Vogel-Fulcher-Tammann Freezing of a Thermally Fluctuating Artificial Spin Ice Probed by X-ray Photon Correlation Spectroscopy
S. A. Morley, D. Alba Venero, J. M. Porro, S. T. Riley, A. Stein, P. Steadman, R. L. Stamps, S. Langridge, C. H. Marrows
aa r X i v : . [ c ond - m a t . d i s - nn ] A p r Vogel-Fulcher-Tammann Freezing of a Thermally Fluctuating Artificial Spin IceProbed by X-ray Photon Correlation Spectroscopy
S. A. Morley, ∗ D. Alba Venero, J. M. Porro, S. T. Riley, A. Stein, P. Steadman, R. L. Stamps, S. Langridge, and C. H. Marrows † School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom ISIS, STFC Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, United Kingdom School of Electronic and Electrical Engineering,University of Leeds, Leeds LS2 9JT, United Kingdom Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973, USA Diamond Light Source, Chilton, Didcot OX11 0DE, United Kingdom SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom (Dated: April 11, 2016)We report on the crossover from the thermal to athermal regime of an artificial spin ice formed froma square array of magnetic islands whose lateral size, 30 nm ×
70 nm, is small enough that they aresuperparamagnetic at room temperature. We used resonant magnetic soft x-ray photon correlationspectroscopy (XPCS) as a method to observe the time-time correlations of the fluctuating magneticconfigurations of spin ice during cooling, which are found to slow abruptly as a freezing temperature T = 178 ± T A =40 ±
10 K, is much less than that expected from a Stoner-Wohlfarth coherent rotation model. Zero-field-cooled/field-cooled magnetometry reveals a freeing up of fluctuations of states within islandsabove this temperature, caused by variation in the local anisotropy axes at the oxidised edges. ThisVogel-Fulcher-Tammann behavior implies that the system enters a glassy state on freezing, whichis unexpected for a system with a well-defined ground state.
PACS numbers: 75.40.Gb, 75.50.Tt, 75.75.Jn, 05.40.-a
In the past decade a new species of magnetic metama-terials has emerged: the artificial spin ices (ASI) [1, 2].They consist of a 2-dimensional array of nanoscale mag-netic islands arranged so that the magnetostatic interac-tions between the islands are geometrically frustrated [3].The size and shape of the individual islands are designedwith the intention that their shape anisotropy means theyact as single-domain Ising-like macrospins, mimicking theatomic spins of their naturally-occurring 3-dimensionalanalogs [4–6], but confined to a plane. They are thus re-alizations of the square ice vertex models solved by Wu [7]and Lieb [8], in which the exact microstate of the statis-tical mechanical system can be inspected using advancedmicroscopy methods. Until very recently, all ASIs studiedhave been athermal–albeit showing an effective thermo-dynamics [9–12]–since the shape anisotropy energy bar-rier E A that must be surmounted to flip the magneticmoment of any one island is orders of magnitude largerthan the thermal energy k B T that can be reached exper-imentally. Whilst convenient for imaging studies, theseathermal, arrested systems lack ergodicity and so fail toexplore phase space in the manner of a true statisticalmechanical system to find thermally equilibrated config-urations.However, there are recent reports of thermalized ASIs,made either by heating the sample close to the Curiepoint of the material from which the islands are fabri-cated in order to drive dynamics [13], whereupon the FIG. 1. Scanning electron micrograph of an artificial spin icewith a lattice spacing of 240 nm, with Permalloy islands oflateral size 30 nm ×
70 nm and 8 nm thickness. arrested state may be imaged upon cooling [14, 15], ora one-shot anneal process that occurs during fabricationhas been used in the same way [16]. Within the last yearor two, studies have been carried out that have dynam-ically imaged real-time thermal fluctuations in artificialspin ice in the square [17], kagom´e [18, 19], and tetris icegeometries [20]. Nevertheless the nature of the crossoverfrom a thermally fluctuating system to the arrested, lessergodic state has so far received little attention. Here wereport on measurements of an ASI with islands of lateralsize 30 nm ×
70 nm [21], shown in Fig. 1. By studyingthe time-dependence of their soft x-ray speckle scatteringpatterns, the ASI is shown to be thermal at room tem-perature and to freeze into a fully arrested state below ∼
178 K. The fluctuation rate follows a Vogel-Fulcher-Tammann law on cooling, implying that the frozen stateis glassy in nature, which is unexpected for a system witha well-defined ground state.The ASI that we studied was fabricated using electronbeam lithography with a lattice constant of 240 nm, asshown in Fig. 1. A lift-off process was used, with 8 nmthick Permalloy (Ni Fe ) evaporated through the re-sist mask to form the islands, followed by a 2 nm Alcap. The substrate was a Si wafer. The ASI array hada 2 mm × L (707 eV) reso-nance. The scattered intensity from the ASI was recordedin the reflection geometry (illustrated in Fig. 2(a)) usinga charge coupled device (CCD) camera, mounted at afixed scattering angle 2 θ = 9 . ◦ , 80 cm from the sample,which was kept at a temperature of 223 K in order to re-duce dark noise. Each image arose from a 40 ms exposureand the images were separated by the 4 s readout time ofthe camera. The transverse coherence length of the beamwas calculated to be 14.9 µ m, and so a 10 µ m diameterpinhole was mounted 23 cm in front of the sample. Airyrings were observed when the direct beam was imagedthrough the pinhole, shown in Fig. 2(b), confirming thecoherence of the beam. The samples were mounted formeasurement on a temperature-controlled stage in theabsence of any applied magnetic field.Since our ASI array has a square unit cell, it produces asquare pattern of diffraction spots. In Fig. 2, the centre ofthe diffracted speckle pattern corresponds to Q x = 2 π/a ,where a is the ASI lattice constant. The calculated lat-tice spacing for the ASI measured from the position onthe array detector and the geometry was 239 ± ± k i ki (a) (b) FIG. 2. (Color online) Coherent soft x-ray scattering mea-surements. (a) Schematic of the experimental XPCS setup,showing the incoming x-ray beam and scattered beams fromthe ASI array, which is represented by an atomic force mi-croscopy image. The three main diffraction spots in the rowabove the specular reflection are shown here. Since the diffrac-tion spots arise from a small, disordered region of the sam-ple that has been coherently illuminated, they contain specklecontrast. (b) Fraunhofer diffraction obtained using the 10 µ mpin hole and straight-through beam on to the CCD detector. direct measurement of magnetic dynamics. Soft x-rays atthe M resonance of Ho revealed antiferromagnetic do-main fluctuations in a thin film of that metal [27], whilstthe jamming of spiral magnetic domains in a Y-Dy-Ytrilayer was revealed in the stretched exponential corre-lations studied using soft X-rays at the Dy M resonance[28]. There have also been speckle scattering studies atthe Co L resonance using small-angle X-ray scatteringgeometry to study the effects of disorder on the domainpattern of multilayer perpendicular Co/Pt films in re-sponse to a field [29].As expected, measurements taken at an energy of700 eV, below the Fe L resonance, showed no changebeyond random noise fluctuations (see SupplementaryMovie 1), since the ASI physical structure is static. Ontuning to the L resonance at 709 eV, magnetic sensi-tivity is achieved and the speckle reconfigures as timepasses, since the magnetic state of the sample is reconfig-uring under thermal activity (see Supplementary Movie2). The XPCS measurements were carried out at dif-ferent temperatures to drive the thermal fluctuations atdifferent rates. In order to quantify the time-dependentbehaviour, we calculated the intensity-intensity temporalautocorrelation function [25], g ( Q , τ ) = h I ( Q , t ′ ) I ( Q , t ′ + τ ) i t ′ h I ( Q , t ′ ) i t ′ = 1 + A | F ( Q , τ ) | , (1)where I ( Q , t ′ ) is the intensity at wave vector Q at a time t ′ , τ is the time delay, and h ... i t ′ indicates a time av-
180 K190 K195 K200 K205 K210 K220 K250 K (a) time delay, t (s) g ( t )
170 190 210 230 250 270
DataVFTNB (b)
Temperature (K) R e l a x a t i on t i m e , t r ( s ) FIG. 3. (Color online) XPCS results. (a) Normalized g ( τ )functions at various temperatures. The lines are fits to Eq.2. (b) Relaxation times t r as a function of temperature, fittedby a Vogel-Fulcher-Tammann (VFT) law. Also plotted is thesuperparamagnetic N´eel-Brown (NB) law, which fits poorly. erage. F ( Q , t ′ ) is the so-called intermediate scatteringfunction, and A is a measure of the degree of specklecontrast. The g function was calculated for each pixelwithin the speckle pattern of the Bragg peak, and aver-aged over all such pixels to give the autocorrelation g ( τ )at each temperature, shown normalized to an initial valuein Fig. 3(a).These g ( τ ) curves were fitted with a heterodynemodel, since the experimental setup means that the fluc-tuating magnetic signal is mixed with a static signal thatcomes from the structural scattering of the array [30].This forms the equivalent of a reference beam in the mea-surement, leading to the following modified form of theKohlrausch-Williams-Watts [31, 32] intermediate scatter-ing function [33]: g ( τ ) = 1 + A cos( ωτ ) exp (cid:0) − ( τ /t r ) β (cid:1) (2)where t r is the characteristic relaxation time, β is astretching exponent, and ω is an oscillation frequencyassociated with the heterodyne mixing. The value of ω was found to be in the range ≈ . − , corre-sponding to a time period, 2 π / ω = 3,140 - 15,700 s ( ≈ A , was found tobe in the range 0.02-0.05, which agrees with that foundfrom the contrast of the Airy pattern [34]. The fittedvalue of β in all our measurements was 1 . ± .
1, indi- cating equilibrated behavior [26]. The function is ratherflat at 180 K, and therefore the relaxation time must bemuch longer than the time of measurement ( ≈ t r lies inthe experimentally accessible range between these twoextremes, shown in Fig. 3b. The simplest thermal ac-tivation behavior in a collection of magnetic nanopar-ticles is the Arrhenius-type N´eel-Brown (NB) law ex-pected for a non-interacting superparamagnetic system, t r = τ exp( T A /T ), where T A = E A /k B is an activationtemperature and τ is an activation time. As can be seenin the figure, this cannot be fitted to the data at all well.On the other hand, a Vogel-Fulcher-Tammann (VFT) law[35–37] captures the low temperature detail much moreaccurately: the data were fitted with the expression t r = τ exp (cid:18) T A T − T (cid:19) , (3)where T is the freezing temperature. The fit yields T A =40 ±
10 K and T = 178 ± T , below which interactions in the system pre-vent any relaxation or fluctuation. Whilst the VFT lawhas been found to fit many different types of data setsvery well, such as spin glasses, super-cooled organic liq-uids, metallic liquids, and glassy (bio)polymer systems[38, 39], it is nevertheless an empirical law not basedon any underlying microscopic picture. Models that at-tempt to provide a VFT-like temperature dependenceinclude–among others–ones based on a time-dependentpercolation process [39], or on the energy distribution ofthe depth of coupled traps [40]. In the spin glass theoryof Shtrikman and Wohlfarth (S-W) [41], the random ex-change interactions between the positionally disorderedspins are represented by a mean magnetic interactionfield H i . The magnetostatic interactions in our ASI arereal magnetic fields, making the S-W theory a naturalone to adapt to our system. In it, the freezing temper-ature T is determined by the characteristic interactionenergy E i . Using the results from the VFT fit, we find E i = p k T T A ≈ . × − J.We can estimate the scale of magnetostatic interactionsin our ASI using a point dipole model to determine theS-W interaction energy: E i = µ H i m island = µ m πa , (4)where m island = M s V is the magnetic moment of an is-land of volume V and the lattice constant a = 240 nm. (b) Temperature (K) H c ( O e )
50 mT35 mT25 mT5 mT (a) (iv)(iii)(ii)(i) Temperature (K) m / m m a x ZF C FIG. 4. (Color online) (a) ZFC and FC magnetization mea-surements under different applied magnetic probe fields of (i)50 mT, (ii) 35 mT, (iii) 25 mT and (iv) 5 mT. The magneticmoment, m , has been normalised to the peak in the ZFCcurve for each field ( m maxZFC ) for comparison. The averageblocking temperature, T B , is also defined by this point and isindicated by the black arrows ( T B >
400 K at 5 mT). Thegrey arrow indicates the ∼
40 K feature in the ZFC data.The field was applied along [01] direction of the ASI lattice.(b) The temperature dependence of the coercivity measuredalong [01]. The line is a fit to Sharrock’s equation, the depar-ture from which is clear below 40 K.
Assuming the nominal island size and the bulk magne-tization of Permalloy ( M s = 860 kA/m) yields E i =3 . × − J, which is of the same order of magnitudeas the experimental value but overestimated by abouta factor of three. Part of the overestimate may be dueto the extrinsic effect of the reduction in effective M S [19] or V [42] due to oxidation. An intrinsic contributionto the overestimate is the fact that the total interactionenergy arises from the sum of several frustrated interac-tions from all the island’s neighbors, leading to partialcancellation of the energy value given above. Thus, themagnetostatic interactions in the system are clearly ofthe right scale to break the Arrhenius-like behaviour andgive rise to a VFT-like freezing of the fluctuations as T is approached.Now we turn to a discussion of the activation tem-perature T A . We can estimate the energy barrier ∆ E for Stoner-Wohlfarth coherent rotation of a single is-land, again assuming the bulk magnetization of Permal-loy, using ∆ E = KV = ln( f t m ) k B T A , with the shapeanisotropy constant K = µ ∆ D M , and the differencein demagnetizing factor along the two relevant directions∆ D ≈ . t m , is taken as two hours, a typicalvalue for 2 π/ω , and we assume f = 10 GHz. This yields T A ≈ T A of40 K, ruling out this simple picture.In order to seek other evidence of thermally-activated processes that might cast light on this large discrepancy,standard zero-field cooled (ZFC) and field-cooled (FC)protocols were carried out on the samples using SQUID-vibrating sample magnetometry. First, the sample washeated to 395 K and a large negative saturating fieldapplied and removed. The sample was then cooled to5 K without any applied field. Next, a probe field wasapplied and the magnetic moment was measured duringheating, which is the lower curve in each measurementin Fig. 4(a). Last, the sample was measured again dur-ing cooling back down with the probe field still applied.The peak in the ZFC, close to where the curves bifur-cate, is defined as the blocking temperature, T B , andmarked with a black arrow. For example, the data withprobe field µ H = 50 mT has T B (50 mT) = 200 ±
10 K.The blocking temperature systematically rises for smallerprobe fields, and exceeds our experiemntal limit of 400 Kat 5 mT. This behavior can be extrapolated to the caseof zero applied probe field, using T B ∝ H / [44], whichgives T B (0 mT) = 460 ±
20 K. This is about a factor offour lower than that expected from the sample parame-ters and nominal volume. Whilst reduced magnetizationor volume can explain some of this discrepancy, it seemsclear once again that a pure coherent rotation mechanismis unlikely to be strictly followed.Nevertheless, there is another feature in the magne-tometry data: there is an initial increase in moment upto 40 K, marked with a grey double-headed arrow, vis-ible in all the ZFC curves. This signifies a much lowerenergy scale for some magnetic relaxation process, onethat is strikingly similar to the T A found from the VFTfit. As previously shown by Ozatay et al. [45], the prop-erties of Py can change below 40 K when native oxidationhas occurred. They showed in elements of a similar sizethat such oxidation is likely around the unprotected edgesand can have significant effects on the reversal properties,where they observed an upturn in coercivity below thistemperature.We performed the same measurement and have plottedthe coercivity of our sample as a function of temperaturein Fig. 4(b). The same characteristic features as thoseseen in Ref. 45 are present. The line in the figure is a fitto Sharrock’s equation [46], which describes the temper-ature dependence expected for the dynamic coercivity ofmagnetic particles. Ozatay et al. showed that for con-formally capped islands this feature disappears and thecoercivity can be fit to Sharrock’s equation for all tem-peratures, whereas here and for their Py particles witha native oxide around the edges there is a sharp depar-ture from this at around 40 K. They attribute this to themodification of the local anisotropy which arises from theexchange coupling between the antiferromagnetic oxideedges and the ferromagnetic Py. This results in differentlocal anisotropy axes which create pinning points for themagnetization, and allow non-uniform reversal modes tobecome energetically accessible. This allows the systemto sample a distribution of metastable states only once atemperature of T A = 40 K is exceeded.To summarize, on cooling an artificial spin ice we haveobserved a dramatic lengthening of the relaxation time asmeasured by magnetic XPCS. The system slows abruptlyas it crosses over from thermal equilibration to an ather-mal, frozen state. This crossover can be described by aVogel-Fulcher-Tammann law, which is typically used todescribe glassy systems. The VFT freezing temperature T ≈
178 K can be accounted for by the magnetostaticinteraction strength through Shtrikman-Wohlfarth the-ory. The activation temperature T A ≈
40 K arising froma fit of this law implies a much lower energy barrier toreversal than is expected from a single-domain coherentrotation picture. The value of 40 K appears to be deter-mined by the onset of fluctuating magnetic states withinthe islands coupled to magnetic oxides at the edges of theislands.The glass-like freezing is remarkable since the squareice system possesses a well-defined ground state, un-like a conventional spin glass. This raises the ques-tion of the true nature of the glassy state that our ar-tificial spin ice freezes into. Related VFT-like freezingbehaviour has recently been observed in the pyrochlorespin ice Dy Ti O [47], prompting speculation aboutthis representing many-body localization of spins in atranslationally-invariant quantum system [48–50]. 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