Nanocrystalline Co40Fe40B20 films, with film thickness tf = 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 4πMs = 15.5 kG, anisotropy field Hk = 0.031 kG, and r = 168 mW.cm. From FMR, we can determine the Kittel mode ferromagnetic resonance (FMR-K) frequency fFMRK = 1,963 MHz. In the h||L case, IM spectra show the quasi-Kittel-mode ferromagnetic resonance (QFMR-K) at f0 and the Walker-mode ferromagnetic resonance (FMR-W) at fn, where n = 1, 2, 3, and 4. In the h||w case, IM spectra show QFMR-K at F0 and FMR-W at Fn. We find that f0 and F0 are shifted from fFMRK, respectively, and fn = Fn. 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
It is known that impedance (IM) of an ferromagnetic (FM) material is closely related to its complex permeability (μ ≡ μR + i μI ), where μR and μI 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 (fc), where μR crosses zero and μI reaches maximum , for each FM material. According to Ref. , fc 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. , we had the situation that the thickness (tF) of the FM ribbon was thick to meet the criterion: tF ≥ δ≅ 10 μm, where δis skin depth (at f = 1 MHz), but in this article, we have a different situation wherein the thickness (tf) of the FM film is thin to meet the criterion: tf = 100 nm << δ≅ 654 nm (at f = 1 GHz). That means the time varying field Hg, generated by the ac current (iac), 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 tf, 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. Two Co40Fe40B20 film samples. L is the length and w the width. iac 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 HE ⊥ 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, iac, 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 = HR = 0.68 kG, and define the half-peak width ΔH = 53 Oe.
Figure 2. Ferromagnetic resonance of the Co40Fe40B20 film with the microwave frequency f = 9.6 GHz. HR is the resonance field.
Other magnetic and electrical properties of the Co40Fe40B20 film were obtained from vibration sample magnetometer measurements: 4πMs = 15.5 kG and the anisotropy field, Hk = 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 >> tf.
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 = f0 and f = fn, where n = 1, 2, 3, 4 in the R-spectrum, and a wiggle (or oscillation) centered around the same f0 and fn in the X-spectrum. To summarize the data in Figures 3 and 4, we have in the h||L case, f0 = 2,081, f1 = 1,551, f2 = 1,291, f3 = 991, and f4 = 781 MHz; and in the h||w case, F0 = 2,431, F1 = 1,551, F2 = 1,281, F3 = 991, and F4 = 721 MHz. From these experimental facts, we reach two conclusions: (1) f0 ≠ F0 and (2) within errors, fn = Fn. Since at either f0 or F0, each corresponding wiggle crosses zero, we believe there is a quasi-FMR-K event. Notice for the moment that because f0 ≠ F0, 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 Co40Fe40B20 film sample with ||L. f0 and fn, 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. F0 and Fn, 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. , 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: fEM(||) = η|| × 7 MHz and fEM(⊥) = η⊥ × 27 MHz, where η|| and η⊥are positive integers. Since based on the experimental findings, fn = fEM(||) should be equal to Fn = fEM(⊥), fn or Fn 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. , 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 HE = 150 G, much larger than the saturation field, was applied to eliminate magnetic domains, those peaks (at f0 to f4 or F0 to F4, 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 fd(Cu) = (1/2π)(LsC)-(1/2) = 2.641 GHz, where Ls is the self-inductance and C is the capacitance of the film . However, we believe that f0 and/or F0 are less likely due to the RLC-circuit resonance mechanism for the reason below. Since Ls = μ × GF ~(102 to 103) × μo × GF for Co40Fe40B20, where GF depends only on the geometrical size and shape of the sample, Ls = 1 × μo × GF for Cu, and CCoFeB ≥ CCu, in principle, we find fd(Co40Fe40B20) ≅ [(1/10) to (1/30)] × fd(Cu) = 0.26 to 0.08 GHz, which is too small to meet the facts, i.e., f0 = 2.081 GHz and F0 = 2.431 GHz.
With regard to the FMR-W mechanism, we have the following discussion. At f = fn and/or Fn, 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 hrf region, no FMR-W modes could be observed. However, as indicated in Ref. , if hrf is sufficiently inhomogeneous to vary over the sample, one will observe various FMR-W modes at H = Hn and Hn < HR. From a simple relationship , such as f = νHeff, where Heff is the effective field and ν = γ/2π is the gyromagnetic ratio, it is easy to recognize that since Hn < HR, we have fn < f0 and/or Fn < F0, which is what has been observed. Second, from Refs.  and , it is known that hrf ≡ Hg = (iacz)/(wtf), where z is a variable parameter along tf. Therefore, in a typical IM measurement, hrf or Hg 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.
By substituting the values of fR = 9.6 GHz, HR, HK, and 4πMs, it is found ν = 2.833 for Co40Fe40B20. Thus, the main (or FMR-K) resonance (at H = 0) would occur at f = fFMRK = ν[HK(HK + 4πMs)]1/2 = 1,963 MHz. According to our previous arguments, this frequency, fFMRK, should be equal to f0 and/or F0 in MI. Obviously, what we have is fFMRK ≠ f0 ≠ F0. The reasons for the frequency shifts of the quasi-FMR-K resonances in IM are given below. According to Refs. [9-11], the quasi-FMR-K-resonance relationship for f0 or F0 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/tf) is the out-of-plane (OFP) standing spin-wave wavevector, p = 0, 1, 2,...etc., θq is the angle between and the surface normal or the z-axis, hence for and , as shown in Figure 1, θq = π/2 always, and τ is the relaxation time , where 1/τ ≡ (αγHR) = 94.3 MHz and α ≡ ν(ΔH)/(2fR) = 0.00777. Therefore, if the relaxation time (1/τ) mechanism dominated in Equation 2, f0 would be equal to 267 MHz, which is much lower than the f0 or F0 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 f0(p = 0) = 1.963 GHz, f0(p = 1) = 4.874 GHz, and f0(p = 2) = 9.136 GHz. Because our Agilent E4991A works only up to 3.0 GHz, f0(p = 1) and f0(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 f0, F0, A, and Hk 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 × tf] = (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 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 jac = (iac)/(tfw) may be required to initiate IPSW, and (2) that jac 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 = fFMRK. However, due to reason (1) above, and the symmetry breaking issue in the IM case, as discussed before, should be shifted from fFMRK to f0 and F0, respectively.
Moreover, if we take the formula Z = (B/As)(1 + i)coth[(t/2As)(1 + i)], where B = (ρL)/(2w), As = [ρ/(πfξμo)][cos(δ/2) + isin(δ/2)], μ ≡ ξμo, and μo = 4π × 10-7 H/m. By using the Newton-Raphson method , we may calculate the f dependence of μR ≡ ξcosδ or μI ≡ -ξsinδ from the R and X data. From the μR vs. f or the μI vs. f plot, as shown in Figure 5, we can define the cutoff frequency fc = 2,051 MHz in the h||L case. Clearly, fc in Figure 5 is almost equal to f0 found in Figure 3.
Figure 5. Permeability μ = μR + i μI. Permeability μ = μR + iμI 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 tf 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 fFMRK = 1,963 MHz, while from the IM data, we find that (1) the quasi-Kittel-mode resonance occurs at f0 = 2,081 MHz in the h||L case and F0 = 2,431 MHz in the h||w case, respectively, and (2) the Walker-mode resonances at fn = Fn for both cases. It is believed that the shift of from fFMRK to f0 or from fFMRK to F0 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.
The authors declare that they have no competing interests.
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.
This work was supported by a grant: NSC97-2112-M-001-023-MY3.
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