3.1. DC Magnetization Curves
As the magnetization of the Cu-rich matrix was negligible compared to that of the NiFe-Cu nanocubes [
22], the measured magnetic property of the Cu
75Ni
20Fe
5 sample was considered to have originated primarily from the NiFe-Cu nanocubes.
Figure 3 shows the magnetization curve (major loop) obtained under the maximal DC magnetic field,
HDC, of 1200 kA/m applied along each crystallographic direction of the NiFe-Cu nanocubes. The values of the measured magnetization are normalized by the saturation magnetization of the sample, and indicated as
M/
MS. The saturation magnetization,
MS, was found to be 116 kA/m. The TEM image shown in
Figure 1 suggests that the NiFe-Cu nanocubes have shape anisotropy in their magnetization, and predicts that the <100> direction is an easy axis along the longer axis of the rectangular aggregates. Nevertheless,
Figure 3b shows a larger magnetization under the applied magnetic field along the <111> direction than that obtained by applying the magnetic field along the <100> direction. The remanent magnetization for the <111> direction was slightly larger than that for the <100> direction, as shown in
Figure 3c. Regardless of the size of the nanocubes, which was in the order of 10 nm, the sample did not exhibit superparamagnetism.
Figure 4 shows the DC minor loops of the sample obtained under various
HDC. The remanent magnetization and coercive force derived from these minor loops are shown in
Figure 5. Both the remanent magnetization and coercive force were independent of the crystallographic direction of the applied DC field for
HDC <64 kA/m. They increased with increasing
HDC, and almost saturated at
HDC = 128 kA/m. The saturated values of the remanent magnetization and coercive force obtained from the DC major loops (
HDC = 1200 kA/m) are also plotted as dotted lines in the figures. The remanent magnetization and coercive force for the <111> direction were larger than those for the <100> direction. The magnetization curves shown in
Figure 3, along with these analyses, suggest the magnetocrystalline anisotropy of the NiFe-Cu nanocubes and the ease of magnetization along the <111> direction. The NiFe-Cu nanocubes have the FCC crystal structure and their composition is rich in Ni, as shown in
Table 1. Accordingly, the easy magnetization axis along the <111> direction in the NiFe-Cu nanocubes is appropriately similar to that of Ni with the FCC crystal structure [
25,
26].
Energy density of the magnetocrystalline anisotropy of cubic crystals,
EA is given by [
26,
27,
28]
where
K0,
K1, and
K2 are the magnetocrystalline anisotropy constants and α
1, α
2, and α
3 are the direction cosine of the magnetization to each axis of
hkl coordinates of the cubic crystal, respectively. The energy density required to magnetize a sample in the crystallographic axis <
hkl>,
is
where,
μ0 is the permeability of free space.
,
, and
. From these equations as well as the measured initial magnetization curves of the sample traced from its virgin state, the magnetocrystalline anisotropy constants are calculated to be
K1 = −1.3 × 10
3 J/m
3 and
K2 = −3.9 × 10
3 J/m
3. Both
K1 and
K2 are negative, similar to the magnetocrystalline anisotropy constants of Ni with an FCC crystal structure, which exhibits easy magnetization along the <111> direction. The atomic ratio of Ni/Fe of the sample is 65/35, which is in the range of that of the NiFe alloy, a permalloy. The
K1 of the permalloy varies depending on the order phase or disorder phase, the composition, and the heat treatment. Nevertheless, it is normally small, of the order of
1 kJ/m
3 [
15].
3.2. AC Magnetization Curves
The evaluation of the dynamic magnetization properties of magnetic nanostructures is essential for developing practical applications as well as for understanding their physics.
Figure 6a shows the DC magnetization curve of the sample obtained with
HDC = 1200 kA/m along the <100> direction. It also shows a schematic indicating the method of AC magnetization measurement with and without the DC bias field,
HB = 1200 kA/m.
Figure 6b,c show the AC magnetization curves of the sample under the bias fields of
HB = 0 and 1200 kA/m, respectively. The highest frequency of
HAC used for the AC magnetization measurement in this study was 100 kHz. The skin depth of Cu at 100 kHz was 0.21 mm, which suggests that
HAC was not substantially applied to the entire volume (depth) of the sample. However, the measured values of magnetization did not decrease under the applied field frequency, up to 100 kHz, as shown in
Figure 6b. The magnetization shown in
Figure 6b is the result of the alternating magnetization reversal of the ferromagnetic NiFe-Cu nanocubes as well as the alternating magnetic field induced by the eddy current, whereas the magnetization shown in
Figure 6c only originated from the alternating magnetic signal induced by the eddy current. This is because the magnetization was saturated by applying a bias field,
HB, of 1200 kA/m, and no modulation of magnetization was facilitated by the applied AC field,
HAC, of 4 kA/m. The magnetic signal induced by the eddy current was almost zero at 1 kHz, and it increased with an increase in the applied field frequency. The effect of the eddy current induced by the applied AC magnetic field was negligible at a low frequency, but was enhanced at a higher frequency.
Figure 7 shows the intrinsic AC magnetization of the NiFe-Cu nanocubes that arose only from the magnetization reversal in response to the applied AC magnetic field. The magnetization curves were calculated by subtracting the waveform of the magnetization measured under
HB = 1200 kA/m from that measured under
HB = 0. The subtraction of magnetization curves is a validated method in extracting specific magnetization dynamics from superposed complex dynamics [
29]. The AC magnetization curves and their dependence on frequency shown in the figure are quite similar to those of the magnetic iron oxide nanoparticles [
30,
31]. The magnetization curves obtained under the applied DC field or AC field at the low frequency of 1 kHz exhibit approximately no remanent magnetization. The dependences of the maximum magnetization at
HAC = 4 kA/m, and of coercive force on the applied field frequency, are shown in
Figure 7b,c. The magnetization decreased while the coercive force increased with increasing frequency. This was caused by a delay in the response of magnetization reversal against the change in the applied field [
32].
The magnetization of magnetic nanoparticles was reversed by the Néel relaxation and Brownian relaxation processes. These are the result of the rotation of magnetization in each particle and the rotation of the particle itself, respectively [
33,
34,
35]. As the NiFe-Cu nanocubes were dispersed in a solid Cu-rich matrix, the magnetization reversal of the nanocubes was accompanied only by the Néel relaxation process. The Néel relaxation time,
τN, of a magnetic nanostructure exhibiting a single magnetic domain is expressed as
where
τ0 (=1.0 × 10
‒9 s),
Ka,
Vc,
kB, and
T (=300 K) denote the attempt time, anisotropy constant, volume of the sample, Boltzmann constant, and temperature, respectively.
Ka is determined to be 580 J/m
3 by
, where <111> and <100> are the easy and hard axes of the sample, respectively. Here, we employed the volume of the NiFe-Cu nanocubes as
Vc, which is reasonable for Equation (3), assuming the magnetization reversal of the single domain structure. The Néel relaxation time was calculated to be 43 ns, which corresponds to the frequency of 3.7 MHz. The magnetization response is delayed with respect to the applied AC magnetic field even at 50 and 100 kHz, as shown in
Figure 7. Nevertheless, this is also commonly observed in the AC magnetization curves of magnetic nanoparticles [
36]. Because of its exponential dependence on the volume and practical distribution of the size of the NiFe-Cu nanocubes, the Néel relaxation time of the nanocubes presumably varies over a wide range of the order of 10 kHz to 100 MHz. The frequency dependence of the obtained magnetization curves clearly indicates the typical feature of a magnetic nanostructure of a single domain.
As shown in
Figure 7b,c, both the maximum magnetization and coercive force are independent of the direction of the applied AC field along the crystallographic axes of <100>, <111>, and <110>. As shown in
Figure 5, the magnetocrystalline anisotropy between these axes is observed under applied magnetic fields larger than 50 kA/m, which cannot be practically achieved with an AC magnetic field of 10 kHz or higher. In general, a larger applied field intensity than the anisotropy energy is necessary to observe the anisotropic properties. Therefore, the magnetocrystalline anisotropy was not observed during the dynamic magnetization process in this measurement under
HAC = 4 kA/m. The anisotropic features in a dynamic magnetization process are observed when the magnetic field larger than the anisotropy energy is applied to the samples [
11,
37,
38,
39].
The SLP is derived by
where
f and
ρ are the frequency of the applied AC magnetic field and the mass density of the sample, respectively. The SLP of the NiFe-Cu nanocube under the applied field of
HAC = 4 kA/m at 100 kHz is calculated to be 1.6 W/g from the magnetization curve along the <100> direction, shown in
Figure 7a, which does not depend on the crystallographic direction of the applied field. This value of SLP is comparable with that of Resovist
® (SLP = 3.2 W/g), calculated for the same AC field condition [
4]. Resovist
® is a contrast agent that is clinically used in magnetic resonance imaging, which also exhibits a high SLP value for hyperthermia. We concluded that the NiFe-Cu nanocubes are an attractive material that can be used as a heating agent, and that a high temperature increase can be expected from them under the applied AC field intensity and frequency sufficient for hyperthermia.