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 hL and the other with hw, 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 mW.cm. From FMR, we can determine the Kittel mode ferromagnetic resonance (FMRK) frequency f_{FMRK }= 1,963 MHz. In the hL case, IM spectra show the quasiKittelmode ferromagnetic resonance (QFMRK) at f_{0 }and the Walkermode ferromagnetic resonance (FMRW) at f_{n}, where n = 1, 2, 3, and 4. In the hw case, IM spectra show QFMRK at F_{0 }and FMRW at F_{n}. We find that f_{0 }and F_{0 }are shifted from f_{FMRK}, respectively, and f_{n }= F_{n}. The inplane spinwave resonances are responsible for those relative shifts.
PACS No. 76.50.+q; 84.37.+q; 75.70.i
Keywords:
spinwave resonance; impedance; magnetic filmsIntroduction
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 highfrequency (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 δis 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 RLCcircuit, the ferromagnetic resonance of the Kittel mode (FMRK), the ferromagnetic resonance of the Walker mode (FMRW), the relaxation time, and the standing spinwave resonance mechanisms. We shall examine all these mechanisms one by one, based on the experimental data collected in this study.
Experimental
The composition of the film sample in this test was Co_{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) hL, and in case (b) hw. 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. 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 inplane spinwave 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 twopoint (ECP18SG1500DP) pico probe, and (3) the peaktopeak 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 Xband 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 inplane 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 FMRK) event at H = H_{R }= 0.68 kG, and define the halfpeak width ΔH = 53 Oe.
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 Co_{40}Fe_{40}B_{20 }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 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
In order to interpret the IM data (or spectrum) of this work, as shown in Figure 3 (the hL case) and in Figure 4 (the hw 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 Rspectrum, and a wiggle (or oscillation) centered around the same f_{0 }and f_{n }in the Xspectrum. To summarize the data in Figures 3 and 4, we have in the hL case, f_{0 }= 2,081, f_{1 }= 1,551, f_{2 }= 1,291, f_{3 }= 991, and f_{4 }= 781 MHz; and in the hw 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 quasiFMRK 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.
Figure 3. Impedance Z = R + iX with  L. Impedance Z = R + iX where R and X are the resistance and reactance of the Co_{40}Fe_{40}B_{20 }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. 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 ≅ 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 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 RLCcircuit 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) = (1/2π)(L_{s}C)^{(1/2) }= 2.641 GHz, where L_{s }is the selfinductance 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 RLCcircuit 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 FMRW mechanism, we have the following discussion. At f = f_{n }and/or F_{n}, we believe each resonance should correspond to a specific FMRW 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 FMRW 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 FMRW 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_{rf }≡ H_{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 FMRW mode, but in Figures 3 or 4, there are various FMRW modes.
With regard to the FMRK mechanism, we propose the following model: When FMRK 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 Co_{40}Fe_{40}B_{20}. Thus, the main (or FMRK) 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_{FMRK }≠ f_{0 }≠ F_{0}. The reasons for the frequency shifts of the quasiFMRK resonances in IM are given below. According to Refs. [911], the quasiFMRKresonance relationship for f_{0 }or F_{0 }at H = 0 and under the exchangedominated condition is expressed as
where A = 1.0 × 10^{11 }J/m is the exchange stiffness, i = L or T, q_{//i }is the inplane (IP) standing spinwave wavevector, (pπ/t_{f}) is the outofplane (OFP) standing spinwave wavevector, p = 0, 1, 2,...etc., θ_{q }is the angle between and the surface normal or the zaxis, 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 spinwave 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 × 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 inplane 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 spinwaves 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 exists in the FMR case, there should be only one , where , by symmetry argument. Nevertheless, for some reasons, such as (1) that a highcurrent 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) + isin(δ/2)], μ ≡ ξμ_{o}, and μ_{o }= 4π × 10^{7 }H/m. By using the NewtonRaphson 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 hL case. Clearly, f_{c }in Figure 5 is almost equal to f_{0 }found in Figure 3.
Figure 5. Permeability μ = μ_{R }+ i μ_{I}. Permeability μ = μ_{R }+ iμ_{I }where μ_{R }and μ_{I }are the real and imaginary parts of the film samples vs. the frequency f.
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 quasiKittelmode resonance occurs at f_{0 }= 2,081 MHz in the hL case and F_{0 }= 2,431 MHz in the hw case, respectively, and (2) the Walkermode 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 hL case and in the hw case, and found that is smaller than as expected.
Competing interests
The authors declare that they have no competing interests.
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.
Acknowledgements
This work was supported by a grant: NSC972112M001023MY3.
References

Jen SU, Chao YD: The fieldannealing effect on magnetoimpedance of a zero magnetostrictive metallic glass.
J Appl Phys 1996, 79:6552. Publisher Full Text

Panina LV, Mohri K, Uchiyama T, Noda M: Giant magnetoimpedance in Corich amorphous wires and films.

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

Berger L: Currentinduced oscillations of a Bloch wall in magnetic thin films.
J Magn Magn Mater 1996, 162:155. Publisher Full Text

Brophy JJ: Basic Electronics for Scientists. New York: McGrawHill; 1977.

Williams HJ, Shockley W: A simple domain structure in an iron crystal showing a direct correlation with the magnetization.
Phys Rev 1949, 75:178. Publisher Full Text

Agarwala AK, Berger L: Domainwall surface energy derived from the complex impedance of Metglas ribbons traversed by ac currents.
J Appl Phys 1985, 57:3505. Publisher Full Text

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

Ding Y, Klemmer TJ, Crawford TM: A coplanar waveguide permeameter for studying highfrequency properties of soft magnetic materials.
J Appl Phys 2004, 96:2969. Publisher Full Text

Kalinilos BA, Slavin AN: Theory of dipoleexchange spin wave spectrum for ferromagnetic films with mixed exchange boundary conditions.
J Phys C 1986, 19:7013. Publisher Full Text

Yabukami S, Ojima R, Yamaguchi M, Arai KI, Kikuchi S: Highly sensitive permeability measurements obtained by electrical impedance.