Abstract
Geometrical frustration occurs when entities in a system, subject to given lattice constraints, are hindered to simultaneously minimize their local interactions. In magnetism, systems incorporating geometrical frustration are fascinating, as their behavior is not only hard to predict, but also leads to the emergence of exotic states of matter. Here, we provide a first look into an artificial frustrated system, the dipolar trident lattice, where the balance of competing interactions between nearestneighbor magnetic moments can be directly controlled, thus allowing versatile tuning of geometrical frustration and manipulation of ground state configurations. Our findings not only provide the basis for future studies on the lowtemperature physics of the dipolar trident lattice, but also demonstrate how this frustrationbydesign concept can deliver magnetically frustrated metamaterials.
Introduction
Artificial square ice^{1}, consisting of dipolar coupled Isingtype nanomagnets, lithographically arranged onto a twodimensional square lattice, was introduced as a twodimensional analog to pyrochlore spin ice^{2} and provided the prospect to directly explore the consequence of geometrical frustration using appropriate imaging techniques^{3,4,5,6,7}. However, due to imbalanced competing interactions of nanomagnets meeting at the fournanomagnet vertices, the spinice degeneracy in artificial square ice is lifted^{8} and the system lacks the macroscopic degeneracy of its threedimensional counterpart, the pyrochlore spin ice. Indeed, artificial square ice, when exhibiting thermally driven moment fluctuations, has been shown to have a clear pathway toward a longrange ordered ground state configuration^{9,10,11,12}. Several concepts have been proposed to overcome this problem, most prominently by introducing a height offset between the corresponding nanomagnets^{8, 13,14,15}. However, a realization that is at the same time technically simple and appropriate for direct realtime imaging of thermal fluctuations remains elusive^{15}. Alternatively, highly frustrated artificial kagome spin ice has been extensively investigated, as it exhibits some analogy to pyrochlore spin ice^{16}, including a spin liquid phase with shortrange correlations^{17,18,19,20,21}. Still, longrange dipolar interactions have been shown to overcome the fixed degree of frustration at lowtemperature regimes, leading the system to access a longrange ordered ground state^{17,18,19}.
This raises the question whether a twodimensional geometrical concept can be proposed that shares some similarities to the square ice geometry, while exhibiting a higher degree of geometrical frustration.
In the following, we address this point by exploring moment configurations achieved in a twodimensional artificial frustrated system consisting of nanomagnets occupying the sites of a socalled trident lattice. Following thermal annealing, we observe how accessible lowenergy configurations can be directly manipulated by tuning the balance of competing interactions.
Results
The dipolar trident lattice
We introduce an artificial frustrated system consisting of threenanomagnet (trident) building blocks periodically arranged in a perpendicular fashion (Fig. 1a). Each nanomagnet is small enough to be singledomain and elongated, so that the magnetization points toward one of two possible directions along the long axis of each nanomagnet, thus representing a single Isingtype moment. As these moments couple via dipolar magnetic fields, we refer to this system as the dipolar trident lattice. Using synchrotronbased photoemission electron microscopy PEEM^{22} (Methods section), we directly visualize thermally induced magnetic relaxation of the trident lattice, and demonstrate the inability of the system to access a fully ordered state down to temperatures around 150 K, when tuning the balance of competing interactions. We show how, above 150 K, the ordering preferences of the system can be altered between two longrange ordered phases via an intermediate disordered state, exhibiting a continuous presence of vertex defects, which through their migration control the relaxation process and configurational fluctuations in thermal equilibrium. Upon cooling, the disordered phase also evolves toward longrange order, exhibiting a mixture of the two magnetic configurations.
Energetically, moment configurations in the trident lattice (Fig. 1b) can be characterized by four vertex types^{1, 9} listed with increasing dipolar energy in Fig. 1c. In addition to the vertex types, the socalled trident types need to be taken into account, which are listed with increasing energy as Type A, B, and C in Fig. 1c. In order to understand the concept of competing interactions in the trident lattice, one has to be aware of the consequence of dipolar interactions: First, at fournanomagnet vertices (for example, α, δ, ε, and η in Fig. 1a), nearest neighbors will preferably exhibit a headtotail moment alignment, giving rise to Type I vertices (Fig. 1c) and vortexlike states (Fig. 1d). Second, the tridents (α, β, and γ in Fig. 1a) favor an antiparallel moment alignment (Fig. 1d). In a longrange picture, a system where vertex interactions are mostly minimized (Type I vertex domination) cannot satisfy all trident interactions, as Type B tridents will dominate the configuration landscape (Fig. 1e). In contrast, if trident interactions are minimized (Type A trident domination), the energetically higher Type II vertices will exhibit a dominating presence (Fig. 1f). In other words, it is impossible to simultaneously satisfy both vertex interactions and trident interactions and the system is expected to be frustrated.
Direct observation of thermal relaxation
As a first step, we aim to visualize the consequence of geometrical frustration on the ordering mechanism in a trident lattice consisting of nanomagnets with length, width, and thickness of 450, 150, and 2.7 nm, respectively (Methods section). The lattice spacing was chosen, so that the two relevant parameters a and b (Fig. 1a), which control the strength of trident and vertex interactions, respectively, are set to be 50 nm each. The blocking temperature T _{B}, which we define as the temperature at which moment reorientations start to occur within the timescale needed to acquire a singlePEEM image (7–9 s per image)^{9, 23} was determined to be 270 K. The sample was kept at a constant temperature of 280 K and a saturating magnetic field (B = 30 mT) was shortly applied along the incoming Xray direction. After the field is switched off, the system undergoes thermally induced magnetic relaxation from a welldefined energetically excited state toward a highly disordered equilibrium state (Fig. 2a–c; Supplementary Fig. 1 and Supplementary Movie 1).
A quantitative analysis of the relaxation mechanism is obtained by looking at the vertextype and tridenttype populations plotted as a function of time (Fig. 2d, e). Starting from a 100% Type II vertex and Type C trident background (saturated state), the system experiences a rapid drop and rise in Type II and Type III vertex populations, respectively, while Type I vertices are moderately on the rise (Fig. 2d). In parallel, the population of Type C tridents decreases rapidly, while Type A and B tridents are showing an almost equal increase (Fig. 2e). As the system continues to relax, the high number of generated Type III vertex defects converts into Type I vertices, while new defects are continuously generated with an ongoing decrease in Type II vertices. Thus, the system stagnates in terms of Type III population during this stage (Fig. 2d). Finally, the system enters a stage where the Type I vertex population rises continuously at the cost of Type II and Type III vertices, until equilibrium is achieved. Kinetic Monte Carlo simulations (Methods section)^{19} are in good agreement with the experimental observations (Supplementary Fig. 2).
Controlling the balance of competing interactions
The dominance of Type I vertices for a = b = 50 nm indicates that the competition between vertex interactions and trident interactions is not perfectly balanced and, as a result, a high degree of frustration is not obtained. This balance of competing interactions can be tuned by varying the b/a ratio. Therefore, a second set of trident arrays are fabricated (Methods section), where a = 50 nm is set to stay constant, while b is varied to be 50, 75, and 100 nm. The sample was kept at a constant temperature of 330 K (T _{B} = 310 K) for ~24 h before it was cooled down to 300 K and magnetic images were obtained (Fig. 3a–c; Supplementary Fig. 3). Plotting the vertex populations and trident populations as a function of b (Fig. 3d, e), we see a transition from a largely ordered phase with Type I vertex and Type B trident domination (b = 50 nm), through a disordered phase with shortrange order and no clear preference for any vertex types (b = 75 nm), to, finally, a phase that shows trends toward Type II vertex and Type A trident preference (b = 100 nm).
This balancing act between competing trident interactions and vertex interactions indicates that accessible lowenergy states can be directly tuned by a variation of the b/a ratio. This is a direct consequence of the enforced lattice constraints, making it impossible for the involved entities to simultaneously minimize both trident interactions and vertex interactions. Similar to previous work^{24, 25}, calculating the energy spectrum of an isolated fivenanomagnet building block (Supplementary Fig. 4), clarifies the degeneracies listed in Fig. 1e, f. For a system with b/a = 1, the building block ground state is fourfold degenerate, with a clear gap (ΔE in Supplementary Fig. 4) to the second energy band, which consists of eight quasidegenrate states. This gap can be tuned by varying the b/a ratio. The critical ratio b/a = 1.5 of the relevant lattice parameters can be further comprehended, when comparing the dipolar energies for fully ordered magnetic configurations of Type A/Type II and Type B/Type I trident types and vertex types as a function of b, while a is kept constant at 50 nm. This is shown in Supplementary Fig. 5, where dipolar energies are equalized, when b reaches a value around 75 nm. In other words, this is the point where the dipolar trident lattice reaches maximum degeneracy with no preference for any of the twelve states listed in Fig. 1e, f.
Lowtemperature configurations and magnetic structure factors
As a next step, we study how the degeneracy of lowenergy building block states affects moment configurations at lower temperatures. Previous work on highly frustrated artificial kagome spin ice^{17, 24, 26} showed that despite an extensive degeneracy and shortrange ordering at higher temperatures, the longrange nature of dipolar interactions gives rise to ordered configurations at lower temperatures. Therefore, it is our purpose here to see whether any signatures of longrange ordering can be observed in the dipolar trident lattice, particularly in the case of highest degeneracy, when b/a = 1.5.
We prepared another set of trident lattices, consisting of nanomagents with lengths L = 300 nm, widths W = 100 nm, and thickness d = 2.4 nm, together with the corresponding lattice parameters a = 33 nm = const. and b = 33, 50, and 66 nm. This reduction of nanomagnet size resulted in a lowering of the blocking temperature down to 160 K. The sample was kept in vacuum at room temperature for 20 days, before it was cooled down to 150 K for Xray magnetic circular dichroism (XMCD) image acquisition (Fig. 4a–c). While longrange ordered patterns are observed for b = 33 and 66 nm, the b = 50 nm array remains disordered at 150 K. Similar to the roomtemperature data (Fig. 3), tuning of geometrical frustration can again be inferred from the evolution of vertex populations and trident populations as a function of lattice parameter b (Supplementary Fig. 6). Our results thus indicate that for the case b/a = 1.5, the system is caught in a shortrange ordered phase, while both the b/a = 1 and 2 cases exhibit longrange ordered ground state configurations.
A deeper quantitative insight into the experimentally accessed lowtemperature configurations is achieved by calculating the respective magnetic structure factors^{15} (Methods section), which are shown in Fig. 4d–f. For both b = 33 nm and b = 66 nm (Fig. 4d, f), we see relatively sharp peaks in the magnetic structure factors. The splitting of the magnetic peaks into four satellites for b = 33 nm reflects multidomain longrange ground state ordering for b = 33 nm (Fig. 4d). Figure 4f (b = 66 nm) shows sharp magnetic peaks that stand for an almost singledomain longrange ordered ground state consisting mostly of a tile of two Type A tridents, as can also be seen from realspace images (Fig. 4c). However, a dramatic change in the magnetic structure factor is observed for the lattice parameter combination of a = 33 nm and b = 50 nm (Fig. 4e), where the diffuse patterns indicate the presence of a disordered phase consisting of a complex arrangements of possible lowenergy configurations (Fig. 1e, f), where neither of these states dominate. Similar patterns are also observed in the structure factor of the simulated b = 50 nm system (Supplementary Fig. 7). To ensure that these simulated configurations are in fact in thermal equilibrium, we use the parallel tempering technique^{27,28,29} (Methods section), where the equilibration time is estimated by calculating the socalled exponential autocorrelation time^{29}, τ _{exp}, which itself is defined by the temporal decay of the autocorrelation function, Γ ∝ exp[−t/τ _{exp}] (Methods section). The agreement between experimental and simulated configurations provides evidence that the experimental observations also represent states in thermal equilibrium. For comparison, we also calculated the magnetic structure factor of a purely paramagnetic trident lattice (Supplementary Fig. 8). In this case, no peaks appear in the structure factor map, providing further evidence that our dipolar trident lattice with a = 33 nm and b = 50 nm is in a disordered phase, where none of the predicted ground states is able to dominate the landscape at 150 K. While only partial order is able to set in at 150 K with relatively weak peak intensities, magnetic structure factor simulations clearly show that the peak intensities rise with decreasing temperature (Supplementary Fig. 9). Eventually, the system evolves toward longrange order at low temperatures, exhibiting a mixture of ordered stripes (Supplementary Fig. 10), which consist of a tile of Type A and Type B tridents and a mixture of Type I and Type II vertices. Compared to classical artificial square ice, which exhibits trivial ground state ordering, the mixed phase patterns in the dipolar trident lattice reflect the high degree of frustration in this geometrically frustrated magnetic metamaterial. For interested readers, a comparison of experimental observations to artificial square ice can be found in Supplementary Note 1.
Discussion
In summary, we presented a magnetically frustrated metamaterial, which provides the possibility to directly control competing dipolar interactions at the nanoscale, thus allowing versatile tuning of geometrical frustration and ground state configurations. The complex phase into which the system gets trapped, when competing interactions are balanced, opens up multiple questions regarding the physics of the dipolar trident lattice, in particular the question regarding possible phase transitions toward complex longrange ordered states at lower temperature regimes^{17,18,19}. Experimentally, this will require the fabrication of trident lattices consisting of nanomagnets with lateral dimensions that go beyond the spatial resolution of known magnetic imaging techniques^{17}, and will therefore rely on emerging scattering and spectroscopic techniques^{17, 26, 30, 31}.
Methods
Sample fabrication
Similar to previous work^{5, 23}, dipolar trident lattices were fabricated by taking advantage of liftoffassisted electronbeam lithography: a silicon (100) substrate was first spincoated with a 70nmthick layer of polymethylmethacrylate resist. Then, trident lattices with various lattice spacings were defined onto the sample with a VISTEC VB300 electron beam writer. Next, using a Semicore SC600 ebeam evaporator, a ferromagnetic permalloy (Ni_{80}Fe_{20}) film was deposited at a base pressure of 1.2 × 10^{−7} Torr, which was followed by liftoff in acetone at a temperature of 50 °C. Thermally driven moment fluctuations in one set of artificial spin ice samples were realized by fabrication of ultrathin nanomagnets with length L = 450 nm and width W = 150 nm. The samples discussed in this work had thicknesses of 2.7 nm and 3 nm, resulting in blocking temperatures of 270 and 310 K, respectively. For lowtemperature measurements, the blocking temperature was moved down to 160 K by preparing nanomagnets with lengths, widths, and thickness of 300, 100, and 2.4 nm, respectively.
Photoemission electron microscopy
Measurements were performed using the cryogenic photoemission electron microscope PEEM3 at beamline 11.0.1 at the Advanced Light Source^{22}. Magnetic images were captured by taking advantage of XMCD at the Fe L_{3}edge^{32}. The obtained contrast is a measure of the projection of the magnetization on the Xray polarization vector, so that nanomagnets with a magnetization parallel or antiparallel to the Xray polarization either appear black or white. Nanomagnets with moments having ±45° and ±135° angles with respect to the incoming Xrays appear dark and bright, respectively.
Magnetic structure factor
The magnetic structure factor is calculated as
where \({\bf{S}}_i^ \bot = {\bf{S}}_i\! \! \left( {\widehat {\bf{q}} \cdot {\bf{S}}_i} \right)\widehat {\bf{q}}\) is the component of the spin vector of each island perpendicular to the reciprocal space vector q, the unit vector is given by \(\widehat {\bf{q}} = {\bf{q}}{\mathrm{/}}\left\ {\bf{q}} \right\\), r _{ i,j } is the vector from island i to j, and N is the total number of islands. Equation (1) has the same form as in neutron scattering experiments and has previously been used to analyze artificial spinice configurations^{15}.
Simulations
We model each nanomagnet as an infinitesimally thin compass needle with a uniform magnetic moment density \(\frac{{\left m \right}}{L}\). The magnetic moment points along the long axis of the island. This description is equivalent to placing a magnetic charge at each end of the island^{5, 19, 33}. The interisland interaction is given by the Hamiltonian
where \(r_{a_{\mathrm{i}}}\) and r _{bi} are the locations of the positive and negative magnetic charge on the ith nanomagnet, μ _{0} is the magnetic permeability, L is the island length, and m = MV is the magnetic moment of each nanomagnet with M being the saturation magnetization and V the nanomagnet volume. The system size is 1200 islands, and only interactions with a magnitude of at least 2% of the nearestneighbor interaction are included in the simulation (~35 neighbors per spin).
To simulate the dynamics of the system, we use the kinetic Monte Carlo method^{9, 23}, which evolves the system through singlespin flips. A particular spin flip move is selected with a probability proportional to its rate. Assuming an Arrheniustype switching behavior, the rate of a spin flip is given by v = v _{0} exp(−E/k _{B} T), where k _{B}=8.62 × 10^{−5} eV K^{−1} is the Boltzmann constant, ν _{0} is the socalled attempt frequency, T is the temperature, and E is the reorientation barrier, which is equal to the intrinsic energy barrier E _{0} plus half the dipolar energy gain associated with moment reorientations (Eq. (2)). The simulation parameters M = 240 kA m^{−1}, E = 0.887 eV, and ν _{0} = 10^{12} s^{−1} were fit using the experimental relaxation results of Fig. 2. These values are in good agreement with previous studies on thermally activated artificial spin ice^{5, 9, 23}. In addition to the assumption of a uniform system, where all nanomagnets have the same intrinsic energy barrier, we also investigated the role of disorder^{9}. This is included by assuming a random variation in E _{0}, which follows a Gaussian distribution with mean E _{0} = 0.893 eV and standard deviation σ = 0.05 eV (Supplementary Fig. 2).
To generate equilibrium configurations, for the results presented in Fig. 3 and structure factor calculations shown in Supplementary Fig. 7, we use the parallel tempering technique^{27, 28}. Replicas of the system are simulated at a number of temperatures simultaneously using kinetic Monte Carlo. After every Monte Carlo sweep a move is proposed which swaps the configuration of a pair of replicas at neighboring temperatures T _{ n } and T _{ m }. This move is accepted with a probability
where E _{ n } is the energy of replica n. The set of temperatures is selected such that the acceptance ratio of a swap move at each temperature is greater than 0.2. A value of M = 362 kA m^{−1} is used to obtain the results in Fig. 3. The equilibration time is estimated with the exponential autocorrelation time^{29}, τ _{exp}. This is defined by the decay of the autocorrelation function, Γ ∝ exp[−t/τ _{exp}]. It is calculated for the autocorrelation function of the spin overlap function between two concurrent independent simulations, its absolute value, and the configuration energy throughout the parallel tempering simulation. Taking the largest of these calculated exponential autocorrelation times, the first 20 × τ _{exp} time steps are treated as equilibration time and discarded.
Code availability
Codes for numerical calculations in this study are available from the corresponding author upon reasonable request.
Data availability
Data supporting the findings in this study are available from the authors upon request.
Change history
12 December 2017
The original version of this article contained an error in the legend to Figure 4. The yellow scale bar should have been defined as ‘~600 nm’, not ‘~600 µm’. This has now been corrected in both the PDF and HTML versions of the article.
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Acknowledgements
We would like to thank Sujoy Roy, Guido Meier, Peter Fischer, Kevin Franke, and Zoe Budrikis for their support and fruitful discussions. This project was funded by the Swiss National Science Foundation and part of this work was performed at the Advanced Light Source and the Molecular Foundry, Lawrence Berkeley National Laboratory, 94720 Berkeley, USA. The Advanced Light Source and the Molecular Foundry are supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DEAC0205CH11231. C.F.P. and M.J.A. are supported by the Academy of Finland through its Centres of Excellence Program (2012–2017) under project no. 251748 and the FiDiPro program, project 13282993. They acknowledge the computational resources provided by the Aalto University School of Science “ScienceIT” project. S.G. was funded by the European Union’s Horizon 2020 research and innovation programme under the Marie SklodowskaCurie grant agreement No. 708674.
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A.F. designed this study and planned the experiments accordingly. A.F., S.D. and Q.H.Q. fabricated the samples. A.F., A.S. and S.G. performed XMCD imaging at the Advanced Light Source. A.F., L.A., C.F.P., M.S., S.V. and C.W. analyzed the obtained data. C.F.P., M.J.A. and P.M. provided the theoretical background to the study. A.S. and S.v.D. supervised the project. All authors contributed to the manuscript.
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Farhan, A., Petersen, C.F., Dhuey, S. et al. Nanoscale control of competing interactions and geometrical frustration in a dipolar trident lattice. Nat Commun 8, 995 (2017). https://doi.org/10.1038/s41467017012384
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