# Impedance of nanometer thickness ferromagnetic Co_{40}Fe_{40}B_{20} films

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## Abstract

Nanocrystalline Co_{40}Fe_{40}B_{20} films, with film thickness *t*_{f} = 100 nm, were deposited on glass substrates by the magnetron sputtering method at room temperature. During the film deposition period, a dc magnetic field, *h* = 40 Oe, was applied to introduce an easy axis for each film sample: one with *h*||*L* and the other with *h*||*w*, where *L* and *w* are the length and width of the film. Ferromagnetic resonance (FMR), ultrahigh frequency impedance (IM), dc electrical resistivity (*ρ*), and magnetic hysteresis loops (MHL) of these films were studied. From the MHL and *r* measurements, we obtain saturation magnetization 4*πM*_{ s } = 15.5 kG, anisotropy field *H*_{k} = 0.031 kG, and *r* = 168 mW.cm. From FMR, we can determine the Kittel mode ferromagnetic resonance (FMR-K) frequency *f*_{FMRK} = 1,963 MHz. In the *h*||*L* case, IM spectra show the quasi-Kittel-mode ferromagnetic resonance (QFMR-K) at *f*_{0} and the Walker-mode ferromagnetic resonance (FMR-W) at *f*_{ n } , where *n* = 1, 2, 3, and 4. In the *h*||*w* case, IM spectra show QFMR-K at *F*_{0} and FMR-W at *F*_{ n } . We find that *f*_{0} and *F*_{0} are shifted from *f*_{FMRK}, respectively, and *f*_{ n } = *F*_{ n } . The in-plane spin-wave resonances are responsible for those relative shifts.

PACS No. 76.50.+q; 84.37.+q; 75.70.-i

### Keywords

spin-wave resonance impedance magnetic films## Introduction

It is known that impedance (IM) of an ferromagnetic (FM) material is closely related to its complex permeability (*μ* ≡ *μ*_{R} + *i μ*_{I} ), where *μ*_{R} and *μ*_{I} are the real and imaginary parts, in the high-frequency (*f*) range [1, 2]. Past experience has also shown that there should exist a cutoff frequency (*f*_{c}), where *μ*_{R} crosses zero and *μ*_{I} reaches maximum [3], for each FM material. According to Ref. [3], *f*_{c} increases as the thickness of the FM sample decreases and finally reaches an upper limit. The thickness dependence is due to the eddy current effect, while the upper limit is due to the spin relaxation (or resonance) effect. Hence, in a sense, we would expect the *f* dependence of impedance *Z* = *R* + *iX*, where *R* is resistance and *X* reactance, behaves similarly. In Ref. [1], we had the situation that the thickness (*t*_{F}) of the FM ribbon was thick to meet the criterion: *t*_{F} ≥ *δ*≅ 10 μm, where *δ* is skin depth (at *f* = 1 MHz), but in this article, we have a different situation wherein the thickness (*t*_{f}) of the FM film is thin to meet the criterion: *t*_{f} = 100 nm << *δ*≅ 654 nm (at *f* = 1 GHz). That means the time varying field *H*_{g}, generated by the ac current (*i*_{ac}), in the IM experiment should penetrate through the film sample even under an ultrahigh frequency condition this time. Moreover, there are various kinds of mechanisms to explain the resonance phenomena: the film size (FZ), the magnetic domain wall (MDW), the RLC-circuit, the ferromagnetic resonance of the Kittel mode (FMR-K), the ferromagnetic resonance of the Walker mode (FMR-W), the relaxation time, and the standing spin-wave resonance mechanisms. We shall examine all these mechanisms one by one, based on the experimental data collected in this study.

## Experimental

_{40}Fe

_{40}B

_{20}. We used magnetron sputtering technique to deposit the film on a glass substrate at room temperature. The film thickness

*t*

_{f}, as mentioned before, was 100 nm. During the deposition period, an external dc field,

*h*≅ 40 Oe, was applied to define the easy axis, as shown in Figure 1, for each nanometer thick sample. In Figure 1, we have length,

*L*= 10.0 mm, and width,

*w*= 500 μm, in case (a)

*h*||

*L*, and in case (b)

*h*||

*w*. Open image in new window is the saturation magnetization of each film. In addition, the nanocrystalline grain structures in our CoFeB films were confirmed from their transmission electron microscope photos.

In a typical IM experiment, there were three features: (1) the rectangular film sample, either as shown in Figure 1a or Figure 1b, was placed at the center of a pair of Helmholtz coils, which could produce a field *H*_{E} ⊥ *L*, (2) *Z* was measured by an Agilent E4991A RF impedance/material analyzer (Agilent Technologies, Santa Clara, CA, USA) with a two-point (ECP18-SG-1500-DP) pico probe, and (3) the peak-to-peak amplitude of the ac current, *i*_{ac}, was fixed at 10 mA, and the frequency *f* of the current was scanned from 1 MHz to 3 GHz.

*f*= 9.6 GHz, and the film sample was oriented such that Open image in new window || Open image in new window and Open image in new window ⊥ Open image in new window , where Open image in new window was an in-plane field which varied from 0 to 5 kG, and Open image in new window was the microwave field. The result is shown in Figure 2, where we can spot an FMR (or FMR-K) event at

*H*=

*H*

_{R}= 0.68 kG, and define the half-peak width Δ

*H*= 53 Oe.

Other magnetic and electrical properties of the Co_{40}Fe_{40}B_{20} film were obtained from vibration sample magnetometer measurements: 4*πM*_{s} = 15.5 kG and the anisotropy field, *H*_{k} = 0.031 kG, and from electrical resistivity (*ρ*) measurement: *ρ* = 168 μΩ. cm. Note that because of the nanocrystalline and the nanometer thickness characteristics, the *ρ* of our Co_{40}Fe_{40}B_{20} films is very high. Here, since δ ∝ (*ρ*)^{1/2}, a larger ρ will lead to a longer δ >> *t*_{f}.

## Results and discussion

*h*||

*L*case) and in Figure 4 (the

*h*||

*w*case), we have the following definitions. First, whenever there is a resonance event, we should find a peak located at

*f*=

*f*

_{0}and

*f*=

*f*

_{ n }, where

*n*= 1, 2, 3, 4 in the R-spectrum, and a wiggle (or oscillation) centered around the same

*f*

_{0}and

*f*

_{ n }in the X-spectrum. To summarize the data in Figures 3 and 4, we have in the

*h*||

*L*case,

*f*

_{0}= 2,081,

*f*

_{1}= 1,551,

*f*

_{2}= 1,291,

*f*

_{3}= 991, and

*f*

_{4}= 781 MHz; and in the

*h*||

*w*case,

*F*

_{0}= 2,431,

*F*

_{1}= 1,551,

*F*

_{2}= 1,281,

*F*

_{3}= 991, and F

_{4}= 721 MHz. From these experimental facts, we reach two conclusions: (1)

*f*

_{0}≠

*F*

_{0}and (2) within errors,

*f*

_{ n }=

*F*

_{ n }. Since at either

*f*

_{0}or

*F*

_{0}, each corresponding wiggle crosses zero, we believe there is a quasi-FMR-K event. Notice for the moment that because

*f*

_{0}≠

*F*

_{0}, we use the prefix "quasi" to describe the event. More explanation will be given later.

Here, we discuss the possibilities of the FZ resonance first. From Ref. [4], we know an electromagnetic (EM) wave may be built up inside the film during IM experiments. In Figure 1a, supposing *L* ≅ *λ*_{||}, where *λ*_{||} is the longitudinal EM wavelength, *w* ≅ *λ*_{⊥}, where *λ*_{⊥}is the transverse EM wavelength, and *μ* ≅ 10^{3}, we find the FZ resonance frequencies: *f*_{EM}(||) = *η*_{||} × 7 MHz and *f*_{EM}(⊥) = *η*_{⊥} × 27 MHz, where *η*_{||} and *η*_{⊥}are positive integers. Since based on the experimental findings, *f*_{ n } = *f*_{EM}(||) should be equal to *F*_{ n } = *f*_{EM}(⊥), *f*_{ n } or *F*_{ n } must be a positive integer number of times of the frequency 189 MHz. Simple calculations show that the above statement cannot be satisfied. Besides, if the statement were true, there would exist at least as many as eight different FZ resonance peaks, instead of only the four resonance peaks observed so far.

Next, the MDW mechanism is discussed. As the size of the sample is large, there are magnetic stripe domains, parallel to Open image in new window in Figures 1a, b. According to Ref. [5], the MDW resonance for the CoFeB film should occur at *f* = 78 MHz. However, we have reasons to believe that this kind of resonance does not exist in our IM spectra. First, in Figures 3 and 4, there is neither a peak nor a wiggle at *f* = 78 MHz. Second, when *H*_{E} = 150 G, much larger than the saturation field, was applied to eliminate magnetic domains, those peaks (at *f*_{0} to *f*_{4} or *F*_{0} to *F*_{4}, respectively) still persisted.

Further, the RLC-circuit resonance mechanism is discussed. If the Co_{40}Fe_{40}B_{20} film is replaced by a Cu film with the same dimensions, there is also one single resonance peak at *f*_{d}(Cu) = (1/2*π*)(*L*_{s}*C*)^{-(1/2)} = 2.641 GHz, where *L*_{s} is the self-inductance and *C* is the capacitance of the film [6]. However, we believe that *f*_{0} and/or *F*_{0} are less likely due to the RLC-circuit resonance mechanism for the reason below. Since *L*_{s} = μ × GF ~(10^{2} to 10^{3}) × *μ*_{o} × GF for Co_{40}Fe_{40}B_{20}, where GF depends only on the geometrical size and shape of the sample, *L*_{s} = 1 × *μ*_{o} × GF for Cu, and *C*_{CoFeB} ≥ *C*_{Cu}, in principle, we find *f*_{d}(Co_{40}Fe_{40}B_{20}) ≅ [(1/10) to (1/30)] × *f*_{d}(Cu) = 0.26 to 0.08 GHz, which is too small to meet the facts, i.e., *f*_{0} = 2.081 GHz and *F*_{0} = 2.431 GHz.

With regard to the FMR-W mechanism, we have the following discussion. At *f* = *f*_{ n } and/or *F*_{ n } , we believe each resonance should correspond to a specific FMR-W mode. The reasons are summarized below. First, in the typical FMR result, as shown in Figure 2 because the sample was placed in the homogeneous *h*_{rf} region, no FMR-W modes could be observed. However, as indicated in Ref. [4], if *h*_{rf} is sufficiently inhomogeneous to vary over the sample, one will observe various FMR-W modes at *H* = *H*_{ n } and *H*_{ n } < *H*_{R}. From a simple relationship [4], such as *f* = *νH*_{eff}, where *H*_{eff} is the effective field and ν = *γ*/2*π* is the gyromagnetic ratio, it is easy to recognize that since *H*_{ n } < *H*_{R}, we have *f*_{ n } < *f*_{0} and/or *F*_{ n } < *F*_{0}, which is what has been observed. Second, from Refs. [7] and [8], it is known that *h*_{rf} ≡ *H*_{g} = (*i*_{ac}*z*)/(w*t*_{f}), where *z* is a variable parameter along *t*_{f}. Therefore, in a typical IM measurement, *h*_{rf} or *H*_{g} cannot be homogeneous all over the sample. That is why in Figure 2, there is no FMR-W mode, but in Figures 3 or 4, there are various FMR-W modes.

*f*

_{R}= 9.6 GHz,

*H*

_{R},

*H*

_{K}, and 4

*πM*

_{s}, it is found ν = 2.833 for Co

_{40}Fe

_{40}B

_{20}. Thus, the main (or FMR-K) resonance (at

*H*= 0) would occur at

*f*=

*f*

_{FMRK}= ν[

*H*

_{K}(

*H*

_{K}+ 4

*πM*

_{s})]

^{1/2}= 1,963 MHz. According to our previous arguments, this frequency,

*f*

_{FMRK}, should be equal to

*f*

_{0}and/or

*F*

_{0}in MI. Obviously, what we have is

*f*

_{FMRK}≠

*f*

_{0}≠

*F*

_{0}. The reasons for the frequency shifts of the quasi-FMR-K resonances in IM are given below. According to Refs. [9, 10, 11], the quasi-FMR-K-resonance relationship for

*f*

_{0}or

*F*

_{0}at

*H*= 0 and under the exchange-dominated condition is expressed as

where *A* = 1.0 × 10^{-11} J/m is the exchange stiffness, *i* = *L* or *T*, *q*_{//i}is the in-plane (IP) standing spin-wave wavevector, (*pπ*/*t*_{f}) is the out-of-plane (OFP) standing spin-wave wavevector, *p* = 0, 1, 2,...etc., *θ*_{ q } is the angle between Open image in new window and the surface normal Open image in new window or the *z*-axis, hence for Open image in new window and Open image in new window , as shown in Figure 1, *θ*_{ q } = *π*/2 always, and *τ* is the relaxation time [9], where 1/*τ* ≡ (*αγH*_{R}) = 94.3 MHz and *α* ≡ ν(Δ*H*)/(2*f*_{R}) = 0.00777. Therefore, if the relaxation time (1/τ) mechanism dominated in Equation 2, *f*_{0} would be equal to 267 MHz, which is much lower than the *f*_{0} or *F*_{0} in Figures 3 and 4.

Next, we consider the OFP standing spin-wave case only, i.e., temporarily assuming *q*_{//i}= 0 or negligible in Equation 2, simple calculations show that *f*_{0}(*p* = 0) = 1.963 GHz, *f*_{0}(*p* = 1) = 4.874 GHz, and *f*_{0}(*p* = 2) = 9.136 GHz. Because our Agilent E4991A works only up to 3.0 GHz, *f*_{0}(*p* = 1) and *f*_{0}(*p* = 2), although existing, were not observed in this work.

*p*= 0 case only. From Equation 2, if

*p*= 0 and the (1/

*τ*) term is negligible, we consider the following two cases: in Figure 1a, Open image in new window ||

*L*, where the azimuth angle

*φ*of Open image in new window is (

*π*/2) and in Figure 1b, Open image in new window ||

*w*, where

*φ*= 0. Then, Equation 2 can be simplified as

By substituting the values of *f*_{0}, *F*_{0}, *A*, and *H*_{k} in Equations 3a, b, respectively, we find *q*_{//L}= 1.326 × 10^{6} (1/*m*) and *q*_{//T}= 3.216 × 10^{6} (1/*m*). Two features can be summarized. First, since [1/(2π)][*q*_{//i}× *t*_{f}] = (0.5 to 1.2) × 10^{-1} << 1, it confirms that we do have a long wavelength in-plane spin wave (IPSW), *q*_{//L}or *q*_{//T}, traveling in each film sample. Second, due to the boundary conditions of the film sample, we should have *q*_{//L}∝ (1/*L*) and *q*_{//T}∝ (1/*w*). Thus, because *L* > *w*, our previous results are reasonable that *q*_{//L}< *q*_{//T}.

Finally, as to why the IP spin-waves can be easily excited in the IM experiment, but cannot be found in the FMR experiment, we have a simple, yet still incomplete, explanation as follows. The film sample used in the latter experiment is circular, which means by symmetry *L* = *w*, while the one used in the former experiment is rectangular, which means that the symmetry is broken, with *L* ≠ *w*. Thus, even if Open image in new window exists in the FMR case, there should be only one Open image in new window , where Open image in new window , by symmetry argument. Nevertheless, for some reasons, such as (1) that a high-current density *j*_{ac} = (*i*_{ac})/(*t*_{f}*w*) may be required to initiate IPSW, and (2) that *j*_{ac} flowing in the FMR experiment may be too low to initiate any IPSW, we think the *q*_{//}term in Open image in new window is likely to be negligible. As a result, in Figure 2, we find only one Open image in new window in the FMR case and Open image in new window = *f*_{FMRK}. However, due to reason (1) above, and the symmetry breaking issue in the IM case, as discussed before, Open image in new window should be shifted from *f*_{FMRK} to *f*_{0} and *F*_{0}, respectively.

*Z*= (

*B*/

*A*

_{s})(1 +

*i*)coth[(

*t*/2

*A*

_{s})(1 +

*i*)], where

*B*= (

*ρL*)/(2

*w*),

*A*

_{s}= [

*ρ*/(

*πfξμ*

_{o})][cos(

*δ*/2) +

*i*sin(

*δ*/2)],

*μ*≡

*ξμ*

_{o}, and

*μ*

_{o}= 4

*π*× 10

^{-7}H/m. By using the Newton-Raphson method [12], we may calculate the

*f*dependence of

*μ*

_{R}≡ ξcos

*δ*or

*μ*

_{I}≡ -ξsin

*δ*from the

*R*and

*X*data. From the

*μ*

_{R}vs.

*f*or the

*μ*

_{I}vs.

*f*plot, as shown in Figure 5, we can define the cutoff frequency

*f*

_{c}= 2,051 MHz in the

*h*||

*L*case. Clearly,

*f*

_{c}in Figure 5 is almost equal to

*f*

_{0}found in Figure 3.

## Conclusion

We have performed IM and FMR experiments on nanometer thickness Co_{40}Fe_{40}B_{20} film samples. Film thickness *t*_{f} was deliberately chosen much smaller than eddy current depth δ in the frequency range 100 MHz to 3 GHz. From the FMR data, we find that the Kittel mode resonance occurs at *f*_{FMRK} = 1,963 MHz, while from the IM data, we find that (1) the quasi-Kittel-mode resonance occurs at *f*_{0} = 2,081 MHz in the *h*||*L* case and *F*_{0} = 2,431 MHz in the *h*||*w* case, respectively, and (2) the Walker-mode resonances at *f*_{ n } = *F*_{ n } for both cases. It is believed that the shift of Open image in new window from *f*_{FMRK} to *f*_{0} or from *f*_{FMRK} to *F*_{0} is due to the existence of IPSWs. Moreover, we have estimated the values of wave vectors of IPSW, Open image in new window in the *h*||*L* case and Open image in new window in the *h*||*w* case, and found that Open image in new window is smaller than Open image in new window as expected.

## Notes

### Acknowledgements

This work was supported by a grant: NSC97-2112-M-001-023-MY3.

## Supplementary material

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