# Off-Diagonal Magnetoimpedance in Annealed Amorphous Microwires with Positive Magnetostriction: Effect of External Stresses

## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. Static Magnetization Distribution

_{0}exp(–iωt) (here t is the time, ω is the angular frequency and i is the imaginary unit) and to the direct bias current I

_{b}, and the external magnetic field H

_{e}is parallel to the microwire axis. The tensile stress σ

_{t}and uniform torsional stress with the angular displacement per unit length τ are applied to the microwire. The microwire cross-section and a sketch of the coordinate system used for analysis are shown in Figure 1. Since all fields lie within the φ–z plane, the magnetization vector is limited to the same plane.

_{a}

_{,j}is the intrinsic magnetic anisotropy fields in the core and shell; θ

_{j}and ψ

_{j}are the equilibrium magnetization angles and the anisotropy axis angles with respect to the azimuthal direction (ψ

_{1}= π/2 for the core region); U

_{b}is the Zeeman energy of the field of the bias current; U

_{ten}and U

_{tor}are the magnetoelastic anisotropy energies induced by the tensile and torsional stresses.

_{b}can be written as follows

_{ten}induced by the tensile stress has the following form [1]:

_{s}> 0 is the magnetostriction coefficient.

_{tor}can be expressed as [1]

_{b}of the bias field and the magnetoelastic anisotropy U

_{tor}due to the torsional stress varies over the microwire cross-section. To simplify the model, we assume further that the contributions of these terms to the free energy density corresponding to the maximal values for the core (at ρ = r) and shell (at ρ = R) regions.

_{j}in the core and shell:

_{b}and U

_{tor}energy terms consists of the integration of the free energy density over the radial coordinate and the following minimization of the resulting energy. Since both the U

_{b}and U

_{tor}terms depend linearly on the radial coordinate, the minimization procedure leads to Equation (2) for the equilibrium magnetization angles, and the only difference is in the numerical coefficients in Equations (6), (7), (9) and (10). Thus, the simplified model described above gives the same basic predictions as compared to the general approach. In addition, the assumption that the contributions of the U

_{b}and U

_{tor}terms correspond to their maximal values in both regions allows one to describe the conditions for the fields at the surface of the metallic nucleus of the microwire.

_{j}. Equation (5) can be rewritten in the following form with effective uniaxial anisotropy:

_{eff,j}and α

_{j}are the effective anisotropy fields and anisotropy angles, correspondingly. After simple mathematical transformations, for α

_{j}and H

_{eff,j}we obtain [31]:

_{t}(at τ = 0) increases the axial effective anisotropy field H

_{eff,1}in the core region. It follows from Equations (12) and (13) that for the shell region, the growth of σ

_{t}leads to a deviation of the effective anisotropy angle α

_{2}from the azimuthal direction and a decrease of the effective anisotropy field H

_{eff,2}. Note that these dependencies are opposite to the behavior of the effective anisotropy in Co-rich amorphous microwires with slightly negative magnetostriction. It is well-known that in Co-rich microwires, the effective anisotropy axis rotates toward the azimuthal direction with a growth of the tensile stress, and the effective anisotropy field increases [1,30,31,32].

_{t}= 0), the angle α

_{1}in the core region changes within the range from 3π/4 at the high negative direction of the stress (counter-clockwise rotation) to π/4 at high positive stress (clockwise rotation). The effective anisotropy field H

_{eff,1}increases monotonically with the stress. Figure 2 illustrates the influence of the torsional stress on the effective anisotropy angle α

_{2}and the effective anisotropy field H

_{eff,2}in the shell region. The angle α

_{2}increases from −π/4 to π/4 with the field H

_{tor,2}. The effective anisotropy in the shell becomes circular (α

_{2}= 0), when H

_{tor,2}= −H

_{a}

_{,2}sin2ψ

_{2}. The dependence of the effective anisotropy field H

_{eff,2}on the field H

_{tor,2}has a minimum (see Figure 2b). The minimum attains at α

_{2}= 0, and the minimal value of the effective anisotropy field is equal to H

_{a}

_{,2}cos2ψ

_{2}.

#### 2.2. Impedance Tensor

_{zz}and off-diagonal Z

_{φ}

_{z}impedance components can be presented as follows [36]:

_{m}and ζ

_{0}are the magnetic and non-magnetic components of the surface impedance tensor and the equilibrium magnetization angle θ

_{2}in the shell is given by Equation (11).

_{m}and ζ

_{0}depend on the magnetic structure of the microwire. The expressions for ζ

_{m}and ζ

_{0}for the microwire with the core–shell magnetic structure were obtained in [19] assuming that the magnetization in the core region is directed along the microwire axis (θ

_{1}= ±π/2). However, in the presence of the bias current or the torsional stress, the equilibrium magnetization angle θ

_{1}deviates from the longitudinal direction (see Equation (5)). Taking into account the deviation of θ

_{1}, the surface impedance component ζ

_{m}can be expressed as

_{n}and Y

_{n}(n = 0, 1) are the Bessel functions of the first and the second kind, respectively; k

_{2}

^{2}= k

_{0}

^{2}μ

_{2}, k

_{0}

^{2}= 2i/δ

^{2}, δ = c/(2πσω)

^{1/2}, σ is the microwire conductivity, μ

_{2}is the transverse permeability in the shell region and the parameter P

_{m}is given by

_{1}

^{2}= k

_{0}

^{2}μ

_{1}, μ

_{1}is the transverse permeability in the core region and

_{1}= ±π/2 from Equation (18) we have Q = J

_{0}(k

_{1}r)/J

_{1}(k

_{1}r), and Equations (16)–(18) transform to expressions obtained in [19]. The non-magnetic component ζ

_{0}can be obtained from Equations (16) and (17) assuming that μ

_{2}= 1 [36]. Thus, for ζ

_{0}we have

_{zz}and off-diagonal Z

_{φ}

_{z}impedance components are governed by the values of the transverse permeability μ

_{j}in the core and shell regions. The transverse permeability can be found from a solution of the linearized Landau–Lifshitz–Gilbert equation. Taking into account the effects of the bias current and external stresses, we can present the values of μ

_{j}in the following form [1,21]:

_{m}= γ4πM, γ is the gyromagnetic constant, κ is the Gilbert damping parameter and

_{j}in the core and shell regions can be found from a solution of Equation (11) taking into account expressions (12) and (13). The values of the transverse permeability μ

_{j}in the core and shell regions are given by Equations (21) and (22). After that, the corresponding values of k

_{1}and k

_{2}can be calculated, and the impedance components Z

_{zz}and Z

_{φ}

_{z}are obtained by means of Equations (14)–(20).

## 3. Results

#### 3.1. Asymmetric Off-Diagonal Magnetoimpedance

_{φ}

_{z}= R

_{φz}− iX

_{φ}

_{z}(here R

_{φ}

_{z}and X

_{φ}

_{z}are the real and imaginary parts of the ODMI) was described previously for a microwire with the core–shell structure assuming that the magnetization in the core region has the longitudinal direction [19]. To obtain the ODMI effect with high field sensitivity, the domain structure in the shell region should be removed. The threshold field H

_{th}of the bias current to eliminate the surface domain structure can be expressed as follows [21,40]: H

_{th}= H

_{a}

_{,2}|sinψ

_{2}|. Taking into account Equation (7), we find for the threshold bias current I

_{th}:

_{2}= −0.05π and H

_{a}

_{,2}= 30 Oe, we find for the value of the threshold bias current I

_{th}≅ 17.5 mA.

_{φ}

_{z}and imaginary parts X

_{φ}

_{z}of the ODMI calculated for different values of I

_{b}> I

_{th}. For convenience, the values of R

_{φ}

_{z}and X

_{φ}

_{z}are reduced to the characteristic off-diagonal impedance Z

_{0}:

_{DC}= l/πσR

^{2}is the resistance in the direct current mode. Assuming that N = 50, 2R = 15 μm and σ = 5 × 10

^{15}s

^{−1}, we obtain Z

_{0}≅ 24 Ohm.

_{b}> I

_{th}, the asymmetry in the field dependence of the ODMI appears The asymmetry arises due to the interaction of the helical anisotropy with the circular magnetic field induced by the bias current [1,21,40]. The real and imaginary parts of the ODMI increase sharply at I

_{b}≅ I

_{th}. With a further increase of the bias current, the field sensitivities of R

_{φ}

_{z}and X

_{φ}

_{z}decrease due to a drop in the transverse permeability in the shell region, however, they remain sufficiently high within a wide range of the bias current.

_{φz}and X

_{φz}on the external field have similar behavior for all values of 2r, and the decrease of the core diameter results in a growth of the ODMI response. Note that the volume parts of the core and shell can be tuned by stress-annealing. In particular, an increase in the annealing temperature and time [12] and tensile stress during the annealing [14,41,42] leads to a decrease in the core region volume.

_{e}≅ −H

_{a}

_{,2}sin2ψ

_{2}, where the ODMI turns to zero. To analyze the frequency dependences of R

_{φ}

_{z}and X

_{φ}

_{z}, let us introduce the field sensitivities of the real S

_{R}and imaginary S

_{X}parts of the ODMI defined as follows [43]:

_{p}and |X

_{p}|are the maximum values of R

_{φ}

_{z}and |X

_{φ}

_{z}| at positive external fields, ΔH = H

_{p}+ H

_{a}

_{,2}sin2ψ

_{2}and H

_{p}is the external field at the peak.

_{R}and S

_{X}calculated for different values of the bias current are presented in Figure 5. The values of S

_{R}and S

_{X}have different frequency behavior. The field sensitivity S

_{R}of the real part of the ODMI increases monotonically with the frequency, whereas S

_{X}has a maximum at a certain frequency. At low frequencies, the field sensitivity S

_{X}is higher than S

_{R}. It also follows from Figure 5 that field sensitivities decrease with a growth of the bias current I

_{b}. However, the magnitudes of S

_{R}and S

_{X}are relatively high within a wide range of the bias current.

#### 3.2. Effect of Tensile Stress on Off-Diagonal Magnetoimpedance

_{φ}

_{z}and X

_{φ}

_{z}calculated for different values of the tensile stress σ

_{t}. Both the real and imaginary parts of the ODMI exhibit similar behavior with an increase in stress. The peaks in R

_{φ}

_{z}and X

_{φ}

_{z}become more pronounced, and the peak fields shift towards zero fields. It follows from Figure 6 also that the field sensitivity of the ODMI increases with the tensile stress.

_{2}in the shell deviates from the azimuthal direction, and the effective anisotropy field H

_{eff,2}decreases (see Equations (12) and (13)). Taking into account the changes in α

_{2}and H

_{eff,2}under the effect of the tensile stress, we can present the expression for threshold field H

_{th}of the bias current to eliminate domain structure in the shell region in the following form: H

_{th}= H

_{eff,2}|sinα

_{2}|. Correspondingly, the expression for the threshold bias current I

_{th}can be rewritten as

_{th}tends to the bias current I

_{b}under the application of the tensile stress. As discussed above, the ODMI increases when the bias current is close to I

_{th}(see Figure 3). Thus, an increase of the tensile stress σ

_{t}results in a growth of the ODMI response (see Figure 6).

_{R}and S

_{X}for different values of the tensile stress σ

_{t}are shown in Figure 7. The values of S

_{R}and S

_{X}are calculated by means of Equations (25) and (26) taking into account that ΔH = H

_{p}+ H

_{eff,2}sin2α

_{2}. The field sensitivities increase monotonically with the value of the applied stress (see Figure 7).

_{t}we introduce the ratios

_{φ}

_{z}(0) and X

_{φ}

_{z}(0) are the real and imaginary parts of the ODMI without tensile stress.

_{φ}

_{z})

_{σ}and (ΔX/X

_{φz})

_{σ}calculated for a fixed external field H

_{e}< H

_{p}at f = 100 MHz for different values of the bias current are shown in Figure 8. Both the ratios of (ΔR/R

_{φ}

_{z})

_{σ}and (ΔX/X

_{φ}

_{z})

_{σ}have high sensitivity to the applied tension stress. As follows from Figure 8, the ratios of (ΔR/R

_{φ}

_{z})

_{σ}and (ΔX/X

_{φ}

_{z})

_{σ}exhibit nearly linear dependence on the tensile stress at low values of σ

_{t}. This fact is attractive for the development of stress sensors.

#### 3.3. Effect of Torsional Stress on Off-Diagonal Magnetoimpedance

_{φ}

_{z}and imaginary parts X

_{φ}

_{z}of the ODMI on the external field are shown in Figure 9 at fixed I

_{b}and different values of the angular displacement per unit length τ > 0. For low τ, the asymmetry between the absolute values of the peaks in R

_{φ}

_{z}and X

_{φ}

_{z}at positive and negative fields decreases. At some critical value of τ = τ

_{cr}, the absolute values of the peaks become equal. At τ > τ

_{cr}, the modulus of the peak in the real and imaginary parts of the ODMI at a negative external field becomes higher (see Figure 9). The critical value τ

_{cr}of the angular displacement per unit length corresponds to the appearance of the effective circular anisotropy in the shell region (α

_{2}= 0). It follows from Equation (12) that this condition satisfies when H

_{tor,2}= –H

_{a}

_{,2}sin2ψ

_{2}. Taking into account Equation (10), we obtain for τ

_{cr}[44]:

_{eff,2}and effective anisotropy angle modulus in the shell. As a result, the threshold bias current I

_{th}increases (see Equation (27)). Correspondingly, the peaks in R

_{φz}and X

_{φz}at negative external field growth. The maximal field sensitivity of the ODMI is achieved when the bias current I

_{b}tends to the threshold one I

_{th}. With a further increase of the stress absolute value, the bias current becomes less than I

_{th}, and the field dependences of R

_{φ}

_{z}and X

_{φ}

_{z}exhibit hysteretic behavior [44].

_{R}of the real part of the ODMI is higher than S

_{X}. Both the sensitivities S

_{R}and S

_{X}decrease monotonically with an increase of the angular displacement per unit length τ.

## 4. Discussion

_{a}

_{,1}and shell H

_{a}

_{,2}, the anisotropy axis angle in the shell ψ

_{2,}and the core diameter 2r as fitting parameters, the calculated results explained the evolution of the field dependence of the GMI response with a frequency increase (the transition from the single-peak to two-peak behavior). In addition, the theoretical results described the positions of the peaks in the GMI field dependence and the magnitude of the GMI ratio.

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The microwire cross-section and unit vectors

**n**

_{ρ},

**n**

_{φ}and

**n**

_{z}of the coordinate system used for analysis. (

**b**) A sketch of the angles in the model lying within the

**n**

_{φ}–

**n**

_{z}plane.

**Figure 2.**(

**a**) The effective anisotropy angle α

_{2}and (

**b**) the effective anisotropy field H

_{eff,2}in the shell region as a function of the field H

_{tor,2}at different values of the intrinsic anisotropy axis angle ψ

_{2}.

**Figure 3.**(

**a**) The real part R

_{φz}and (

**b**) the imaginary part X

_{φz}of the ODMI as a function of the external field H

_{e}at f = ω/2π = 100 MHz for different values of the bias current I

_{b}. Parameters used for calculations are 2R = 15 μm, 2r = 8 μm, M = 900 G, σ = 5 × 10

^{15}s

^{−1}, κ = 0.15, H

_{a}

_{,1}= 5 Oe, H

_{a}

_{,2}= 30 Oe and ψ

_{2}= −0.05π.

**Figure 4.**(

**a**) The real part R

_{φz}and (

**b**) the imaginary part X

_{φz}of the ODMI as a function of the external field H

_{e}at f = 100 MHz and I

_{b}= 30 mA for different values of the core region diameter 2r. Other parameters used for calculations are the same as in Figure 3.

**Figure 5.**The field sensitivity of the real part S

_{R}(

**a**) and imaginary part S

_{X}(

**b**) of the ODMI as a function of the frequency f for different values of the bias current I

_{b}. Other parameters used for calculations are the same as in Figure 3. The values of S

_{R}and S

_{X}are reduced to the characteristic field sensitivity S

_{0}= Z

_{0}/H

_{a}

_{,2}.

**Figure 6.**(

**a**) The real part R

_{φz}and (

**b**) the imaginary part X

_{φz}of the ODMI as a function of the external field H

_{e}at f = 100 MHz and I

_{b}= 30 mA for different values of the tensile stress σ

_{t}. Magnetostriction coefficient λ

_{s}= 40 × 10

^{−6}, other parameters used for calculations are the same as in Figure 3.

**Figure 7.**The field sensitivity of the real part S

_{R}(

**a**) and imaginary part S

_{X}(

**b**) of the ODMI as a function of the frequency f at I

_{b}= 30 mA for different values of the tensile stress σ

_{t}. Magnetostriction coefficient λ

_{s}= 40 × 10

^{−6}, other parameters used for calculations are the same as in Figure 3.

**Figure 8.**The ratio of (ΔR/R

_{φz})

_{σ}(

**a**) and (ΔX/X

_{φz})

_{σ}(

**b**) as a function of the tensile stress σ

_{t}at H

_{e}= 10 Oe and f = 100 MHz for different values of the bias current I

_{b}. Magnetostriction coefficient λ

_{s}= 40 × 10

^{−6}, other parameters used for calculations are the same as in Figure 3.

**Figure 9.**(

**a**) The real part R

_{φz}and (

**b**) the imaginary part X

_{φz}of the ODMI as a function of the external field H

_{e}at f = 100 MHz and I

_{b}= 30 mA for different values of the angular displacement per unit length τ. Magnetostriction coefficient λ

_{s}= 40 × 10

^{−6}, shear modulus G = 50 GPa, other parameters used for calculations are the same as in Figure 3.

**Figure 10.**The field sensitivity of the real part S

_{R}(

**a**) and imaginary part S

_{X}(

**b**) of the ODMI as a function of the frequency f at I

_{b}= 30 mA for different values of the angular displacement per unit length τ. Magnetostriction coefficient λ

_{s}= 40 × 10

^{−6}, shear modulus G = 50 GPa, other parameters used for calculations are the same as in Figure 3.

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## Share and Cite

**MDPI and ACS Style**

Buznikov, N.A.
Off-Diagonal Magnetoimpedance in Annealed Amorphous Microwires with Positive Magnetostriction: Effect of External Stresses. *Magnetism* **2023**, *3*, 45-60.
https://doi.org/10.3390/magnetism3010005

**AMA Style**

Buznikov NA.
Off-Diagonal Magnetoimpedance in Annealed Amorphous Microwires with Positive Magnetostriction: Effect of External Stresses. *Magnetism*. 2023; 3(1):45-60.
https://doi.org/10.3390/magnetism3010005

**Chicago/Turabian Style**

Buznikov, Nikita A.
2023. "Off-Diagonal Magnetoimpedance in Annealed Amorphous Microwires with Positive Magnetostriction: Effect of External Stresses" *Magnetism* 3, no. 1: 45-60.
https://doi.org/10.3390/magnetism3010005