8.2.1 Effect of composition: Properties relevant for applications

As a rule, better soft magnetic properties are observed for nearly zero magnetostrictive cobalt (Co)-rich compositions. It is worth mentioning that the magnetostriction constant λ_s in a system (Co_xFe_1 − x)₇₅Si₁₅B₁₀ changes with x, from − 5 × 10^− 6 at x = 1 to λ_s ≈ 35 × 10^− 6 at x ≈ 0.2, achieving nearly zero values at Co-to-iron (Fe) ratios of about 70:5 (Konno and Mohri, 1989; Zhukov et al., 2003). On the other hand, microwires with an Fe-rich metallic nucleus present rather different magnetic properties and exhibit rectangular hysteresis loops related to a large and single Barkhausen jump.

In general, magnetic properties and the overall shape of the hysteresis loops of amorphous microwires depend on the composition of the metallic nucleus, as well as the composition and thickness of the glass coating. This is illustrated in Figure 8.1, where the hysteresis loops of three main groups of amorphous microwires are Fe-rich, Co-rich, and Co–Fe-rich wires with positive, negative, and vanishing magnetostriction constants, respectively.

The composite character of glass-coated microwires affects magnetic properties mostly through the internal stresses induced by the glass coating (Zhukov et al., 2014a). The origin of these stresses is determined by the difference in the thermal expansion coefficients of the glass and metallic alloy and by rapid solidification of the composite microwire. It is worth mentioning that the strength of such internal stresses depends on the ρ ratio between the metallic nucleus diameter d, and the total microwire diameter D. Consequently, the strength of these stresses can be controlled by the ρ ratio: As the strength of the internal stresses increases, the ρ ratio decreases, that is, it increases with an increase in the glass volume (Velázquez et al., 1996; Chiriac et al., 2003; Antonov et al., 2000). Therefore, within each group of the aforementioned amorphous microwire compositions (Fe-rich with a rectangular hysteresis loop or Co–Fe-rich with a vanishing magnetostriction coefficient), the considerable effect of the ρ ratio on hysteretic magnetic properties can be appreciated from Figures 8.2 and 8.3. Thus, increasing of the magnetic anisotropy field (Figure 8.2) and increasing of the coercivity (Figure 8.3) with a decrease in the ρ ratio is observed for Co–Ni–Fe- and for Fe-based microwires, respectively.

All Co-rich microwires exhibit almost unhysteretic loops with a low enough coercivity (4–5 A/m) (Figure 8.2). The magnetic anisotropy field H_k increases as the ρ ratio increases. Low coercivity, remanent magnetization, and magnetic anisotropy field and high magnetic permeability are attributed to the considerable contribution of magnetization rotation at a low magnetic field. This denotes the existence of a circumferential magnetic anisotropy throughout the whole volume of the metallic nucleus, practically without an inner core (Vázquez, 2007; Zhukov et al., 2000a,b), indicating that almost the entire cross section should consist of a circumferentially magnetized shell with a well-defined transverse magnetic anisotropy k_φ, whose anisotropy field $H_{k_{\phi }}$ can be experimentally evaluated as the field required to reach saturation in the axial direction.

Magnetic bistability exhibited by amorphous, glass-coated microwires with a positive magnetostriction constant is associated with a perfectly rectangular hysteresis loop, which is typical of microwires related to axial orientation of magnetization in the majority of the metallic nucleus (Vázquez and Zhukov, 1996; Vázquez, 2007; Zhukov et al., 2014a). All hysteresis loops of Fe-rich microwires presented in Figure 8.3 exhibit a perfectly rectangular character when the hysteresis loops are measured at low enough values of frequency f and/or amplitude H₀ of an applied magnetic field. The magnetization reversal is characterized by the switching field H_s, which corresponds to the beginning of the magnetization switching. For low enough f and/or H₀ values, H_s practically coincides with the coercivity H_c. Increasing the f and/or H₀, a considerable change in the slope of the vertical regions of the hysteresis loops can be observed. This phenomena should be attributed to the counterbalance between the sweeping rate dH/dt = 4fH₀ (i.e., increasing H₀ results in a faster change of magnetic field dH for the same time interval dt) and the switching time related to the time needed for domain wall propagation along the whole wire (Zhukov et al., 1997).

As can be observed in Figure 8.3, for the same Fe-rich composition of the metallic nucleus, quite different coercivity values can be obtained. The switching field H_s, that is, the magnetic field required to reverse magnetization (creating a rectangular hysteresis loop), increases for the same metallic nucleus composition (Fe₆₅B₁₅Si₁₅C₅) with a decrease in the geometric ratio ρ. Such increase in the switching field has been attributed to an increase in the strength of internal stresses as the thickness of the glass coating increases. The switching field H_s should be proportional to the energy required to form the domain wall γ involved in the bistable process. The domain wall energy is related to the magnetoelastic anisotropy and, therefore, to the applied tensile stress, as given by (Zhukov and Zhukova, 2009):

$H_{\mathrm{s}}\approx \gamma \propto {\left[3A{\lambda }_{\mathrm{s}}\left({\sigma }_{\mathrm{a}}+{\sigma }_{\mathrm{i}}\right)/2\right]}^{1/2}/cos\alpha \text{,}$

where α is the angle between the magnetization and axial directions, A is the exchange energy constant, λ_s is the saturation magnetostriction constant, and σ_i is the internal stress. Consequently, H_s must be proportional to (σ_a + σ_r)1/2 for cos α ≈ 1. Note that, as mentioned above, the strength of the internal stresses increases as the ρ ratio decreases.

Consequently, a considerable increase (almost one order of magnitude) in the switching field (from about 80 to 700 A/m) is observed when the ferromagnetic metallic nucleus diameter decreases from 15 to 1.4 μm (Zhukov et al., 2012a,b). The rectangular shape of the hysteresis loop is maintained, even for the smallest microwires diameters.

Thus one more relevant parameter affecting the strength of internal stresses and the magnetoelastic energy is the ρ ratio. Such strong dependence of the hysteresis loops on these parameters should be attributed to the magnetoelastic energy K_me, given by

$K_{\mathrm{me}}\approx 3/2{\lambda }_{\mathrm{s}}{\sigma }_{\mathrm{i}}\text{,}$

where λ_s is the saturation magnetostriction and σ_i is the internal stress (Zhukov and Zhukova, 2009; Zhukov et al., 2014a). The estimated values of the internal stresses in these glass-coated microwires are on the order of 100–1000 MPa, depend strongly on the ρ ratio, and increase with a decrease in the ρ ratio (Chiriac and Ovari, 1996; Velázquez et al., 1996). Such large internal stresses give rise to a drastic change in the magnetoelastic energy K_me, even for small changes in the glass-coating thickness at a fixed metallic core diameter.

The aforementioned magnetic bistable behavior is related to the presence of a single large Barkhausen jump, which was interpreted as the magnetization reversal in a large single domain (Vázquez and Zhukov, 1996). This peculiar domain structure is observed above a certain critical length that correlates well with the demagnetizing factor (Zhukov et al., 1995), indicating that the closure domains penetrate from the wire ends inside the internal axially magnetized core, destroying the single domain structure. Use of glass-coated microwires allows drastic reduction of the demagnetizing factor and, consequently, of the critical length l_c, from a few centimeters (l_c ≈ 7 cm for conventional Fe-rich amorphous wires with a diameter of ~ 120 μm) to a few millimeters (l_c ≈ 2 mm for Fe-rich glass-coated microwires with a diameter of ≈ 10 μm). This drastic reduction in critical length is essentially important for microsensor applications.

As mentioned above, the current main technological interest in magnetically soft microwires is related to the excellent magnetic softness and GMI effect observed in nearly zero magnetostriction composition (Zhukov et al., 2014a). This GMI effect consists of the large change in the electric impedance of a magnetic conductor when it is subjected to an axial DC magnetic field. The GMI effect is usually characterized by the MI ratio ΔZ/Z, defined as

$\mathrm{\Delta }Z/Z=\left[Z\left(H\right)-Z\left(H_{\text{max}}\right)\right]/Z\left(H_{\text{max}}\right)\text{,}$

where H_max is the maximum DC longitudinal magnetic field on the order of few kiloamperes per meter, usually supplied by long solenoid and/or Helmholtz coils. The example of ΔZ/Z (H) dependence measured in a Co-rich glass-coated microwire is shown in Figure 8.4. The large sensitivity of the total impedance of a soft magnetic conductor at low magnetic fields and a high-frequency driven alternating current (AC) originates from the dependence of the transverse magnetic permeability on the axial DC magnetic field and the skin effect (Beach and Berkowitz, 1994; Panina and Mohri, 1994). The main interest in GMI effect is that aforementioned large magnetic field sensitivity is suitable for inexpensive and quite sensitive magnetic field detection. A few laboratory and industrial magnetic field sensors allowing 1- to 10-pT magnetic field resolution have been reported to date (Uchiyama et al., 2011; Gudoshnikov et al., 2014). In addition to high magnetic field resolution, low price and small dimensions are other advantages of proposed GMI magnetic field sensors (Uchiyama et al., 2011; Honkura, 2002).

8.2.2 GMI effect and enhanced magnetic softness: Tailoring magnetic properties and GMI

As already mentioned in Section 8.2.1, the GMI effect usually observed in soft magnetic materials phenomenologically consists of a change in the AC impedance, Z = R + iX (where R is the real part, or resistance, and X is the imaginary part, or reactance), when submitted to an external magnetic field H. The GMI effect was interpreted well in terms of the classical skin effect in a magnetic conductor assuming the dependence of the penetration depth of the AC flowing through the magnetically soft conductor on the DC applied magnetic field (Beach and Berkowitz, 1994; Panina and Mohri, 1994). The cylindrical shape and high circumferential permeability typical of amorphous wires are quite favorable for achieving a high GMI effect (Panina and Mohri, 1994; Zhukov et al., 2014a). Dependence of the GMI ratio on the magnetic field is intrinsically related to the magnetic anisotropy features: the circumferential anisotropy leads to the observation of the maximum of the real component of wire impedance (and, consequently, of the GMI ratio) as a function of the external magnetic field. For axial magnetic anisotropy, the maximum value of the GMI ratio corresponds to zero magnetic fields (Usov et al., 1998), that is, it results in a monotonic decay of the GMI ratio with the axial magnetic field.

Like magnetic permeability, the GMI effect presents a tensor character (Sandacci et al., 2004; Zhukov et al., 2008). Moreover, from the point of view of industrial applications, antisymmetrical magnetic field dependence of the output voltage with a linear region obtained for a pulsed GMI effect based on detection of the off-diagonal GMI component of amorphous wires is very useful (Sandacci et al., 2004; Zhukov et al., 2008). Consequently, this pulsed GMI effect is actually used for industrial applications.

The GMI effect was initially interpreted assuming a scalar character for the magnetic permeability, as a consequence of a change in the penetration depth of the AC caused by the DC applied magnetic field. The electrical impedance Z of a magnetic conductor in this case is given by (Panina and Mohri, 1994):

$Z=R_{\mathrm{D}\mathrm{C}}kr{J}_0\left(kr\right)/2J_1\left(kr\right)\text{,}$

with k = (1 + j)/δ, where J₀ and J₁ are the Bessel functions, r is the wire’s radius, and δ is the penetration depth given by:

$\delta =1/\sqrt{\pi \sigma {\mu }_{\phi }f}\text{,}$

where σ is the electrical conductivity, f is the frequency of the current along the sample, and μϕ the circular magnetic permeability, which is assumed to be scalar. The DC applied magnetic field introduces significant changes in the circular permeability μϕ. Therefore, the penetration depth also changes and finally results in a change in Z (Panina and Mohri, 1994).

The main feature of the GMI effect that makes it suitable for applications is a large change in the total impedance (in most cases > 100%). In the case of amorphous wires with high circumferential permeability, the highest GMI effect is usually reported (Zhukov et al., 2014a). Thus, a few researchers reported an achievement of an ~ 600% GMI ratio in Co-rich microwires with a vanishing magnetostriction constant (Zhukova et al., 2002; Pirota et al., 2000). In this case, it is quite promising for the application of magnetic sensors.

Moreover, the GMI materials—whether wires, ribbons, or films—are usually extremely soft magnetic materials. In addition, the AC plays an important part in the GMI effect, mainly because the AC current flowing through the sample creates a circumferential magnetic field. This circumferential magnetic field affects the magnetic permeability, which, similar to the GMI effect, presents a tensor character (Aragoneses et al., 2000). The AC current can also produce Joule heating (Zhukova et al., 2001).

The surface impedance tensor in magnetic wires with circumferential anisotropy has been expressed as (Makhnovskiy et al., 2001; Sandacci et al., 2004; Zhukov et al., 2008)

$e=\hat{\varsigma }h\mathrm{or}\left\{\begin{array}{l}{e}_z={\varsigma }_{zz}{h}_{\phi }-{\varsigma }_{z\phi }{h}_z\\ e_{\phi }={\varsigma }_{\phi z}{h}_{\phi }-{\varsigma }_{\phi \phi }{h}_z\end{array}\right\}\text{.}$

The circular magnetic field h_φ is produced by the currents i_w running through the wire. At the wire’s surface, h_φ = i/2πr, where r is the wire radius. The longitudinal magnetic field h_z is produced by the currents i_c running through the exciting coil, h_z = N₁i_c, where N₁ is the number of turns of the exciting coil.

Various excitation and measurement methods are required to reveal the impedance matrix elements. The longitudinal and circumferential electrical fields on the wire’s surface can be measured as a voltage drop along the wire v_w and voltage induced in the pickup coil v_c wound around it (Makhnovskiy et al., 2001; Sandacci et al., 2004; Zhukov et al., 2008).

$v_{\mathrm{w}}\equiv e_z{l}_{\mathrm{w}}=\left({\varsigma }_{zz}{h}_{\phi }-{\varsigma }_{z\phi }{h}_z\right)l_w\text{,}$

$v_{\mathrm{c}}\equiv e_{\phi }{l}_{\mathrm{t}}=\left({\varsigma }_{\phi z}{h}_{\phi }-{\varsigma }_{\phi \phi }{h}_z\right)l_{\mathrm{t}}\text{,}$

where l_w is the wire length and l_t = 2πrN₂ the total length of the pickup coil turns N₂ wound directly around the wire.

The methods for revealing the different elements of an impedance tensor are shown in Figure 8.5. The longitudinal diagonal component ς_zz is defined as the voltage drop along the wire and corresponds to the definition of impedance in a classical model (Figure 8.5a):

${\varsigma }_{zz}\equiv \frac{v_{\mathrm{w}}}{h_{\phi }{l}_{\mathrm{w}}}=\left(\frac{2\pi a}{l_{\mathrm{w}}}\right)\left(\frac{v_{\mathrm{w}}}{i_{\mathrm{w}}}\right)\text{.}$

The off-diagonal components ς_zφ and ς_φz (Figure 8.5b and c) and the circumferential diagonal component ς_φφ (Figure 8.5d) arose from a cross-sectional magnetization process (h_φ → m_z and h_z → m_φ) (Makhnovskiy et al., 2001; Sandacci et al., 2004; Zhukov et al., 2008; Ipatov et al., 2008a,b).

As mentioned above, a pulsed excitation scheme allows the antisymmetrical magnetic field dependence of the output voltage with a linear region to be obtained (Makhnovskiy et al., 2001; Sandacci et al., 2004; Zhukov et al., 2008; Ipatov et al., 2008a,b). The shape of the magnetic field dependence of the GMI effect (including off-diagonal components) is intrinsically related to the magnetic anisotropy and peculiar surface domain structure of amorphous wires (Usov et al., 1998). The magnetic anisotropy of amorphous microwires in the absence of magnetocrystalline anisotropy is determined mostly by the magnetoelastic term (Zhukov et al., 2014a). Therefore, the magnetic anisotropy can be tailored by thermal treatment (Zhukov and Zhukova, 2009; Zhukov and Zhukova, 2014; Zhukov et al., 2014a–c). On the other hand, considerable GMI hysteresis, which was recently observed and analyzed in microwires (Ipatov et al., 2010), has been explained considering the helical magnetic anisotropy.

The magnetic field H and dependence of the real part R on the longitudinal wire impedance Z (Z = R + iX), measured up to 4 GHz in Co₆₆Cr_3.5Fe_3.5B₁₆Si₁₁ and Co₆₇Fe_3.85Ni_1.45B_11.5Si_14.5Mo_1.7 microwires, are shown in Figure 8.6. General features of these dependences are the existence of two maximums, ΔZ/Z_m, that shift to higher magnetic fields with increasing the frequency f. A considerable GMI effect has been observed even at gigahertz frequencies.

Values of ΔZ/Z_m at high enough frequencies usually decrease starting from some frequency. Most microwires show the highest GMI ratio at frequencies between 100 and 300 MHz (Figures 8.6 and 8.7). Other interesting features observed in Figure 8.7a and b are that the frequency dependence of the maximum GMI ratio ΔZ/Z_m(f), measured in microwires with the same composition but different diameters, presents an optimum frequency (at which ΔZ/Z_m versus f exhibits the maximum value) at different frequencies. Thus, for metallic nucleus diameters ranging between 8.5 and 9.0 μm, the optimum frequency is about 100 MHz, whereas for microwires with metallic nucleus diameters between 9 and 11.7 μm, the optimum frequency is about 200 MHz.

One more factor that must be avoided for applications of the GMI effect is GMI hysteresis. Both off-diagonal and diagonal components of GMI present considerable hysteresis, as shown in Figure 8.8 for a Co₆₇Fe_3.85Ni_1.45B_11.5Si_14.5Mo_1.7 microwire. This GMI hysteresis does not depend on frequency: hysteresis persists with increases in the frequency of the GMI (Figure 8.8b).

Our recent studies revealed that observed GMI hysteresis is related to considerable deviation of the anisotropy easy axis from the transverse direction (Ipatov et al., 2010). Consequently, we explained the nature of observed low field hysteresis on R(H) and Z_ϕz(H) (Figure 8.8a and b) considering the existence of helical magnetic anisotropy (Ipatov et al., 2010). Application of the circular bias magnetic field H_b produced by the DC I_B running through the wire affects the hysteresis and asymmetry of the GMI dependence, suppressing this hysteresis when I_B is high enough (see Figure 8.9, which shows the effect of bias voltage on diagonal impedance R and on the S₂₁ parameter, proportional to off-diagonal GMI component).

As shown above, the strength of internal stresses can be controlled by the ρ ratio: The strength of internal stresses increases with a decreasing ρ ratio (i.e., it increases with an increase in the glass volume). The DC magnetic field that corresponds to the maximum GMI ratio H_m is attributed to the static magnetic anisotropy field H_k (at frequencies < 1 GHz, where the influence of FMR is not significant). Figure 8.2 shows the influence of the ρ ratio on the hysteresis loops and magnetic anisotropy field of Co_67.1Fe_3.8Ni_1.4Si_14.5B_11.5 microwires with the same composition of metallic nucleus but different ρ ratios.

Consequently, the ρ ratio also affects the GMI effect of glass-coated microwires, as shown in Figure 8.10. Both maximum values of the GMI ratio ΔZ/Z_m and the magnetic anisotropy considerably depend on the sample’s geometry. It is worth mentioning that for microwires with thinnest glass coating (the largest ρ ratio), ΔZ/Z_m ≈ 600% has been observed (Zhukova et al., 2002).

Considering that the magnetoelastic energy K_me is determined by both internal (σ_i) and applied stresses (σ_a), the GMI effect has been measured under tensile stresses in various Co-rich microwires. Figure 8.11 presents tensile stress dependence measured in Co_67.05Fe_3.85Ni_1.4B_11.33Si_14.47Mo_1.69 and Co_68.5Mn_6.5Si₁₀B₁₅ microwires. ΔZ/Z and H_m are quite sensitive to the application of external tensile stresses σ_a; here the magnetic field H_m, corresponding to the maximum of ΔZ/Z, shows a roughly linear increase with σ (Figure 8.11c).

As mentioned above, the value of the DC axial field that corresponds to the maximum GMI ratio H_m should be attributed to the static circular anisotropy field H_k. Consequently, using ΔZ/Z(H) dependences measured at different σ_a, we obtained H_m(σ) dependence (presented in Figure 8.11c) and evaluated the magnetostriction coefficient value considering well-known expression for the stress dependence of the anisotropy field (Cobeño et al., 2001), given by

${\lambda }_{\mathrm{s}}=\left({\mu }_{\mathrm{o}}{M}_{\mathrm{s}}/3\right)\left(\mathrm{d}{H}_{\mathrm{k}}/\mathrm{d}\sigma \right)\text{,}$

where μ_ο M_s is the saturation magnetization.

Estimated values of λ_s ≈ − 2 × 10^− 7 have been obtained, which are rather reasonable compared with the reported values measured from the stress dependence of initial magnetic susceptibility (λ_s ≈ − 3 × 10^− 7 for such composition) and the magnetostriction values measured in amorphous wires of similar compositions (Cobeño et al., 2001).

Moreover, the observed tendency of H_m to change under application of tensile stresses (Figure 8.11c) and the change in H_m and H_k with a decreasing ρ ratio (Figures 8.2 and 8.10) are the same, confirming the effect of magnetoelastic anisotropy on hysteresis loops and GMI.

Similarly, the ρ ratio affects the antisymmetrical magnetic field dependence of the output voltage typical for a pulsed excitation regime. In this case, the output voltage from the pickup coil surrounding the microwire is proportional to the off-diagonal GMI component of amorphous wires. Figure 8.12 shows the field dependence of the off-diagonal voltage response V_out, measured using a pulsed scheme, in a Co_67.1Fe_3.8Ni_1.4Si_14.5B_11.5Mo_1.7 (λ_s ≈ − 3 × 10^− 7) microwire with different geometry: a metallic nucleus diameter of 6 μm, a total diameter 10.2 μm (ρ ≈ 0.59); a metallic nucleus diameter of 7 μm and a total diameter of 11 μm (ρ ≈ 0.64); and a metallic nucleus diameter of 8.2 μm and a total diameter of 13.7 μm (ρ ≈ 0.6). As can be observed, the amplitude of the V_out increases with an increase in the metallic nucleus diameter d, and the H_m value depends on the ρ ratio. As in the case of the conventional GMI effect, the effect of the ρ ratio on V_out(H) (Figure 8.12) should be attributed to the magnetoelastic anisotropy related to internal stresses.

The conventional way to tailor the magnetoelastic anisotropy is relaxing internal stress by heat treatment. Under DC annealing the H_m decreases from 480 A/m in an as-cast state to 230 A/m after 5 min annealing with a 50-mA current (Figure 8.13). After the Joule heat treatment, the V_out(H) curve becomes sharper, exhibiting higher magnetic field sensitivity and showing a lower maximum field H_m related to the magnetic anisotropy field. We explained the observed changes by considering that the Joule heating of nearly zero magnetostriction microwires results in a decrease in the internal stresses and an increase in the magnetic softness (Zhukov et al., 2008).

Similarly, in Co₇₄B₁₃Si₁₁C₂ microwire with high negative magnetostriction (λ_s ≈ − 10^− 6; d ≈ 10 μm), Joule heating strongly affects the off-diagonal MI curve (Zhukova et al, 2008); the hysteretic MI curve transforms into an unhysteretic one with a large enough nearly linear region (Figure 8.14). Low sensitivity of microwires with negative magnetostriction should be attributed to a high enough magnetoelastic energy, related to high negative magnetostriction and stresses induced in the metallic nucleus by the glass coating during simultaneous quenching. Joule heating reduces internal stresses and enhances the V_out.

Similarly, current annealing (caused by Joule heating) induced changes in the GMI ratio (Figure 8.15). This effect should be mostly attributed to stress relaxation (although the electrical current also induces a circular magnetic field).

We recently observed drastic changes induced by annealing in nearly zero magnetostrictive Co-rich microwires. These microwires showed a low, negative magnetostriction coefficient in an as-prepared state. As-prepared Co_69.2Fe_4.1B_11.8Si_13.8C_1.1 amorphous microwires present soft magnetic behavior with quite low coercivity (about 4 A/m; Figure 8.16a), similar to the other Co-rich amorphous microwires with a nearly zero negative magnetostriction constant and similar compositions. Annealing even for only 5 min at different temperatures induced considerable changes in the hysteresis loops (Figure 8.16b and c). After annealing at temperatures above 200 °C, the hysteresis loop of a Co_69.2Fe_4.1B_11.8Si_13.8C_1.1 microwire drastically changes: we observed a considerable increase in coercivity, from 4 to 40 A/m. It is worth mentioning that an annealed (annealing temperature [T_ann] of 300 °C for 5 min) Co_69.2Fe_4.1B_11.8Si_13.8C_1.1 amorphous microwire presents a rectangular hysteresis loop similar to that of an as-prepared amorphous Fe_73.8Cu₁Nb_3.1B_9.1Si₁₃ microwire. The differences between these hysteresis loops are the coercivity (about 15% higher for Fe-rich microwires) and the magnetic permeability of the flat branches of hysteresis loop (higher for Co-rich microwires).

As-prepared Co_69.2Fe_4.1B_11.8Si_13.8C_1.1 microwires present a rather large GMI effect (GMI ratio ΔZ/Z up to 300%; Figure 8.17a). After annealing at T_ann = 300 °C for 5 min, a decrease in ΔZ/Z, from 300% to 150%, is observed (Figure 8.17b). A comparison of the GMI ratios of amorphous Fe_73.8Cu₁Nb_3.1B_9.1Si₁₃ microwires measured at the same conditions is provided in Figure 8.17c. As can be appreciated, Co-rich microwires showing rectangular hysteresis loops present a much larger GMI ratio than Fe-rich microwires, which also show a rectangular hysteresis loop. For Fe-rich microwires, we observed ΔZ/Z ≈ 8%.

An off-diagonal GMI effect represented by S₂₁ without bias current is rather low for as-prepared and annealed samples (Figure 8.18). As can be appreciated from Figure 8.18, and similar to other Co-rich microwires, the bias field H_b, produced by the bias current I_B, strongly affects the off-diagonal impedance and GMI hysteresis. But considerable hysteresis is observed even when I_B = 50 mA, reflecting the considerable helical magnetic anisotropy of the sample (Ipatov et al., 2010).

To interpret an unusual increase in the coercivity and the overall change in the hysteresis loop, both stress relaxation and the considerable effect of internal stresses on the magnetostriction constant have been considered. It was assumed that stress relaxation induced by annealing results in a change in the magnetostriction sign. However, a considerable GMI effect allows the assumption that the outer domain shell of the annealed Co-rich microwire, exhibiting a rectangular hysteresis loop and a GMI effect, has high circumferential magnetic permeability.

It was previously demonstrated that application of stress and/or magnetic field during annealing of amorphous materials may induce strong additional magnetic anisotropy. In the case of microwires, this induced anisotropy can be reinforced by strong internal stresses; in this case even conventional annealing must be considered as stress annealing (Zhukov et al., 2000a,b).

In some cases, this results in drastic changes in hysteretic magnetic properties and GMI behavior (Zhukov, 2006; Zhukov et al., 2006; Zhukova et al., 2003a,b). As an example, application of an axial magnetic field during annealing induces axial magnetic anisotropy in Co-rich microwires (Figure 8.19). Here, hysteresis loops of Co₆₇Fe_3.85Ni_1.45B_11.5Si_14.5Mo_1.7 microwires (d = 22.4 μm, D = 22.8 μm) annealed by Joule heating without magnetic field, current annealing (CA) and with application of an axial magnetic field, field current annealing (FCA) are shown. As can be appreciated, application of a magnetic field during annealing resulted in the complete opposite tendency in the change of magnetic properties induced by annealing: An increase in the remanent magnetization and a decrease in coercivity are observed after FCA, whereas CA treatment induced a decrease in the remanence and the coercivity.

In the case of stress-annealed Fe-rich microwires, the character of the hysteresis loops completely changed when compared with as-prepared samples (Figure 8.20). Stress annealing of Fe₇₄B₁₃Si₁₁C₂ microwires caused considerable stress-induced anisotropy (Zhukov, 2006; Zhukov et al., 2006; Zhukova et al., 2003a,b). The shape of the hysteresis loop changes completely, and the degree of induced changes depends on the duration and temperature of annealing (Figure 8.20a). In this case, the easy axis of magnetic anisotropy has been changed from axial to transverse (Zhukov, 2006; Zhukov et al., 2006; Zhukova et al., 2003a,b). In addition, the application of stress during measurements of stress-annealed microwires with well-defined transverse anisotropy resulted in a drastic change in the hysteresis loop (Figure 8.20b).

The origin of such stress-induced anisotropy has been explained by considering so-called back stresses that originate from the composite origin of glass-coated microwires annealed under tensile stress. Compressive stresses compensate for the axial stress component, and transversal stress components are predominant in stress-annealed microwires under these conditions (Zhukov, 2006; Zhukov et al., 2006; Zhukova et al., 2003a,b).

Consequently, these stress-annealed samples exhibit a stress impedance effect, that is, an impedance change (ΔZ/Z) under applied stress σ is observed in samples with stress-induced transversal anisotropy (Zhukov, 2006; Zhukov et al., 2006; Zhukova et al., 2003a,b). After annealing at proper conditions, considerable stress impedance is achieved, as shown in Figure 8.21.

8.2.3 Effect of partial crystallization and nanocrystallization on magnetic properties and GMI

Although crystallization of amorphous materials usually results in the degradation of their magnetic softness, in some cases crystallization can improve magnetically soft behavior. This is the case of so-called nanocrystalline alloys obtained by suitable annealing of amorphous metals. These materials were introduced in 1988 by Yoshizawa et al. (1988) and later were intensively studied by a number of research groups (Herzer, 1990). A nanocrystalline structure of partially crystalline amorphous precursors is observed, in particular in Fe–silicon (Si)–boron (B) with small additions of copper (Cu) and niobium (Nb). Small additions of Cu and Nb inhibit the grain nucleation and decrease the grain growth rate (Herzer, 1990). The main interest in such nanocrystalline alloys is related to extremely soft magnetic properties combined with high saturation magnetization. Such soft magnetic character is thought to originate when the magnetocrystalline anisotropy vanishes and the very small magnetostriction value achieved when the grain size approaches 10 nm (Herzer, 1990). In addition to the suppressed magnetocrystalline anisotropy, low magnetostriction values provide the basis for the superior soft magnetic properties observed in particular compositions (Herzer et al., 2010).

Considerable magnetic softening and enhancement of the GMI effect have previously been reported for FINEMET-type ribbons and conventional wires (Guo et al., 2001; Hernando et al., 2006). Magnetic softening was reported in glass-coated, FINMET-type microwires (Dudek et al., 2007; Arcas et al., 1996; Chiriac et al., 1998). However, considerable improvement in the GMI effect in the case of glass-coated microwires was reported quite recently (Zhukov et al., 2014c; Talaat et al., 2014a,b).

As-prepared FINMET-type Fe_73.4Cu₁Nb_3.1Si_xB_22.5 − x (x = 11.5, 13.5, and 16.5) and Fe_73.4 − xCu₁Nb_3.1Si_13.4 + xB_9.1 (0 ≤ x ≤ 1.1) microwires present rectangular hysteresis loops similar to other Fe-rich amorphous microwires (Zhukov et al., 2014c; Talaat et al., 2014a,b). For illustration, in Figure 8.22 we provide hysteresis loops of Fe_70.8Cu₁Nb_3.1Si_14.5B_10.6 microwires with different ρ ratios (ρ = 0.79 and ρ = 0.38).

The coercivity H_c of as-prepared, Finemet-type microwires depends on the ratio ρ = d/D (Figure 8.23). After annealing we observed magnetic softening of Finemet-type microwires followed by abrupt magnetic hardening at T_ann > 873°K (Figure 8.24). This magnetic hardening is usually associated with the second crystallization process.

Magnetic softening with a rather low H_c value is obtained in samples treated at T_ann of 773–873°K. This magnetic softening is related to the nanocrystallization process due to the precipitation of fine grains (10–15 nm) of an α-Fe(Si) phase within the amorphous matrix. Such interpretation has been confirmed by X-ray diffraction studies of as-prepared samples and samples annealed at different temperatures (Talaat et al., 2014a; Zhukov et al., 2014c; Talaat et al., 2014a; Figure 8.25). As-prepared Fe_70.8Cu₁Nb_3.1Si_14.5B_10.6 microwires present an amorphous structure. The first crystallization process was observed at a temperature ≥ 823 K (Figure 8.25) and corresponds to the precipitation of α-Fe (Si) body-centered cubic crystal structure, similar to the nanocrystallization of conventional FeCuNbSiB materials (Herzer, 1990; Herzer et al., 2010). We estimated the grain size D_g using the Debye–Sherrer formula. As shown in Figure 8.26, the average crystallite size at T_ann = 823°K is 12 nm, increasing to 27 nm at T = 923°K. Similar dependence of the average crystallite size on T_ann has been observed for the Fe_70.8Cu₁Nb_3.1Si_14.5B_10.6 microwires with different ρ ratios.

The GMI effect in as-prepared Fe-rich microwires is rather small (Figure 8.27a–c): about 1% at a measured frequency of ~ 100 MHz. After annealing, a considerable increase in the GMI ratio ΔZ/Z has been observed (Figure 8.27a–c). The GMI effect has been measured after annealing at T_ann below that of the first crystallization process (723°K) and right after the beginning of the nanocrystallization. As can be appreciated from Figure 8.27, after annealing at temperatures at which the nanocrystalline structure of the samples was achieved, we observed a drastic enhancement of the ΔZ/Z effect. This considerable growth of the GMI ratio must be related to the magnetic softening of microwires after annealing and related to the internal stress relaxation and especially the nanocrystallization.

Another factor that affects the GMI ratio is the frequency at which the GMI ratio is measured. Consequently, ΔZ/Z(H) has been measured at different frequencies (Figure 8.28). In as-prepared samples, we did not observe any appreciable frequency dependence, and the GMI effect is low (< 3%; Figure 8.28a). In samples annealed at 823°K (Figure 8.28b), however, the GMI ratio achieves values around 90% at 500 MHz for the same Fe_70.8Cu₁Nb_3.1Si_14.5B_10.6 microwire.

It is well known that magnetic anisotropy considerably affects the ΔZ/Z(H) dependence (Usov et al., 1998). A double-peak ΔZ/Z(H) observed in nanocrystalline microwires is typical for samples with a low negative magnetostriction constant. Therefore, we can assume that annealing results in the formation of a double-phase structure with a vanishing magnetostriction constant.

As observed in Figure 8.27, the GMI ratio increases even after annealing at temperatures less than the initial crystallization temperatures. The GMI effect is affected by the internal stresses distribution (Cobeño et al., 2002). Consequently, this GMI ratio increase must be attributed to stress relaxation.

Analyzing the dependence of as-prepared microwires on ΔZ/Z(H), we found that some as-prepared samples present an anomalously high GMI effect (Figure 8.29). This is the case for an Fe_73.8Cu₁Nb_3.1Si₁₃B_9.1 microwire, which exhibits a maximum GMI ratio of ~ 45%. X-ray diffraction studies show that as-prepared Fe_73.8Cu₁Nb_3.1Si₁₃B_9.1 microwires exhibiting a high GMI ratio in fact have a nanocrystalline structure, although as-prepared Fe_70.8Cu₁Nb_3.1Si_14.5B_10.6 microwires with a low GMI effect are amorphous.

Studies of magnetic properties of FINEMET-type FeCuNbSiB microwires reveal that annealing considerably affects the hysteresis loops and the GMI effect of this family of microwires. In as-prepared microwires, the reduction of the ρ ratio results in the increase in coercivity. Magnetoelastic anisotropy affects the soft magnetic properties of as-prepared FeCuNbSiB microwires. We observed magnetic softening and a considerable increase in the GMI effect in FINEMET-type FeCuNbSiB with a nanocrystalline structure, even in as-prepared microwires. After adequate annealing of FINEMET-type microwires, we observed a GMI ratio of ~ 100%. The nanocrystallization of FeCuNbSiB microwires is a key for optimizing the GMI effect.

8.3.1 Effect of magnetoelastic anisotropy

The perfectly rectangular shape of the hysteresis loop observed in Fe-rich microwires has been related to the very high speed of domain wall propagation. A few methods demonstrate that the remagnetization process of such magnetic microwires starts from the ends of the sample as a consequence of the domain walls being depinned from the closure domains and subsequent domain wall propagation from the closure domains (Ekstrom and Zhukov, 2010; Ipatov et al., 2009; Zhukov et al., 2012a,b; Vázquez et al., 2012). The magnetization process runs in an axial direction through the propagation of the single head-to-head domain wall. It is worth mentioning that the micromagnetic origin of a rapidly moving head-to-head domain wall in microwires is still unclear, although there is evidence that this domain wall is relatively thick and has a complex structure (Ekstrom and Zhukov, 2010; Gudoshnikov et al., 2009).

In the case of microwires, the domain wall dynamics were measured using a modified Sixtus–Tonks technique (Sixtus and Tonks, 1932), as described recently elsewhere (Ipatov et al., 2009). Using a system consisting of three coaxial pickup coils is essential. Details about this are provided in Section 8.3.2. In addition, one end of the sample is placed outside the solenoid to ensure domain wall nucleation always occurs near one of the microwire ends. In this way, contrary to classical Sixtus–Tonks experiments, the nucleation coils are not needed to nucleate the domain wall because the closure domain wall already exists.

As described above, a characteristic feature of microwires is the additional magnetoelastic anisotropy that considerably affects the soft magnetic properties of glass-coated microwires. Consequently, considerable attention has been paid to the effect of magnetoelastic anisotropy on domain wall dynamics in amorphous, magnetically bistable microwires (Zhukov et al., 2012a,b).

The magnetoelastic energy given by Eq. (8.2) depends on the magnetostriction coefficient and on the value of internal stress. It is worth mentioning that the magnetostriction constant λ_s in a (Co_xFe_1 − x)₇₅Si₁₅B₁₀ system changes with x, from − 5 × 10^− 6 at x = 1 to λ_s ≈ 35 × 10^− 6 at x ≈ 0.2 (Konno and Mohri, 1989). Therefore, by producing microwires with various Fe-/Co-rich compositions, we were able to change the magnetostriction constant from λ_s ≈ 35 × 10^− 6 for Fe-rich compositions (Fe_72.75Co_2.25B₁₅Si₁₀ and Fe₇₀B₁₅Si₁₀C₅) to λ_s ≈ 10^− 7 for a Co₅₆Fe₈Ni₁₀Si₁₀B₁₆ microwire. In addition, within each metallic nucleus composition we also produced microwires with different ratios of metallic nucleus diameter and total diameter D, that is, with different ratios ρ = d/D. Because the strength of internal stresses is determined by ratio ρ, this allowed us to control residual stresses. In this way the effect of the magnetoelastic contribution on domain wall dynamics has been experimentally studied, controlling the magnetostriction constant and applied and/or residual stresses.

It is usually assumed that a domain wall propagates along the wire with a velocity:

$v=S\left(H-H_0\right)\text{,}$

where S is the domain wall mobility, H is the axial magnetic field, and H₀ is the critical propagation field.

The dependence of domain wall velocity v on the magnetic field H for Fe₁₆Co₆₀Si₁₃B₁₁ and Co_41.7Fe_36.4Si_10.1B_11.8 amorphous microwires with the same ρ ratio is shown in Figure 8.30. In this case, because the internal stresses determined by the ρ ratio are the same, the effect of only the magnetostriction coefficient must be considered. Comparison of v(H) dependence for a Co_41.7Fe_36.4Si_10.1B₁₁ microwire (λ_s ≈ 25 × 10^− 6) with that of an Fe₁₆Co₆₀Si₁₃B₁₁ microwire (λ_s ≈ 15 × 10^− 6) allows us to deduce that a slower domain wall velocity at the same magnetic field and less domain wall mobility S are observed for a Co_41.7Fe_36.4Si_10.1B₁₁ microwire with a higher λ_s.

The dependence of domain wall velocity on the applied field for Fe₅₅Co₂₃B_11.8Si_10.1 microwires with different ratios is shown in Figure 8.31. As in Figure 8.30, at the same applied field value H, the domain wall velocity is higher for microwires with a higher ρ ratio, that is, when the internal stresses are lower.

Another way to manipulate the magnetoelastic energy is to apply stresses during measurements. Figure 8.32 shows v(H) dependences for a Co_41.7Fe_36.4Si_10.1B_11.8 microwire (ρ ≈ 0.55) measured under applied tensile stresses. A considerable decrease in domain wall velocity v at the magnetic field value H have been observed under application of stress. In addition, increasing the applied stress σ_a results in a decrease in the domain wall velocity (Zhukov et al., 2013).

Consequently, in low magnetostrictive Co₅₆Fe₈Ni₁₀Si₁₀B₁₆ microwires, the domain wall velocity values achieved at the same values of applied field (Figure 8.33) are considerably higher (almost two times) than those observed for microwires with a higher magnetostriction constant (compare with Figures 8.30–8.32).

The domain wall dynamics in a viscous regime are determined by a mobility relation (8.11), where S is the domain wall mobility given by:

$S=2{\mu }_0{M}_{\sigma }/\beta \text{,}$

where β is the viscous damping coefficient and μ₀ is the magnetic permeability of vacuum. Damping is the most relevant parameter determining the domain wall dynamics. Various contributions to viscous damping β have been considered, and two are generally accepted (Varga et al., 2006; Zhukov et al., 2013): Microeddy currents circulating near moving domain walls are the more obvious cause of damping in metals. However, the eddy current parameter β_ɛ is considered to be negligible in highly resistive materials such as thin amorphous microwires. The second generally accepted contribution of energy dissipation is magnetic relaxation damping β_ρ, which is related to the delayed rotation of electron spins. This damping is related to the Gilbert damping parameter and is inversely proportional to the domain wall width δ_ω (Zhukov et al., 2013):

${\beta }_{\mathrm{r}}\approx \alpha M_{\mathrm{s}}/\gamma \Delta \approx M_{\mathrm{s}}{\left(K_{\mathrm{me}}/A\right)}^{1/2}\text{,}$

where γ is the gyromagnetic ratio, A is the exchange stiffness constant, and K_me is the magnetoelastic anisotropy energy given by Eq. (8.2). Consequently, the magnetoelastic energy can affect domain wall mobility S, as we experimentally observed in a few Co-/Fe-rich microwires.

Considering the aforementioned results, we can suggest that domain wall velocity v should decrease with stress and an increase in the magnetostriction constant, and only if the magnetoelastic energy affects the domain wall dynamics should v show an inverse square root dependence on stress or magnetostriction. Therefore, we tried to evaluate the v(σ_app) dependence. An example of v(σ_app) dependence obtained for Fe₅₅Co₂₃B_11.8Si_10.2 microwires (d = 13.2 μm; D = 29.6 μm) is shown in Figure 8.34a. Qualitatively, we observed a decrease in domain wall velocity v with an applied stress σ_app. To evaluate whether obtained dependence v(σ_app) fits the inverse square root dependence on applied stress σ_app, we expressed obtained dependence as σ_app(v^− 2). From Figure 8.34b, we can conclude that v(σ_app) dependence cannot be described by a single v(σ_app^− 1/2) dependence. On the other hand, at a high enough σ_app, observed v(σ_app) dependence probably can be fitted by two v(σ_app^− 1/2).

To summarize this section, we experimentally observed that manipulating the magnetoelastic energy through the application of tensile stress and changing the magnetostriction constant and internal stresses of studied microwires, we significantly affected the domain wall dynamics in magnetically bistable microwires. Considering this, we assume that to achieve a faster domain wall propagation velocity at the same magnetic field and enhanced domain wall mobility, special attention should be paid to decreasing the magnetoelastic energy.

8.3.2 Role of defects

Microwires (like any materials) present various kinds of defects. The presence of defects can be detected indirectly through measurements of the local nucleation fields of reversed domains, as well as directly using optical and electron microscopy (Zhukov et al., 2014b).

Considering a number of unusual effects found while studying domain wall propagation in amorphous microwires, we tried to reveal the contribution of local defects to peculiarities of domain wall propagation and the correlation of the linearity of v(H) dependence with defects existing in amorphous microwires. A comparative study of single domain wall dynamics and local nucleation fields in Fe-rich amorphous, glass-coated microwires was performed (Ipatov et al., 2008b, 2009).

To detect the possible nucleation and subsequent propagation of several domain walls, the setup using three pickup coils, described above, was used. To measure local nucleation fields, a special setup was developed (Ipatov et al., 2008b). To create a reversal domain with opposite magnetization far from the wire ends, a short microcoil that produces a magnetic field strong enough to nucleate a reversed domain has been used. Thus, a pair of head-to-head and tail-to-tail domain walls can be created in a controllable manner, avoiding the nucleation of domain walls at the wire end, as usually occurs in experiments. The magnetic field necessary for nucleation of the reversed domain gives a local value of the wire nucleation field at various points along the wire length. These measurements can be made under applied tension stress created by a small load attached to the bottom end of the wire (Ipatov et al., 2008b). In this way, the remagnetization dynamics in magnetically bistable Fe-rich microwires have been studied.

The distributions of the H_N(x) for Fe-rich samples are shown in Figure 8.35. A number of dip holes on H_N(x) curves for both samples have been previously attributed to the positions of localized defects existing within the microwire (Ipatov et al., 2008b). The overall minimum H_N observed for both microwires from H_N(x) has been compared with the v(H) dependence of the same samples. Figure 8.36 shows the measured dependence of domain wall velocity on the applied magnetic field in the same bistable, Fe-rich, amorphous, glass-coated microwires. The linear v(H) regime has been observed in both samples up to 294 and 340 A/m for samples 1 and 2, respectively. The domain wall mobility S of ~ 5 m²/As and a maximum domain wall velocity v of 1.7 km/s have been observed. The mechanism of such ultrafast magnetization switching in the second regime of the sample magnetization reversal is considered below in detail. It was assumed that such a drastic change in the remagnetization process and deviation from linear v(H) dependence is caused by possible nucleation and the consequent growth of an additional reversed domain with the lowest local nucleation field H_N. This new reverse domain can be located at any place inside the sample (Ipatov et al., 2009).

Considering the aforementioned details, it was assumed that when the applied magnetic field has reached H_N, the new domain is nucleated and two more domain walls start to propagate toward the wire’s ends. As noted above, in such a situation it is not possible to measure correctly a single domain wall’s velocity, and we can consider only the effective domain wall velocity.

Consequently, the existence of defects can lead to considerable acceleration of the microwire remagnetization process. On the other hand, neglecting this factor can result in exaggerated estimation of domain wall velocity from Sixtus–Tonks experiments when using just two pickup coils. Therefore, at a magnetic field above the H_N, the contribution of defects can be essential. The appearance of these additional domains at H > H_N can accelerate remagnetization switching, resulting in a faster effective domain wall velocity.

The essence of this process is clearly shown in Figure 8.37. There the left front of the propagating head-to-head domain wall dw2 moves toward dw1. These two reversed domains finally clamp and the right front dw3 becomes unique. This process, which can be nominated as tandem remagnetization, obviously results in a significant decrease of the magnetization switching time and acceleration of magnetization switching in magnetically bistable microwires. The proposed mechanism of ultrafast magnetization switching can explain the nonlinearity of v(H) dependence and ultrafast domain wall propagation reported in magnetically bistable microwires (Ipatov et al., 2009).

A local nucleation field is typical, describing the defects distribution in a given sample. Figure 8.38 shows the distribution of the local nucleation fields H_N along the sample length x, measured using a short, movable magnetizing coil placed far from the sample’s ends and moving in the opposite directions. To distinguish the microwire’s ends, we have marked one as “end 1” and other as “end 2.” The black solid line shows the distribution of the local nucleation fields measured by moving a short magnetization coil between end 1 and end 2. The red dotted line indicates the movement in the opposite direction. As we can observe in Figure 8.38, the nucleation field near the wire’s ends is considerably smaller. This characteristic feature is responsible for the origin of a large Barkhausen jump, which occurs when fast switching of the magnetization runs by depinning and consequent fast domain wall propagation from one of the wire ends. Based on the H_N(x) distributions, one can also see that the dip holes keep the same position, moving the short magnetizing coil from end 1 to end 2 and back, that is, this nucleation field profile is specific for a given microwire sample.

Figure 8.39 demonstrates a correlation between the H_N(x) and domain wall dynamics in an Fe₇₄B₁₃Si₁₁C₂ microwire. Figure 8.39a shows H_N(x) dependence. The deepest minimum is observed between pickup coils 1 and 2 at H_N ≈ 170 A/m. Figure 8.39b represents the dependence of domain wall velocity v on the applied magnetic field H, measured by a pair of coils 1–2 (v_1–2) and 2–3 (v_2–3). These dependences exhibit a significant difference: both v_1–2(H) and v_2–3(H) at low fields show linear growth with H, but at H ≈ 190 A/m we observed an abrupt jump on v_1–2(H), while v_2–3 continued growing with H. The field of jump on v_1–2(H) corresponds to the minimum nucleation field H_N ≈ 170 A/m observed in Figure 8.39b.

Consequently, the microwire’s inhomogeneities sufficiently affect the remagnetization process (Ipatov et al., 2009; Zhukova et al., 2014), as was observed through the measurement of the distribution of the local nucleation field H_N along the sample length L. The origin of the defects might be related to stress inhomogeneities, shape irregularities, or oxides. It is assumed that at least some of these defects might have a magnetoelastic origin and therefore might be affected by a heat treatment.

8.3.3 Manipulation of domain wall dynamics

Considering various ways to achieve a higher domain wall velocity in microwires, we can assume that at least some of the defects might have a magnetoelastic origin and therefore might be affected by heat treatment. The effect of annealing on domain wall dynamics, coercivity, and local nucleation field distributions has recently been reported (Chichay et al., 2013).

Dependence of the coercivity H_c on annealing time t_ann for an Fe₇₄B₁₃Si₁₁C₂ microwire measured at two different temperatures T_ann is shown in Figure 8.40. The considerable difference in H_c(t_ann) at T_ann = 250 and 300 °C observed in the Fe₇₄B₁₃Si₁₁C₂ microwire might be explained by considering the main contribution of stress relaxation at lower T_ann and t_ann, as well as the increasing influence of the first crystallization processes at a higher T_ann and an elevated t_ann.

The dependence of domain wall velocity between coils 2–3 (V_2–3) for the Fe_66.7Cr_11.4B₁₂Si₉Ni_0.9 microwires, and between coils 1–3 (V_1–3) for the Fe₇₄B₁₃Si₁₁C₂ microwires are presented in Figure 8.41. We observed a typical, almost linear v(H) dependence in both as-prepared samples. After heat treatments at both temperatures, we observed an extension of the magnetic field range, where linear v(H) dependence takes place (corresponding to a single domain wall propagation regime) and the domain wall velocity increases. At the longest annealing time (> 120 min) both parameters (domain wall velocity and range of magnetic field for linear v(H) dependence) are almost insensible on t_ann, exhibiting a tendency toward saturation.

Dependence of the domain wall velocity value measured at the same magnetic field after different heat treatments as a function of annealing time is presented in Figure 8.42a and b. We can observe a clear tendency of increasing domain wall velocity with an increase in t_ann at fixed H values. The increase in the velocity observed in the Fe_66.7Cr_11.4B₁₂Si₉Ni_0.9 sample annealed at T_ann = 250 °C (Figure 8.42a) is less pronounced when compared with the Fe₇₄B₁₃Si₁₁C₂ microwire annealed at T_ann = 300 °C (Figure 8.42b). Moreover, for the as-prepared samples, the difference for the velocities V_1–2, V_2–3, and V_1–3 is larger than that of annealed samples with different t_ann values. Local nucleation field distributions in as-prepared and annealed (T_ann = 250 °C for t_ann = 150 min) Fe_66.7Cr_11.4B₁₂Si₉Ni_0.9 microwires, as well as in as-prepared and annealed (T_ann = 300 °C) Fe₇₄B₁₃Si₁₁C₂ microwires, are shown in Figure 8.43a and b. As can be appreciated in these images, a decrease in oscillation amplitudes in H_N(L) dependence takes place after annealing. Observed decreases in the amplitude of local nucleation fields, an increase in domain wall velocity, and extension of the magnetic field range of a single domain wall propagation regime could be attributed to stress relaxation after annealing.

Consequently, we can assume that at least some of the samples’ inhomogeneities that can be observed through the oscillations of the local nucleation fields and limiting single domain wall regime have an origin related to inhomogeneities in the internal stress distribution. In addition, based on the results presented above, we may assume two possible mechanisms influencing the defects on domain wall dynamics: domain wall nucleation and the pinning of propagating domain wall. Annealing of microwires, giving rise to internal stress relaxation and/or the release of some defect, can enhance the domain wall velocity at a given magnetic field, as well as allow the range of magnetic fields in which a single domain wall propagation takes place to be extended.

It is worth mentioning that application of a transverse magnetic field can enhance the domain wall velocity (Zhukova et al., 2009b). On the other hand, by moving the sample with the holder inside the magnetization coil and/or applying the bias field at different angles with respect to the wire axis, domain wall propagation from the opposite end of the microwire can be activated and the domain wall collision can be observed. Examples of the manipulation of domain wall dynamics are presented in Figure 8.44, where a temporal dependence of the voltage peaks in coils 1, 2, and 3 wound over a 10-cm-long wire and separated by 27 mm is seen.

As seen in Figure 8.44a–d, by changing value of H_b (and, consequently, the axial H_b projection), we can activate in a controllable way the domain wall propagation from the opposite end of the sample and realize a domain wall collision at various locations in the sample. If this collision takes place in the vicinity of a pickup coil, an increase in the voltage peak amplitude will be observed. Consequently, we are able to engineer domain wall collisions between two moving domain walls in different places on the magnetic microwire by controlling an applied bias field. Manipulating the bias field enables the release of the domain walls at targeted locations along the microwire.

Analyzing the data in Figure 8.44c, we find that at H_b ≈ 252 A/m, the domain wall collision takes place near the position of the second coil. In this case, the height of the signal from coil 2 drastically increases (Figure 8.44b and c). In addition, when the H_b value is high enough, the domain wall from the opposite end of the sample and the domain wall propagating from the nearest end of the sample arrive at coil 1 simultaneously (Figure 8.44d). In this case we observed the domain wall collision by coil 1 (Figure 8.44d).

To summarize, annealing of microwires allows the magnetoelastic anisotropy of glass-coated microwires to be manipulated and therefore enhances the domain wall velocity by extending the field range for single domain wall propagation, as well as enhancing domain wall velocity at a given magnetic field through internal stress relaxation. The correlation between the annealing influence on the local nucleation field distribution and a change in the magnetic field dependence of domain wall velocity have been observed in amorphous Fe-based microwires with two different compositions and geometric parameters. Consequently, the nature of local defects limiting the single domain wall propagation regime and damping the domain wall might be related to the internal stress distribution.

Under certain experimental conditions, manipulating the domain wall dynamics in a magnetic wire in a field-driven regime and observing controllable domain wall collisions are possible. This controllable domain wall collision can be realized in different parts of the wire. Such domain wall collisions can be used to release pinned domain walls with weak external fields. The control of domain wall dynamics is essential for advanced racetrack memory devices.