Characterization of switching field distributions in Ising-like magnetic arrays
Robert D. Fraleigh, Susan Kempinger, Paul Lammert, Vincent H. Crespi, Nitin Samarth, Sheng Zhang, Peter Schiffer
CCharacterization of switching field distributions in Ising-likemagnetic arrays
Robert D. Fraleigh, Susan Kempinger, PaulLammert, Vincent H. Crespi, and Nitin Samarth ∗ Department of Physics, The Pennsylvania State University,University Park, Pennsylvania 16802-6300, USA
Sheng Zhang
Materials Science Division, Argonne National Laboratory,9700 S. Cass Avenue, Argonne, IL 60439, USA
Peter Schiffer
Department of Physics and the Frederick Seitz Materials Research Laboratory,University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA (Dated: November 6, 2018)
Abstract
The switching field distribution within arrays of single-domain ferromagnetic islands incorpo-rates both island-island interactions and quenched disorder in island geometry. Separating thesetwo contributions is important for disentangling the effects of disorder and interactions in themagnetization dynamics of island arrays. Using sub-micron, spatially resolved Kerr imaging inan external magnetic field for islands with perpendicular magnetic anisotropy, we map out theevolution of island arrays during hysteresis loops. Resolving and tracking individual islands acrossfour different lattice types and a range of inter-island spacings, we extract the individual switchingfields of every island and thereby determine the relative contributions of interactions and quencheddisorder in the arrays. The width of the switching field distribution is well explained by a simplemodel comprising the sum of an array-independent contribution (interpreted as disorder-induced),and a term proportional to the maximum field the fully polarized array could exert on a singleisland. We conclude that disorder in these arrays is primarily a single-island property. a r X i v : . [ c ond - m a t . m e s - h a ll ] J un rdered arrays of nanoscale single-domain ferromagnetic islands provide a well-definedIsing system at a spatial scale where it is possible to resolve every Ising degree of free-dom. Geometries with frustrated interactions, such as artificial realizations of “spin ice,”are particularly interesting because they allow direct visualization of magnetic frustrationin a well-controlled model environment [1–3]. Recent experiments have used artificial spinice arrays to study the nature of the frustrated ground state [4–6], the effect of thermal fluc-tuations [7], the emergence of effective magnetic charges[8, 9], and disorder [10]. However,a deeper understanding of the relationship between the experimental arrays and theoreticalmodels requires a more precise quantification of the relative strengths of disorder and inter-actions in the experimental systems. The recent development of ferromagnetic island arrayswith perpendicular anisotropy [11, 12] provides an important opportunity in this regard, inthat these arrays are amenable to polar magneto-optical Kerr effect (MOKE) studies. Kerrimaging can potentially resolve array dynamics at the individual-island level, imaged acrossan entire array, during field sweeps: individual Ising degrees of freedom can be tracked ex-haustively not only in space, but also in time. Furthermore, the pairwise interaction betweentwo perpendicular moments depends only on the separation between them, unlike the morecomplex anisotropic interactions in systems with in-plane moments.A variety of methods have been used to investigate the static and dynamic magneticbehavior of both in-plane and perpendicular anisotropy arrays. For example, magneticforce microscopy (MFM) imaging has been used to study how individual islands behavewithin small bit-patterned media arrays[13] ( ∼
100 islands), using the remanent states andcoarse field bins. Thermal fluctuations of in-plane islands have been imaged using X-raymagnetic circular dichroism photoemission electron microscopy (XMCD-PEEM) to resolvethe individual islands[7]. However, only magneto-optical methods can spatially map theevolution of an array’s magnetization continuously in an external magnetic field at timescaleswhich allow for a quasi-dynamic exploration of a system’s microstates[14]. In particular,exhaustive statistical analysis of the switching field distributions of magnetic nano-arrayscan precisely quantify the role of static disorder in island reversal dynamics – this informationis critical to understanding the relative roles of island disorder and island-island interactionin the dynamics of artificial spin ice.We use high-resolution polar MOKE to isolate and detect the magnetic state of individualislands within arrays in a continuously varying external field. We thereby directly measure2he distribution of switching fields in situ for arrays of several thousand islands. The widthof the switching field distribution (here called σ , with the dimension of magnetic field) shouldhave contributions from both quenched disorder and dipolar interactions between islands.We thus try the simple model σ ( L ) = αKB ( L ) + σ d . (1)for the width σ ( L ) of the switching field distribution of an array of inter-island spacing L . Here, B ( r ) = πµ (cid:126)r ( (cid:126)r · (cid:126)m ) − (cid:126)m | (cid:126)r | is the magnetic field strength of a single point dipole ata distance r and σ d represents quenched disorder. K is an effective coordination numberimplicitly defined by the condition that KB ( L ) is the field the entire rest of a polarizedarray would exert on a given island. K depends only on the geometry, and therefore givessigma a geometry dependence suppressed in the notation.The effective coordination numbers for hexagonal, kagome, square and triangular latticesare 4.53, 5.52. 5.91, and 7.58 respectively; these values exceed the nearest-neighbor latticecoordination numbers due to the contribution from further neighbors. Here, we approximatethe field experienced by a given island by the value at the island center, assuming pure dipolarfields from nearby islands. Finally, the term α is a correction factor: if the fitting form ofEquation 1 is physically well-grounded, then α will be a simple constant of proportionalityof value close to 1. Several questions present themselves upon consideration of this scalingform. A priori, it is not clear whether the disorder contribution can be entirely identifiedwith effective individual-island properties (e.g. variations in shape and edge roughness) orwhether random local variations in lattice geometry may also enter in an essential way.Also, it is not clear a priori whether this effective coordination number K fully capturesthe effect of lattice geometry on the switching field distribution: i.e. is α actually constantacross different lattice geometries? Since αKB ( r ) ∝ r − and α is the only component thatcannot be calculated from other physical properites, functional fits to Eqn. 1 use the form ar − + σ d , and α is then extracted from the parameter a .Electron beam (e-beam) lithography was used to define 450 nm diameter circular islandsin both non-frustrated (square and hexagonal) and frustrated (kagome and triangular) ge-ometries, using standard lift-off of a bilayer PMMA/PMGI resist stack. Magnetostaticdipolar interaction strength was tuned by varying the array inter-island spacing. Pt/Co mul-tilayer stacks in the sequence Ti(2 nm)/Pt(10 nm)/[Co(0 . were deposited3sing DC sputtering; such multilayers have strong perpendicular anisotropy and nearlysquare hysteresis loops[15]. Bulk properties were measured using superconducting quantuminterference device magnetometry (SQUID). Hysteresis loops up to ± M s = 375 × A/m and anisotropy constant K = 94 × J/m . Scanning electronmicroscopy shows that the individual nanomagnets in the array have an edge roughness of ∼ ∼ d = λ n sin θ , where n sin θ is the numerical aperture (N.A.) of the objectivelens and λ is the wavelength of light. We use a 100x oil objective lens (1.3 N.A.) to at-tain diffraction-limited spatial resolution (150 nm – 300 nm) using white light filtered to thevisible spectrum (400 nm – 700 nm), which lets us clearly resolve the 450 nm diameter nano-magnets in each array. Island arrays were fabricated to be approximately 35 µm × µm sothat the entire array fits within the 35 µm × µm field of view of the Kerr imaging setup.Lateral drift of the array is inevitable for field sweeps lasting several tens of minutes. Thecentroid for the collection of pixels associated with each island was isolated and trackedthroughout a saturating magnetic field sweep. Field sweeps range from −
800 to 800 G. Inthe switching region from 150 to 500 G, we use 2 G steps, and outside this region, we use40 G steps.By the Kerr effect, island reversal manifests as a fractional change in island intensity,linear in magnetization. The representative image in Fig . . . . σ , a global property of the array. Figure3 shows values of σ for each lattice examined and the corresponding fits to Equation 1. Fitswere carried out using the Levenberg-Marquardt algorithm, and have reduced χ values of7.53, 2.54, 2.28, and 0.71 with 5 degrees of freedom for hexagonal, kagome, square, andtriangular arrays respectively.The initial curve fitting finds a value of σ d for each lattice type; however, since σ d arisesfrom physical properties of an island and all islands are fabricated simultaneously, it isreasonable to assume σ d is a constant across all lattices on the same chip. The calculatedvalues of σ d from the curve fits are consistent with this assumption. To treat σ d as a globalproperty of all arrays, we average the values of σ d obtained from fits to each lattice type andthen recalculate each fit with σ d fixed to this average value. The results are shown in Fig . H ( M, ∆ M ) = I − (cid:18) − M (cid:19) − I − (cid:32) − ( M + ∆ M )2 (cid:33) . (2)5ere, I − is given by: I − (cid:18) − M (cid:19) = −√ σ I erf − ( M )1 + γM − w π M )1 + βM . (3)The fit parameters σ I and w describe the intrinsic switching distribution of the array,with γ and β allowing for distribution asymmetry. Both Gaussian and Lorentzian formsare allowed in the model, to capture contributions that originate from both the compositionof local variations (Gaussian terms originating in the central limit theorem) and possiblelinewidth broadening effects (Lorentzian terms). In our fits, the Lorentzian term w is severalorders of magnitude smaller than the Gaussian contribution σ I , which underlines the originof the switching field distribution in the composition of multiple local variations (i . e . disorder)in individual island structure.The values of σ I from these fits agree well with the values of σ d obtained from fitting toEqn .
1. These values are consistent within a margin of error across all lattice types and inter-island spacings, which supports the hypothesis that the disorder contribution arises fromlocal variations in individual island properties and does not contain significant contributionsdue to variations arising from lattice geometry.Equations 1 and 2 give consistent values for the width due to physical island properties,but it remains to be verified that Eqn . B ( r ): M s = 375 × A/m from SQUIDmeasurements, V = πr h and for our islands r = 225 nm and h = 10 . K for eachlattice is listed previously. Using these parameters and the value of a from fitting σ , we cancalculate the value of the proportionality constant α . Hex Kag Squ Tri σ . ± .
96 23 . ± .
92 25 . ± .
20 31 . ± . σ I . ± .
44 13 . ± .
32 15 . ± .
97 15 . ± . α F . ± .
21 0 . ± .
17 0 . ± .
18 0 . ± . σ and α as a function of lattice geometry for the second fabricated samplewith 600 nm inter-island spacing. σ and σ I represent the distribution widths for the full switchingfield distribution and the calculated intrinsic distribution, and σ I is interchangable with σ d . α F arevalues of α with fixed values of the parameter σ d = 15 . ± . α are shown in Figure 3. Values from the initial fit with variable σ d are denoted α V and values from the second fit treating σ d as a fixed global parameterare denoted α F . They are constant within the margin of error. This suggests that all sig-nificant differences in the switching field distribution due to variations in lattice geometryare adequately accounted for by the effective coordination number K . Physically, this sup-ports the idea that, at least in this range of interaction strengths, the dominant cause ofdistribution broadening is the overall magnitude of the field experienced by an island fromits neighbors and is unrelated to precise details of the geometric arrangement. Averagingacross the different geometries we find (cid:104) α V (cid:105) = 0 . ± .
04 and (cid:104) α F (cid:105) = 0 . ± . . α is a general quantity, we applied our analysis to adifferent set of fabricated arrays, considering now only the most strongly interacting arraysfrom that chip. These arrays have different parameters, specifically M s = 346 × A/mand r = 200 nm. The smallest inter-island spacing is L = 600 nm. Because these arrayswere fabricated at a different time with different parameters, we expect the value of σ d forthese lattices to differ from the previous set. However, since we verified with the previoussamples that σ I gives a reasonable approximation for σ d , we can find this parameter usingthe minor loops method described previously. Again, we find the average value of σ d fromall geometries and treat it as a global variable. The values of σ for these different arrays areshown in Table I, along with the value of α calculated using Eqn 1 with the new parametersand fixed inter-island spacing L . It is not surprising that the error in this measurement islarger, since we are including information from different lattice geometries at a fixed spacing,instead of fitting each lattice type across a range of spacings. To find the average value of α F , taking into account the different errors, a weighted average is used. The values areweighted by the variance, (cid:104) α F (cid:105) = (cid:80) i α F i σ − i / (cid:80) i σ − i . This value, (cid:104) α F (cid:105) = 0 . ± .
09, agreeswell with that calculated for the other chip.In summary, we have demonstrated that using diffraction-limited MOKE imaging com-bined with appropriate image processing techniques, we can reliably find the switching fieldsof individual islands within a large array of perpendicular nanomagnets. This informationallows us to directly measure the switching field distribution, which we can then analyticallyinterpret to isolate the contributions from dipolar interactions and disorder due to individ-7al island properties. By confirming the efficacy of our numerical analysis using the refinedhysteresis loops from individual island switching fields, we have verified a global analysismethod that allows for quick characterizion of the strength of disorder. Quantifying thisdisorder strength is important for understanding the quality of the samples. Interactionscan be designed in an idealized way during the fabrication process, but the actual disor-der present is an important factor in how the islands will physically behave and how closethe arrays can be brought to the ground state, and this disorder can only be determinedpost-fabrication. Moreover, accessing individual island information from a quasi-dynamicmeasurement with an in situ applied field opens the door to further studies of dynamicsand correlations that could lead to a much richer understanding of the behavior of systemsgoverned by dipolar interactions.This project was funded by the US Department of Energy, Office of Basic Energy Sciences,Materials Sciences and Engineering Division under Grant No. de-sc0010778. ∗ [email protected][1] R. F. Wang, C. Nisoli, R. S. Freitas, J. Li, W. McConville, B. J. Cooley, M. S. Lund,N. Samarth, C. Leighton, V. H. Crespi, and P. Schiffer, Nature , 303 (2006).[2] Y. Qi, T. Brintlinger, and J. Cumings, Phys. Rev. B , 1 (2008).[3] C. Nisoli, R. Moessner, and P. Schiffer, Rev. Mod. Phys. , 1473 (2013).[4] X. Ke, J. Li, C. Nisoli, P. E. Lammert, W. McConville, R. F. Wang, V. H. Crespi, andP. Schiffer, Phys. Rev. Lett. , 1 (2008).[5] J. P. Morgan, A. Stein, S. Langridge, and C. H. Marrows, New J. Phys. , 105002 (2011).[6] J. P. Morgan, A. Bellew, A. Stein, S. Langridge, and C. H. Marrows, Frontiers in CondensedMatter Physics , 28 (2013).[7] V. Kapaklis, U. B. Arnalds, A. Farhan, R. V. Chopdekar, A. Balan, A. Scholl, L. J. Heyder-man, and B. Hj¨orvarsson, Nature Nano. , 514 (2014).[8] S. Ladak, D. E. Read, W. R. Branford, and L. F. Cohen, New J. Phys. , 359 (2011).[9] E. Mengotti, L. J. Heyderman, A. F. Rodr´ıguez, F. Nolting, R. V. H¨ugli, and H.-B. Braun,Nature Phys. , 68 (2011).[10] Z. Budrikis, in Solid State Physics, Vol. 65 , edited by R. E.Camley and R. L. Stamps (2014) p. 109–236.[11] E. Mengotti, L. J. Heyderman, A. Bisig, A. F. Rodr´ıguez, L. Le Guyader, F. Nolting, andH. B. Braun, J. Appl. Phys. , 113113 (2009).[12] S. Zhang, J. Li, I. Gilbert, J. Bartell, M. J. Erickson, Y. Pan, P. E. Lammert, C. Nisoli, K. K.Kohli, R. Misra, V. H. Crespi, N. Samarth, C. Leighton, and P. Schiffer, Phys. Rev. Lett. , 087201 (2012)..[13] W. M. Li, Y. J. Chen, T. L. Huang, J. M. Xue, and J. Ding, J. Appl. Phys. , 1 (2011).[14] M. Cormier, J. Fer, a. Mougin, J. P. Crom`ıres, and V. Klein, Rev. Sci. Instrum. , 1 (2008).[15] C.-J. Lin, G. Gorman, C. Lee, R. Farrow, E. Marinero, H. Do, H. Notarys, and C. Chien, J.Magn. Magn. Mater. , 194 (1991).[16] Th. Gerrits, H. A. M. van den Berg, J. Hohlfeld, L. B¨ar, and Th. Rasing, Nature , 509(2002).[17] A. Berger, Y. Xu, B. Lengsfield, Y. Ikeda, and E. E. Fullerton, IEEE Trans. Mag. , 3178(2005). IG. 1. Left: Scanning electron microscopy images of arrays with different geometries. Partialarrays are shown. Full arrays measure 35 µm × µm . All images shown are of arrays with ainter-island spacing of 500 nm. Right: MOKE contrast in a 500 nm inter-island spacing squarearray, near the coercive field during a hysteresis loop. IG. 2. (a) Raw hysteresis loop calculated across an entire array (red) compared to the refinedhysteresis loop calculated by combining contributions from individually resolved island switchingfields (blue). The inset is a MOKE image of the entire array. (b) Hysteresis loop of an individualisland and its pixelated image. (c) Histogram of switching fields averaged over several runs, withGaussian fit. The histogram is centered at the coercive field H c . All data in this figure were takenon a 500 nm square array. IG. 3. (a) Inter-island spacing dependent values of σ (closed circles) along with the “intrinsic”values of sigma from the ∆ H method (open circles). The value of σ d from fitting to Eqn . . σ d held to its global average are shown as black dashed lines. The fitted values of α for both variable ( α V ) and fixed σ d ( α F ) are shown as numerical values.) are shown as numerical values.