Magnetostriction experiment

In this study, we measured the saturation magnetostriction of a thin ribbon by the strain gauge method was used for our λ measurements[22, 34-36]. The gauge is cemented directly onto the surface of the ribbon sample under test, so that a small change in length are detected the change in the resistance (R) of the gauge. Bearing these considerations in mind, we used the experimental set-up shown in Fig. 3.22, which is similar to that used in Ref. 37. The ribbon sample was clamped by both ends on a brass sample holder; the xy-plane was parallel to

62

the ribbon. A horizontal in-plane field HE was applied along the + x directions from an electromagnet. The lower-end clamp could slide freely downward or along the y direction by hooking various standard weights over the aluminum ring. Thus, while the ribbon sample was stretched by the weight, it was also guided by two vertical brass rods alongside the ribbon. In this way, we could minimize the twisting and/or bending effects, which can cause spurious signals during the λS

K = 1 + ν + 𝐿𝑑𝜌

𝑑𝐿 (3.4)

measurements. The gauge sensitivity is defined as the relative change in resistance per relative change in length, or the gauge factor, K=

(∆R/R)/(∆L/L). The gauge factor is given by

where L is the total length of the metal or semi-conductor wire the gauge, ∆R is the resistance change, due to the length ∆L, ν is Poisson’s ratio for the wire material, ρ is the resistivity. The strain gauge used was a CEA-13-125UN-120 type with K=2.0-2.2, purchased from Vishay Intertechnology, Inc. When HE=0, we could measure the external stress strain (σ-∆ε) curve plot for each ribbon sample. From the obtained σ-∆ε curves, it is easy to obtain the ES and

∆E/E0, where ∆E = ES - E0, ES is the Young’s modulus of the sample in the magnetically saturated state, and E0 is that in the demagnetized state[3, 38].

63

Fig. 3.22 A photo of the experimental set-up for the stress-strain (σ-∆ε) and magnetostriction (λ) measurements.

64

4. Results and discussion for films

4.1 XRD data

Table 4.1 lists the x-ray structure data of the (FeNi)81Ga19 films deposited on glass substrates. Briefly speaking, when y = 22 at.%Ni, there is only one single A2 phase, and when y =0, 4, 11, and 17 at.%Ni, there are mixed phases with A2 (major) and D019 and/or L12 phases (minor). It is evident that all the FeNiGa films are highly (110) textured.

Table 4.1 Structural properties, the x-ray diffraction peaks, of the

Fe81-yNiyGa19/glass films. I/Imax is the peak intensity ratio. a is the lattice constant.

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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

66

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 f_C 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, t_f =100 nm < δ ≈ 650 nm (at f ~ 1 GHz), thus we expect fC = f_FMR

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],

(μ_R − 1)f_FMR² = (ν4πM_s)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

67

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 = _ν4πM^Δf1/2

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/2f_FMR , (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],

68

where the angle φ is the rotating field vector h�⃑_rf made a finite azimuthal angle with the magnetization vector M���⃑_S, λ ≡ (ανMS)/(4πµ^o), 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 = 90^o

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=

45^o, which makes the value of χ" in half of its maximum value.

69

In other words, if Eqs. 4.5 and 4.6 are combined, we obtain,

μ_I ∝ 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

70

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

71

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.

72

Fig. 4.2(a) Magnetic anisotropy field (HK) decreases, as at% Ni increases of the FeNiGa (x or y).

73

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.

74

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 = 90^o, and the width of the absorption curve at half the maximum value or a half-value width correspond at φ = 45^o.

75

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.

76

0 5 10 15 20 25 30

110 120 130 140 150

160 Fe

_81-y

Ni

_y

Ga

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.

77

4.3 Magnetostrictions data

4.3.1 Young′s modulus (Ef) data

Fig. 4.7 shows that for the Fe81-yNiyGa19/glass films, as y increases, Ef is almost constant, 120 GPa, except when y = 17, the E_f is up to 133 GPa, and for the Fe81-xNixGa19/Si(100) films, as x increases from 0 to 17, Ef first increases from 170 to 182 GPa, then decreases from 182 to 154 GPa, and finally when x > 17, Ef increases again. Different film-growth mechanisms, due to uses of different substrates, cause the former being less denser than the latter.

Thus, even under the same film composition condition (i.e., x = y), Ef of the x-film is stiffer than that of the y-film.

Fig. 4.7 This is Young′s modulus of the two series of Fe81-xNixGa19/Si(100) and Fe81-yNiyGa19/glass films, respectively.

78

According to Ref. 22, the D019 phase and L12 phase are detrimental to saturation magnetostriction. In this study, we found the addition of Ni into Fe81Ga19 alloy films oppresses the formation of D019 and L12 phases. Even when y = 22 at.%Ni, there is only one single A2 phase. So we get magnetostriction constants in the FeNiGa ternary alloys higher than those of the Fe81Ga19

The t

binary alloys. In addition, notice that the saturation field of these films is very low, about 15 Oe. Hence, their magnetostriction sensitivity can be quite high, about 1.9 – 2.7 ppm/Oe, which is suitable for the low field and high frequency application.

f dependence of λS of Fe59Ni22Ga19/Si(100) and Fe59Ni22Ga19/glass films is shown in Fig. 4.9. The trend for Fe59Ni22Ga19/Si(100) films is that as tf increases from 65 to 195 nm, λS increases from 25 to 30 ppm and, and as 195

< tf ≤ 260 nm, λS remains constant. The trend for Fe59Ni22Ga19/glass films is similar. These results can be explained by the surface oxidation and/or the film/substrate interface effects [42].

79

0 5 10 15 20 25 30 5

10 15 20 25 30

35 _Fe

81-x Ni _x Ga ₁₉ /Si(100) Fe _81-y Ni _y Ga ₁₉ /Glass

x, y (at. %Ni) λ s ( ppm )

Fig. 4.8 Saturation magnetostriction (λS) reaches maximum, when x or y = 22 at.%Ni.

Fig. 4.9 The tf dependence of λS of the Fe59Ni22Ga19/Si(100) and Fe59Ni22Ga19/glass films.

80

5. Results and discussion for ribbons

5.1 XRD data

According to the x-ray diffraction patterns (Table 5.1) of the melt-spun Fe81-zNizGa19 and the JCPD information[43], we can identify the A2(110), A2(200), and A2(211) diffraction peaks. From these findings, lattice constant (a0) of melt-spun Fe81-zNizGa19 can be calculated respectively[44]. For

example, Fig. 5.1 shows the information of θ angle and the corresponded lattice of the melt-spun Fe81Ga19 x-ray diffraction pattern peaks of A2(110), A2(200), and A2(211). From these three points, the value of a0 can be found by plotting ahkl (lattice) against (cos²θ/sinθ+cos²θ/θ), which approaches zero as θ

approaches 90^o; a fitting line runs through three points and crosses the a-axis[45].

That can make sure the lattice constant of the melt-spun Fe81Ga19 ribbon.

Each melt-spun Fe81-zNizGa19 ribbon samples are shown in Fig. 5.2. The lattice constant first increases from 2.925 Å to 2.931 Å, then decreases to 2.900 Å finally thenstabilizes around 2.900 Å, as z of the FeNiGa ribbons increases.

The radii of atomic Fe, Ni are about 1.23Å, 1.23Å, and the radii of ion Fe²⁺, Ni²⁺

are about 0.74Å, 0.69Å,respectively[46]. Thus, the lattice constants of these ribbons showed dependence on z. Furtherance exhibited the similar trends in z-dependence, as the lattice constant and electrical resistivity, shown in section 5.3. However, for z=24 at.% Ni, the diffraction peaks are not all identified, indicating these might exist some other phases at in these ribbon. We will come back on this in chapter 7.

81

Table 5.1 Structural properties, the x-ray diffraction peaks, of the Fe81-zNizGa19 ribbons. I/Imax is the peak intensity ratio. a0

z

82

Fig. 5.1 The lattices constant of melt-spun Fe81Ga19 be calculated by the red line cross y-axis. The lattice constant of this ribbon is 2.925 Å.

Fig. 5.2. The lattice constant first decreases from 2.925 Å to 2.900 Å and then stabilizes around 2.900 Å for the FeNiGa ribbons.

83

5.2 VSM results

In general, addition Ni replace Fe at.% in alloys to be caused refined magnetic in rich-Fe alloys. Especially in the magnetic anisotropy energy and saturation magnetization, there would be shown later. The VSM results of the Fe81Ga19

ribbon are shown in Fig. 5.3. When θ=0^o, the in-plane external field HE is parallel (//) to EA, and when θ=90^o, HE is perpendicular (┴) to EA. We can obtain MS, SQR, HC, and Hs from these two plots. In addition, upon the addition of Ni in the series of Fe81-zNizGa19 ribbon, the magnetic anisotropy energy disappeared completely. Namely, the magnetic anisotropy energy because isotropic for the series Fe81-zNizGa19 ribbons, except when z is 0.

In soft ferromagnetic alloy devices, the basic requirements are that low coercivity (HC), and high saturation magnetization (MS). Fig. 5.4(a) shows MS

plotted as a function of z. As z increases from 0 to 24 in Fe81-zNizGa19, MS

decreases from 170 emu/g to 116 emu/g. Fig. 5.4(b) indicates that HC

increases from 4.8 Oe not a lot, as z increases except z=24. As will be discussed later in Chap. 7, this implies that there are other phases in the Fe57Ni24Ga19 alloy ribbon. At last, Fig. 5.4(c) shows that Hs does not change too much with increasing z.

84

-1000 -500 0 500 1000

-200 -150 -100 -50 0 50 100 150

200 M _s =170.7 (emu/g) H _c =4.84 (Oe)

SQR=4.7 (%) H _s =634 (Oe)

θ=0 ^o

Fe

₈₁

Ga

₁₉

M ( em u/ g)

H (Oe)

Fig. 5.3 The easy-axis (θ=0^o) hysteresis loops of the melt-spun Fe81Ga19.

85

0 5 10 15 20 25

110 120 130 140 150 160 170

(a) 180 _Fe

81-z Ni _z Ga ₁₉

z (at. %) M s (em u/ g )

0 5 10 15 20 25 4

6 8 10

(b) 12 Fe _81-z Ni _z Ga ₁₉

z (at. %)

H c ( Oe )

86

0 5 10 15 20 25

500 550 600 650 700 750

(c) 800 _Fe

81-z Ni _z Ga ₁₉

z (at. %) H s ( Oe )

Fig. 5.4 shows Saturation magnetization (Ms), coercivity (Hc), and saturation field (Hs) hysteresis of the melt-spun Fe81-zNizGa19.

87

5.3 Electrical resistivity

Fig. 5.5 illustrates the ρ vs. x plot, as z increases from 0 to 13, ρ of the melt-spun Fe81-zNizGa19 decreases little from 111 µΩcm to 95 µΩcm, as z increases from 0 to 13, ρ of the melt-spun Fe81-zNi_zGa₁₉ increases little from 95 µΩcm to 108 µΩcm. The resistivity and lattice constants reaches minimum simultaneously for the melt-spun Fe68Ni13Ga19, indicating that substituting larger Fe²⁺ with smalls Ni²⁺ may introduced more carriers. The rapid increase of ρ of the Fe57Ni₂₄Ga₁₉, however, might be due to the existence of other phases arising from the excessive doping of Ni.

0 5 10 15 20 25

90 100 110

Fe _81-z Ni _z Ga ₁₉

z (at. %)

ρ (µΩ cm )

Fig. 5.5 Electrical resistivity (ρ) of the melt-spun Fe81-zNizGa19.

88

5.4 Young’s modulus of the ribbon

The magnetostriction is a dependence of Young’s modulus E of a magnetic material on its state of magnetization. When an originally demagnetized specimen is saturated, its modulus increases by an amount ∆E. The value of

∆E/E0

Fig. 5.6 shows the normal σ vs. ∆ε plot, which plot is called normal, because it is concave up, as shown in Refs. 2, 3, and 38. That is, the slope of the dotted fitting line (slope I) is smaller than that of the solid fitting line (slopeII). At the intersection of the slopes I and II lines, we can define the critical internal stress (σ

depends greatly on the way in which it is measured, as will be explained below[2].

ic) of the ribbon[3,38]. The physical meaning of σic can be considered as the critical transition point for the ribbon sample from the demagnetized state to the saturation state through the magneto-elastic mechanism; e.g., Fig. 8.26 of Ref.

14. Moreover, from slope II, we can determine the Young’s modulus (ES) in the saturated state. Also, from slope II and the intercept of the solid fitting line, we can find the elastic strain ∆εel and magneto-elastic strain ∆εme

𝐸₀ = σ

∆𝜀_𝑒𝑙 + ∆𝜀_𝑚𝑒 (5.1)

[14, 22]. As a result of these two kinds of strain, the modulus in the demagnetized state is[2]

and the modulus in the saturated state is

89

𝐸_𝑠 = σ

∆𝜀_𝑒𝑙 (5.2)

These two relations lead to

∆𝐸

𝐸₀ = 𝐸_𝑠 − 𝐸₀

𝐸₀ =∆𝜀_𝑚𝑒

∆𝜀_𝑒𝑙 (5.3)

For sample of the Fe₈₁Ga₁₉ ribbon, we fixed σ = 25 Mpa to calculate ES and

∆E/E0 by Eq. 5.2 and Eq. 5.3. In Fig. 5.7, we find ES of the Fe81-zNizGa19

ribbons is in the range 115 to 52 GPa. In Fig. 5.8, we find ∆E/E0 is in the range 14% to 115%. In Fig. 5.9, σic not change a lot between Fe81Ga19 and Fe₇₈Ni₃Ga₁₉, as z increases from 3 to 19 in Fe_81-zNi_zGa₁₉, σic increases, and σic

decreases at z = 24. There are two points to be noted here. Firstly, the

Fe57Ni24Ga19 has phases other than A2 phase existing in the ribbon, which may account for the deviation of the general trend in this series of ribbon. (As can be seen in Fig. 5.7- 5.9 and Fig. 5.12 in the next section.) Secondly, we were unable to obtain reliable data from Fe68Ni13Ga19 ribbon. The reason is not clear at present, and further investigations are need.

90

Fig. 5.6 The stress (σ) vs. strain (∆ε) curves of the as-spun Fe81Ga₁₉

0 5 10 15 20 25

40 50 60 70 80 90 100 110 120

Fe

_81-z

Ni

_z

Ga

₁₉

z (at. %) E s (Gp a )

ribbon.

Fig. 5.7 The Young’s modulus in the saturated state (Es) plotted as a function of

Fig. 5.7 The Young’s modulus in the saturated state (Es) plotted as a function of

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