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Disorder-induced magnetic memory: Experiments and theories M. S. Pierce1,2, C. R. Buechler1, L. B. Sorensen1, S. D. Kevan3, E. A. Jagla4, J. M. Deutsch5,T. Mai5, O. Narayan5, J. E. Davies6, Kai Liu6, G. T. Zimanyi6, H. G. Katzgraber7, O. Hellwig8, E. E. Fullerton8, P. Fischer9, and J. B. Kortright9. 1Department of Physics, University of Washington, Seattle, Washington 98195, USA 7 2Materials Science Division, Argonne National Laboratory, Argonne Illinois 60439, USA 0 3Department of Physics, University of Oregon, Eugene, Oregon 97403, USA 0 4Centro Atómico Bariloche, Comisión Nacional de Energ´ıa Atómica, (8400) Bariloche, Argentina 2 5Department of Physics, University of California, Santa Cruz, California 95064, USA 6Department of Physics, University of California, Davis, California 95616, USA n 7Theoretische Physik, ETH Zu¨rich, CH-8093 Zu¨rich, Switzerland a J 8Hitachi Global Storage Technologies, San Jose, California 95120, USA and 9Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 6 (Dated: February 6, 2008) 2 Beautiful theories of magnetic hysteresis based on random microscopic disorder have been de- ] veloped over the past ten years. Our goal was to directly compare these theories with precise i c experiments. To do so, we first developed and then applied coherent x-ray speckle metrology to a s seriesofthinmultilayerperpendicularmagneticmaterials. Todirectlyobservetheeffectsofdisorder, - l we deliberately introduced increasing degrees of disorder into our films. We used coherent x-rays, r t produced at the Advanced Light Source at Lawrence Berkeley National Laboratory, to generate m highlyspeckled magnetic scattering patterns. Theapparently “random”arrangement of thespeck- . les is dueto theexact configuration of the magnetic domains in thesample. In effect, each speckle at pattern acts as a unique fingerprint for the magnetic domain configuration. Small changes in the m domain structure change the speckles, and comparison of the different speckle patterns provides a quantitativedeterminationofhowmuchthedomainstructurehaschanged. Ourexperimentsquickly - d answeredonelongstandingquestion: Howisthemagneticdomainconfigurationatonepointonthe n majorhysteresislooprelatedtotheconfigurationsatthesamepointontheloop duringsubsequent o cycles? This is called microscopic return-point memory (RPM). We found the RPM is partial and c imperfectinthedisorderedsamples,andcompletelyabsentwhenthedisorderwasbelowathreshold [ level. We also introduced and answered a second important new question: How are the magnetic domains at one point on the major loop related to the domains at the complementary point, the 2 inversionsymmetricpointontheloop,duringthesameandduringsubsequentcycles? Thisiscalled v microscopiccomplementary-pointmemory(CPM).WefoundtheCPMisalsopartialandimperfect 2 in the disordered samples and completely absent when the disorder was not present. In addition, 4 we found that the RPM is always a little larger than the CPM. We also studied the correlations 5 between the domains within a single ascending or descending loop. This is called microscopic half- 1 loop memory (HLM) and enabled us to measure the degree of change in the domain structure due 1 6 to changes in the applied field. No existing theory was capable of reproducing our experimental 0 results. Sowedevelopednewtheoreticalmodelsthatdofitourexperiments. Ourexperimentaland / theoretical results set new benchmarksfor future work. t a PACSnumbers: 07.85.+n,61.10.-i,78.70.Dm,78.70.Cr m - d I. INTRODUCTION developed to explain them. n o Magnetic hysteresis is fundamental to all magnetic c storagetechnologiesandconsequentlyisacornerstoneof Whatcausesmagnetichysteresisandhowisitinduced : v and influenced by coexisting microscopic disorder? This the present informationage. The magnetic recordingin- i dustry deliberately introduces carefully controlled disor- X is the questionthatwe addressand providenew answers der intoits materialsto obtainthe desiredhystereticbe- about in this paper. We are able to provide new in- r havior and magnetic properties. Over the past 40 years, a formation about this venerable old question because we such magnetic hardening has developed into a high art have developed a new way to directly probe the effect form. However,despite decadesofintensestudy andsig- of disorder on the spatial structure of the microscopic nificantrecentadvances,westilldonothaveacompletely magnetic domain configuration as a function of the ap- satisfactory microscopic understanding of magnetic hys- plied magnetic field history. When we finished our ex- teresis. perimental study, we discovered that our results could not be explained by any existing microscopic theories of The exponential growth of computing power that fu- magnetic hysteresis. So we developed severalviable the- eled the information age has been driven by two techno- oretical models. In this paper, we present our detailed logical revolutions: 1) The integrated circuit revolution experimental results and the theoretical models that we anditsexponentialgrowthdescribedbyMoore’sLaw. 2) 2 The magnetic disk drive revolution and its exponential foreaddingtheadditionalphysics,andcomplications,as- growthwhich for the past decade has surpassed Moore’s sociatedwithhighsweeprates. Asweexplainbelow,the Law. Both of these mature technologies are rapidly ap- disorderdependenceoftherate-independenthysteresisin proaching their fundamental physical limits. If the in- oursystemturnedouttoberemarkablyrichandinterest- credible growthrate of storagecapacity in magnetic me- ing. We do not discuss our results for rate-independent dia is to continue, new advances in our fundamental un- minorloopmemoryinthispaper,butwebrieflyreported derstanding of magnetic hysteresis are needed. them recently [1]. For the past 20 years, magnetic films with perpendic- The best modern microscopic disorder-based theories ular anisotropy have been extensively studied for their of magnetic hysteresis were built on the foundations of potential to extend the limits of storage capacity. Early Barkhausen noise measurements [2]. Even in the rate- in 2005, the first commercial disk drives using perpen- independentlimit, the magnetizationofadisorderedfer- dicular magnetic media became available. The system romagnetdoesnotchangesmoothlyastheappliedfieldis that we study here is a model for these new perpendicu- sweptup and down. Instead, there aremagnetic domain larmagneticmedia. Inthispaper,wepresentourresults avalanches that produce magnetization jumps. These on the effect of disorder on the correlations between the avalanches exhibit power-law size distributions indicat- domain configurations in these systems. ing that many different size regions change their magne- Tostudythedetailedevolutionofthemagneticdomain tization in jumps as the field is swept around the major configuration correlations in our samples, we developed hysteresis loop. a new x-ray scattering technique, coherent x-ray speckle A comprehensive, recent review of Barkhausen noise metrology (CXSM). We illuminate our samples with co- studies—including a translation of Barkhausen’s 1919 herent x-rays tuned to excite virtual 2p to 3d resonant paper—is given in Ref. [3]. For some materials in the transitionsincobalt. Theresultingresonantexcitationof rate-independentlimit,the Barkhausennoiseisindepen- the cobalt provides our magnetic signal. The coherence dent of the magnetic sweep rate; these avalanches occur of the x-rays produces a magnetic x-ray speckle pattern. at fixed values of the applied field, independent of the The positions and intensity of the speckles provides a sweep rate [4]. detailed fingerprint of the microscopic magnetic domain Barkhausen measurements provide exquisite informa- configuration. Changes in the magnetic domain config- tionaboutthetimestructureoftheavalanches,butthey uration produce changes in the speckle pattern. So by usuallydonotprovideanyspatialinformationaboutthe comparing these magnetic fingerprints versus the mag- location of the avalanches. Because we directly measure netic field history—by cross-correlating speckle patterns the nanometer scale spatial structure of the magnetic withdifferentmagneticfieldhistories—weobtainaquan- domain configuration changes, we obtain detailed infor- titative measureofthe appliedfield-history-inducedevo- mation about the configuration evolution that cannot lution of the magnetic domain configuration. be obtained directly from the best classical Barkhausen HerewereportourresultsobtainedbyapplyingCXSM noise studies or from their modern optical implemen- to investigate the effects of controlled disorder on the tations [5]. Because there has been extensive theoreti- magnetic domain evolution in a series of Co/Pt mul- cal work on Barkhausen noise, the corresponding field- tilayer samples with perpendicular anisotropy. We in- history-dependentmicroscopic morphologiesof the mag- troduced disorder into the samples by systematically in- neticdomainconfigurationshavebeenindirectlyinferred creasingthe interfacialroughnessof the Co/Ptmultilay- from the Barkhausen time-signals via detailed computer ers during the growthprocess. We found that this disor- simulations. Forexample,Sethna,Dahmenandtheirco- derinducesmemoryinthemicroscopicmagneticdomain workers have shown that the morphology for their ran- configurationsfromonecycleofthehysteresislooptothe dom field Ising model (RFIM) is fractal in space. They next, despite taking the samples through magnetic satu- providea comprehensivereviewoftheir workinRef. [2]. ration. Our lowest disorder samples have no detectable Taken together, the detailed fractal-in-time structure cycle-to-cyclememory;their domainpatternsareunique measured via the Barkhausen noise, and the extensive each time the sample is cycled around the major loop. computer simulations by Sethna, Dahmen, and others, As we increase the disorder, the cycle-to-cycle memory implythattheirmagneticdomainconfigurationsarefrac- develops and grows to a maximum value, but never be- tal in space. Therefore, why not simply measure the comes perfect or complete at room temperature. correlationsbetweenthemagneticdomainconfigurations In this paper, we only present our results for micro- directly? That is precisely what we do in this paper. scopic magnetic memory along the major loop in the There has been very little systematic, ensemble-level ex- slow field sweep limit. In this limit, the measured hys- perimentalworkonthe spatialevolutionofthe magnetic teresis loop is the same over many decades of sweep domainconfigurations[5,6],butthisinformationisread- rate. The hysteresis in this limit is often called rate- ily available from the existing simulations. However, up independent hysteresis, or quasi-static hysteresis. There until now almost all of the work has been done for pure are, of course, also interesting and important hysteresis RFIMs. Our experimental system and the new gener- effects that occur at high sweep rates. Our strategy was ation of perpendicular magnetic disk drive media have tostudythesimplerrate-independenthysteresiscasebe- long-rangedipoleinteractions. Thismeansthatnewthe- 3 ories that include the dipolar interactions[7] will be re- most all models have perfect memory at T = 0 and im- quired to understand these materials. perfect memory for T > 0. And it seems likely that the During our work, we unearthed three interesting as- imperfect memory that we observe could be caused by pects of our magnetic domain wall evolution. The first, temperatureeffects,butthishasnotyetbeenestablished. calledmajorloopreturn-pointmemory(RPM),describes On the other hand, no viable theoretical model for the the magnetization for each point on the major loop. If slightRPM-CPMsymmetry breakingexisted. So we de- this magnetization is precisely the same for each cycle veloped viable models. The key idea behind each of our around the major loop, then we have macroscopic ma- modelswastocombinephysicswithspin-reversalsymme- jor loop RPM. If, in addition, the microscopic magnetic try with physics without spin-reversal symmetry. Then domain configuration is also identical, then we have mi- the spin-reversal-symmetric physics produces symmetric croscopic major loop RPM. Our experiments show that memory RPM=CPM and the non-symmetric physics our samples have perfect macroscopic major loop RPM, produces symmetry-brokenmemory RPM=CPM. 6 butimperfectmicroscopicmajorloopRPMatroomtem- perature. Within the standard RXIM models—viz., RAIM, The second, called complementary-point memory RBIM, and RCIM, and RFIM where A denotes (CPM), describes the inversion symmetry of the major anisotropy, B denotes bond, C denotes coercivity, and F loop through the origin. If the magnetization at field H denotes field—the first three have spin-reversal symme- onthe descendingbranchis equalto the minus the mag- try,but the fourth(RFIM) does not. So one wayto pro- netization at field H on the ascending branch,then we − duceslightlysymmetry-brokenmemoryistocombinethe have perfect macroscopic major loop CPM. If, in addi- RFIM with one of the symmetric models. Surprisingly, tion, the magnetic domains are precisely reversed, then another way is to combine one of the symmetric mod- wehaveperfectmicroscopicmajorloopCPM.Ourexper- els with vector spin dynamics because vector dynamics iments show that our samples have perfect macroscopic breaks the spin-reversal symmetry. We report our work major loop CPM, but imperfect microscopic major loop on three viable models: Model 1 combines a pure RFIM CPM at room temperature. In addition, we find that withapureRCIM.Model2combinesapureRAIMwith our measured values for the microscopic RPM are con- vector spin dynamics. Model 3 combines a pure RFIM sistently a little larger than those for our microscopic with a pure spin-glass model CPM—thustheRPM-CPMsymmetryisslightlybroken. The third, called half-loop-memory(HLM), describes the degree of change in the magnetic domain configura- We explored Model 1 and Model 2 in the most de- tions along a single branch of the major hysteresis loop. tail. By tuning the model parameters, we were able to Ourexperimentsshowthatdisorderhasadirecteffecton semi-quantitatively match our experimentally observed howthedomainsevolve. Thegreaterthedisorderpresent disorder-dependence and magnetic-field-dependence of in the sample, then the greater the observed changes in (i)thedomainconfigurations,(ii)theshapeofthemajor the domain configurations as the applied field is slowly loops,(iii)thevaluesoftheRPMandCPM,and(iv)the adjusted to take the system along the major hysteresis slight RPM-CPM symmetry breaking. loop. Our measuredvalues for the HLM are consistently higher in the low disorder samples than those present in the disordered samples. Note that in order to properly describe our observed We were inspired to do our experimental study by the magnetic domain configurations, we had to include the beautiful work on the RFIM by Sethna, Dahmen, and long-range dipolar interactions. In contrast to the coworkers [2]. We were therefore very surprised to dis- Sethna-Dahmen RFIMs that predict spatially fractal coverthattheirmodelcouldnotdescribeourexperimen- magnetic domain configurations [2], our samples exhibit tal results. Their pure zero-temperature RFIM predicts labyrinthine domain configurations due to their long- perfect macroscopic and microscopic major loop RPM, range dipolar interactions. but it does not agree with our experiments because it predicts essentially no microscopic major loop CPM. It seemsreasonablethattheirT >0RFIMwillpredictper- The remainder of this paper is organized as follows. fect macroscopic RPM but imperfect microscopic RPM Section 2 describes the physics of return point memory like that observed in our experiments, but this has not and complementary point memory. Section 3 describes been tested. However their model cannot predict our our experiments, sample fabrication, structural charac- observed microscopic CPM and therefore it also cannot terization,magneticcharacterization,andcoherentx-ray predict the slightly brokenmicroscopic RPM-CPMsym- speckle metrology (CXSM) characterization. Section 4 metry that our experiments observe. describes our data analysis methodology. Section 5 de- So, what physics is required to produce imperfect mi- scribes the results of our data analysis. Section 6 de- croscopic RPM and CPM with the slightly broken sym- scribes the theoretical models that we developed to ac- metry? There are two aspects to this question—the im- countfor the observedbehaviorofoursystem. Section 7 perfection and the RPM-CPM symmetry breaking. Al- presents our conclusions. 4 II. MACROSCOPIC AND MICROSCOPIC Madelungformulatedhisrulesbasedonhiscarefulex- RETURN-POINT MEMORY AND perimental studies of different alloys of steel and pub- COMPLEMENTARY POINT MEMORY lished them in 1905 and 1912 [8]. Because Madelung formulatedhis rules before the existence of magnetic do- In his 1903 dissertation at Göttingen entitled “On the mains was known, he only considered the macroscopic magnetization produced by fast currents and the oper- magnetization. Nevertheless, his rules still predict the ation of Rutherford-Marconi magnetodetectors,” Erwin macroscopic magnetization of “any typical” sample ver- Madelung presented his rules for magnetic hysteresis as sus its applied field history. Madelung’s rules have truly illustrated in Figure 1: beenthefoundationforallmoderntheoriesofhysteresis. It is therefore surprising that Madelung’s rules are so 1. Major-loopreturn-point memory. rarely cited. Apparently this is because essentially all of the subsequent work has been focused on the Preisach The magnetizationof the sample atevery point on model. The obscurity of Madelung’s magnetic hystere- the major loop is completely determined only by sis work is particularly surprising because the Preisach the applied field, and all first-order reversal curves model has been well known to be unphysical for a very starting from the major loop and going to satu- long time due to heavy reliance on phenomenology. ration are uniquely determined by their starting Ofcourse,Madelung’srulesdonotapplytoeverymag- point. The curve 1 +S in Fig. 1 illustrates → netic system. For example, many systems exhibit acco- a first-order reversal curve (forc). modation, reptation and magnetic viscosity effects, and all systems exhibit dynamic hysteresis effects. However, 2. Minor-loop return-point memory. on the other hand, Madelung’s rules do apply to an in- The magnetizationof the sample atevery point on credible number of magnetic systems under a vast range the major loop is completely determined solely by of conditions. the value of the applied field, even when the point Now that we know that the microscopic magnetic do- on the major loop is reached starting from a point mains are intimately involved in the production of mag- inside the major loop. This holds for every order netic hysteresis, we immediately come to the first ques- reversal curve. The curve 2 1 illustrates this tion at the core of our investigation: How do the mag- → property for a second-order reversal curve (sorc). netic domains behave on the microscopic level. Do the domains remember—viz., return precisely to—their ini- 3. The memory deletion property, a.k.a. the wiping tial states, or does just the ensemble averageremember? out property. We show below that, at room temperature, some of the domainsinoursamplesreturntotheiroriginalconfigura- The magnetizationof the sample atevery point on tions andsome do not, but nevertheless the macroscopic areversalcurveispreciselythesameasthatforits magnetization—set by the ensemble average—does re- parent curve as soon the reversal curve returns to turn to its original value. its parent. In this way, all memory of the previ- ousfieldhistorybetweenthe initialdeparturefrom In other words, we find that our samples have per- the parent and the return to the parent has been fect macroscopic RPM, but they have imperfect micro- erased. This holds for every order reversal curve. scopicRPMatroomtemperature. Infact,ourmeasured The curve3 +S illustrates this for a third-order RPM values for each sample demonstrate a rich, com- reversalcurv→e (torc). plex behavior reflecting the fundamental physics of the magneticdomains. Wequantitativelymeasuredthefrac- 4. The congruency property. tion of the domains that remember and thereby demonstrated that the disorder has a profound impact on the All return curves that start from reversal at the microscopic RPM. As we tune the disorder, our sam- samevalueoftheappliedfieldhavethesameshape ples develop microscopic RPM that starts from zero in thereafter independent of the entire previous ap- the low-disorder limit and jumps to a saturated value plied field history. in the high-disorder limit, but never becomes perfect at room temperature. Consequently, our experimental 5. The similarity property for initial magnetization system is a finite-temperature realization of the “mi- curves. croscopic disorder-induced phase transition between no When any initial magnetization curve is reversed memoryandperfectmemory”predictedbySethan,Dah- at point a, the reversed return curve to saturation men, and coworkers[2]. will pass through the inversionsymmetric point to Themajorloopfor“anytypical”magneticsystemusu- a as it proceeds to saturation. As discussed be- ally has an additional symmetry—it is symmetric about low, we call the analogous property to the similar- inversion through the origin. This inversion symmetry ity property—for reversal curves that do not start immediatelyraisesthe secondquestionatthecoreofour from a point on the initial magnetization curve— investigation: How are the domains at the complemen- the complementary-point memory property. tary points of the major loop related? Do the magnetic 5 FIG. 1: The topology of Madelung’s rules. (a) If the magnetization curves from point 1 to saturation are uniquelyde- termined bytheapplied field at theirdeparturepoint 1from FIG. 2: The Geometry of complementary-point memory themajor loop, thenthesystem exhibitsmacroscopic major- (CPM). (a) If the magnetization at point 1′ is equal to that loopreturn-pointmemory. Ifafterreturningtopoint1onthe at point 1, then the system exhibits macroscopic major-loop ′ major hysteresis loop, the system continues along the major CPM. If the domain configuration at point 1 is highly cor- loop, thenthesystemexhibitsmacroscopic major-loop mem- related with that at point 1, then thesystem exhibits micro- ′ orydeletion(wipingout). (b)Ifthefirst-orderreversalcurve scopic major-loop CPM. (b) If the magnetization at point 2 from point 2 back to the major loop arrives at its original is equal to that at point 2, then the system exhibits first- departure point, then the system exhibits first-order macro- order macroscopic minor-loop CPM. If the domain configu- ′ scopic minor-loop return-point memory. (c) If the second- ration at point 2 is highly correlated with that at point 2, order reversal curve from point 3 back towards saturation then the system exhibits first-order microscopic minor-loop ′ passesthroughpoint2,thenthesystemexhibitssecond-order CPM. (c) If themagnetization at point 3 is equal tothat at macroscopicminor-loopreturn-pointmemory. Ifthereafterit point 3, then the system exhibits macroscopic second-order continues along the original curve from 1 to saturation, then minor-loop CPM. If the domain configuration at point 3′ is thesystemexhibitsmacroscopicminor-loopmemorydeletion highly correlated with that at point 3, then the system ex- (wiping out). hibitssecond-ordermicroscopic minor-loop CPM. Ingeneral, CPM can occur for any order of reversal. domainsatthe opposingpointsonthe majorloopevolve in a similar, but perhaps mirror correlated fashion? We III. EXPERIMENTAL ASPECTS callthiseffectmicroscopicmajorloopCPM.Thegeome- tryofcomplementary-pointmemoryisillustratedinFig. Tomeasurethefield-historyinducedchangesinthemi- 2. croscopic magnetic domain configurations,we developed Despiteanincredibleamountofeffortsince1905,ithas coherent x-ray speckle metrology (CXSM) [1, 11, 12]. provenimpossibletodevelopasimple,yetadequate,phe- OurCXSMexperimentswereperformedattheAdvanced nomenological model that can be used to treat all mag- LightSourceatLawrenceBerkeleyNationalLaboratory. netic materials. Westilldonothaveaphenomenological A schematic diagram of the experimental apparatus is model for modern magnetic technology. In addition, al- showninFigure3. Weusedlinearlypolarizedx-raysfrom though there has also been tremendous effort expended the third and higher harmonics of the beamline 9 undu- andprogressachieved,ithassimilarlyprovenimpossible lator. The raw undulator beam was first reflected from to developa generalpurpose micromagnetics model. We a nickel-coated-bremmstrahlung-safety mirror and then now know,based on recent theoreticalwork [2], that the passedthroughawater-cooledBewindowtodecreaseun- detailed magnetic hysteresis properties of real materials wantedlight. The partially coherentincidentbeam from cannot be treated using standard mean-field methods. the undulator was passed through a 35-micron-diameter This is because the hysteresis depends on the interac- pinhole to select a transversely coherent portion. The tions between each domain and a limited number of its sample was located 40 centimeters downstream of the neighbors, as well as between each domain and its local coherence-selection pinhole. This provided transversely disorder. Consequently, our approachhas been to deter- coherentilluminationofabouta40microndiameterarea mine to what extent the nanoscale domain-level physics of the sample. The transversely coherent x-ray beam of our experimental system obeys Madelung’s rules, and was incident perpendicular to the sample surface and thentoexplorewhetherwecanbetterunderstandtheob- was scattered in transmission by the sample. The res- servedbehaviorusingtraditional(overly)simplifiedIsing onant magnetic scattering was detected by a soft x-ray models. CCD cameralocated 1.1meters downstreamofthe sam- 6 FIG.3: (coloronline)Schematicdiagramoftheexperimental apparatus. Soft x-rays from the undulator passed through a pinhole and were perpendicularly incident on the thin film samples. The x-rays were scattered in transmisson and were detected by a soft x-ray CCD camera. Not shown in this diagramistheelectromagnetusedtoapplyuniformmagnetic FIG. 4: (color online) The measured rms roughness from fields perpendicularto the sample. AFM and x-ray reflectivity measurements plotted versus the Argon sputtering pressures. The interfacial roughness increases as the sample growth pressure increases. Below ple. Betweenthe sampleandthe CCDcameraweuseda 8.5mTorr, the roughness increases slowly as indicated by the small blocker to prevent the direct beam from damaging left (green) fit line. Above 8.5mTorrthe roughness increases the CCD. much more rapidly as indicated by the included by the right The photon energy was set to the cobaltL3 resonance (red) fit line. This behavior is very similar to that observed byRef. [13] in sputtered Nb/Simultilayers. at 778 eV. These photons resonantly excited virtual 2p to 3d transitions in the cobalt atoms and thereby provided our magnetic sensitivity. The intensity of the raw undulator beam was 2 1014 photons/sec, the intensity cal multilayer structure they were grown at different ar- of the coherent beam w×as 2 1012 photons/sec, and the gonsputteringpressurestotunethe disorderinthesam- × intensity of the scatteredbeamwas 2 107 photons/sec. ples. Duringgrowth,weadjustedthedepositiontimesto × Wetypicallymeasuredeachspecklepatternfor10to100 keepthe Co andPt layerthickness constantoverthe en- seconds, so the total number of photons in each CCD tire series. For low argon pressures, the sputtered metal image was 108 to 109. atoms arrive at the growth substrate with considerable The applied magnetic field was provided by an in- kinetic energy which locally heats and anneals the grow- vacuum water-cooled electromagnet allowing in situ ad- ingfilm. ThisleadstosmoothCo/Ptinterfacesproduced justment of the magnetic field during the experiment. at a low sputtering pressure. For higher argon sputter- The return path for the electromagnet consists of an ex- ing pressures, the sputtered atoms arrive at the growth ternal soft Fe yoke that feeds field to vanadium perman- substrate with minimal kinetic energy thereby resulting durpolepiecesthatareintegraltothe vacuumchamber. in rougher Co/Pt interfaces. The resulting roughness is The pressureinside the chamber during our experiments cumulative through the samples [13]. The magnetocrys- was typically 10−8 Torr. The in-vacuum electromagnet taline anisotropy at the Co/Pt interface forces the mag- provided magnetic fields up to 11 kOe. netization to align perpendicularly to the surface of the film. Our samples were grown at six different sputtering pressures: 3, 7, 8.5, 10, 12, and 20 mTorr. Due to A. Sample Fabrication and Structural the importantandinterestingmagneticproperties,these Characterization samples and others very similar in form and structure have been studied in different experiments[14, 15]. Ourthin-film samples weregrownbymagnetronsput- The rms roughness for the samples was measured in tering in the San Jose Hitachi Global Storage Technol- the Almaden Hitachi Global Storage Technology Labo- ogyLaboratoryon smooth,low-stress,160-nm-thicksili- ratory using two different methods. First, we measured connitridemembranes. Thesampleshad20-nm-thickPt the roughness by scanning the sample surface with an buffer layers,and2.3-nm-thickPtcaps to preventoxida- atomic force microscope (AFM) and calculating the rms tion. Between the buffer layer and the cap, the samples roughnessfromtheAFMimages. Sinceoursampleshave had 50 repeating units of a 0.4-nm-thick Co layer and a conformal roughness, the rms roughness of the surface 0.7-nm-thick Pt layer. While the six samples had identi- is a reasonable measure of the internal rms roughness. 7 FIG. 5: (color online) The measured MFM images, AFM images, and x-ray reflectivity curves for the six samples. The MFM images evolve from clear labyrinthine patterns for the low rms roughness samples to visually highly disordered patterns for the high rms roughness samples. However, the persistence of the annular shape of the speckle patterns—even for the highest roughness samples—reveals an underlying labyrinthineorder. The MFM images show 3 micron by 3 micron areas. The AFM images show that the top surface of the samples becomes more rougher at higher pressures. The AFM images show 1 micron by1 micron areas. Both thex-ray reflectivitycurvesand theAFMimages were used to determinethe rms roughness for each sample. However, to directly probe the internal rms roughness, TABLEI:TheMeasuredMagneticCharacteristicsofOurSix we alsodidthe x-rayreflectivity measurementsshownin Samples Fig. 5. ThereflectivitydatawasfitusingaDebye-Waller factortodetermine the roughness. Insteadofthe system Samplea σrms b Ms c Hn d Hc e Hs f possessing thermal fluctuations, the displacements from 3 mTorr 0.48 1360 1.58 0.16 3.7 the average height are randomly distributed and fixed. 7 mTorr 0.57 1392 0.64 0.68 5.0 The rms roughness values from the x-ray measurements 8.5 mTorr 0.62 1136 1.68 1.42 5.5 10 mTorr 0.69 1069 1.45 1.87 6.5 agreed with those from the AFM measurements, con- 12 mTorr 0.90 1101 1.23 2.74 9.5 firming the conformal roughness of our samples. The 20 mTorr 1.44 918 -1.81 5.89 14.2 rms roughness values are shown in Fig. 4 and are listed in Table 1. We found that the rms roughness for the 3 aOursamplesarelabeledbytheirgrowthpressureinmTorr mTorr sample is about 0.48 nm and that it increases to bThemeasuredrmsinterfacialroughnessinnm 1.44 nm for the 20 mTorr sample. cThemeasuredsaturationmagnetization ofCoinemu/cm3 dThenucleation fieldmeasuredfrompositivesaturation eThemeasuredcoercivefieldinkOe fThemeasuredsaturationfieldinkOe B. Magnetic Characterization We measured the major hysteresis loops for all of our Fig. 6 exhibit clear changes that are related to the in- samples using both Kerr magnetometry at the San Jose creasingroughness. The twolowdisorderfilms (3 mTorr HitachiGlobalStorageTechnologyLaboratoryandalter- and7mTorr)exhibited“classicKooy-Enz[16]softloops” nating gradient magnetometry (AGM) at the University with low remanence and abrupt nucleation transitions. ofCalifornia-Davis. The measuredmajorloopsshownin Between7mTorrand8.5mTorr,thereisanabrupttran- 8 FIG.7: (coloronline)Themeasuredmagneticcharacteristics foroursamplesplottedversustheirmeasuredrmsroughness. The coercive, nucleation, and saturation fields are denoted by Hc, Hn, and Hs, respectively. Note the apparently linear dependenceof these properties on therms roughness. The AGM magnetometer was used to measure the saturation magnetization of the samples. The measured saturation magnetizations are reported in Table 1; they should be compared against the value of M = s 1400emu/cm3 for pure cobalt. It is interesting to note that with increasing interfacial roughness, that the coercivity and saturation field increase and the nucleation fielddecreaseslinearlywiththe roughness;the measured saturation magnetizations also decrease as the disorder increases. FIG.6: (coloronline)Themeasuredmagnetichysteresisloops for our samples. Note that the shape of the major hysteresis C. Coherent X-Ray Speckle Metrology loopschangeabruptlyabovethe“critical roughness”value— whichoccursbetweenthe7and8.5mTorrsamples—andthat To measure the field-history induced changes in the the areas inside the major loop increase as the disorder increases past the “critical roughness” value. The two low rms correlations between the microscopic magnetic domain roughness samples possess exhibit “classic Kooy-Enz behav- configurations, we developed coherent x-ray speckle ior” characterized by a sharp nucleation region and low rem- metrology (CXSM). The magnetic sensitivity of CXSM nantmagnetization,whereasthehighrmsroughnesssamples is provided by virtual 2p to 3d resonant magnetic scat- exhibit an almost constant slope. tering. We produce a transversely coherent beam by passing the partially coherent beam from the undulator througha35-micron-diametercircularpinholetoselecta sition to loops which do not show a clear nucleation re- highly transverselycoherent portion. The beam selected gion. Between 8.5 mTorr and 20 mTorr, the ascending by the spatial filter is largely coherent over the entire il- and descending slopes of the loop remain the approxi- luminated area. Due to this largeuniformtransverseco- matelythesame,buttheloopsgraduallywidenuntilthe herence, the resonant magnetic scattering produces the fullmagneticmomentisleftatremanence. Thevaluesof speckle patterns that we use to track the field-history- thenucleation,coerciveandsaturationfieldseachexhibit induced evolution of the magnetic domains. We explain a roughlylinear dependence upon the sample roughness. our analysis methodology for the magnetic speckle pat- This behavior is shown in Fig. 7. In addition, we also terns in the next section. foundviamagnetometrythatallofourfilmsexhibitper- What information does x-ray speckle metrology pro- fect macroscopic major loop and minor loop RPM and videaboutthemagneticdomains,andwhydon’twesim- CPM. ply study the magnetic domains in real space? This is 9 the venerable old question about diffraction versus mi- where the diffuse scattering is measurable, namely be- croscopy. The conventionalanswer is that they are com- tween the inside and outside radii of the annulus. For plementary: use “conventional beam diffraction” to ob- our samples this was from about 110 to 260 nm in real tain information about the ensemble average, and use space. microscopy to obtain information about the individual As argued above, all of the physical information that defects. There are two limiting cases of conventional can be obtained using our incident wavelength is con- diffraction studies. When the diffraction pattern con- tainedwithinthelimitedrangethatcontainsmeasurable sists of Bragg peaks, then the information that conven- scattering intensity. Our incident wavelength is fixed by tional diffraction provides is the ensemble averageof the the magnetic resonant scattering condition for cobalt so long-range order. When the diffraction pattern consists λ 1.6 nm. For this wavelength, diffraction provides ≃ of diffuse scattering, then the information that conven- informationrangingfrom0.8nmfor backscatteringwith tional diffraction provides is the ensemble averageof the 2θ = 180 degrees up to 40 microns set by the illumina- short-range order. In our labyrinthine systems, there tionarea. Atourusualsample-to-cameraseparation,the is no long-range magnetic order, and consequently the pixel size of our cameratranslates into a real-spacereso- diffractionisdiffuse. Withanunfilteredbeamweobserve lutionof13micronsandthetotalcoverageofthecamera only a diffuse annulus which contains information about translatesintoareal-spaceresolutionof270nm. Thean- the strength (amplitude) of the magnetic domains, the gular size of our beamstop translates into 70 nm. Since mean spacing of the magnetic domains, and the correla- 70 < 110 nm, 260 < 270 nm, and 27 < 40 microns, our tionlengthofthemagneticdomains. Wehaveperformed camera and our beamstop do not limit the spatial scales suchstudiesalreadyandtheresultsareinpreparationfor that we can access. Instead, the limits are set only by publication. the disorder levels in our samples. Fully coherent diffraction changes that paradigm be- The intensity I(qr,qθ) of each speckle located at po- cause the precise configuration of the speckles provide sition (qr,qθ) is proportional to the square-modulus detailed information about the defects, or in our case aq 2 ofthe scatteringamplitude aq of the corresponding| | about the configuration of the magnetic labyrinths. In Fourier component of the magnetic density ρmag(qr,qθ). the Bragg case, the information is about the defects in So by taking the square root of the intensity of our the crystallineorder. Inthe diffuse case,the information speckle pattern, we can first calculate and then visual- is about the defects in the short-range order. In fact, in ize the result as a map of the magnetic density ampli- two- or three-dimensions, if the speckle pattern is sam- tudes for all of the most important Fourier components. pledwith sufficientwavevectorresolution,thenallofthe Each component located at q = (qr,qθ) tells us the am- real-space information is contained in the speckle pat- plitudeofaninfinite-spatial-extentcomplex-valuedexpo- tern and can be recovered using “oversampling speckle nential density component exp(iq r) multiplied by our · reconstruction” [17]. However, no successful oversam- illuminationfunctionwhichisroughlyequaltooneinside pling speckle reconstructions have yet been reported for the illumination circle and zero outside. Imagine a large magnetic domains, though holography methods have re- numberofthesecomplex-valuedoscillatingexponentials, cently been demonstrated in similar systems[18]. Con- each one oriented along the direction θ with amplitude sequently, our objective was not to extract the complete √I and with wavevector qr. real-spaceinformation,but insteadto directlydetermine So,howmanyoftheseFouriercomponentsdowemea- the changes between the correlations of the magnetic sure? The area of our observed annulus in reciprocal domain configurations prepared via different applied- space is given by field histories—specifically without requiring the oversampling speckle reconstruction of our speckle image A =πq2 πq2 magnetic fingerprints. q max− min Sowhatinformationdoesourmagneticspecklemetrol- and the area of each one of our speckles in reciprocal ogy provide? We illuminate a 40 micron diameter circle space is given by onthesamplesoourensemble-averageisoverthatregion. Consequently,eachspeckleinourspecklepatternconsists ofanAirypatternwithacharacteristicsizeinreciprocal A =πδq2 speckle speckle space of 2π/40 inverse microns. The second important length scale in our problem is set by the width of the so the number of speckles inside the annulus is given by magnetic domains in the labyrinth state. This width is 200 nm, and consequently the corresponding character- istic size in reciprocal space is 2π/200 inverse nm; this A /A 30,000. annulus speckle ≃ sets the mean-radius of our annular speckle patterns. In principle, our speckle patterns can provide spatial infor- In other words, we directly measure this many Fourier mation down to λ/2 = 0.8 nm, but in practice—due to components of the magnetization density. Because the the strong disorder in our labyrinths—our speckle pat- speckle intensity outside the annulus is negligible, the terns really only contain information set by the region corresponding Fourier components outside the annulus 10 are also negligible. So we directly obtain information magnetic speckle patterns—our magnetic speckle finger- aboutallofthenon-negligibleFouriercomponentsofthe prints. However, this comparison can be done in recip- magnetic density that produce the magnetic scattering rocal space—as we have primarily been doing up until within the speckled annular region that we measure. now—or in real space as we are just beginning to do. Ontheotherhand,moderncomputercontrolandcom- Some of our initial experimental work in real space is puter image analysis should enable modern magnetic x- illustrated Figure 10 which shows the magnetic domains raymicroscopyto obtainensemble-averagedinformation in our 8.5 mTorr sample measured using x-ray magnetic about the magnetic domains. This is certainly worth microscopy [20]. These images were recorded at the Co pursuing, and we are just beginning such studies. L3 edge using XM-1 at the ALS and were taken on the Speckle contrast, the normalized standard deviation descending majorloop; the left panelshowsthe domains of the intensity, is generally used as a measure of the atH = 0.50kOeandtherightpanelshowsthedomains − quality of the produced speckle patterns. As the diffuse atH = 1.00kOe. The correlationcoefficients obtained − scatteringenvelopeisazimuthally symmetricaboutq = from our real-space normalized cross-correlation analy- r 0, it is correct to define the speckle contrast σ2 (q ) as sis of the domain patterns agrees with our correlation con r coefficients obtained via our standard reciprocal-space normalized cross-correlation analysis of the correspond- 1 N (I I )2 ing speckle patterns. Therefore we believe that our real σ2 (q )= v k−h i con r I u N 1 spaceandreciprocalspacemethodswillprovetobecom- h iutXk − plementary. Ournormalizedcross-correlationanalysisprocedurein for small steps of q where the sum is carried out over r reciprocal space is illustrated in figures 11-13. Figure allN is the number ofdata points included in eachstep. 11 shows the speckle fingerprints measured in recipro- Usingthiscalculation,thecontrastpresentinourspeckle cal space; again the left panel shows the fingerprint at patterns typically ranges from 0.6 to 0.4, with a small H = 0.5 and the right panel shows the fingerprint at dip in values over the peak scattering. This interesting − H = 1.0 kOe. Figure 12 shows the calculated autocor- variation of the speckle contrast as it depends upon qr − relationfunctionsforthesetwospecklefingerprints. Note is quite reminiscent of the speckle contrast studies by thatbothoftheseconsistofabroadsmooth“mountain” Retsch and McNulty [19] across absorption edges and with a sharp “tree” on top of it. could provide useful information if properly understood. Themountaincorrespondstothediffusescatteringen- velope from the short range magnetic ordering and the tree corresponds to the coherent scattering from the en- IV. DATA AND DATA ANALYSIS tire illuminated area. Figure 13 shows the calculated cross-correlationfunctionforthetwospecklefingerprints The typical evolution of the speckle patterns for one- shown in Fig. 12. Again there is an “diffuse mountain” halfcyclearoundthemajorhysteresisloopforthe3sam- with a “coherent tree” on top of it. ple is schematically illustrated in Fig. 8. and the corre- Wewanttousethecoherentcomponentsoftheseauto- sponding measured speckle patterns are shown in Fig. and cross-correlation functions to compute the normal- 9. Starting at positive saturation there is no magnetic ized correlation coefficient. We extract the volume of contrast—allthe magnetic domains are aligned with the each tree and then we calculate the ratio field—consequently there is no magnetic scattering. As we descend from positive saturation the magnetic domains nucleate and produce a magnetic speckle pattern volume(a b) ρ(a,b)= ⊗ . that is shaped like a cookie (disk). When we reach zero 1/2 volume(a a) volume(b b) applied field, the magnetic domains have grownso much ⊗ ⊗ (cid:2) (cid:3) that they fill the entire sample; in this limit they must The resulting normalizedcorrelationcoefficient ρ(a,b) interact,andtheirinteractionproducesthedonut(annu- measures the normalized degree-of-correlation between lar) shaped speckle pattern. When we reachthe reversal any two speckle patterns. We use it to quantify the region,the domaindensity is againlow,and so the asso- degree-of-correlation between pairs of speckle patterns ciated speckle pattern is again cookie shaped. which in turn is our measure of the degree-of-correlation between the corresponding magnetic domain patterns. When ρ(a,b)=1 the two magnetic domain patterns are A. Correlation Coefficients identical,andwhenρ(a,b)=0the twomagnetic domain patterns are completely different. To quantitatively compare the magnetic domain con- In general, the value of ρ specifies the degree-of- figurations versus the applied magnetic field history, we correlation between the two speckle patterns which in calculate the normalized correlationcoefficients between turn are proportional to the Fourier coefficients of the pairs of our measured images acquired for different ap- magnetization density for the two magnetic domain plied field histories. To date, our work has been pri- configurations. Because our correlations are based on marilybasedonthenormalizedcross-correlationofthese the intensity, we are unable to determine the sign of