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Impedance of nanometer thickness ferromagnetic Co40Fe40B20 films


Nanocrystalline Co40Fe40B20 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 4πM s = 15.5 kG, anisotropy field H k = 0.031 kG, and r = 168 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


It is known that impedance (IM) of an ferromagnetic (FM) material is closely related to its complex permeability (μμ 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 Fδ 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.


The composition of the film sample in this test was Co40Fe40B20. 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. 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.

Figure 1
figure 1

Two Co 40 Fe 40 B 20 film samples. L is the length and w the width. i ac is the ac current sent through each sample. is the saturation magnetization and is the deposition field. and are the in-plane spin-wave wave vectors. (a) The ||L case and (b) the ||w case.

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.

A circular film sample was taken for the FMR experiment. The cavity used was a Bruker ER41025ST X-band resonator (Bruker Optics Taiwan Ltd., San Chung, Taiwan, Republic of China) which was tuned at f = 9.6 GHz, and the film sample was oriented such that ||and , where was an in-plane field which varied from 0 to 5 kG, and 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 ΔH = 53 Oe.

Figure 2
figure 2

Ferromagnetic resonance of the Co 40 Fe 40 B 20 film with the microwave frequency f = 9.6 GHz. H R is the resonance field.

Other magnetic and electrical properties of the Co40Fe40B20 film were obtained from vibration sample magnetometer measurements: 4πM 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 Co40Fe40B20 films is very high. Here, since δ (ρ)1/2, a larger ρ will lead to a longer δ >> t f.

Results and discussion

In order to interpret the IM data (or spectrum) of this work, as shown in Figure 3 (the 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 F4 = 721 MHz. From these experimental facts, we reach two conclusions: (1) f 0F 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 0F 0, we use the prefix "quasi" to describe the event. More explanation will be given later.

Figure 3
figure 3

Impedance Z = R + iX with || L. Impedance Z = R + iX where R and X are the resistance and reactance of the Co40Fe40B20 film sample with ||L. f 0 and f n , with n = 1, 2, 3, 4, are the frequency peaks associated with various kinds of resonances.

Figure 4
figure 4

Impedance Z = R + iX but with || w. F 0 and F n , with n = 1, 2, 3, 4 are the frequency peaks associated with various kinds of resonances.

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 103, 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 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 Co40Fe40B20 film is replaced by a Cu film with the same dimensions, there is also one single resonance peak at f d(Cu) = (1/2π)(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 ~(102 to 103) × μ o × GF for Co40Fe40B20, where GF depends only on the geometrical size and shape of the sample, L s = 1 × μ o × GF for Cu, and C CoFeBC Cu, in principle, we find f d(Co40Fe40B20) [(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/2π 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 rfH g = (i ac z)/(wt 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.

With regard to the FMR-K mechanism, we propose the following model: When FMR-K occurs in Figure 2, we have [9].


By substituting the values of f R = 9.6 GHz, H R, H K, and 4πM s, it is found ν = 2.833 for Co40Fe40B20. Thus, the main (or FMR-K) resonance (at H = 0) would occur at f = f FMRK = ν[H K(H K + 4πM 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 FMRKf 0F 0. The reasons for the frequency shifts of the quasi-FMR-K resonances in IM are given below. According to Refs. [911], 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, (/t f) is the out-of-plane (OFP) standing spin-wave wavevector, p = 0, 1, 2,...etc., θ q is the angle between and the surface normal or the z-axis, hence for and , 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(ΔH)/(2f 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.

In the following, we shall refer to the 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, ||L, where the azimuth angle φ of is (π/2) and in Figure 1b, ||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 × 106 (1/m) and q //T = 3.216 × 106 (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 Lw. Thus, even if exists in the FMR case, there should be only one , where , 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 is likely to be negligible. As a result, in Figure 2, we find only one in the FMR case and = f FMRK. However, due to reason (1) above, and the symmetry breaking issue in the IM case, as discussed before, should be shifted from f FMRK to f 0 and F 0, respectively.

Moreover, if we take the formula Z = (B/A s)(1 + i)coth[(t/2A s)(1 + i)], where B = (ρL)/(2w), A s = [ρ/(πfξμ o)][cos(δ/2) + i sin(δ/2)], μξμ o, and μ o = 4π × 10-7 H/m. By using the Newton-Raphson method [12], we may calculate the f dependence of μ R ≡ ξcosδ or μ I ≡ -ξsinδ 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.

Figure 5
figure 5

Permeability μ = μ R + i μ I . Permeability μ = μR + I where μR and μI are the real and imaginary parts of the film samples vs. the frequency f.


We have performed IM and FMR experiments on nanometer thickness Co40Fe40B20 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 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, in the h||L case and in the h||w case, and found that is smaller than as expected.


  1. Jen SU, Chao YD: The field-annealing effect on magnetoimpedance of a zero magnetostrictive metallic glass. J Appl Phys 1996, 79: 6552. 10.1063/1.361944

    Article  Google Scholar 

  2. Panina LV, Mohri K, Uchiyama T, Noda M: Giant magneto-impedance in Co-rich amorphous wires and films. IEEE Trans MAG 1995, 31: 1249.

    Article  Google Scholar 

  3. Boll R: Soft Magnetic Materials. Berlin: Simens Aktiengesellschaft; 1979.

    Google Scholar 

  4. Chikazumi S: Physics of Magnetism. New York: Krieger; 1978.

    Google Scholar 

  5. Berger L: Current-induced oscillations of a Bloch wall in magnetic thin films. J Magn Magn Mater 1996, 162: 155. 10.1016/S0304-8853(96)00286-7

    Article  Google Scholar 

  6. Brophy JJ: Basic Electronics for Scientists. New York: McGraw-Hill; 1977.

    Google Scholar 

  7. Williams HJ, Shockley W: A simple domain structure in an iron crystal showing a direct correlation with the magnetization. Phys Rev 1949, 75: 178. 10.1103/PhysRev.75.178

    Article  Google Scholar 

  8. Agarwala AK, Berger L: Domain-wall surface energy derived from the complex impedance of Metglas ribbons traversed by ac currents. J Appl Phys 1985, 57: 3505. 10.1063/1.335042

    Article  Google Scholar 

  9. Morrish AH: The Physical Principles of Magnetism. New York: Krieger; 1980.

    Google Scholar 

  10. Ding Y, Klemmer TJ, Crawford TM: A coplanar waveguide permeameter for studying high-frequency properties of soft magnetic materials. J Appl Phys 2004, 96: 2969. 10.1063/1.1774242

    Article  Google Scholar 

  11. Kalinilos BA, Slavin AN: Theory of dipole-exchange spin wave spectrum for ferromagnetic films with mixed exchange boundary conditions. J Phys C 1986, 19: 7013. 10.1088/0022-3719/19/35/014

    Article  Google Scholar 

  12. Yabukami S, Ojima R, Yamaguchi M, Arai KI, Kikuchi S: Highly sensitive permeability measurements obtained by electrical impedance. J Magn Magn Mater 2003, 254–255: 111.

    Article  Google Scholar 

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

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Correspondence to Shien Uang Jen.

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Authors' contributions

SUJ designed and directed this research, analyzed and interpreted the data, and wrote the paper.

TYC carried out the IM and FMR experiments.

CKL set up and provided the FMR equipments.

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Jen, S.U., Chou, T.Y. & Lo, C.K. Impedance of nanometer thickness ferromagnetic Co40Fe40B20 films. Nanoscale Res Lett 6, 468 (2011).

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  • spin-wave resonance
  • impedance
  • magnetic films