Accordingly, the observed difference in the hysteresis loops must be attributed to different λs values and sign.
Therefore, the modification of the magnetic anisotropy of glass-coated microwires is expected upon heating.
The experimental results on the temperature dependence of the magnetic properties and GMI effect are provided below.
3.1. Effect of Temperature on Magnetic Properties and GMI of Fe-Rich Amorphous Microwires
The temperature dependence of the hysteresis loops of Fe75B9Si12C4 is provided in Figure 3. A substantial hysteresis loops modification from rectangular into inclined upon heating is clearly seen (see Figure 3a–j). With increasing temperature, T, the hysteresis loop becomes more inclined. However, a change in this tendency is observed at T > 250 °C (see Figure 3d,e,k).
This change can be better seen in Figure 3l, which shows the temperature dependence of the magnetic anisotropy field, Hk, evaluated from the hysteresis loops. At low temperatures (T < 150 °C), the Hk,(T) dependence appears linear. One of the reasons for the temperature dependence of Hk must be attributed to the change in the internal stresses [43]. Indeed, the linear dependence of Hk versus applied stress was previously reported for Co-rich microwires with vanishing negative λs [28]. Accordingly, the observed Hk,(T) roughly correlates with Equation (3) for T ≤ 150 °C, while at T > 150 °C, the Hk,(T) dependence becomes more complex. Such a deviation from the linear tendency in Hk,(T) correlates with the change in the character of the hysteresis loops at T ≥ 150 °C. As can be observed from Figure 4, the characters of the hysteresis loops measured at room temperature and at T = 150 °C are rather different: the magnetic bistability related to the perfectly rectangular hysteresis loop disappears upon heating to 150 °C (see Figure 4a–c).
On the other hand, the hysteresis loop measured after heating to 300 °C and cooling to room temperature restores its rectangular shape (see Figure 5). Accordingly, the observed changes in the hysteresis loops upon heating are almost completely reversible.
The origin of the GMI effect is satisfactorily explained from the point of view of the skin effect of magnetic conductors with high magnetic permeability [7,9,18]. As discussed elsewhere [9,11,44,45,46], the relationship between skin depth Δ and the circumferential magnetic permeability μϕ of magnetic wires is expressed as follows:
where σ is the electrical conductivity and f is the electrical current frequency.
where σ is the electrical conductivity and f is the electrical current frequency.
Therefore, it can be assumed that the observed change in the hysteresis loops upon heating should be related to a modification of the GMI effect.
The temperature dependence of the GMI effect of the Fe75B9Si12C4 microwire measured at different frequencies, f, is depicted in Figure 6. The main feature observed at all f is a remarkable increase in the GMI ratio at T = 300 °C. The maximum GMI ratio, ΔZ/Zmax, of about 180% observed at T = 300 °C and f = 110 MHz is comparable to ΔZ/Zmax reported at room temperature for Co-rich microwires elsewhere [45].
The temperature dependencies of ΔZ/Zmax evaluated from ΔZ/Z(H) dependencies measured at different T for 10, 50, and 110 MHz are summarized in Figure 7. From the ΔZ/Zmax(T) dependencies, an increase in ΔZ/Zmax at T ≥ 250 °C is clearly appreciated.
The observed increase in ΔZ/Zmax upon the heating of Fe75B9Si12C4 microwire should be associated with several processes during heating.
One of such factors is the transformation of the hysteresis loops from rectangular into inclined upon the heating of the Fe75B9Si12C4 sample. Generally, it is assumed and experimentally demonstrated that the GMI effect is usually rather small in microwires with a rectangular character of hysteresis loops owing to low circumferential magnetic permeability [45]. The domain structure of such microwires exhibiting spontaneous magnetic bistability (with high positive λs) is assumed to be consisting of an inner single domain magnetized axially and a radial magnetized outer domain shell [8,12,45]. As mentioned above, the axial character of the magnetic anisotropy of glass-coated microwires with positive λs is attributed to the highest axial component of the internal stresses induced by the simultaneous rapid solidification of metallic nucleus and glass coating with rather different αm and ag values (see Equation (2)) [45,47,48,49]. As discussed above, the σi values associated with the different αm and ag values are expected to decrease during heating.
The Hopkinson effect should also be involved in a noticeable increase in the GMI ratio observed at T = 300 °C. The origin of the Hopkinson effect is explained by considering a faster decrease in the magnetic anisotropy constant with temperature as compared to magnetization [50]. Therefore, a sharp maximum in magnetic permeability at temperatures slightly below the Curie temperature Tc observed in magnetic materials is commonly explained by the Hopkinson effect [50,51]. Such interpretation looks reasonable considering a decrease in the magnetic anisotropy field at T ≥ 250 °C observed in Hk(T) dependence (see Figure 3c).
Another factor that can influence the temperature dependence of the hysteresis loops and the GMI effect is the internal stresses’ relaxation associated with heating. As shown in Figure 5, the hysteresis loops measured at room temperature after heating to 300 °C remain almost unchanged. However, the skin depth at 100 MHz in magnetically soft microwires can be substantially smaller than the microwire diameter [52].
The attempt to separate the effect of heating from the effect of the internal stresses’ relaxation is provided in Figure 8, where the ΔZ/Z(H) dependencies measured at room temperature before and after heating up to 300 °C and at ΔZ/Z(H) dependencies measured at T = 300 °C are presented. Some increase in the ΔZ/Zmax value at room temperature after heating to 300 °C is evident from the provided ΔZ/Z(H) dependences (see Figure 8). The observed difference in ΔZ/Zmax at room temperature before and after heating to 300 °C must be associated with the relaxation of internal stresses. However, the main contribution to the increase in ΔZ/Zmax is related to the heating itself, since the ΔZ/Zmax values obtained at T = 300 °C are almost twice as high. This difference in ΔZ/Zmax is observed at various frequencies (see Figure 8a,b).
Accordingly, the observed temperature dependence of the GMI effect and magnetic properties of Fe-rich microwires can be useful for temperature monitoring. However, the effect of heating must be separated from the internal stresses’ relaxation upon heating.
3.2. Effect of Heating on Magnetic Properties and GMI of Co-Rich Amorphous Microwires
As can be appreciated from Figure 2b, Co69.2Fe3.6Ni1B12.5Si11Mo1.5C1.2 microwires present rather good magnetic softness with coercivity, Hc, of about 4 A/m and a magnetic anisotropy field, Hk, of about 180 A/m at room temperature.
Similarly to Fe-rich microwires, in Co69.2Fe3.6Ni1B12.5Si11Mo1.5C1.2 microwires upon heating, a significant change in the hysteresis loops is observed (see Figure 9): the shape of the hysteresis loops changes from almost linear to nearly rectangular. The observed modification in the hysteresis loop character is directly opposite to that observed in the Fe75B9Si12C4 microwire.
The origin of the observed modification in hysteresis loop shape upon heating can be understood considering the similar transformation of linear hysteresis loops into rectangular upon the annealing of Co-rich microwires with nearly zero λs [39,46,52]. The origin of such a transformation in the shape of hysteresis loops from linear to rectangular is explained in terms of the change in the λs value and sign due to internal stresses’ relaxation [39,46,52].
As discussed elsewhere [45,53], the ΔZ/Z magnitude and shape of ΔZ/Z(H) dependencies are related to the magnetic anisotropy of the magnetic wires. Therefore, one can expect changes in the ΔZ/Z value and ΔZ/Z(H) dependencies upon heating. As expected, the observed change in the magnetic anisotropy of Co69.2Fe3.6Ni1B12.5Si11Mo1.5C1.2 microwires correlates with the changes in the ΔZ/Z value and ΔZ/Z(H) dependencies. The effect of temperature on the ΔZ/Z(H) dependencies can be observed in Figure 10, where the ΔZ/Z(H) dependencies measured at different temperatures T are provided. As can be appreciated from Figure 10 and Figure 11, a gradual change from double-peak to single-peak ΔZ/Z(H) dependence takes place upon heating at 150 ≤ T ≤ 200 °C (see Figure 10). Thus, at T = 150 °C, a double peak ΔZ/Z(H) dependence can still be observed for all the frequencies (see Figure 11b). However, as compared to the ΔZ/Z(H) dependence measured at room T, the magnetic field of ΔZ/Z(H) maximum Hm becomes smaller. Finally, at T ≥ 200 °C, single-peak ΔZ/Z(H) dependencies are observed for all f values (see Figure 11c).
The aforementioned field of maximum Hm on ΔZ/Z(H) dependencies is attributed to the magnetic anisotropy field at MHz frequencies [45,53]. Thus, single-peak ΔZ/Z(H) dependencies correspond to the case of magnetic wires with axial magnetic anisotropy [45,53]. Accordingly, the observed modification in ΔZ/Z(H) dependencies upon heating correlates with the temperature dependence of the hysteresis loops, shown in Figure 9. Such changes must be attributed to the change in the magnetic anisotropy from weak transverse magnetic anisotropy to axial magnetic anisotropy upon heating.
As can be seen from Figure 11, despite the axial character of the hysteresis loop upon heating, the ΔZ/Zmax values remain relatively high. However, from a comparison of the ΔZ/Z(H) dependencies measured in Co-rich samples with the same character of hysteresis loops (see Figure 12), some increase in the ΔZ/Zmax values is visible with increasing T.
The aforementioned features of the ΔZ/Z(H) dependencies are summarized in Figure 13, where the dependencies of ΔZ/Zmax and Hm versus temperature are provided. A decrease in ΔZ/Zmax values between 150 and 200 °C is followed by an increase in ΔZ/Zmax values at T ≥ 250 °C (see Figure 13a). Generally, a decrease in ΔZ/Zmax values can be associated with the lower magnetic permeability of microwires with rectangular hysteresis loops. Typically, a lower GMI effect is reported elsewhere for magnetic wires with rectangular hysteresis loops [46,52]. However, the ΔZ/Zmax values obtained at T = 300 °C (for f = 110 MHz) are even higher than for room T (see Figure 13). Accordingly, similarly to that discussed above for the case of Fe-rich microwires, an increase in ΔZ/Zmax values at T ≥ 250 °C can be attributed to the Hopkinson effect. Such an interpretation looks reasonable, since the Curie temperature Tc of the studied Co69.2Fe3.6Ni1B12.5Si11Mo1.5C1.2 sample is about 325 °C [54,55].
A gradual decrease in Hm upon heating reflects the change in both hysteresis loops and ΔZ/Z(H) dependencies upon heating associated with the magnetic anisotropy change from transverse to axial (see Figure 13b).
Accordingly, similarly to the case of Fe-rich microwire, three main factors that affect the temperature dependence of hysteresis loops and the GMI effect must be underlined: (i) the Hopkinson effect, (ii) internal stresses’ relaxation, and (iii) temperature dependence of the internal stresses.
As mentioned above, the Hopkinson effect is characterized by a sharp maximum in magnetic permeability at temperatures slightly below Tc [50,51,56,57]. Therefore, an increase in ΔZ/Zmax and a decrease in Hm observed in both of the studied samples at T ≥ 250 °C (see Figure 3l, Figure 7 and Figure 13a,b) can be attributed to the Hopkinson effect.
The difference between the influence of the internal stresses’ relaxation and the temperature dependence of internal stresses is that in the latter case the changes are reversible, while the internal stresses’ relaxation produces irreversible changes in magnetic properties.
In the case of the studied Co69.2Fe3.6Ni1B12.5Si11Mo1.5C1.2 sample, the contribution of the internal stresses’ relaxation upon heating looks more relevant than for the Fe75B9Si12C4 sample microwire. This contribution is evidenced by an irreversible change in the ΔZ/Z(H) dependencies and ΔZ/Zmax values of the studied Co-rich microwire after heating to 300 °C, followed by its cooling to room temperature (see Figure 14). Both the ΔZ/Z(H) dependencies and ΔZ/Zmax values of the studied Co-rich microwire are substantially affected by heating up to 300 °C.
A change in ΔZ/Z(H) dependencies from double-peak to single-peak ΔZ/Z(H) dependence is observed after heating to 300 °C at all the measured frequencies (see Figure 14a,b). Additionally, a substantial decrease in ΔZ/Zmax values is observed. This decrease is more noticeable at f = 10 MHz (see Figure 14a).
As discussed before, the single-peak ΔZ/Z(H) dependence is theoretically predicted and experimentally confirmed for magnetic wires with an axial character of magnetic anisotropy [45,46,53]. Therefore, the observed irreversible changes in ΔZ/Z(H) dependencies after heating must be attributed to the magnetic anisotropy changes after heating. The hysteresis loops measured at room temperature after heating to 200 °C and 300 °C are provided in Figure 15. Observed substantial changes after heating must be attributed to the change in the magnetic anisotropy from weak transverse magnetic anisotropy to axial magnetic anisotropy after heating.
It must be noticed that a modification of hysteresis loops in various Co-rich microwires from linear to rectangular upon annealing and even upon heating was recently observed and interpreted considering the change in the magnetostriction coefficient value and even sign upon annealing due to the stress dependence of the magnetostriction [38,58,59]. The relevant influence of internal stresses’ relaxation on the magnetostriction coefficient value and sign in Co-rich amorphous alloys with vanishing magnetostriction was reported and discussed in terms of the changes in the local atomic environment, clustering, and the internal stresses’ relaxation [58,59,60].
Accordingly, in contrast to the Fe-rich sample, the contribution of the irreversible changes in ΔZ/Z(H) dependencies related to stresses relaxation is more relevant in Co-rich microwire. Therefore, previous annealing of Co-rich microwires allowing internal stresses’ relaxation can be considered to avoid the observed irreversibility in ΔZ/Z(H) dependencies upon heating. Recently, such an assumption has been confirmed by measurements of the temperature dependence of the GMI effect in Co-rich microwires with stress annealing-induced magnetic anisotropy; in such microwires, the irreversibility in the GMI effect and hysteresis loops upon heating was substantially reduced [60].
In both of the studied microwires, the ΔZ/Zmax(T) dependencies present similar features: a decrease in ΔZ/Zmax at T ≈ 200 °C followed by an increase in ΔZ/Zmax with T increasing (see Figure 7 and Figure 13a). To explain the observed similarity in the ΔZ/Zmax(T) dependencies for both of the studied microwires, it is necessary to recall that a high GMI effect can be realized if δ is substantially affected by H through the µφ(H) dependence (see Equation (3)). It is generally accepted that the most favorable conditions for the implementation of the GMI effect are realized in magnetic wires with low transverse magnetic anisotropy and high magnetic permeability. Therefore, in both extreme cases, Co-rich microwires with axial magnetic anisotropy or Fe-rich wires with high transverse magnetic anisotropy, a high GMI effect is not expected. As discussed above, the Hopkinson effect is linked to a sharp magnetic permeability maximum at temperatures slightly below the Curie temperature, Tc. In both studied microwires, a decrease in Hk upon heating (at T ≥ 200–250 °C) is observed (see Figure 3c and Figure 9). Therefore, an increase in ΔZ/Zmax at temperatures close to the Curie temperature, i.e., at T = 300 °C, must be attributed to the Hopkinson effect.
Finally, heating leads to both relaxation of internal stresses, σi, and a change in σi due to their temperature dependence.
As discussed above, a linear σi(T) dependence is expected in glass-coated microwires (given by Equation (3)) [33]. In the case of the studied Fe-rich microwire, the linear Hk(T) observed at T ≤ 150 °C can be related to the σi(T) dependence. Below, we will try to evaluate the contribution linked with the change in internal stresses due to their temperature dependence in the studied Co-rich microwire.
While the applied stress dependence of hysteresis loops of amorphous microwires with positive λs is studied in detail [26,27,61], there are only very few previous studies of the dependence of hysteresis loops and the magnetic anisotropy field, Hk, of Co-rich microwires with inclined hysteresis loops on applied tensile stress, σ [28]. As experimentally demonstrated [28], the Hk(σ) dependence of Co-rich microwires with inclined hysteresis loops has been well described by a linear increase in Hk with σ. The origin of such linear Hk(σ) dependence was attributed to the λs(σ) dependence, previously reported in Co-rich wires with low and negative λs, given as follows [58,59]:
λs,o—the magnetostriction coefficient at σ = 0, λs,σ—the magnetostriction coefficient at a given σ, and B—the positive coefficient of order 10−10 MPa.
Therefore, a linear decrease in Hk(T) is expected for the contribution linked to the temperature dependence of the internal stresses.
Similarly to what is provided in Figure 3l for the Fe75B9Si12C4 sample, we tried to evaluate Hk(T) dependence for the Co69.2Fe3.6Ni1B12.5Si11Mo1.5C1.2 sample from the hysteresis loops shown in Figure 9. As can be appreciated from Figure 16, a linear Hk(T) dependence can be roughly observed for T ≤ 150 °C. It is worth noting that the same tendency (linear Hk(T) dependence) is observed for the Fe75B9Si12C4 sample at the same temperature range (see Figure 3l). Therefore, we can propose the following interpretation for the observed temperature dependencies of hysteresis loops and the GMI ratio in both of the studied samples:
The main contribution of the temperature dependence of internal stresses is at T ≤ 150 °C. In this temperature range, the changes in magnetic properties are reversible.
There is a noticeable contribution of the internal stresses’ relaxation at T ≥ 150 °C. This process is associated with irreversible changes in magnetic properties.
There is an improvement in the GMI affect at T ≥ 250 °C due to the Hopkinson effect.
Regarding the Hk(T) dependence for the Co69.2Fe3.6Ni1B12.5Si11Mo1.5C1.2 sample, for comparison in the inset of Figure 16, we provided Hm(T) dependence for the same sample evaluated from the ΔZ/Z(H) dependence. Such a comparison is provided considering that the field of maximum, Hm, observed in the ΔZ/Z(H) dependencies is commonly attributed to the magnetic anisotropy field at MHz frequencies [44,53]. Generally, both Hk(T) and Hm(T) dependencies have the same trend. However, substantially higher (almost twice) Hm values are observed (see Figure 16). Such difference in Hk and Hm values was previously discussed in terms of rather different magnetic anisotropy in the surface layer and in the bulk layer due to the interface layer between the metallic nucleus and glass coating [38]. On the other hand, the origin of the GMI effect at elevated frequencies was discussed in terms of ferromagnetic resonance, FMR [62,63], and higher Hm values are expected.
Accordingly, the observed thermal dependence of the GMI effect in the studied microwires must be attributed to the interplay of the Hopkinson effect, relaxation of the internal stresses, temperature dependence of internal stresses, and related change in the magnetostriction coefficient.
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