Session V8: Focus Session: Frustrated Magnetism - Kagome II

Monte Carlo simulations of the fcc Kagom\'{e} lattice

For many years, Ir-Mn alloys have been widely used by the magnetic storage industry in thin-film form as the antiferromagnetic pinning layer in GMR and TMR spin valves [1]. Despite the technological importance of this structure, it has not previously been noted that the magnetic Mn-ions of fcc IrMn$_3$ reside on Kagom\'{e} layers ABC stacked along $<$111$>$ axes normal to the film plane [2,3]. Results of Monte Carlo simulations will be reported on the bulk fcc Kagom\'{e} lattice for both XY and Heisenberg models including the eight NN exchange interactions. Degeneracies persist in the 3D case and there is strong evidence for a fluctuation-driven first-order transition to well-defined long-range order characterized as the layered ``$q$=0'' 120-degree spin structure. Effects of varying the inter-layer coupling are also examined. \\[4pt] [1] M. Tsunoda et al, Appl. Phys. Lett. {\bf 97}, 072501 (2010).\\[0pt] [2] I. Tomeno et al, J. Appl. Phys. {\bf 86}, 3853 (1999).\\[0pt] [3] L. Szunyogh et al, Phys. Rev. B {\bf 79}, 020403(R) (2009).   

Phase Control of Magnetic Order in (Y,Lu)BaCo$_{4}$O$_{7}$

The RBaCo$_{4}$O$_{7}$ (R=Ca, Y, Tb-Lu) system provides a novel topology for studying geometric frustration, in which face-sharing tetrahedra of magnetic ions link to form trigonal bipyramids on a Kagom\'{e} lattice. Here we describe the structural and magnetic behavior of the Lu member and the solid solution joining Lu to Y as a chemical means to tune between magnetically ordered and disordered ground states. Mean-field models of the generic magnetic phase diagram of RBaCo$_{4}$O$_{7}$ determined recently by our group (D. D. Khalyavin et al. Physical Review B 82, 094401 (2010)) show a variety of magnetic states as a function of two exchange parameters: J$_{ab}$ and J$_{c}$, where J$_{ab}$ links Co ions in the Kagom\'{e} planes and J$_{c}$ links Co ions from the Kagome plane to the interleaving triangular layer. Experimentally, we find that YBaCo$_{4}$O$_{7}$ has a long-range ordered antiferromagnetic ground state, while LuBaCo$_{4}$O$_{7}$ appears to be disordered above 2 K with very slow dynamics measured by neutron scattering. We use the solid solution to interpolate between these endpoints and discuss these results with respect to the mean-field phase diagram.   

Spin dynamics in the extended kagom\'{e} YBaCo$_{4}$O$_{7}$

The extended kagom\'{e} systemYBaCo$_{4}$O$_{7}$ consists of antiferromagnetically coupled Co$^{2+}$ and Co$^{3+}$ ions arranged in stacks with alternating kagom\'{e} and triangular layers in ab planes in the orthorhombic lattice. The oxygen-stoichiometric material orders below 110 K. The system exhibits interesting exchange topology with both trigonal bipyramids and triangular kagom\'{e} clusters of Co ions. Model calculations and neutron scattering experiments, made by other workers, have provided considerable information on the magnetic structure. Ordered chains are found for the apical ions along the c-axis with neighbor chains having oppositely directed polarizations perpendicular to c in an antiferromagnetic configuration. Only short range order is present in the kagom\'{e} planes at temperatures as low as 2 K. The present pulsed NMR measurements, made on a single crystal, both in zero magnetic field and in low applied fields, distinguish the Co ion sites and provide information on the evolution of the spin dynamics for the plane and chain sites as a function of temperature in the range 1.7 - 50 K above which signal wipe-out occurs. A dramatic change in the spin dynamics is found below 5 K.   

Spectral Signature of Neodymium Dopants in Frustrated Gadolinium Gallium Garnet Lattice

We investigate the spectral emission of Nd3+ dopant ions (1{\%} at. wt) in the frustrated magnet Gadolinium gallium Garnet (GGG) as a function of temperature and magnetic field. We concentrate on the low energy excitations centered at 1064 nm and 935 nm, which show a multiplet structure at room temperature. As temperature decreases the emission spectra demonstrate changes in relative intensities that undergo a cross-over at 122 K under zero field cooled conditions. This cross-over is magnetic field dependent and changes as we field-cool the sample. Typically, with decreasing temperature the line widths of the spectral peaks decrease, as is expected. However, when cooled below 10 K selective peaks start exhibiting broadening, even when zero-field cooled. We follow this line broadening as a function of magnetic field and dopant concentration and speculate it is a result of the dopant ions coupling to the internal magnetic fields of the host lattice.   

Investigating the nature of magnetic correlations in the anti-ferromagnetic hyper-kagome material, Yb$_3$Ga$_5$O$_{12}$

The magnetic Yb$^{3+}$ ions in Yb$_3$Ga$_5$O$_{12}$ (YbGG) reside on a hyper-kagome lattice, which has the same connectivity as the planar kagome lattice but in higher dimensions. For anti-ferromagnetically (AFM) coupled spins the hyper-kagome lattice provides a highly-frustrated geometry in three-dimensions. YbGG is isostructural with the well-studied Gd$_3$Ga$_5$O$_{12}$ (GGG), which enjoys an exotic magnetic phase diagram. In GGG, the effects of geometric frustration manifest as a disordered, partial spin-glass ground state down to 25mK in zero-field. The application of an external magnetic field first induces an intermediate spin liquid state, then a long range AFM ordered phase. Much less is known about YbGG, though all experimental evidence indicates a lack of LRO in zero-field down to 30mK. We have recently produced single crystals of YbGG and have performed neutron scattering experiments over a range of temperatures (80mK - 10K) and magnetic field strengths (0T - 8T). The results indicate low-energy, fluctuating spin correlations at 80mK, 0T, as well as a dramatic response to an applied magnetic field. At zero-field, we also observe a low-energy dispersionless spin excitation that softens as the temperature is increased above the Schottky anomaly in the specific heat.   

Spatially anisotropic kagome antiferromagnet with Dzyaloshinskii-Moriya interaction

We theoretically study the spatially anisotropic spin-1/2 kagome antiferromagnet with Dzyaloshinskii-Moriya (DM) interaction using a renormalization group analysis in the quasi-one-dimensional limit. We identify the various temperature and energy scales for ordering in the system. For very weak DM interaction, we find a low-temperature spiral phase with the plane of the spiral selected by the DM interaction. This phase is similar to a previously identified phase in the absence of the DM interaction. However, above a critical DM interaction strength we find a transition to a phase with coexisting antiferromagnetic and dimer order, reminiscent of one-dimensional antiferromagnetic systems with a uniform DM interaction. Our results help shed light on the fate of two dimensional systems with both strong interactions and significant spin-orbit coupling.   

Dimensional reduction, avalanches and disorder in artificial kagome spin ice

In collaboration with an experimental team at the Swiss Light Source we have recently demonstrated that emergent monopoles and associated Dirac strings can directly be observed in real space via x-ray circular dichroism in a kagome lattice geometry. Here we build on the fact that the experimental results are in excellent agreement with MC simulations of a lattice of point dipoles with disorder realized in the form of random switching fields. We demonstrate that within a large range of physical parameters such as interdipolar coupling and randomness, magnetization reversal proceeds via a novel 1D avalanche behaviour whose hallmark is an exponential avalanche size distribution. After presenting simple arguments for the origin of such dimensional reduction we demonstrate that such 1D avalanche behavior also occurs in a model where the dipoles are stretched into magnetic charge dumbbells which provides a more realistic model for nanolithographic islands. Finally we demonstrate how a judicious design of the island anisotropy can be used to achieve controlled switching and avalanche propagation which paves the way for spintronic applications   

Propagation of monopole defects and flux channels in an artificial square spin-ice lattice

The recent development of artificial lattices of magnetic islands in which competing interactions give rise to macroscopic analogs of atomically frustrated spin ices has opened up a new field of research, in which the interaction, frustration and evolution of individual magnetic elements can be directly observed in real space. We investigate the magnetic reversal along the (11) symmetry axis of permalloy islands in an artificial ``square'' spin-ice geometry with in-situ Lorentz transmission electron microscopy. Novel differential transport-of-intensity allows for the identification of ``monopole''-like defects and flux channels, similar to Dirac strings, that link them. We track the growth and propagation of these defects and flux channels throughout the reversal process. Simulations are used to compare with experiment to show how nucleation and propagation of defects affect the reversal of the lattice as a whole. We find that interactions between defects and flux channels can explain the saturation of defect populations at low net magnetizations. This work was supported by U.S. Department of Energy, Office of Basic Energy Science, Material Sciences and Engineering Division, under Contract No. DE-AC02-98CH10886.   

Magnetization dynamics in artificial spin ice lattices

Artificial spin ice lattices (ASIL) consist of regular arrays of single-domain nanomagnets displaying ice rule ordering. Frustration is introduced through shape anisotropy. ASILs have been shown to exhibit complex behavior, with rich phase diagrams and quasi-static magnetization reversal. In particular, topological defects, such as Dirac monopoles and Dirac strings, play a fundamental role in the quasi-static behavior of ASILs. In this work, we use micromagnetic simulations to investigate the resonant frequencies of square lattice ASILs consisting of stadium-shaped nanomagnets. We calculate the evolution of the fundamental modes of a single element when elements are combined in four-stadia configurations and large lattices. In a cross-shaped four-stadium configuration for example, the Dirac monopole splits the frequencies of the lowest (near)-degenerate symmetric and antisymmetric edge modes of a single stadium. This splitting increases in a 24-stadium system with two monopoles. We also calculate the evolution of the spectral characteristics as the monopoles move farther apart in the lattice, but stay connected through a Dirac string. Our work suggests that these topological defects have distinct spectral signatures that can be detected experimentally.   

Study of system-size effects on the emergent magnetic monopoles and Dirac strings in artificial kagome spin ice

In this work we study the dynamical properties of a finite array of nanomagnets in artificial kagome spin ice at room temperature. The dynamic response of the array of nanomagnets is studied by implementing a ``frustrated celular aut\'{o}mata'' (FCA), based in the charge model. In this model, each dipole is replaced by a dumbbell of two opposite charges, which are situated at the neighbouring vertices of the honeycomb lattice. The FCA simulations, allow us to study in real-time and deterministic way, the dynamic of the system, with minimal computational resource. The update function is defined according to the coordination number of vertices in the system. Our results show that for a set geometric parameters of the array of nanomagnets, the system exhibits high density of Dirac strings and high density emergent magnetic monopoles. A study of the effect of disorder in the arrangement of nanomagnets is incorporated in this work.   

Magnetic reversal of an artificial square ice: dipolar correlation and charge ordering

Artificial spin ices are lithographically patterned arrays of single domain nanomagnets [1-4]. The elongated elements form a 2D system of interlinked vertices at which Ising-like dipole moments meet with incompatible interactions. They are directly analogous to 3D bulk spin ice materials [5]. We report on the magnetic reversal of an athermal artificial square ice pattern subject to a sequence of magnetic fields applied slightly off the diagonal symmetry axis, investigated via magnetic force microscopy of the remanent states that result [1]. From an initial diagonally polarised state, sublattice independent reversal is observed via bulk-nucleated incrementally-pinned flipped moment chains along parallel channels of magnetic elements, as evident from analysis of vertex populations and dipolar correlation functions. Weak dipolar interactions between adjacent chains favour antialignment and give rise to weak charge ordering of ``monopole'' vertices during reversal. \\[4pt] [1] J. P. Morgan, A. Stein, S. Langridge \& C.H. Marrows, New Journal of Physics (2011), 13, 105002.\\[0pt] [2] R. F. Wang et al., Nature (2006), 439, 303-306.\\[0pt] [3] E. Mengotti et al., Nature Physics (2011), 7, 68-74.\\[0pt] [4] J. P. Morgan et al., Nature Physics (2011), 7, 75-79.\\[0pt] [5] M. J. Harris et al., PRL (1997), 79, 2554-255   

Experimental Realization of Isotropic Ising Spins in Frustrated and Unfrustrated Artificial Spin Ice with Perpendicular Anisotropy

We have studied lithographically defined arrays of magnetostatically interacting single domain ferromagnetic islands with moments normal to the plane, leading to fully isotropic magnetostatic interactions. Probing both frustrated kagome and unfrustrated honeycomb array geometries, we find that the spin configurations can be reproduced with models based on only nearest-neighbor correlations. While the honeycomb geometry displays ordering of moments in well-defined domains, the kagome geometry has only short range correlations that show striking similarities to those of analogous in-plane systems and are closely comparable to expectations for a simple Ising system.   

Hysteresis and Return Point Memory in Artificial Spin Ice Systems

We investigate hysteresis loops and return point memory for artificial square and kagome spin ice systems by cycling an applied bias force and comparing microscopic effective spin configurations throughout the hysteresis cycle. Return point memory loss is caused by motion of individual defects in kagome ice or of grain boundaries in square ice. In successive cycles, return point memory is recovered rapidly in kagome ice. Memory is recovered more gradually in square ice due to the extended nature of the grain boundaries. Increasing the amount of quenched disorder increases the defect density but also enhances the return point memory since the defects become trapped more easily.   

Collective Magnetic Behavior of Geometrically Frustrated Arrays with Perpendicular Anisotropy

We use the magneto-optical Kerr effect (MOKE) to study the global and local magnetic behavior of geometrically frustrated arrays of single domain ferromagnetic islands with perpendicular anisotropy. MOKE measurements over macroscopic length scales probe the global properties of arrays with different lattice geometries and island spacings. The variation of switching field as a function of island spacing gives us insight into the influence of local frustration on the collective magnetic response of the arrays. The experimental results are compared with mean field calculations. Finally, we use spatially resolved Kerr microscopy to probe nucleation and domain propagation in the magnetization reversal process. Supported by U.S. Department of Energy Award DE-SC0005313. Lithography performed with the support of the National Nanotechnology Infrastructure Network