island material and the distance between the islands. This allows us to determine the equilibrium states in the artificial spin ice structures and to show that these exhibit a strong temperature dependence relating to the interaction between the elements. Thermal excitations result in monopole-like magnetic charges, which, for the first time, can be observed in thermal equilibrium. Photoemission electron microscopy with X-ray magnetic circular dichroism (PEEM-XMCD) was used to determine the magnetization of the individual elements comprising the arrays, allowing for the determination of their vertex configurations and ordering. The islands are shaped into elongated structures, as shown in Fig. 1a, with a long axis of 470 nm and a width of 170 nm. The investigated arrays differ only in their periodicity (600 and 640 nm, respectively; Fig. 1b). A typical PEEM-XMCD image of the 600 nm array is shown in Fig. 1c. A clear magnetic contrast is visible for most
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Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120, Uppsala, Sweden, 2 Laboratory for Micro- and Nanotechnology, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland, 3 Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich, CH-8093 Zurich, Switzerland, 4 Swiss Light Source, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland, 5 Lawrence Berkeley National Laboratory (LBNL), 1 Cyclotron Road, Berkeley, California 94720, USA. * e-mail: [email protected] NATURE NANOTECHNOLOGY | ADVANCE ONLINE PUBLICATION | www.nature.com/naturenanotechnology
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Figure 1 | Geometry and magnetic imaging. a, The extended artificial square spin ice arrays are composed of 470 nm × 170 nm magnetic islands of 1.4 monolayers d-doped Pd(Fe). b, Two arrays of different periodicity (600 and 640 nm). c, A typical PEEM-XMCD image showing the 600 nm array, with a clear magnetic contrast for the d-doped Pd(Fe) islands at a temperature of 30 K. The array has been rotated 458 with respect to the X-ray sensitivity axis to fully resolve the magnetic configuration. In the PEEM-XMCD image, islands with magnetization pointing towards the right are visualized with a black contrast and islands pointing to the left with a white contrast. d, Determination of the magnetic configuration of part of the 600 nm square spin ice array at 30 K. The image illustrates the magnetic moment directions determined from the magnetic contrast of the PEEM-XMCD image on the left (region within the white frame), as shown by the black arrows. The black outlines of the nanomagnets indicate their position. An extended domain of ground state vertices can be seen, with the typical alternating chirality pattern depicted by blue and red circular arrows.
islands, enabling the determination of the magnetization direction. As a result, the state configuration for a vertex, where all neighbouring islands are assigned a magnetization vector, can be defined (Fig. 1d). A bias towards ground state ordering, with vertices following the ice rule (two in, two out) is seen, although excited vertex configurations, not obeying the ice rule, are also observed. In artificial square spin ice, each vertex can have one of 16 possible vertex configurations, which can be sorted in ascending energy (Supplementary Fig. 1). Types I and II comprise vertex states that obey the ice rule, with type II possessing a dipole magnetic moment. Types III and IV do not obey the ice rule. As the coupling strength increases, so does the energy difference between the 2
different vertex states. As a consequence, slow cooling to 30 K results in dissimilarities in the ordering of the two arrays, as seen in Fig. 2. Larger domains of type I vertices are present in the 600 nm array (Fig. 2a,b) than in the more weakly coupled 640 nm array (Fig. 2c,d). Furthermore, the populations of excited vertex states (types III and IV) are also significantly smaller in the strongly coupled array, which demonstrates the importance of the strength of the inter-island coupling for the obtained ordering25. The results are unambiguous, as both arrays were patterned on the same substrate and have undergone the same cooling protocol. The temperature dependence of the array states presented in Fig. 3 was determined from images at different temperatures
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Figure 2 | Determination of vertex states from magnetic imaging. a–d, PEEM-XMCD images and magnetic configurations of the 600 nm (a,b) and 640 nm (c,d) arrays at 30 K. The arrows in b and d depict the islands’ magnetization vectors. Green dots indicate ground state vertices of type I, blue dots indicate type II, and yellow and red cones types III and IV, respectively. The up and down cones depict the positive and negative charge of type III and IV vertices (Supplementary Fig. 1). Missing arrows in b and d indicate insufficient magnetic contrast to safely assign a magnetization direction. The 600 nm array has a higher number of ground state vertices than the 640 nm array, where the dipolar interaction between islands is weaker.
(Supplementary Fig. 2). In thermally active nanomagnetic systems the possibility of observing thermal fluctuations is determined by the timescale of thermal fluctuations and the timescale of imaging. The total acquisition time at each temperature was tm ¼ 500 s, consisting of fifty single frames of 10 s duration. As the temperature is raised, more islands will fluctuate between the two low-energy states, resulting in a lower number of islands that can be assigned a magnetization vector. As seen in Fig. 3, a difference in the temperature dependence is also obtained: the predominantly type I background breaks up and disappears at a substantially lower temperature for the weakly coupled 640 nm array than for the more strongly coupled 600 nm array. The temperature dependence of the number of identifiable island states for both arrays is shown in Fig. 4. A difference in the onset of thermal fluctuations is obtained, providing support for the effect of geometry on the energy landscape. The timescale t of the individual island fluctuations is given by t = t0 eEr /(kB T), where t0 is the inverse attempt frequency22, which dictates the temperature range over which PEEM measurements can be used to resolve the magnetic configuration of the islands. The probability of the magnetization not having switched after a time t is therefore given26 by P(t) ≈ e 2t/t. A Gaussian distribution of the island sizes is
assumed, which is used to capture the effect of imperfections in the arrays resulting in a distribution of the island moments and therefore the reversal energy barriers. The temperature dependence of the moment of the islands is included in the calculations (see Methods). A median value of the reversal energy barrier (E0) at 0 K is obtained from this analysis. The inverse attempt frequency t0 is assumed to be 1 × 10210 s for the islands in both arrays14,22, and the analysis yields E0 ¼ 9.7 × 10220 J (E0/kB ≈ 7.0 × 103 K) for the 600 nm array and 7 × 10220 J (E0/kB ≈ 5 × 103 K) for the 640 nm array with a standard deviation s equal to 10% of the median value of E0. As the temperature is lowered, P(t) does not reach unity because of a residual number of islands for which a magnetization vector cannot be unambiguously assigned, even at the lowest temperatures (Fig. 2). The very small magnetic moment from the buried 1.4 monolayers of Fe combined with the Pd top layer (see Methods) leads to a weak magnetic contrast. This results in an offset that is included in the fits presented in Fig. 4. The reversal energy barrier at 0 K is given by E0 ¼ Er(0 K) þ Ei (0 K), where Er is the intrinsic barrier and Ei arises from the inter-island interactions. All energies are temperature-dependent, because the magnetic moment decreases with increasing temperature. The intrinsic reversal barrier is determined using
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Figure 3 | Effect of inter-island interactions on the onset of vertex state fluctuations. a,b, Vertex states of the two arrays determined from PEEM-XMCD imaging, shown as a function of temperature for the 600 nm (a) and 640 nm (b) arrays. Green dots refer to ground state vertices of type I, blue dots to type II vertices, and yellow and red cones to types III and IV, respectively. The up and down cones depict the positive and negative charge of type III and IV vertices (Supplementary Fig. 1). Missing vertices result from the inability to unambiguously assign a magnetization direction for neighbouring islands on a timescale defined by the PEEM imaging. The weakly coupled array (640 nm) in b undergoes a transition to a fluctuating state at a lower temperature than the strongly coupled array (600 nm) in a.
magnetostatic analysis, applying a methodology proposed by Osborn27. The interaction energy was obtained from micromagnetic calculations assuming a collinear magnetization of the islands (see Methods). The contribution of the interaction energy to the reversal barrier can be estimated by considering the reversal of one element in a ground state configuration, resulting in the creation of two type III vertices. The energy cost for such a transition is Ei ¼ 2ET3–2ET1 , 1.0
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Figure 4 | Temperature dependence of magnetic fluctuations. Probability Pl of the island magnetization not having switched during the PEEM-XMCD acquisition time, determined by the number of islands that can be assigned a magnetization vector: 600 nm (blue squares) and 640 nm (red circles) arrays. Lines represent fits to a thermally activated magnetization reversal model, yielding median values of E0 ¼ 9.7 × 10220 J for the 600 nm array and E0 ¼ 7 × 10220 J for the 640 nm array and a standard deviation of s ¼ 10% for both E0 values. All error bars represent one standard deviation. 4
where ET3 and ET1 are the energies of type III and I vertices, respectively. For non-interacting islands, ln(tm /t0 )kB TB = 1/2m0 M 2 (TB ) DNV = Er (TB ) at the blocking temperature28 TB , where m0 is the vacuum magnetic permeability, M(TB) is the island magnetic moment at that temperature, DN represents the demagnetizing factors calculated using the Osborn methodology27, V is the island volume and tm is the measurement time. Using this approach, the blocking temperature of non-interacting was determined islands array to be 95 K. For interacting islands, ln tm /t0 kB TB = 1/2m0 M 2 array array (TB )DNV + Ei (TB ), where T array is the blocking temperature B of an island in the array. This results in an estimate of TBarray(640 nm) ¼ 108 K, which is identical to the experimentally obtained value of 108 K (Fig. 4). For the 600 nm array we obtain TBarray(600 nm) ¼ 116 K, which is lower than the experimental value of 130 K. The effective island reversal barrier is not only defined by the lattice spacing but also by the magnetic configuration of the vertices, altering the reversal energy barriers and therefore the fluctuation frequency for the different vertex states. Excited vertex states of types III and IV are therefore more short-lived than states of types I and II at any given temperature, as these have higher energy. The role of the energy landscape on the thermal behaviour of the arrays can therefore be explored by looking at the observed probabilities PT of the four vertex states. As shown in Fig. 5a the 600 nm array displays a high probability of type I vertex states over the examined temperature range. The onset of fluctuations is most prominently reflected in the decrease of type II and III probabilities above 110 K for the 600 nm array. The transition occurs at a lower temperature for the 640 nm array, enabling us to follow the full order-to-disorder transition. Here, type I vertices are revealed to have a reduced probability, almost equal to that for types II and III. Taking into account the degeneracy of the vertex states, this array still exhibits significant ground state ordering. The onset of vertex state specific transitions is revealed in Fig. 5b, with the higher-energy type III states beginning to fluctuate at lower
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Figure 5 | Thermal distributions, fluctuations and energy statistics of vertex states. a,b, Observed probabilities for all types of vertex state for the 600 nm (a) and 640 nm (b) arrays as a function of temperature, revealing the different decays of the vertex states. Dashed lines serve as guides to the eye, denoting the vertex specific onset observed for the 640 nm array. The distribution of vertex states unveils the boundary between a frozen and a thermally active state (boundary depicted by the vertical dashed lines) where investigations of thermal dynamics in artificial spin ice arrays are possible. c, Probability distributions for the vertex energy states. The energy values (ETn) for the strongly coupled (600 nm, blue open circles) and weakly coupled (640 nm, red open squares) arrays have been determined using micromagnetic calculations for the experimentally observed blocking temperatures. The number of vertices of each type are taken from images recorded at the lowest temperature used for measuring arrays. The probabilities are given by Boltzmann factors (exp(bEi)) and the related slopes of the fitted lines reflect the blocking temperature Tarray of each array. The coloured regions highlight the vertex types: I (green), II (blue), B III (yellow) and IV (red). All error bars represent one standard deviation.
temperatures than for types II and I. Furthermore, the rate at which vertex states decay while raising the temperature increases with increasing vertex energy. The results also illustrate the existence of a boundary between two distinct states: a frozen low-temperature state, where the population distribution of the vertex configurations is predominantly dictated by the coupling strength and cooling conditions, and a high-temperature state with dynamic and evolving vertex configurations reflecting the energy landscape of the array. In this dynamic state, excited vertices carrying a magnetic charge are mobile, hinting at new routes for achieving magnetic charge mobility, with implications for future device design. At this stage, it cannot be excluded that pattern defects or inhomogeneities can act as localization sites for such excitations. Intentional introduction of these in artificial spin ice structures could provide further information on their role and potential for magnetic charge manipulation (see Supplementary Information in ref. 16). Correcting the observed abundances of the different vertex states for their degeneracy reveals their probability with respect to energy. The observed probabilities are presented in Fig. 5c and can be described by a Boltzmann distribution. The presented vertex
populations are taken at the lowest temperatures for the 640 nm and 600 nm arrays. These correspond to frozen configurations of the arrays, well below their respective blocking temperatures, and the slopes relate to the state distributions at the array blocking temp. The slopes result in a ratio for the blocking temperaeratures Tarray B array array tures (TB (600 nm)/TB (640 nm)) of 1.6(2), which compares reasonably with the ratio of 1.2(1) extracted from the observations presented in Fig. 4 (numbers in parentheses denote standard deviations), demonstrating once again how the interaction strength modifies the thermal equilibrium state of artificial spin ice arrays. We therefore conclude that the energy landscape of two-dimensional nanomagnetic systems can be altered through the design of the lattice geometry and island material, thereby defining their thermal behaviour. The use of real-space imaging techniques allows unprecedented and detailed information on the magnetic states to be obtained, enabling observations of real thermal dynamics at mesoscopic length scales and over time13–15,19,29. In the case of artificial spin ice, this sets the scene for the design of systems where manipulation of magnetic charges is feasible. Control of the excitation populations and magnetic charges by
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means of the array geometry and temperature can serve as a platform for studies of charge mobility under externally applied magnetic fields or temperature gradients. In future investigations, this will provide insights into the utilization of magnetic charges in novel technologies such as magnetic cellular automata30,31. Finally, thermal artificial spin ice, with its intrinsic multiplicity of states, could be an ideal testbed for addressing the physics of emergence in magnetic systems.
Methods The extended artificial square spin arrays were patterned by a combination of electron-beam lithography and an argon-ion milling process13. The islands were made from d-doped Pd(Fe) thin films, consisting of 40 nm Pd, 1.4 monolayers of Fe and finally 2 nm Pd on top, all grown on a V seeding layer19,32. The remanent magnetization of the films vanishes at TC ¼ 250 K, while the in-field magnetization (at an applied field of m0H ¼ 50 mT) extends to T0 ¼ 290 K, as measured using magneto-optical Kerr effect magnetometry32. The temperature dependence of the in-field magnetization can be successfully described by the function M(T ) ¼ M0(1 2 T/T0)0.5. For comparison with the experimental data an island moment value of m0 ¼ 4 × 106mB at T ¼ 0 K was used. The magnetic moment of an individual island can also be determined from its area and the magnetization of the continuous film, measured using a superconducting quantum interference device (SQUID) magnetometer. The resulting island magnetic moment using this method at 0 K corresponds to 6 × 106mB. Micromagnetic calculations were carried out for individual islands and vertex configurations, assuming collinear magnetization and using the object oriented micromagnetic framework (OOMMF) for a thickness of the magnetic layer of 2 nm. The resulting energies are significantly higher than those obtained using simple point-dipole calculations. The obtained values were scaled for the temperature dependence of the magnetic moment M(T ) using the equation stated above, and the energies for all vertex states can thus be calculated for a temperature range from 0 K to T0. The orientation of the magnetization of the islands was determined using PEEM and XMCD33 at the Fe L3 edge. The experiments were performed at the PEEM3 station of the Advanced Light Source at the Lawrence Berkeley National Laboratory. The arrays were cooled in situ using a liquid-helium cryostat, and all measurements were performed in the absence of an externally applied magnetic field (Supplementary Fig. 2). As all the patterns were fabricated on the same substrate they also undergo the same cooling rates in the PEEM microscope stage.
Received 27 January 2014; accepted 2 May 2014; published online 8 June 2014
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Acknowledgements The authors acknowledge support from the Knut and Alice Wallenberg Foundation, the Swedish Research Council, the Swedish Foundation for International Cooperation in Research and Higher Education and the Swiss National Science Foundation. The Advanced Light Source (ALS) is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the US Department of Energy (contract no. DE-AC02-05CH11231). The authors thank A. Young of the ALS for support during the PEEM experiments and A. Weber for development of the patterning processes and sample manufacture. The authors are also grateful to V. Guzenko for support with electron-beam lithography. V.K. would like to thank P.E. Jo¨nsson for discussions.
Author contributions B.H. and L.J.H. initiated the work. V.K., U.B.A., A.F., R.V.C., A.B. and A.S. performed the PEEM-XMCD imaging. V.K. and U.B.A. analysed the data. V.K., U.B.A. and B.H. wrote the paper. All authors discussed the results and commented on the manuscript.
Additional information Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to V.K.
Competing financial interests The authors declare no competing financial interests.
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