New characteristic parameter of energy loss in permalloy

Characteristic parameters of energy storage and loss in soft magnetic materials (SMMs), product S of initial susceptibility χi and resonance frequency ωr and product L of maximum χm′′ and half width Δω of imaginary part, were introduced in permalloy films. Expressions of characteristic parameters were derived from Landau–Lifshits–Gilbert equation with a general anisotropic free energy model. It was found S = L equals to intrinsic quantity of SMMs Q≡γMs1+H⊥/H∥ , where γ is the gyromagnetic ratio, Ms is the saturation magnetization, and H⊥ and H∥ are mutually orthogonal effective magnetic fields. In the presence of eddy current, it was found S remains equal to Q while L < Q results from inhomogeneity of microwave magnetic field caused by eddy current. Moreover, a linear scale of L/S versus Gilbert damping factor α was found in permalloy films with different thicknesses. Hence, our findings provide a new perspective on energy storage and loss of SMMs.


Introduction
High storage and low loss of energy are the key performance of soft magnetic materials (SMMs) [1][2][3][4][5][6][7][8].SMMs enable high magnetic flux and low loss inductive switching, which is useful in transformers to achieve the conversion of electrical energy in different circuits as well as in electric motors and generators to achieve the mutual conversion of mechanical energy and electrical energy.The energy storage of SMMs characterizes how much energy can be converted [9], while the energy loss of SMMs characterizes the energy consumption in the conversion process [10,11].The two properties determine the energy conversion efficiency of SMMs within working frequency.As a result, metallic SMMs with high storage and loss of energy such as Si steel [12,13], are widely used in the Hz range, while metallic soft magnetic composites [14][15][16] and amorphous nanocrystals [17] can be used in the kHz range.With the development of sustainable and electrified world, none of SMMs can release full potential of efficient power conversion based on wide bandgap semiconductors [18,19].To find the metallic SMMs with high storage and low loss of energy in the MHz range [20] has become the available way to achieve more than 98% [21] energy conversion efficiency.
Susceptibility χ represents the response of the internal magnetization M in magnetic materials to an microwave magnetic field h and is defined as M/ h.Real and imaginary parts of susceptibility are typically used to describe the energy storage and loss of metallic SMMs [8,22].Cut-off frequency defined as the frequency at where the value of permeability is half of initial value determines the highest working frequency of SMMs.In order to achieve high storage and low loss in the MHz range, it is necessary to break through the cut-off frequency limit 100 kHz of susceptibility in metallic SMMs.First of all, the intrinsic cut-off frequency of susceptibility without eddy current should be answered.It is known that there are three mechanisms [23] of susceptibility, which are diffusion relaxation, domain wall replacement, and nature resonance.The three mechanisms result from atomic diffusion and/or electron diffusion, growth of the parallel domains and shrink of the antiparallel domains, coherent precession of magnetization, respectively.Their cut-off frequency are in kHz, MHz, and GHz range, respectively.It can be seen that the last one of metallic SMMs is the most promising mechanism to realize excellent performance in the MHz range [24].Experimental cut-off frequency determined by domain wall replacement and nature resonance can not be observed in susceptibility spectra of metallic SMMs [25], but in that of ferrite [26][27][28].Apparently, 100 kHz cut-off frequency is not completely determined by above mechanisms.Our research found that the susceptibility of metallic SMMs results from the modulation of eddy current on nature resonance, rather than domain wall replacement and diffusion relaxation.
However, the real and imaginary parts of susceptibility spectrum in the presence of eddy current [29,30] are no long antisymmetric and symmetric Lorentz lines respectively which can describe the typical spectrum of intrinsic susceptibility [31].Consequently, it is hard to describe the spectrum with intrinsic quantities such as saturation magnetization M s , anisotropic field H K , and damping factor α easily.Refer to the importance of maximum magnetic energy product (BH) max [32] in hard magnetic materials, characteristic parameters of energy storage and loss are urgent to be introduced.It is known that the initial susceptibility χ i determines the susceptibility of metallic SMMs within working frequency range, and the resonance frequency ω r determines the highest working frequency of metallic SMMs as shown in figure 1.Hence, a characteristic parameter, S = χ i ω r , is able to describe the energy storage capacity of metallic SMMs as shown by blue shadow area in figure 1.On the other hand, the maximum χ ′ ′ m and half width ∆ω of imaginary part represents the maximum energy loss and the frequency range of the main energy loss of metallic SMMs, respectively.Characteristic parameter of energy loss can be defined as L = χ ′ ′ m ∆ω as shown by gray shadow area in figure 1.
Actually, as early as 1947, Snoek [33,34] introduced S to describe the relationship between susceptibility versus frequency and the intrinsic parameters of ferrites, and found S = γM s where γ is gyromagnetic ratio.Subsequently, it was found that S = γM s H θ /H φ where H θ and H φ are magnetic anisotropic fields in polar and azimuth angle direction of planar ferrite [35,36].Later, Archer found ω r S = γ 2 M 2 s in ferromagnetic thin film [37].We have developed a more general bi-anisotropy model [38,39] which unify the above three models and is verified in planar rare earth [40,41], flake-shaped transition metal alloys [42,43] and soft magnetic films [44,45].Therefore, to investigate the relationship between L and S in bi-anisotropy model especially with eddy current becomes a key issue of energy loss and storage of metallic SMMs.
In this paper, a new characteristic parameter of loss, L = χ ′ ′ m ∆ω, was constructed based on the imaginary part of susceptibility.The relationship between characteristic parameters and intrinsic quantities of metallic SMMs, S = L = γM s 1 + H ⊥ /H ∥ ≡ Q, were derived from a general bi-anisotropy model by solving Landau-Lifshits-Gilbert (LLG) equation [46] without eddy current, which is conform to Kramers-Kronig (K-K) relation [47,48].The susceptibility modulated by eddy current was further established based on the model.Experimental results of characteristic parameters were obtained by measuring susceptibility spectra of permalloy films under different static magnetic fields and found L < S = Q in the presence of eddy current.A linear scale between ratio L/S and Gilbert damping factor α was found in permalloy films with different thicknesses.

Theory
The real and imaginary of susceptibility is the response of magnetization M to microwave magnetic field h.The dynamic process of M described by LLG equation is determined by total effective magnetic field H eff = −dF/d(µ 0 M) besides damping torque.For the sake of simplicity and without losing generality, in Cartesian coordinate system o-xyz we constructed a bi-anisotropy model with free energy density where anisotropy coefficient K ⊥ represents a plane-symmetric anisotropy, K ∥ represents an axisymmetric anisotropy, M i (i = y, z) are the components of M. The stabilized direction of magnetization is z axis.First and second terms on the right-hand side of equation ( 1) are plane-symmetric and axisymmetric which are the lowest and highest rotational symmetry of easy magnetization axis, respectively.With decreasing K ⊥ /K ∥ , the trajectory of M changes from oblate elliptic cone to cone.This is similar with change of in-plane pendulum to cone pendulum.The landscape of F with K ⊥ = 2 × 10 5 Jm −3 and K ∥ = 1 × 10 5 Jm −3 is shown as inset of figure 1 when magnetization deviates from z axis at a small angle.Supposing h = h 0 e −iωt e x , by solving LLG equation the complex susceptibility χ = χ ′ + i χ ′ ′ can be obtained as where is resonance frequency, and χ i = M s /H ∥ is the initial susceptibility χ ′ (ω = 0).According to equation (2a), characteristic parameter of energy storage, S, can be obtained as as shown by blue shadow area in figure 1.
When ω = ω r , the maximum value of the imaginary part of susceptibility can be obtained from equation (2b) Set ω = ω 1/2 and satisfying χ ′ ′ (ω 1/2 ) = χ ′ ′ m /2 and considering 4ω 2 r ≫ α 2 γ 2 (2H ∥ + H ⊥ ) 2 , the half width ∆ω of the imaginary part of susceptibility can be obtained as Combined with equations ( 4) and ( 5), characteristic parameter L of energy loss can be defined as as shown by gray shadow area in figure 1.
Comparing equations ( 3) and ( 6), it is found where Q ≡ γM s 1 + H ⊥ /H ∥ is the intrinsic quantity of metallic SMMs, which represents the blue S and gray L shadow areas are equal in figure 1 and is consistent with K-K relation.

Results and discussion
The metal films with large permeability represent an ideal system for basic investigations.Compared with other films, permalloy film is more easily transformed into single domain system to simplify mechanism of susceptibility due to the weak in-plane anisotropic field and strong out-of-plane demagnetization field.Therefore, permalloy film was chosen as our main research object.To investigate characteristic parameters experimentally, 400 nm permalloy (Fe 20 Ni 80 ) film was deposited by rf magnetron sputtering on Si substrate at room temperature.The out-of-plane effective anisotropic field H ⊥ = 9.042 × 10 5 A m −1 and saturation magnetization M s = 7.691 × 10 5 A m −1 can be obtained from static hysteresis loop characterized by vibrating sample magnetometer [50].Figures 2(a) and (b) show the real and imaginary parts of susceptibility spectra of the film measured by vector network analyzer with short-circuit microstrip line [49].Each susceptibility spectrum was measured ten times and the average is presented.To eliminate the influence of domain structure, the spectra were measured under a static in-plane magnetic field H which is larger than the saturation magnetic field about 20 Oe.Replacing χ i with χ ′ (0.13 GHz), experimental characteristic parameters S ′ = χ ′ (0.13 GHz)ω r and L ′ = χ ′ ′ m ∆ω can be obtained from figure 2 directly as shown by red hollow circle and blue hollow triangle respectively.It can be seen that S ′ > L ′ obviously, which mainly results from χ ′ (0.13 GHz) > χ i .
It is known that significant skin effect caused by eddy current in metal around GHz will result in an inhomogeneous microwave magnetic field along thickness direction.
In order to obtain accurate experimental characteristic parameters, the influence of the inhomogeneous field on the susceptibility of permalloy film should be taken into account.Considering an infinite film, the susceptibility χ exp = χ ′ exp + i χ ′ ′ exp can be described with χ modulated by eddy current where are the real and imaginary parts of complex modulation factor G generated by the inhomogeneous field, t is the film thickness, k and κ are the real and imaginary parts of wave vector   A linear scaling of L/S as a function of α can be applied to permalloy films with different thicknesses.With the increase of film thickness, the slope of the linear scale increases from 2.04 to 8.40, and the intercept of the linear scale decreases from 0.99 to 0.83.
consistent with theoretical results of characteristic parameters Q.The accurate experimental results of L can be obtained by fitting with equation (8b).
With the increase of the external magnetic field, not only S and L are decreasing, but also the Gilbert damping factor α of 400 nm permalloy film obtained by fitting susceptibility spectra with equations (8a) and (8b) is reduced from 0.017 to 0.013.It was found that S = Q due to eddy current has no effect on χ i and ω r as well as L < Q resulting from the inhomogeneous field.The linear scale between L/S and α in 400 nm permalloy film was found as shown by orange hexagon in figure 4. The relationship between experimental characteristic parameters S and L in the presence of eddy current can be expressed as where a = 0.91 and b = 5.62.In order to demonstrate universality of the linear scale in films, 40-600 nm permalloy films were investigated.The ratios L/S of permalloy films with different thicknesses are all linearly dependent on Gilbert damping factor α as shown in figure 4. It can be seen that when the films are less than 200 nm, L/S ≈ 1, which is due to microwave magnetic field is approximately uniform in the thinner films and L ≈ S = Q, which is consistent with theoretical result i.e. equation (7).With the increase of film thickness, the inhomogeneity of microwave magnetic field increases.As a consequence the intercept a decreases from 0.99 to 0.83 and the slope b of the linear scale increases from 2.04 to 8.40.This means that the thicker permalloy  film, the greater energy loss, and the more remarkable dependence of L/S on damping, which is consistent with the nature of eddy current loss.The change of effective Gilbert damping caused by the inhomogeneity of microwave magnetic field needs further study.We conducted the same research in 60 nm CoZr granular film with strong uniaxial anisotropic field H K and similar results as permalloy film were obtained.The CoZr granular film was deposited by rf magnetron sputtering on Si substrate at room temperature.During sputtering, an oblique incidence angle of 50 • was used to introduce in-plane uniaxial anisotropy.The out-of-plane effective anisotropic field H ⊥ = 1.225 × 10 6 A m −1 and M s = 1.103 × 10 6 A m −1 is obtained from hysteresis loops.The real and imaginary parts of susceptibility spectra of CoZr granular film were measured under a static in-plane magnetic field H which is applied in the direction of easy axis as shown by figures 5(a) and (b), respectively.Fitting data of resonance frequency versus H as shown by inset of figure 5(b) γ = 2.238 × 10 5 Hz (A m −2 ) and H K = 1.219 × 10 4 A m −1 were obtained.Experimental characteristic parameters S ′ and L ′ can be obtained by fitting the susceptibility spectra with equations (2a) and (2b) as shown by figure 6.It can be seen that S ′ and L ′ are roughly equal under different external magnetic fields and consistent with theoretical results Q because the influence of eddy current on susceptibility can be ignored in 60 nm CoZr granular film.Therefore, our results can also be applied in CoZr granular film.

Conclusion
In conclusion, characteristic parameters S and L in metallic SMMs were derived from real and imaginary parts of susceptibility based on a general bi-anisotropy model.In the absence of eddy current, the relationships between two characteristic parameters, S = L = Q, were established, which is consistent with

Figure 1 .
Figure 1.Diagram of characteristic parameters of energy storage and loss, S = χ i ωr (blue shadow area) and L = χ ′ ′ m ∆ω (gray shadow area).The susceptibility spectrum was calculated based on a free energy density as shown by inset.

Figure 2 .
Figure 2. Susceptibility of 400 nm permalloy film versus ω under different in-plane static magnetic fields H which is larger than the saturation magnetic field.(a) Real part of spectra, (b) imaginary part of spectra.Inset shows resonance frequency versus H.
considered in our work, and σ = 2.58 × 10 6 S m −1 is the electrical conductivity.In-plane anisotropic effective field H ∥ = H + H K .Fitting data of resonance frequency versus H as shown by inset of figure 2(b) γ = 2.156 × 10 5 Hz (A m −2 ) and H K = 50.929A m −1 were obtained.The accurate experimental results of S = χ i ω r , can be obtained by fitting susceptibility spectra with equation (8a) as shown by solid red ball in figure 3. S and L ′ are roughly equal under different external magnetic fields and

Figure 3 .
Figure 3. Characteristic parameters S ′ , L ′ , and S versus external field H.The results of S ′ = χ ′ (0.13 GHz)ωr (red hollow circle) and L ′ = χ ′ ′ m ∆ω ( blue hollow triangle) obtained from susceptibility spectra of 400 nm permalloy film directly.Accurate value of characteristic parameter S = χ i ωr can be obtained by fitting susceptibility spectra with equation (8a).The black line is the theoretical result of Q ≡ γMs √ 1 + H ⊥ /H ∥ .

Figure 4 .
Figure 4. L/S versus Gilbert damping factor α in 40-600 nm permalloy films.Because the eddy current loss becomes small and can be ignored in permalloy films thinner 200 nm, L/S ≈ 1 can be found, which is consistent with theoretical results S = L = Q.A linear scaling of L/S as a function of α can be applied to permalloy films with different thicknesses.With the increase of film thickness, the slope of the linear scale increases from 2.04 to 8.40, and the intercept of the linear scale decreases from 0.99 to 0.83.

Figure 5 .
Figure 5. Susceptibility of 60 nm CoZr granular film versus ω under different in-plane static magnetic fields Hparallel easy axis.(a) Real part of spectra, (b) imaginary part of spectra.Inset shows resonance frequency versus H.

Figure 6 .
Figure 6.Characteristic parameters S ′ and L ′ versus H + HK.The black line is the theoretical result of Q ≡ γMs √ 1 + H ⊥ /H ∥ .

Figure 8 .
Figure 8. Out-of-plane and in-plane (inset) hysteresis loops of 60 nm CoZr granular film, where in-plane hysteresis loops were measured along easy axis (EA) and hard axis (HA), respectively.