Microsoft Word - DumasLV10695BRrevised.doc
Magnetic Fingerprints of sub-100 nm  
Fe Nanodots 
Randy K. Dumas,1 Chang-Peng Li,2 Igor V. Roshchin,2 Ivan K. Schuller2 and Kai Liu1,*  
1Physics Department, University of California, Davis, California 95616 
2Physics Department, University of California - San Diego, La Jolla, California 92093 
Abstract 
Sub-100 nm nanomagnets not only are technologically important, but also exhibit 
complex magnetization reversal behaviors as their dimensions are comparable to typical 
magnetic domain wall widths. Here we capture magnetic “fingerprints” of 109 Fe 
nanodots as they undergo a single domain to vortex state transition, using a first-order 
reversal curve (FORC) method. As the nanodot size increases from 52 nm to 67 nm, the 
FORC diagrams reveal striking differences, despite only subtle changes in their major 
hysteresis loops.  The 52 nm nanodots exhibit single domain behavior and the coercivity 
distribution extracted from the FORC distribution agrees well with a calculation based on 
the measured nanodot size distribution. The 58 and 67 nm nanodots exhibit vortex states, 
where the nucleation and annihilation of the vortices are manifested as butterfly-like 
features in the FORC distribution and confirmed by micromagnetic simulations. 
Furthermore, the FORC method gives quantitative measures of the magnetic phase 
fractions, and vortex nucleation and annihilation fields.  
PACS’s: 75.60.Jk, 75.60.-d, 75.70.Kw, 75.75.+a 
Phys. Rev. B, in press. 
I. Introduction 
Deep sub-100 nm magnetic nanoelements have been the focus of intense research 
interest due to their fascinating fundamental properties and potential technological 
applications.1-6 At such small dimensions, comparable to the typical magnetic domain 
wall width, properties of the nanomagnets are rich and complex. It is known that well 
above the domain wall width, in micron and sub-micron sized patterns, magnetization 
reversal often occur via a vortex state (VS).7-12 At reduced sizes, single domain (SD) 
static states are energetically more favorable.13 However, even in SD nanoparticles, the 
magnetization reversal can be quite complex, involving thermally activated incoherent 
processes.14 The VS-SD crossover itself is fascinating. For example, recently Jausovec et 
al. have proposed that a third, metastable, state exists in 97 nm permalloy nanodots, 
based on minor loop and remanence curve studies.15 To date, direct observation of the 
VS-SD crossover, especially in the deep sub-100 nm regime, has been challenging. 
Fundamentally, the vortex core is expected to have a nanoscale size, comparable to the 
exchange length. Magnetic imaging techniques face resolution limits and often are 
limited to remanent state and room temperature studies. Practically, collections of 
nanomagnets inevitably have variations in size, shape, anisotropy, etc.16 The ensemble-
averaged properties obtained by collective measurements such as magnetometry no 
longer yield clear signatures of the nucleation / annihilation fields. Furthermore, 
quantitatively capturing the distributions of magnetic properties are essential to the 
understanding and application of magnetic and spintronic devices, which may consist of 
billions of nanomagnets. How to qualitatively and quantitatively investigate the 
properties of such nanomagnets remains a key challenge for condensed matter physics 
and materials science.  
 In this study, we investigate the VS-SD crossover in deep sub-100 nm Fe 
nanodots. We have captured “fingerprints” of such nanodots using a first-order reversal 
curve (FORC) method,17-21 which circumvents the resolution, remanent state and room 
temperature limits by measuring the collective magnetic responses of the dots. The 
“fingerprints”, shown as FORC diagrams, reveal remarkably rich information about the 
nanodots. A qualitatively different reversal pattern is observed as the dot size is increased 
from 52 to 67 nm, despite only subtle differences in their major hysteresis loops.  The 52 
nm nanodots behave as SD particles; the 67 nm ones exhibit VS reversal; and the 58 nm 
ones have both SD and VS characteristics. Quantitatively, the FORC diagram shows 
explicitly a coercivity distribution for the SD dots, which agrees well with calculations; it 
yields SD and VS phase fractions in the larger dots; it also extracts unambiguously the 
nucleation and annihilation fields for the VS dots and distinguishes annihilations from 
opposite sides of the dots. 
II. Experimental 
Samples for the study are Fe nanodots fabricated using a nanoporous alumina 
shadow mask technique in conjunction with electron beam evaporation.22,23 This method 
allows for fabrication of high density nanodots (~1010 /cm2) over macroscopic areas (~1 
cm2).  Three different types of samples have been made on Si and MgO substrates with 
mean nanodot sizes of 52±8, 58±8, and 67±13 nm, and a thickness of 20 nm, 15 nm, and 
20 nm, respectively. The nanodot center-to-center spacing is typically twice its diameter. 
The Fe nanodots thus made are polycrystalline, capped with an Al or Ag layer. A 
scanning electron microscopy (SEM) image of the 67 nm sample is shown in Fig. 1. A 
survey of the size distribution is illustrated in Fig. 1 inset.  
Magnetic properties have been measured using a Princeton Measurements Corp.  
2900 alternating gradient and vibrating sample magnetometer (AGM/VSM), with the 
applied field in the plane of the nanodots. Samples have been cut down to ~ 3 × 3 mm2 
pieces, which contains ~ 109 Fe nanodots each. Additionally, the FORC technique has 
been employed to study details of the magnetization reversal. After saturation, the 
magnetization M is measured starting from a reversal field HR back to positive saturation, 
tracing out a FORC. A family of FORC’s is measured at different HR, with equal field 
spacing, thus filling the interior of the major hysteresis loop [Figs. 2(a)-2(c)]. The FORC 
distribution is defined as a mixed second order derivative:17-21  
( ) ( )
−≡ ,     (1) 
which eliminates the purely reversible components of the magnetization. Thus any non-
zero ρ  corresponds to irreversible switching processes.19-21  The FORC distribution is 
plotted against (H, HR) coordinates on a contour map or a 3-dimensional plot. For 
example, along each FORC in Fig 4(a) with a specific reversal field HR, the 
magnetization M is measured with increasing applied field H; the corresponding FORC 
distribution ρ in Fig. 4(b) is represented by a horizontal line scan at that HR along H.  
Alternatively ρ can be plotted in coordinates of (HC, HB), where HC is the local coercive 
field and HB is the local interaction or bias field.  This transformation is accomplished by 
a simple rotation of the coordinate system defined by: HB=(H+HR)/2 and HC=(H-HR)/2.  
Both coordinate systems are discussed in this paper.  
III. Results 
Families of the FORC’s for the 52, 58, and 67 nm nanodots are shown in Figs. 
2(a)-2(c). The major hysteresis loops, delineated by the outer boundaries of the FORC’s, 
exhibit only subtle differences. The 52 nm nanodots show a regular major loop, with a 
remanence of 57 % and a coercivity of 475 Oe [Fig. 2(a)]. The 67 nm nanodots have a 
slight “pinching” in its loop near zero applied field, with a remanence of 27 % and a 
coercivity of 246 Oe [Fig. 2(c)]. The unique shape, small values of coercivity and 
remanence suggest that the magnetization reversal is via a VS.  Indeed, the VS is 
confirmed by polarized neutron reflectivity measurements on similarly prepared 65 nm 
Fe nanodots, which find an out of plane magnetic moment corresponding to a vortex core 
of 15 nm.24 However, due to the relatively gradual changes in magnetization along the 
major loop, averaged over signals from ~ 109 Fe nanodots, it is difficult to determine the 
vortex nucleation and annihilation fields.  In contrast, the relatively fuller major loop of 
the 52 nm nanodots is suggestive of a SD state. The loop of the 58 nm nanodots appears 
to have combined features from those of the other two samples [Fig. 2(b)].   
 The subtle differences seen in the major hysteresis loops manifest themselves as 
striking differences in the corresponding FORC distributions, shown in Figs. 2(d)-2(i).  
For the 52 nm nanodots, the only predominant feature is a narrow ridge along the local 
coercivity HC-axis with zero bias [Figs. 2(d) and 2(g)].  The ridge is peaked at HC = 525 
Oe, near the major loop coercivity value of 475 Oe. This pattern is characteristic of a 
collection of non-interacting SD particles.25  Given that the nanodot spacing is about 
twice its diameter and the random in-plane easy axes, dipolar interactions are expected to 
be small.26 The relative spread of the FORC distribution along the HB-axis actually gives 
a direct measure of the interdot interactions, as we have shown in single domain 
magnetite nanoparticles with different separations.27 The sharp ridge shown in Fig. 2(g) is 
similar to that of an assembly of well–dispersed magnetite nanoparticles with little 
dipolar interactions. In the present case, the ridge is localized between bias field of HB ~ 
±100 Oe and has a narrow FWHM (full width at half maximum) of about 136 Oe [Fig. 
3(a)].   A simple calculation of the dipolar fields yields a value similar to the FWHM.     
As the nanodot size is increased, the FORC distribution becomes much more 
complex.  The 67 nm sample is characterized by three main features, as shown in Figs. 
2(f) and 2(i): two pronounced peaks at HC= 650 Oe and HB = ± 750 Oe, and a ridge along 
HB = 0, forming a butterfly-like contour plot [Fig. 2(i)]. The ridge has changed 
significantly from that of the 52 nm sample: a peak corresponding to the coercivity of the 
major loop is now virtually absent; instead a large peak at HC =1500 Oe has appeared, 
accompanied by two small negative regions nearby.  The 58 nm sample shows a FORC 
pattern representative of both the 52 and 67 nm samples [Figs. 2(e) and 2(h)].  The 
overall distribution resembles that of the 67 nm sample, with two peaks centered at HC = 
650 Oe and HB = ±400 Oe.  A ridge along HB = 0, peaking at roughly the major loop 
coercivity, is similar to that seen in the 52 nm sample.  Note that the 58 nm sample, being 
thinner, would tend to inhibit the formation of a VS in the smaller dots and therefore 
show magnetic reversal via a SD.  However, significant fractions of the nanodots in the 
ensemble are apparently reversing via a VS.    
The FORC distribution also allows us to extract quantitative information about 
the reversal processes.  Since each sample measured consists of ~ 1 billion nanodots with 
a distribution of sizes, a coercivity spread is contained in the FORC distribution.  For the 
52 nm sample, we have indeed extracted this distribution by projecting the ridge in Fig. 
2(g) onto the HC-axis (HB = 0), as shown as the open circles in Fig. 3(b).  The relative 
height gives the appropriate weight of nanodots with a given coercivity.  This extracted 
coercivity distribution can be compared with a simple theoretical calculation.  The 
coercivity of a SD particle undergoing reversal via a curling mode increases strongly with 
decreasing particle size d, according to   
HC −∝ ,     (2) 
where C1 and C2 are constants.28  Based on the mean nanodot size and the size 
distribution determined from SEM, we have calculated a coercivity distribution [solid 
circles in Fig. 3(b)]. A good agreement is obtained with that determined from the FORC 
distribution, after a rescaling of the latter by an arbitrary weight.  Thus for other non-
interacting single-domain particle systems with unknown size distributions, the FORC 
method may be used to extract that information. This is particularly important in 3D 
distributions of nanostructures where there is no direct image access to the individual 
dots, as is the case here for a 2D distribution. 
As we have demonstrated earlier, the FORC distribution ρ is extremely sensitive 
to irreversible switching.19 This is most convenient to see in the (H, HR) coordinate 
system (meaningful data is in H>HR), as non-zero values of ρ correspond to the degree of 
irreversibility along a given FORC. We have employed this capability to analyze the VS 
nucleation and annihilation for the 67 nm sample.  The complex butterfly-like pattern of 
Fig. 2(i) now transforms into irreversible switching mainly along line scans 1 and 2 in 
Fig. 4(b), which correspond to FORC’s starting at HR= 100 Oe and -1450 Oe, 
respectively [marked as bold with large open circles as starting points in Fig. 4(a)]. Along 
line scan 1 (HR=100 Oe), when applied field H=100 Oe, vortices have already nucleated 
in most of the nanodots. With increasing field H, ρ becomes non-zero and increases with 
H and peaks at 1320 Oe.  This corresponds to the annihilation of the vortices in majority 
of the nanodots, and eventually ρ returns to zero near positive saturation.  Line scan 2 
starts at HR= -1450 Oe, where the majority, but not all, of the nanodots have been 
negatively saturated.  As H is increased, a first maximum in ρ is seen at H= -100 Oe, 
corresponding to the nucleation of vortices within the nanodots. Between -100 Oe < H < 
1450 Oe, ρ is essentially zero, indicating reversible motion of the vortices through the 
nanodots.  A second ρ  maximum is found at H = 1450 Oe, as the vortices are 
annihilated. This is again followed by reversible behavior near positive saturation.  Note 
that along line scan 1, the vortices are annihilated from the same side of the nanodot from 
which they first nucleated, and thus the net magnetization remains positive; along line 
scan 2, the vortices nucleate on one side of the dot and are annihilated from the other, and 
consequently the net magnetization changes sign [Fig. 4(a)].  Interestingly the 
annihilation field along line scan 2, 1450 Oe, is larger than that along line scan 1, 1320 
Oe.  It seems more difficult to drive a vortex across the nanodot and then annihilate it. 
The peaks in Fig. 4(b) are rather broad, which is a manifestation of vortex nucleation and 
annihilation field distributions.  Also note that the interactions among the VS dots are 
expected to be negligible due to the high degree of flux closure, as confirmed by 
simulations.29 
IV. Simulations and Discussions 
For comparison, micromagnetic simulations have been carried out on nearly 
circular nanodots with 60 nm diameters and 20 nm thicknesses.30  We have used 
parameters appropriate for Fe (exchange stiffness A = 2.1 × 10-11 J/m, saturation 
magnetization Ms = 1.7 × 106 A/m, and anisotropy constant K = 4.8 × 104 J/m3).  Each 
polycrystalline nanodot is composed of 2 nm square cells that are 20 nm thick, where 
each cell is a different grain with a random easy axis.  A small cut on one side of the 
nanodot generates two distinct annihilation fields that depend on which side of the 
nanodot the vortex annihilates from. This exercise models the fact that our fabricated dots 
are not perfectly circular.31 We have simulated FORC’s generated by two nanodots with 
different orientations [Fig. 4(c)]: the edge-cut in one is parallel, and in the other at a 45º 
angle, to the applied field.  The simulated M-H curves show abrupt magnetization 
changes, corresponding to the nucleation, propagation, and annihilation of vortices.  
The corresponding FORC distribution is shown in Fig. 4(d).  Peaks in the 
simulated FORC distribution clearly indicate the nucleation and annihilation fields of the 
vortices which are apparent in Fig 4(c).  Along line scan 1 of Fig. 4(d), a vortex is already 
nucleated at HR = 100 Oe and subsequently annihilated at H = 2300 Oe (upper right 
corner).  Along line scan 2 with HR = -2450 Oe, a vortex is nucleated at H = -100 Oe and 
finally annihilated at H = 2450 Oe (lower right corner).  It is clear that the simulated 
FORC reproduces the key features of the experimentally obtained one in Fig. 4(b).  Here 
the asymmetric dot shape is essential to obtain a different annihilation field along scan 2 
than that along scan 1.  We have simulated the angular dependence of nucleation and 
annihilation fields in such circular dot with a small cut as the cut orientation is varied in a 
field.  We find that for most angles it is harder to annihilate a vortex from the opposite 
side of its nucleation site.  However, for a small range of angles near 45° it is actually 
slightly easier to annihilate from the opposite side.  It is the combination of these two 
behaviors that gives rise to the negative-positive-negative trio of features in the lower 
right portion of the FORC distribution.  The presence of similar features in the 
experimental data shown in Fig. 4(b) thus illustrates that the FORC distribution is also 
sensitive to variations of dot shapes in the array. Because only two dots are simulated, the 
features in the FORC distribution are much sharper than the experimental data where 
distributions of vortex nucleation and annihilation fields are present.  Including more dots 
in the simulation with different applied field orientations and size distributions would 
tend to broaden the features generated by the two dots simulated.      
Additionally, by selectively integrating the normalized FORC distribution20,21
corresponding to the SD phase (the aforementioned ridge centered at low coercivity 
values in Fig. 2), we can quantitatively determine the percentage of nanodots in SD state 
for each sample.  The SD phase fraction is 100%, 43%, and 10% for the 52, 58, and 67 
nm sample, respectively. Thus the 58 nm nanodots have a significant co-existence of both 
SD and VS states. However, we do not observe clear evidence of any additional 
metastable phase.15 
V. Conclusions 
In summary, we have used the FORC method to “fingerprint” the rich 
magnetization reversal behavior in arrays of 52, 58, and 67 nm sized Fe nanodots. 
Distinctly different reversal mechanisms have been captured, despite only subtle 
differences in the major hysteresis loops.  The 52 nm nanodots are in SD states.  A 
coercivity distribution has been extracted, which agrees with calculations. The 67 nm 
dots reverse their magnetization via the nucleation and annihilation of vortices. Different 
fields are required to annihilate vortices from opposite sides of the dots.  Quantitative 
measures of the vortex nucleation and annihilation fields have been obtained. OOMMF 
simulations confirm the experimental FORC distributions.   The 58 nm sample shows 
coexistence of SD and VS reversal, without evidence of additional reversal mode.  These 
results further demonstrate the FORC method as a simple yet powerful technique for 
studying magnetization reversal, due to its capability of capturing distributions of 
magnetic properties, sensitivity to irreversible switching, and the quantitative phase 
information it can extract. 
Acknowledgements 
This work has been supported by ACS (PRF-43637-AC10), AFOSR, and the 
Alfred P. Sloan Foundation. We thank J. E. Davies, J. Olamit, M. Winklhofer, C. R. Pike, 
H. G. Katzgraber, R. T. Scalettar, G. T. Zimányi, and K. L. Verosub for helpful 
discussions. R.K.D. acknowledges support from the Katherine Fadley Pusateri Memorial 
Travel Award. 
References 
*   Corresponding author, email address: kailiu@ucdavis.edu. 
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94, 157202 (2005). 
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M. D. Baró, B. Dieny, and J. Nogués, Phys. Rev. Lett. 97, 067201 (2006). 
13 We use "single domain state" to refer to all magnetization configurations with no 
domain wall and a non-zero net magnetization. 
14 Y. Li, P. Xiong, S. von Molnár, Y. Ohno, and H. Ohno, Phys. Rev. B 71, 214425 
(2005). 
15 A.-V. Jausovec, G. Xiong, and R. P. Cowburn, Appl. Phys. Lett. 88, 052501 (2006). 
16 Even in nanomagnets with identical size and shape, there may exist a distribution of 
intrinsic anisotropy. See e.g., T. Thomson, G. Hu and B. D. Terris, Phys. Rev. Lett. 
96, 257204 (2006). 
17 C.R. Pike and A. Fernandez, J. Appl. Phys. 85, 6668 (1999). 
18 H. G. Katzgraber, F. Pazmandi, C. R. Pike, K. Liu, R. T. Scalettar, K. L. Verosub, 
and G. T. Zimanyi, Phys. Rev. Lett. 89, 257202 (2002). 
19 J. E. Davies, O. Hellwig, E. E. Fullerton, G. Denbeaux, J. B. Kortright, and K. Liu, 
Phys. Rev. B 70, 224434 (2004). 
20 J. E. Davies, J. Wu, C. Leighton, and K. Liu, Phys. Rev. B 72, 134419 (2005). 
21 J. Olamit, K. Liu, Z. P. Li, and I. K. Schuller, Appl. Phys. Lett. 90, 032510 (2007). 
22 K. Liu, J. Nogues, C. Leighton, H. Masuda, K. Nishio, I. V. Roshchin, and I. K. 
Schuller, Appl. Phys. Lett. 81, 4434 (2002).  
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30   OOMMF code. http://math.nist.gov/oommf.  
31   Simulations done on perfectly circular and slightly elliptical dots show only a single 
annihilation field.  To reproduce the two distinct annihilation fields the symmetry of 
the dot must be broken by some type of shape defect.   
Figure Captions 
Fig. 1. (color online). Scanning electron micrograph of the 67 nm diameter nanodot 
sample. Inset is a histogram showing the distribution of nanodot sizes. 
Fig. 2. First-order reversal curves and the corresponding distributions. Families of 
FORC’s (a-c), whose starting points are represented by black dots for the 52, 58, 
and 67 nm Fe nanodots, respectively. The corresponding FORC distributions are 
shown in 3-dimensional plots (d-f) and contour plots (g-i).  
Fig. 3. (color online). Projection of the FORC distribution ρ of the 52 nm nanodots onto 
(a) the HB-axis, showing weak dipolar interactions; and (b) the HC-axis (open 
circles), showing a coercivity distribution that agrees with a calculation based on 
measured size distribution (solid circles).   
Fig. 4. (a) A family of measured FORC’s for the 67nm diameter dots. (b) The 
corresponding experimental FORC distribution plotted against applied field H 
and reversal field HR.  (c) A family of simulated FORC’s generated using the 
OOMMF code.  Inset shows the orientations of the two dots simulated.  (d) The 
FORC distribution calculated from the simulated FORC’s shown in (c).  The two 
white dashed lines in (b) and (d) correspond to the two bold FORC’s whose 
starting points are large open circles in (a) and (c), respectively.  
Fig. 1, Dumas, et al. 
Fig. 2, Dumas, et al. 
Fig. 3, Dumas, et al. 
Fig. 4, Dumas, et al.

  Sub-100 nm nanomagnets not only are technologically important, but also
exhibit complex magnetization reversal behaviors as their dimensions are
comparable to typical magnetic domain wall widths. Here we capture magnetic
"fingerprints" of 1 billion Fe nanodots as they undergo a single domain to
vortex state transition, using a first-order reversal curve (FORC) method. As
the nanodot size increases from 52 nm to 67 nm, the FORC diagrams reveal
striking differences, despite only subtle changes in their major hysteresis
loops. The 52 nm nanodots exhibit single domain behavior and the coercivity
distribution extracted from the FORC distribution agrees well with a
calculation based on the measured nanodot size distribution. The 58 and 67 nm
nanodots exhibit vortex states, where the nucleation and annihilation of the
vortices are manifested as butterfly-like features in the FORC distribution and
confirmed by micromagnetic simulations. Furthermore, the FORC method gives
quantitative measures of the magnetic phase fractions, and vortex nucleation
and annihilation fields.

Introduction 
Deep sub-100 nm magnetic nanoelements have been the focus of intense research 
interest due to their fascinating fundamental properties and potential technological 
applications.1-6 At such small dimensions, comparable to the typical magnetic domain 
wall width, properties of the nanomagnets are rich and complex. It is known that well 
above the domain wall width, in micron and sub-micron sized patterns, magnetization 
reversal often occur via a vortex state (VS).7-12 At reduced sizes, single domain (SD) 
static states are energetically more favorable.13 However, even in SD nanoparticles, the 
magnetization reversal can be quite complex, involving thermally activated incoherent 
processes.14 The VS-SD crossover itself is fascinating. For example, recently Jausovec et 
al. have proposed that a third, metastable, state exists in 97 nm permalloy nanodots, 
based on minor loop and remanence curve studies.15 To date, direct observation of the 
VS-SD crossover, especially in the deep sub-100 nm regime, has been challenging. 
Fundamentally, the vortex core is expected to have a nanoscale size, comparable to the 
exchange length. Magnetic imaging techniques face resolution limits and often are 
limited to remanent state and room temperature studies. Practically, collections of 
nanomagnets inevitably have variations in size, shape, anisotropy, etc.16 The ensemble-
averaged properties obtained by collective measurements such as magnetometry no 
longer yield clear signatures of the nucleation / annihilation fields. Furthermore, 
quantitatively capturing the distributions of magnetic properties are essential to the 
understanding and application of magnetic and spintronic devices, which may consist of 
billions of nanomagnets. How to qualitatively and quantitatively investigate the 
properties of such nanomagnets remains a key challenge for condensed matter physics 
and materials science.  
 In this study, we investigate the VS-SD crossover in deep sub-100 nm Fe 
nanodots. We have captured “fingerprints” of such nanodots using a first-order reversal 
curve (FORC) method,17-21 which circumvents the resolution, remanent state and room 
temperature limits by measuring the collective magnetic responses of the dots. The 
“fingerprints”, shown as FORC diagrams, reveal remarkably rich information about the 
nanodots. A qualitatively different reversal pattern is observed as the dot size is increased 
from 52 to 67 nm, despite only subtle differences in their major hysteresis loops.  The 52 
nm nanodots behave as SD particles; the 67 nm ones exhibit VS reversal; and the 58 nm 
ones have both SD and VS characteristics. Quantitatively, the FORC diagram shows 
explicitly a coercivity distribution for the SD dots, which agrees well with calculations; it 
yields SD and VS phase fractions in the larger dots; it also extracts unambiguously the 
nucleation and annihilation fields for the VS dots and distinguishes annihilations from 
opposite sides of the dots. 
II. Experimental 
Samples for the study are Fe nanodots fabricated using a nanoporous alumina 
shadow mask technique in conjunction with electron beam evaporation.22,23 This method 
allows for fabrication of high density nanodots (~1010 /cm2) over macroscopic areas (~1 
cm2).  Three different types of samples have been made on Si and MgO substrates with 
mean nanodot sizes of 52±8, 58±8, and 67±13 nm, and a thickness of 20 nm, 15 nm, and 
20 nm, respectively. The nanodot center-to-center spacing is typically twice its diameter. 
The Fe nanodots thus made are polycrystalline, capped with an Al or Ag layer. A 
scanning electron microscopy (SEM) image of the 67 nm sample is shown in Fig. 1. A 
survey of the size distribution is illustrated in Fig. 1 inset.  
Magnetic properties have been measured using a Princeton Measurements Corp.  
2900 alternating gradient and vibrating sample magnetometer (AGM/VSM), with the 
applied field in the plane of the nanodots. Samples have been cut down to ~ 3 × 3 mm2 
pieces, which contains ~ 109 Fe nanodots each. Additionally, the FORC technique has 
been employed to study details of the magnetization reversal. After saturation, the 
magnetization M is measured starting from a reversal field HR back to positive saturation, 
tracing out a FORC. A family of FORC’s is measured at different HR, with equal field 
spacing, thus filling the interior of the major hysteresis loop [Figs. 2(a)-2(c)]. The FORC 
distribution is defined as a mixed second order derivative:17-21  
( ) ( )
−≡ ,     (1) 
which eliminates the purely reversible components of the magnetization. Thus any non-
zero ρ  corresponds to irreversible switching processes.19-21  The FORC distribution is 
plotted against (H, HR) coordinates on a contour map or a 3-dimensional plot. For 
example, along each FORC in Fig 4(a) with a specific reversal field HR, the 
magnetization M is measured with increasing applied field H; the corresponding FORC 
distribution ρ in Fig. 4(b) is represented by a horizontal line scan at that HR along H.  
Alternatively ρ can be plotted in coordinates of (HC, HB), where HC is the local coercive 
field and HB is the local interaction or bias field.  This transformation is accomplished by 
a simple rotation of the coordinate system defined by: HB=(H+HR)/2 and HC=(H-HR)/2.  
Both coordinate systems are discussed in this paper.  
III. Results 
Families of the FORC’s for the 52, 58, and 67 nm nanodots are shown in Figs. 
2(a)-2(c). The major hysteresis loops, delineated by the outer boundaries of the FORC’s, 
exhibit only subtle differences. The 52 nm nanodots show a regular major loop, with a 
remanence of 57 % and a coercivity of 475 Oe [Fig. 2(a)]. The 67 nm nanodots have a 
slight “pinching” in its loop near zero applied field, with a remanence of 27 % and a 
coercivity of 246 Oe [Fig. 2(c)]. The unique shape, small values of coercivity and 
remanence suggest that the magnetization reversal is via a VS.  Indeed, the VS is 
confirmed by polarized neutron reflectivity measurements on similarly prepared 65 nm 
Fe nanodots, which find an out of plane magnetic moment corresponding to a vortex core 
of 15 nm.24 However, due to the relatively gradual changes in magnetization along the 
major loop, averaged over signals from ~ 109 Fe nanodots, it is difficult to determine the 
vortex nucleation and annihilation fields.  In contrast, the relatively fuller major loop of 
the 52 nm nanodots is suggestive of a SD state. The loop of the 58 nm nanodots appears 
to have combined features from those of the other two samples [Fig. 2(b)].   
 The subtle differences seen in the major hysteresis loops manifest themselves as 
striking differences in the corresponding FORC distributions, shown in Figs. 2(d)-2(i).  
For the 52 nm nanodots, the only predominant feature is a narrow ridge along the local 
coercivity HC-axis with zero bias [Figs. 2(d) and 2(g)].  The ridge is peaked at HC = 525 
Oe, near the major loop coercivity value of 475 Oe. This pattern is characteristic of a 
collection of non-interacting SD particles.25  Given that the nanodot spacing is about 
twice its diameter and the random in-plane easy axes, dipolar interactions are expected to 
be small.26 The relative spread of the FORC distribution along the HB-axis actually gives 
a direct measure of the interdot interactions, as we have shown in single domain 
magnetite nanoparticles with different separations.27 The sharp ridge shown in Fig. 2(g) is 
similar to that of an assembly of well–dispersed magnetite nanoparticles with little 
dipolar interactions. In the present case, the ridge is localized between bias field of HB ~ 
±100 Oe and has a narrow FWHM (full width at half maximum) of about 136 Oe [Fig. 
3(a)].   A simple calculation of the dipolar fields yields a value similar to the FWHM.     
As the nanodot size is increased, the FORC distribution becomes much more 
complex.  The 67 nm sample is characterized by three main features, as shown in Figs. 
2(f) and 2(i): two pronounced peaks at HC= 650 Oe and HB = ± 750 Oe, and a ridge along 
HB = 0, forming a butterfly-like contour plot [Fig. 2(i)]. The ridge has changed 
significantly from that of the 52 nm sample: a peak corresponding to the coercivity of the 
major loop is now virtually absent; instead a large peak at HC =1500 Oe has appeared, 
accompanied by two small negative regions nearby.  The 58 nm sample shows a FORC 
pattern representative of both the 52 and 67 nm samples [Figs. 2(e) and 2(h)].  The 
overall distribution resembles that of the 67 nm sample, with two peaks centered at HC = 
650 Oe and HB = ±400 Oe.  A ridge along HB = 0, peaking at roughly the major loop 
coercivity, is similar to that seen in the 52 nm sample.  Note that the 58 nm sample, being 
thinner, would tend to inhibit the formation of a VS in the smaller dots and therefore 
show magnetic reversal via a SD.  However, significant fractions of the nanodots in the 
ensemble are apparently reversing via a VS.    
The FORC distribution also allows us to extract quantitative information about 
the reversal processes.  Since each sample measured consists of ~ 1 billion nanodots with 
a distribution of sizes, a coercivity spread is contained in the FORC distribution.  For the 
52 nm sample, we have indeed extracted this distribution by projecting the ridge in Fig. 
2(g) onto the HC-axis (HB = 0), as shown as the open circles in Fig. 3(b).  The relative 
height gives the appropriate weight of nanodots with a given coercivity.  This extracted 
coercivity distribution can be compared with a simple theoretical calculation.  The 
coercivity of a SD particle undergoing reversal via a curling mode increases strongly with 
decreasing particle size d, according to   
HC −∝ ,     (2) 
where C1 and C2 are constants.28  Based on the mean nanodot size and the size 
distribution determined from SEM, we have calculated a coercivity distribution [solid 
circles in Fig. 3(b)]. A good agreement is obtained with that determined from the FORC 
distribution, after a rescaling of the latter by an arbitrary weight.  Thus for other non-
interacting single-domain particle systems with unknown size distributions, the FORC 
method may be used to extract that information. This is particularly important in 3D 
distributions of nanostructures where there is no direct image access to the individual 
dots, as is the case here for a 2D distribution. 
As we have demonstrated earlier, the FORC distribution ρ is extremely sensitive 
to irreversible switching.19 This is most convenient to see in the (H, HR) coordinate 
system (meaningful data is in H>HR), as non-zero values of ρ correspond to the degree of 
irreversibility along a given FORC. We have employed this capability to analyze the VS 
nucleation and annihilation for the 67 nm sample.  The complex butterfly-like pattern of 
Fig. 2(i) now transforms into irreversible switching mainly along line scans 1 and 2 in 
Fig. 4(b), which correspond to FORC’s starting at HR= 100 Oe and -1450 Oe, 
respectively [marked as bold with large open circles as starting points in Fig. 4(a)]. Along 
line scan 1 (HR=100 Oe), when applied field H=100 Oe, vortices have already nucleated 
in most of the nanodots. With increasing field H, ρ becomes non-zero and increases with 
H and peaks at 1320 Oe.  This corresponds to the annihilation of the vortices in majority 
of the nanodots, and eventually ρ returns to zero near positive saturation.  Line scan 2 
starts at HR= -1450 Oe, where the majority, but not all, of the nanodots have been 
negatively saturated.  As H is increased, a first maximum in ρ is seen at H= -100 Oe, 
corresponding to the nucleation of vortices within the nanodots. Between -100 Oe < H < 
1450 Oe, ρ is essentially zero, indicating reversible motion of the vortices through the 
nanodots.  A second ρ  maximum is found at H = 1450 Oe, as the vortices are 
annihilated. This is again followed by reversible behavior near positive saturation.  Note 
that along line scan 1, the vortices are annihilated from the same side of the nanodot from 
which they first nucleated, and thus the net magnetization remains positive; along line 
scan 2, the vortices nucleate on one side of the dot and are annihilated from the other, and 
consequently the net magnetization changes sign [Fig. 4(a)].  Interestingly the 
annihilation field along line scan 2, 1450 Oe, is larger than that along line scan 1, 1320 
Oe.  It seems more difficult to drive a vortex across the nanodot and then annihilate it. 
The peaks in Fig. 4(b) are rather broad, which is a manifestation of vortex nucleation and 
annihilation field distributions.  Also note that the interactions among the VS dots are 
expected to be negligible due to the high degree of flux closure, as confirmed by 
simulations.29 
IV. Simulations and Discussions 
For comparison, micromagnetic simulations have been carried out on nearly 
circular nanodots with 60 nm diameters and 20 nm thicknesses.30  We have used 
parameters appropriate for Fe (exchange stiffness A = 2.1 × 10-11 J/m, saturation 
magnetization Ms = 1.7 × 106 A/m, and anisotropy constant K = 4.8 × 104 J/m3).  Each 
polycrystalline nanodot is composed of 2 nm square cells that are 20 nm thick, where 
each cell is a different grain with a random easy axis.  A small cut on one side of the 
nanodot generates two distinct annihilation fields that depend on which side of the 
nanodot the vortex annihilates from. This exercise models the fact that our fabricated dots 
are not perfectly circular.31 We have simulated FORC’s generated by two nanodots with 
different orientations [Fig. 4(c)]: the edge-cut in one is parallel, and in the other at a 45º 
angle, to the applied field.  The simulated M-H curves show abrupt magnetization 
changes, corresponding to the nucleation, propagation, and annihilation of vortices.  
The corresponding FORC distribution is shown in Fig. 4(d).  Peaks in the 
simulated FORC distribution clearly indicate the nucleation and annihilation fields of the 
vortices which are apparent in Fig 4(c).  Along line scan 1 of Fig. 4(d), a vortex is already 
nucleated at HR = 100 Oe and subsequently annihilated at H = 2300 Oe (upper right 
corner).  Along line scan 2 with HR = -2450 Oe, a vortex is nucleated at H = -100 Oe and 
finally annihilated at H = 2450 Oe (lower right corner).  It is clear that the simulated 
FORC reproduces the key features of the experimentally obtained one in Fig. 4(b).  Here 
the asymmetric dot shape is essential to obtain a different annihilation field along scan 2 
than that along scan 1.  We have simulated the angular dependence of nucleation and 
annihilation fields in such circular dot with a small cut as the cut orientation is varied in a 
field.  We find that for most angles it is harder to annihilate a vortex from the opposite 
side of its nucleation site.  However, for a small range of angles near 45° it is actually 
slightly easier to annihilate from the opposite side.  It is the combination of these two 
behaviors that gives rise to the negative-positive-negative trio of features in the lower 
right portion of the FORC distribution.  The presence of similar features in the 
experimental data shown in Fig. 4(b) thus illustrates that the FORC distribution is also 
sensitive to variations of dot shapes in the array. Because only two dots are simulated, the 
features in the FORC distribution are much sharper than the experimental data where 
distributions of vortex nucleation and annihilation fields are present.  Including more dots 
in the simulation with different applied field orientations and size distributions would 
tend to broaden the features generated by the two dots simulated.      
Additionally, by selectively integrating the normalized FORC distribution20,21
corresponding to the SD phase (the aforementioned ridge centered at low coercivity 
values in Fig. 2), we can quantitatively determine the percentage of nanodots in SD state 
for each sample.  The SD phase fraction is 100%, 43%, and 10% for the 52, 58, and 67 
nm sample, respectively. Thus the 58 nm nanodots have a significant co-existence of both 
SD and VS states. However, we do not observe clear evidence of any additional 
metastable phase.15 
V. Conclusions 
In summary, we have used the FORC method to “fingerprint” the rich 
magnetization reversal behavior in arrays of 52, 58, and 67 nm sized Fe nanodots. 
Distinctly different reversal mechanisms have been captured, despite only subtle 
differences in the major hysteresis loops.  The 52 nm nanodots are in SD states.  A 
coercivity distribution has been extracted, which agrees with calculations. The 67 nm 
dots reverse their magnetization via the nucleation and annihilation of vortices. Different 
fields are required to annihilate vortices from opposite sides of the dots.  Quantitative 
measures of the vortex nucleation and annihilation fields have been obtained. OOMMF 
simulations confirm the experimental FORC distributions.   The 58 nm sample shows 
coexistence of SD and VS reversal, without evidence of additional reversal mode.  These 
results further demonstrate the FORC method as a simple yet powerful technique for 
studying magnetization reversal, due to its capability of capturing distributions of 
magnetic properties, sensitivity to irreversible switching, and the quantitative phase 
information it can extract. 
Acknowledgements 
This work has been supported by ACS (PRF-43637-AC10), AFOSR, and the 
Alfred P. Sloan Foundation. We thank J. E. Davies, J. Olamit, M. Winklhofer, C. R. Pike, 
H. G. Katzgraber, R. T. Scalettar, G. T. Zimányi, and K. L. Verosub for helpful 
discussions. R.K.D. acknowledges support from the Katherine Fadley Pusateri Memorial 
Travel Award. 
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Figure Captions 
Fig. 1. (color online). Scanning electron micrograph of the 67 nm diameter nanodot 
sample. Inset is a histogram showing the distribution of nanodot sizes. 
Fig. 2. First-order reversal curves and the corresponding distributions. Families of 
FORC’s (a-c), whose starting points are represented by black dots for the 52, 58, 
and 67 nm Fe nanodots, respectively. The corresponding FORC distributions are 
shown in 3-dimensional plots (d-f) and contour plots (g-i).  
Fig. 3. (color online). Projection of the FORC distribution ρ of the 52 nm nanodots onto 
(a) the HB-axis, showing weak dipolar interactions; and (b) the HC-axis (open 
circles), showing a coercivity distribution that agrees with a calculation based on 
measured size distribution (solid circles).   
Fig. 4. (a) A family of measured FORC’s for the 67nm diameter dots. (b) The 
corresponding experimental FORC distribution plotted against applied field H 
and reversal field HR.  (c) A family of simulated FORC’s generated using the 
OOMMF code.  Inset shows the orientations of the two dots simulated.  (d) The 
FORC distribution calculated from the simulated FORC’s shown in (c).  The two 
white dashed lines in (b) and (d) correspond to the two bold FORC’s whose 
starting points are large open circles in (a) and (c), respectively.  
Fig. 1, Dumas, et al. 
Fig. 2, Dumas, et al. 
Fig. 3, Dumas, et al. 
Fig. 4, Dumas, et al.
