First-order reversal curve diagram analysis of a perpendicular nickel nanopillar array

We apply a first-order reversal curve (FORC) diagram analysis to a perpendicular nickel nanopillar array. We find that the FORC diagram signature of this system consists of a two-branch “wishbone” structure. Two distinct negative regions are also observed, along with a prominent reversible ridge. The objective of this paper is to find a qualitative physical understanding or interpretation of these features. To accomplish this, we employ an interacting hysteron model. We find that a collection of symmetric hysterons with distributed coercivities and antiparallel mean field can account for the wishbone signature and one of the negative regions. By employing curvilinear hysterons, we can account for the reversible ridge and the other of the negative regions. Through a comparison of modeling and experimental work, we obtain a quantitative estimate of the dipolar interaction strength.

Figure 1

A first-order reversal curve (FORC) is acquired after saturating the sample in a positive applied field. The applied field is lowered to a reversal field $H_r$. A FORC is the magnetization curve that results when the applied field $H_a$ is increased back to saturation. The magnetization at the applied field $H_a$ on a FORC with reversal field $H_r$ is denoted by $M(H_a,H_r)$.

Figure 2

A hysteron $s$ with coercivity $h_c$ and bias $h_b$. A hysteron with zero bias is said to be symmetric.

Figure 3

(a) A FORC diagram for Sony high density floppy disk sample. In the legend for the contour shading, Max denotes the value of the FORC distribution at its “irreversible” peak located at about $H_c=90\mathrm{mT}$. A prominent reversible ridge appears on the $H_c=0$. Note that the high density of vertical contour lines near the $H_c=0$ axis makes the shading there appear darker than it actually is. A negative region occurs adjacent to the reversible ridge at about $H_b=-85\mathrm{mT}$. The shape of the reversible ridge is conveyed by the two cross sections in (b) and (c). In (b) we plot a horizontal cross section passing though the irreversible peak at $H_b=-5\mathrm{mT}$. In (c) we plot a vertical cross section through the reversible ridge at $H_c=0$.

Figure 4

Scanning electron micrograph of pillar sample.

Figure 5

(a) The first-order reversal curve (FORC) data for nickel pillar sample. To make it easier for the eye to resolve individual curves, only 70 of the 140 measured FORCs are shown. (b) The FORC distribution generated from this data. Max denotes the value of the distribution at the “irreversible” peak located at about $H_c=23\mathrm{mT}$, $H_b=20\mathrm{mT}$. On the $H_c=0$ vertical axis is a sharply peak ridge due to reversible magnetization.(c) The vertical cross section through the “reversible” ridge at $H_c=0$ as a function of $H_b$.

Figure 6

The FORCs and the FORC distribution of a collection of hysterons with mean field interactions: (a) and (b) a narrow Gaussian coercivity distribution and $J=21.1\mathrm{mT}$; a broader Gamma distribution with $J=0$ [(c) and (d)] and with $J=21.1\mathrm{mT}$ [(e) and (f)]. Only 70 of the 140 calculated FORCs are shown in (a), (c), and (e). Many of these FORCs coincide and cannot be distinguished in this plot.

Figure 7

The calculated FORC distribution of a 2-D array of hysterons with gamma distribution of coercivities and local antiparallel interactions $(J=21.1\mathrm{mT})$. Two types of interactions were used. In (a), a $3\times 3$ local interaction neighborhood with uniform coupling [see Eq. (10)]. In (b), “dipolar” interactions with a $11\times 11$ interaction neighborhood [see Eq. (11)].

Figure 8

The calculated FORC distribution with “dipolar” interactions and disorder. A gamma distribution of coercivities is used and $J=21.1\mathrm{mT}$. In (a), a random term is inserted into the coupling with ${\sigma }_j=0.35$ [see Eq. (12)]. In (b), a random bias field is inserted at each hysteron site with ${\sigma }_b=0.076$ [see Eq. (13)]. In (c), a random bias field and a random term in the coupling are combined, with ${\sigma }_b=0.049$ and ${\sigma }_j=0.24$ [see Eq. (14)].

Figure 9

(a) The curvilinear hysterons superimposed upon the distribution of coercivities. (b) The numerically calculated FORC distributions with “dipole” interactions and curvilinear hysterons. (c) The vertical cross section through the reversible ridge of the FORC distributions at $H_c=0$. The dotted line is from the experimental data in Fig. 5c for comparison.

Figure 10

The overlay of modeled and experimental FORC distributions. The contour lines of the experimental result are widened and lightened (a) with “optimal” parameters, (b) with a broader distribution of coercivities, and (c) with decreased coupling strength. The insets are to emphasize that the decreased coupling lowers the vertical location of the upper peak.