Chapter 4 Results and discussion for films
4.2 FMR data
Fig. 4.1 shows the main resonance field (HR) at fR = 9.6 GHz as a function of the Ni concentration (x or y) for the two series of (FeNi)81Ga19 films, respectively. From this figure, we find that HR increases as x or y increases.
In general, addition Ni replace Fe at.% in alloys to be caused reduced magnetic in rich-Fe alloys. Then needs higher resonance field at the same fR
When at the Kittel mode resonance, the relationship among H
as x or y
where g = fR/(1.40*x) = ν/1.40 represents the g-factor of the material, and ν = γ/2π is the gyromagnetic ratio. Hence, based on Eq. 4.1, the natural (or FMR-K) resonance (at H=0) would occur at fFMR = ν[HK(HK+4πMS)]1/2 ≒ ν[HK4πMS]1/2 with HK << 4πMS. The addition of Ni into Fe81Ga19 alloys causes reduced the magnetic anisotropy energy which let the HK decreases, leads to decreases in HK is evident from Fig. 4.2(a). The plot of fFMR vs. x or y is shown in Fig. 4.2(b), which exhibits similar trend of HK(x,y) shown in Fig.
4.2(a). Because 4πMS of the FeNiGa films is almost independent of x or y, it implies that the fFMR of the FeNiGa films depend solely on HK as suggested by the equation fFMR ≒ ν[HK4πMS]1/2. The fact that both have the same trend with the variations of Ni concentration is consistent with this conjecture. As
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other discussed in next paragraph, fFMR should serve as the cut-off (or limiting) frequency (fC
In general, the complex permeability, µ = µ
) for these series of ferromagnetic films.
R – iµI, has an anomalous behavior, with the real part, µR, drops off and the imaginary part, µI, exhibits an absorption peak at a certain high frequency, e.g. fC. For a non-metallic ferromagnet, fC is mainly determined by FMR. For a metallic ferromagnet, according to Fig. 4.17 of Ref. [1], fC should depend on the dimensions, e.g., tf, of the sample. For example, if tf < δ, where δ is the skin depth, fC, the µR or µI
anomaly at fC is related the FMR effect in the µ vs. f spectrum. On the other hand, if tf > δ, fC, will be dominated by the eddy-current effect[1]. In our case, tf =100 nm < δ ≈ 650 nm (at f ~ 1 GHz), thus we expect fC = fFMR
At low frequencies, the definition of rotation permeability for an un-plane uniaxial film, such as the FeNiGa film studied has, is µ ~ µ
for all the FeNiGa films. However, since the FeNiGa films are in each metallic, the eddy-current effect might also play a role and should be considered in the resonance cases. We will come back to this point later.
R = 4πMS/HK. Fig.
4.3(a) shows the calculated µR of the FeNiGa films for variance Ni concentrations. It is evident that in both series of FeNiGa film µR increases with increasing Ni concentrations. This general tendency can be understood as follows. The Snoek’s law derived from the Landau-Lifshitz equation, gives[2, 3],
(μR − 1)fFMR2 = (ν4πMs)2. (4.2)
In Eq. 4.2, 4πMS of the FeNiGa films is almost independent of x or y; i.e. it only decreases by 5%, as x or y increases from 0 to 26 at.%Ni. However, based on
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Fig. 4.2, (fFMR)2 decreases by 64%, as x or y increases similarly. From Eq. 4.2, there should be a trade-off between µR and fFMR for the two series of FeNiGa films. Indeed, as can be seen from Fig 4.3(b), the value of µR×(fFMR)2
For a ferromagnetic film, high speed magnetization switching means low Gilbert damping parameter (α). From Landau-Lifshitz-Gilbert (LLG) equation, the magnetic damping parameter (α) can be written as[40],
remains nearly constant over the range of Ni concentration for FeNiGa films investigated in this study.
α = ν4πMΔf1/2
S , (4.3)
where ∆f1/2 is the full width at half maximum for the absorption peak of µI at resonance. The form is used in a shorted microstrip transmission line perturbation experiment. Alternatively, Eq. 4.3 can also be written as[16],
α = ν(ΔH)S/2fFMR , (4.4)
where (∆H)S is the theoretical full width at half maximum of the absorption peak around the main resonance field (HR). Notice that the subscript “s” of
∆H in Eq. 4.4 means that this theoretical ∆H should be, in principle, symmetric with respect to the central peak, HR. In the following, we shall give a reason for this argument. The LLG equation with Kittel mode can obtain the equation[2, 3, 41],
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where the angle φ is the rotating field vector h�⃑rf made a finite azimuthal angle with the magnetization vector M���⃑S, λ ≡ (ανMS)/(4πµo), and H��⃑0T is an in-plane field with EA//H��⃑. From Eq. 4.5, tan(φ) reaches the maximum value, when H=HR or φ = 90o
That we can write down the real and imaginary parts of the susceptibility as .
Fig 4.4 shows the result inrepresentation. The width of the absorption curve at half the maximum value or a half-value width can be calculated by putting φ=
45o, which makes the value of χ" in half of its maximum value.
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In other words, if Eqs. 4.5 and 4.6 are combined, we obtain,
μI ∝ sin2ϕ ∝ H2
H2+(α1)2(H−HR)2 (4.8)
In Eq. (4.8), µI is a Lorentzian function of H, which means that µI must be symmetric with respect to HR, and by definition the symmetric width (∆H)S ≡ 2αHR, i.e. Eq. 4.4. However, in reality, as shown in Fig. 3.13, the experimental width (∆H)exp is not always ideally symmetric. Thus, we consider that (∆H)exp=(∆H)S+(∆H)A, where (∆H)A is the asymmetric parts in (∆H)exp. In general, there are three sources which may make contribution to (∆H)A: one is from the structural inhomogeneity, and the other two are from the magnetic inhomogeneites[41]. As discussed previously, Table 4.2 shows that if y = 0, 22, and 26 at.%Ni (range A), the films are structurally homogenous (i.e., containing only one A2 phase) and if y = 4 – 17 at.%Ni (range B), they are structurally inhomogenous (i.e., containing mixed phases). Thus, we expect for the former films thei0r (∆H)A’s are smaller, while for the latter films their (∆H)A’s are larger. It is in agreement with the experimental data from FMR experiments that for films in range A, the degree of asymmetry of the peak width, (∆H)A/(∆H)exp = 0 – 5.5%, is smaller, while for those in range B, the degree of asymmetry, (∆H)A/(∆H)exp = 8.8 – 20.2%, is larger. As to the magnetic inhomogeneity, one mechanism is due to the asymmetric distributions of the magnitude and/or angle dispersion of H��⃑K. The other is associated with the local inhomogenous demagnetizing field (Hd) near edges of the film sample.
Here, we are unable to assess how much the magnetic-inhomogeneity
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mechanisms would affect (∆H)A/(∆H)exp
Fig. 4.5 shows that as y increases, α decreases from 0.076 to 0.018 for the FeNiGa films deposited on glass substrates, and as x increases, α first decreases from 0.051 to 0.018 and then increases from 0.018 to 0.053 for the FeNiGa films deposited on Si(100) substrates. Yager, Galt, and Merritt had pointed out that ΔH is related to H
quantitatively.
K. They argued that the anisotropy forces magnetization to rotate parallel to the direction of easy magnetization, and, for the same reason, rotation of magnetization may cause a change in the arrangement of two kinds of ions such as to rotate the easy direction toward the direction of magnetization[2]. In addition, we also find the addition of Ni into Fe81Ga19 alloy films on glass substrates to reduce the D019 phase and L12 phase in XRD results. Briefly speaking, when y = 22 at.%Ni, there is only one single A2 phase. That centralized the HR and narrow ΔH in the alloys. Notice that in equation 4.4, α of the FeNiGa alloy is calculated from ΔH; α decreases, as ΔH decreases. Another one may notice that there is discrepancy between α data of the x- and y-series films in the range 17 ≤ x or y ≤ 26 at.%Ni. In the following, we shall explain why the α data from the y-series (Fe81-yNiyGa19/glass) films should be more reliable. The p-doped Si(100) semiconductor is conducting with electrical resistivity (ρ), about 5-10 Ωcm. On the other hand, ρ is about 120 - 150 µΩcm for FeNiGa films deposited on insulating glass, it is shown in Fig. 4.6. Further, the ratio of tSi/tf is about 103. A simple calculation would show that the electrical resistance ratio, RSi/Rf, for the x-series (Fe81-xNixGa19/Si(100)) films is of the order of one. Thus, the current shunting effect must be significant in the case of FeNiGa films deposited on Si(100). As observed, the (apparent) ρ of the x-series films is general smaller than that of the
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y-series films. In an FMR situation, the eddy current, iac induced by h�⃑rf, must be at least flowing in the conducting FeNiGa film. For the y-series film, because glass is an insulator, iac is mainly limited inside the film region.
However, for the x-series films, due to the current shunting effect, iac will flow across the film/Si interface. Moreover, the spin injection across the interface indicates that the proximity region on the Si side, which is partially magnetized, will also absorb microwave, and make an extra contribution to the main FMR signal from the film. As a result, for the x-series (Fe81-xNixGa19/Si(100)) films, there is an additional broadening of the FMR peak width due to the extraneous eddy-current effect[41]. Thus, in general α of the Fe81-xNixGa19/Si(100) films, as shown in Fig. 4.4, may be less accurate than that of the Fe81-yNiyGa19/glass films.
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Fig. 4.2(a) Magnetic anisotropy field (HK) decreases, as at% Ni increases of the FeNiGa (x or y).
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Fig. 4.2(b) Natural resonance frequency (fFMR) of the FeNiGa films plotted vs.
the Ni concentration (x or y).
Fig. 4.3(a) Static rotational permeability (µR) increases, x or y increases in the FeNiGa films.
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0 5 10 15 20 25 30
1000 1500 2000 2500 3000 3500
x, y (at. %Ni) µ R x f F MR
2 ( GHz ) 2 Fe 81-x Ni x Ga 19 /Si(100)
Fe 81-y Ni y Ga 19 /Glass
Fig. 4.3(b) The products, µR×(fFMR)2≒constant, x or y increases in the FeNiGa films.
Fig 4.4 shows dependence of the real and imaginary part of the rotational susceptibility on the intensity of the dc field about the resonance field. The maximum χ" occurred at φ = 90o, and the width of the absorption curve at half the maximum value or a half-value width correspond at φ = 45o.
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Table 4.2 (∆H)A/(∆H)exp
x or y (at% Ni)
is the degree of asymmetry of the FMR linewidth.
0 3 11 17 22 26
Fig. 4.5 Gilbert damping parameter (α) plotted as a function of the Ni concentration (x or y) for the FeNiGa films.
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0 5 10 15 20 25 30
110 120 130 140 150
160 Fe
81-yNi
yGa
19/Glass
y (at. %)
ρ (µΩ cm )
Fig. 4.6 Electrical resistivity (ρ) is about 120 - 150 µΩcm for FeNiGa films deposited on insulating glass.
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