Configuration dependent demagnetizing field in assemblies of interacting magnetic particles

The effective or total demagnetizing factor can be expressed using the interpolation procedure introduced by Netzelmann [24]. To simplify the notation, in the following the effective or total demagnetizing factor (N_eff) will be denoted as N_T which can be expressed in terms of the packing fraction P as [24, 25]

$$\begin{eqnarray} &&{N}_{\mathrm{T}}=(1-P)N+P{N}^{+}. \end{eqnarray} \tag{ 4 }$$

However, for the treatment of an assembly of magnetic particles it is more useful to rewrite this last expression as

$$\begin{eqnarray} &&{N}_{\mathrm{T}}=N+({N}^{+}-N)P. \end{eqnarray} \tag{ 5 }$$

The first term on the right-hand side corresponds to the self-demagnetizing factor of the particles that makeup the assembly, i.e. N_self, while the second term contains all the contributions that depend on the packing and corresponds to the dipolar interaction contribution N_dip, then

$$\begin{eqnarray} &&{N}_{\mathrm{T}}={N}_{\mathrm{self}}+{N}_{\mathrm{dip}}. \end{eqnarray} \tag{ 6 }$$

When P = 0 the demagnetizing factor reduces to that of an isolated non-interacting particle. The opposite limit, P → 1, leads to the outer demagnetizing factor. If both the inner and outer demagnetizing factors are diagonal and their trace is Tr(N) = 4π, then from equation (5)

$$\begin{eqnarray} &&\mathrm{Tr}({N}_{\mathrm{T}})=\mathrm{Tr}(N)+[\mathrm{Tr}({N}^{+})-\mathrm{Tr}(N)]P, \end{eqnarray} \tag{ 7 }$$

and from the form of (6) the following properties of N_T are obtained:

$$\begin{eqnarray} \mathrm{Tr}({N}_{\mathrm{T}})&=&4 \pi , \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} \mathrm{Tr}({N}_{\mathrm{dip}})&=&0. \end{eqnarray} \tag{ 9 }$$

The ith component of the effective demagnetizing field in the saturated state for an assembly of particles follows from equation (5)

$$\begin{eqnarray} &&{H}_{\mathrm{T}}^{\mathrm{D}i}={M}_{\mathrm{ s}}{N}_{i}+({N}_{i}^{+}-{N}_{i}){M}_{\mathrm{s}}P, \end{eqnarray} \tag{ 10 }$$

where the first term is the usual self-demagnetizing field of the particle, ${H}_{i}^{\mathrm{D}}={M}_{\mathrm{s}}{N}_{i}$, and the second term is the dipolar interaction field, this is

$$\begin{eqnarray} &&{H}_{\mathrm{dip}}^{i}=({N}_{i}^{+}-{N}_{i}){M}_{\mathrm{s}}P. \end{eqnarray} \tag{ 11 }$$

The total or effective magnetostatic anisotropy field in the saturated state is defined as ${H}_{\mathrm{T}}^{\mathrm{S}}={M}_{\mathrm{s}}({N}_{\mathrm{T}}^{x}-{N}_{\mathrm{T}}^{z})$, where the hard axis is along the x-direction and the easy axis is taken along the z-axis, then

$$\begin{eqnarray} &&{H}_{\mathrm{T}}^{\mathrm{S}}={M}_{\mathrm{ s}}\Delta N+(\Delta {N}^{+}-\Delta N){M}_{\mathrm{ s}}P. \end{eqnarray} \tag{ 12 }$$

The first term on the right-hand side is the shape anisotropy field of the individual particles (H_S), while the second term corresponds to the total or effective dipolar field in the saturated state:

$$\begin{eqnarray} &&{H}_{\mathrm{dip}}=(\Delta {N}^{+}-\Delta N){M}_{\mathrm{ s}}P. \end{eqnarray} \tag{ 13 }$$

The total shape (magnetostatic) anisotropy energy is given by ${K}_{\mathrm{T}}^{\mathrm{S}}={M}_{\mathrm{s}}{H}_{\mathrm{S}}^{\mathrm{T}}/2$, which includes both the shape anisotropy of the particle and the contribution of the dipolar interaction. This anisotropy constant now regroups K_S and E_int in equation (3).

Finally, from the expressions obtained for the interaction field there are two other relations of interest that follow when both the inner and outer volumes have in-plane symmetry, so that N_x = N_y. Due to the symmetry, the in-plane components of the interaction field are equal, ${H}_{\mathrm{dip}}^{x}={H}_{\mathrm{dip}}^{y}$, and since Tr(N_dip) = 0, according to equation (9), the following relation between the components of the dipolar fields is obtained:

$$\begin{eqnarray} &&{H}_{\mathrm{dip}}^{z}=-2{H}_{\mathrm{ dip}}^{x}. \end{eqnarray} \tag{ 14 }$$

This shows that when both the inner and outer volume have rotational symmetry, the dipolar interaction field along the symmetry axis is twice the dipolar field in the hard axis with opposite sign. This, for example, has been obtained by several authors [26, 27]. A consequence of this result is that the dipolar part of the total or effective anisotropy field (${H}_{\mathrm{dip}}={H}_{\mathrm{dip}}^{x}-{H}_{\mathrm{dip}}^{z}$) is

$$\begin{eqnarray} &&{H}_{\mathrm{dip}}=-\frac{3}{2}{H}_{\mathrm{dip}}^{z}, \end{eqnarray} \tag{ 15 }$$

which provides a relation between the component of the interaction field along the easy axis, with the net contribution of the interaction field to the total anisotropy field.

To extend the previous expressions to the non-saturated case, consider that the particles are bistable, this is, they can only be magnetized along their easy axis in both the positive or negative direction. The normalized magnetization, m = M(H)/M_s, so that −1 ≤ m ≤ 1. Any magnetic state, called hereafter m, can be written in terms of the fraction of particles magnetized in the positive (m₊) and negative (m₋) directions, with 0 ≤ m_± ≤ 1, and m = m₊ − m₋. Moreover, since the number of particles is constant, m₊ + m₋ = 1, and m_± can be obtained from the value of m as

$$\begin{eqnarray} &&{m}_{\pm }=\frac{1\pm m}{2}. \end{eqnarray} \tag{ 16 }$$

A recent FMR study done on non-saturated arrays of bistable nanowires led to the expressions for the effective field, the dipolar interaction term and the FMR dispersion relation that include explicitly the magnetic configuration of the system [28, 29]. In particular, it was shown that to extend the known expressions for the saturated case to the non-saturated case, it is necessary to rewrite the dipolar interaction term as a parametric expression of m_± [28, 29]. Assuming a dipolar interaction field of the form αm, it implies that the dependence of the interaction field on m is not as a direct product but instead requires using equation (16); this is

$$\begin{eqnarray} &&{H}_{\mathrm{dip}}{m}_{\pm }=\frac{{H}_{\mathrm{dip}}}{2}\pm \frac{{H}_{\mathrm{dip}}}{2}m, \end{eqnarray} \tag{ 17 }$$

which at saturation reduces to the expected value. The ± sign is introduced to include both positive (m = 1) and negative (m =− 1) saturation.

Using equation (10), the ith component of the effective demagnetizing field is

$$\begin{eqnarray} &&{H}_{\mathrm{T}}^{\mathrm{D}i}={M}_{\mathrm{ s}}{N}_{i}+({N}_{i}^{+}-{N}_{i})\frac{{M}_{\mathrm{s}}P}{2}\pm ({N}_{i}^{+}-{N}_{i})\frac{{M}_{\mathrm{s}}P}{2}m. \end{eqnarray} \tag{ 18 }$$

The dipolar contribution now contains two terms, the first one is constant while the second one is proportional to m and corresponds to the magnetization dependent part of the demagnetizing field. If the magnetization dependent interaction field is taken as αm, then from the last term on the right-hand side, the ith component of the dipolar interaction field coefficient α can be identified as

$$\begin{eqnarray} &&{\alpha }_{i}=\frac{{H}_{\mathrm{dip}}^{i}}{2}=({N}_{i}^{+}-{N}_{i})\frac{{M}_{\mathrm{s}}P}{2}. \end{eqnarray} \tag{ 19 }$$

Then equation (18), can be written as

$$\begin{eqnarray} &&{H}_{\mathrm{T}}^{\mathrm{D}i}={H}_{i}^{\mathrm{D}}+{\alpha }_{i}\pm {\alpha }_{i}m, \end{eqnarray} \tag{ 20 }$$

where ${H}_{i}^{\mathrm{D}}$ is the demagnetizing field of the isolated particle. From equation (12), the effective or total magnetostatic anisotropy field is

$$\begin{eqnarray} &&{H}_{\mathrm{T}}^{\mathrm{S}}={H}_{\mathrm{ S}}+{\alpha }_{\mathrm{T}}\pm {\alpha }_{\mathrm{T}}m, \end{eqnarray} \tag{ 21 }$$

and α_T = α_x − α_z is the total dipolar field coefficient. At saturation m = 1, both equations (20) and (21) reduce to equations (10) and (12), respectively.

As seen from these expressions, the effective demagnetizing field accounts for both the shape anisotropy of a particle in the assembly and the dipolar interaction it experiences. However, the existence of other anisotropy contributions, such as magnetocrystalline or magnetoelastic ones, will simply result in an additive term to the particle shape anisotropy. Indeed, the shape anisotropy as well as other anisotropy contributions will result in the effective anisotropy of the particle and will define the effective energy barrier of each of the particles in the assembly, and corresponds to the coefficient K_A in equation (3) while the interaction field is accounted for by another term in the energy.

An important class of magnetic assemblies are 2D arrays of exchange decoupled nanoparticles, ideally a monolayer of single domain particles, with perpendicular magnetization. These correspond, for example, to systems obtained by lithography, including perpendicular bit patterned media, also granular thin films with columnar structure like those used for perpendicular recording media, self-assembled monolayers and nanowire arrays [8, 9].

For these systems one can assume that the height of the particles is very small compared with the lateral dimensions of the entire array and the volume containing the particles can be considered as an infinite thin film, so that ${N}_{x}^{+}={N}_{y}^{+}=0$ and ${N}_{z}^{+}=4 \pi$, as depicted in figures 2(a) and (b). Using these values in equation (10) and assuming particles with in-plane symmetry so that N_x = N_y and using N_z = 4π − 2N_x to express quantities in terms of N_x, the effective demagnetizing field perpendicular to the plane (z) is

$$\begin{eqnarray} &&{H}_{\mathrm{T}}^{\mathrm{D}z}=4 \pi {M}_{\mathrm{ s}}-2{N}_{x}(1-P){M}_{\mathrm{s}}. \end{eqnarray} \tag{ 22 }$$

This shows that the effective demagnetizing field of a thin film made of exchange decoupled entities is less than 4πM_s by a quantity equal to 2N_x(1 − P)M_s.

The total shape anisotropy field (equation (21)) is

$$\begin{eqnarray} &&{H}_{\mathrm{T}}^{\mathrm{S}}=(3{N}_{x}-4 \pi ){M}_{\mathrm{s}}-\frac{3}{2}{N}_{x}{M}_{\mathrm{s}}P\mp \frac{3}{2}{N}_{x}{M}_{\mathrm{s}}P m. \end{eqnarray} \tag{ 23 }$$

The first term on the right-hand side is the shape anisotropy of the particle, while the second and third terms correspond to the dipolar part of the total anisotropy field, for m = 1,

$$\begin{eqnarray} &&{H}_{\mathrm{dip}}=-3{N}_{x}{M}_{\mathrm{s}}P, \end{eqnarray} \tag{ 24 }$$

which is a particular case of equation (15). As seen from this expression H_dip is only a function of N_x and P. Since the particles have in-plane symmetry, then N_z = 4π − 2N_x, from where it follows that if 0 ≤ N_z ≤ 4π, then 0 ≤ N_x ≤ 2π. And from the previous expression for H_dip, it is possible to establish an upper bound for the dipolar part of the total or effective anisotropy. Taking N_x = 2π,

$$\begin{eqnarray} &&{H}_{\mathrm{dip}}^{\mathrm{max}}=-6 \pi {M}_{\mathrm{ s}}P. \end{eqnarray} \tag{ 25 }$$

The dipolar contribution to the total anisotropy field is antiferromagnetic, inferred by the negative sign in equation (24), as expected for a 2D array of nanomagnets with perpendicular anisotropy.

Regarding the influence of the geometrical arrangement of the particles in a 2D assembly, this can be taken into account by using a proper definition for the packing fraction, since as seen in equations (23) and (24) the dipolar interaction is proportional to P. For example, the packing fraction for 2D hexagonal and square arrays are $P=(\pi {\phi }^{2})/(2\sqrt{3}{d}^{2})$ and P = (πϕ²)/(4d²), respectively. Since these packings are not equal the dipolar interaction does depend on the arrangement of the particles.

In the particular case of a 2D array of circular cylinders of arbitrary height aligned parallel to each other so their axes are along the z axis, as shown in figure 2(b), their effective demagnetizing fields along the easy and hard directions are given by equation (10). As in the previous section, the outer demagnetizing factor is taken as an infinite thin film. For a cylinder of arbitrary aspect ratio (wire height divided by its diameter), N_z can be determined numerically [30]. However, for the nanowires considered in this study, the wire height is typically of the order of 20 μm so the aspect ratio is very high, and they can be considered as infinite so their demagnetizing factors are N_z = 0 and N_x = N_y = 2π and from equation (10) the demagnetizing field along the easy and hard axes are

$$\begin{eqnarray} {H}_{\mathrm{T}}^{\mathrm{D}z}&=&4 \pi {M}_{\mathrm{ s}}P, \end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray} {H}_{\mathrm{T}}^{\mathrm{D}x}&=&2 \pi {M}_{\mathrm{ s}}-2 \pi {M}_{\mathrm{s}}P. \end{eqnarray} \tag{ 27 }$$

Due to the symmetry properties of both inner and outer demagnetizing factors, ${H}_{\mathrm{T}}^{\mathrm{D}x}={H}_{\mathrm{T}}^{\mathrm{D}y}$ and the dipolar terms satisfy both equations (9) and (14). The effective anisotropy field is

$$\begin{eqnarray} &&{H}_{\mathrm{T}}^{\mathrm{S}}=2 \pi {M}_{\mathrm{ s}}-6 \pi {M}_{\mathrm{s}}P, \end{eqnarray} \tag{ 28 }$$

which corresponds to the known expression for the effective field for an array of infinite cylindrical nanowires in the saturated state [18]. The effective dipolar interaction field is H_dip =− 6πM_sP, and using equations (19) and (21), the magnetization dependent part of the effective interaction field is

$$\begin{eqnarray} &&{\alpha }_{\mathrm{T}}=\frac{{H}_{\mathrm{dip}}}{2}=3 \pi {M}_{\mathrm{s}}P, \end{eqnarray} \tag{ 29 }$$

in agreement with the expression obtained in [29].

On the other hand, the measurement of the interaction field using equation (2) and the IRM and DCD remanence curves is done solely along the wire axis, this is, the measurement is related to the component of the interaction field along this direction, as pointed out in [23]. From equation (26), and equations (19) and (20), the component of the magnetization dependent part of the interaction field along the wire axis is

$$\begin{eqnarray} &&{\alpha }_{z}=2 \pi {M}_{\mathrm{s}}P. \end{eqnarray} \tag{ 30 }$$

Then equations (28), (29) and (30) provide different expressions for the interaction field which have different physical meanings. Moreover, depending on the measuring technique, each of these three quantities can be measured independently [18, 29, 23], so it is important to distinguish between them.

Since the effective anisotropy field ${H}_{\mathrm{T}}^{\mathrm{S}}$ can be determined from the dispersion relation obtained from the FMR measurements, using equation (1), then from equation (28), ${H}_{\mathrm{dip}}={H}_{\mathrm{T}}^{\mathrm{S}}-2 \pi {M}_{\mathrm{s}}$. So H_dip has been determined for each sample from the FMR measurements using the nominal values of M_s for Co (1400 emu cm⁻³), Ni₈₁Fe₁₉ (800 emu cm⁻³), and Co₅₅Fe₄₅ (1900 emu cm⁻³) and the values of H_eff given in table 1. These values where then divided by 3 so they correspond numerically to equation (30) which is the value of the component of the interaction field along the wire axis measured with the IRM and DCD remanence curves. Figure 3 shows the values obtained using the IRM and DCD remanence curves, α_z, plotted as a function of the values obtained by FMR, H_dip/3.

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These results show a very good agreement, which shows that equations (28) and (30) both provide correct results for the interaction field. Moreover, these two expressions provide expressions for quantities that can be measured independently, so it is possible to find a self-consistent method to determine both M_s and P. Since ${H}_{\mathrm{T}}^{\mathrm{S}}$ and α_z are determined from FMR and magnetometry measurements, respectively, then combining equations (28) and (30) leads to the following expressions for M_s and P:

$$\begin{eqnarray} {M}_{\mathrm{s}}&=&\frac{{H}_{\mathrm{T}}^{\mathrm{S}}+3{\alpha }_{z}}{2 \pi }, \end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray} P&=&\frac{{\alpha }_{z}}{{H}_{\mathrm{T}}^{\mathrm{S}}+3{\alpha }_{z}}. \end{eqnarray} \tag{ 32 }$$

The values of ${H}_{\mathrm{T}}^{\mathrm{S}}$ and α_z measured by FMR and the remanence curves, respectively, that are given in table 1 have been used as input in equations (31) and (32) to obtain the corresponding values of M_s and P. The results are given in table 1 in the columns labeled P_m and ${M}_{\mathrm{s}}^{\ast }$, respectively. As a first point, the values of ${M}_{\mathrm{s}}^{\ast }$ show a very good agreement with the known values for Co, NiFe and CoFe. Furthermore the values for NiFe and CoFe alloys are in good agreement with those determined solely by FMR and reported elsewhere [29, 31]. Regarding the values of the template porosity, comparing the nominal values determined by scanning electron microscopy P_N with those determined by the measurements P_m, a very good agreement is also found in practically all the samples. This procedure is general and can be extended to other assemblies by solving for M_s and P using equations (12) and (19).

So far, only 2D systems with perpendicular magnetization have been considered. Another interesting example is that of the linear chain of particles, and particularly for circular cylinders of different aspect ratio and spheres, like those shown in figures 2(c) and (d), respectively. For an infinite number of particles, the outer demagnetizing factor is an infinite cylinder of circular cross section whose axis is assumed along the z axis so ${N}_{x}^{+}={N}_{y}^{+}=2 \pi$ and ${N}_{z}^{+}=0$. While for both spheres and cylinders, the components of the inner demagnetizing factor are N_x = N_y and N_z = 4π − 2N_x. From equation (12), the effective anisotropy field is

$$\begin{eqnarray} &&{H}_{\mathrm{T}}^{\mathrm{S}}={M}_{\mathrm{ s}}({N}_{x}-{N}_{z})+{M}_{\mathrm{s}}(6 \pi -3{N}_{x})P. \end{eqnarray} \tag{ 33 }$$

For the calculations, the demagnetizing factors and the packing fraction are required. For the chain of spheres, N_x = N_y = N_z = 4π/3 and the packing fraction is P = (2/3)ϕ/d, where d is the center to center distance and ϕ is the diameter of the sphere. For the cylinders, N_z is determined using known expressions as a function of the aspect ratio τ = h/ϕ, where h is the height of the cylinder [30], and in this case P = h/d. Figure 4(a) shows the reduced effective anisotropy $\Delta N={H}_{\mathrm{T}}^{\mathrm{S}}/{M}_{\mathrm{s}}$ for linear chains of cylinders with different aspect ratio (τ = 0.5,0.8,2.5 and 5) as well as for a chain of spheres (dashed line).

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Consider first the case of the linear chain of cylinders. For a single, isolated and non-interacting circular cylinder, the critical aspect ratio is τ = 0.9065; above this value, N_x > N_z and the easy axis is along the cylinder axis and at lower values N_x < N_z it is perpendicular to the axis. As seen in figure 4(a) at large distances the anisotropy tends asymptotically to the expected value of the isolated non-interacting cylinder, which yields negative values for τ = 0.5 and 0.8, and positive for τ = 2.5 and 5.

As the particles are brought closer the interaction increases. For τ > 0.9065, the interaction is ferromagnetic as it favors head to tail alignment of the magnetization, which results in an increase of the effective anisotropy as the packing fraction increases. For low aspect ratios (τ < 0.9065) the easy axis of a given cylinder is perpendicular to the chain axis and the first term in equation (33) is negative, so the interaction becomes antiferromagnetic as it has the opposite sign, and competes against it. As the packing fraction is increased, the value of the interaction increases and overcomes the anisotropy of the particle leading to a reorientation of the magnetization easy axis.

In other words, to have a reorientation of the magnetization easy axis due to the dipolar interaction, the interaction has to be of the demagnetizing type (antiferromagnetic) with respect to the easy axis direction of the particle when considered isolated. For the linear chain of cylinders there is a critical aspect ratio for the easy axis reversal which depends on the interparticle distance that corresponds to those values for which the effective anisotropy vanishes. Setting ${H}_{\mathrm{T}}^{\mathrm{S}}=0$ in equation (33) and using 2N_x = 4π − N_z, the critical packing fraction can be expressed as

$$\begin{eqnarray} &&{P}_{\mathrm{c}}=1-\left (\frac{4 \pi }{3}\right )\frac{1}{{N}_{z}}, \end{eqnarray} \tag{ 34 }$$

which is only valid for N_z > 4π/3. Here N_z = N_z(τ) gives the dependence on the aspect ratio of the cylinder which is given by well known expressions [30]. Figure 4(b) shows the variation of the critical aspect ratio with the reduced center to center distance for a linear chain of cylinders. Above this curve, the magnetization lies along the cylinder axis and the interaction between them is ferromagnetic. Below this line, the magnetization easy axis is perpendicular to the cylinder axis and the dipolar interaction is antiferromagnetic. At large distances, this curve tends asymptotically to τ = 0.9065 (horizontal dashed line) which is the critical aspect ratio for an isolated non-interacting cylinder with N_z = 4π/3 [30]. The highest packing fraction required to reverse the easy axis is obtained when the axial demagnetizing factor of the disk, N_z, tends to 4π, which, from equation (34), corresponds to P_c = 2/3.

Finally, the linear chain of spheres presents features that are interesting, as seen in figure 4(a), in particular the existence of an effective anisotropy and its increase as the packing fraction increases, which in light of equation (33) is entirely due to the dipolar interaction. Indeed, as seen in the figure, the effective anisotropy tends to zero as the distance between particles is increased, consistent with a single sphere with no shape anisotropy. Moreover, the maximum value of the anisotropy is attained when the spheres come in contact and this value is lower than 2π, which shows that there is not a full equivalence between a cylindrical wire and the chain of spheres. This point has been brought to light recently and is important since modeling of cylindrical wires is commonly done by considering them as a chain of spheres [32, 33]. From equation (33), the maximum value of the effective anisotropy for the chain of spheres is reached at P = 2/3 and

$$\begin{eqnarray} &&{H}_{\mathrm{T}\mathrm{max}}^{\mathrm{S}}=\frac{4 \pi }{3}{M}_{\mathrm{s}}, \end{eqnarray} \tag{ 35 }$$

as indicated in figure 4(a).