Motivation

In this study, we chose one series of the Fe81-xNixGa19/Si(100) and another one series of the Fe81-yNiyGa19/glass films, where x or y ranges from 0 to 26%, and other series of magnetic metallic ribbons, Fe_81−zNizGa19 with z = 0, 3, 7, 13, and 24. The TEM photos are shown in Fig. 1.5 from Ref. [13], Bormio-Nunes and Sato Turtelli added nickel (Ni) element into the FeGa alloys leading to refined grain size and/or a more disordered lattice. Besides, it has been found that incorporation of the nickel (Ni) element, into the Fe85Ga15

Hopefully, we can get the combinations of the following favorable features, such as low H

alloy would improve magnetostriction, as shown in Fig. 1.6 and Fig. 1.7.

C or HS, high 4πMS, high λS, from one of these FeNiGa ribbons and/or films for magneto-electric device. On the other hand, we want get the high µr, low ∆H or α, from one of these FeNiGa films for magneto-electric microwave device.[13 – 15]

7

Fig.1.5 The cross-section TEM photos for Fe85Ga15 and Fe78Ni7Ga15

ribbons[13].

8

Fig. 1.6 Magnetostriction measured on stacked Fe85Ga15 ribbon, applying the field parallel to the ribbon thickness[13].

Fig. 1.7 Magnetostriction measured on stacked ribbons of Fe78Ni7Ga15 applying the field parallel to the ribbon thickness[13].

9

2. Brief review of magnetism and relevant effects

2.1 Magnetism

The first writings of magnetism appeared with a kind of mineral called magnetite (Fe3O4

The discovery of two regions named magnetic poles, or sometimes just

“poles,” which attracted a piece of iron more strongly than the rest of the magnetite, this discovery was made by P. Peregrines about 1269 A.D.

Sometime later, Coulomb (1736-1806) found that there were two types of poles, now called positive or north poles, and negative or south poles. There is always with magnets and felt the mysterious forces of attraction and repulsion between two magnetic poles[3, 16]. The mysterious forces of attraction and repulsion between the two magnetic poles can be felt. This force of attraction and repulsion is proportional to the product of the strength of the poles and inversely proportional to square of the distance between them. This is Coulomb’s law, which can be written mathematically as,

), which has been claimed that the Chinese used it in compasses sometime before 2500 B.C., but the precise date still remained unknown[3, 16].

𝐹⃑ = 𝑘𝑝₁𝑝₂

𝑟² 𝑟⃑₀ (2.1)

10

where 𝐹⃑ is the force, p¹ and p₂

When a magnetic pole creates a magnetic field around it, and this field will then produces a force on second pole nearby. Experiment shows that this force is directly proportional to the product of the pole strength and field strength or field strength or field intensity 𝐻��⃑,

the pole strengths, r the distance between the poles, and 𝑟⃑₀ one unit vector directed along r. The constant of proportionality

k that occurs permits a definition of pole strength, and the proportionality

constant k is equal 1 in the cgs-emu unit.

𝐹⃑ = 𝑘𝑝𝐻��⃑. (2.2)

The equation 2.2 then defines 𝐻��⃑, a field of unit strength is one which exerts a force 1 dyne on a unit pole. A field of unit strength has an intensity of one oersted (Oe).

Besides, a magnet with poles of strength p located near each end and separated by distance l. Suppose the magnet is placed at an angle

θ

to a

11

where m is the magnetic moment of the magnet. It is the moment of torque exerted on the magnet when it is at right angles to a uniform field of 1 Oe[2, 3, 16].

Consider a piece of iron is subjected to a magnetic field, it becomes magnetized, and the level of its magnetism depends on the strength of the field.

We therefore need a quantity to describe the degree to which a body is magnetized. The application of an external magnetic field causes both an alignment of the magnetic moments of the spinning electrons and an induced magnetic moment due to a change in the orbital motion of electrons In order to obtain a formula for determining the quantitative change in the magnetic flux density caused by the presence of a magnetic material, we let 𝑚��⃑_𝑘 ^Rbe the magnetic moment of an atom. If there are n atoms per unit volume, we define a magnetization vector, 𝑀��⃑, as

𝑀��⃑ = lim_∆𝑣→0∑^𝑛∆𝑣_𝑘=1𝑚��⃑_𝑘

∆𝑣 (2.5)

Where ∆ν is the volume and n is the number of ∆ν[2, 16].

The magnetic properties of a material are characterized not only by the magnitude and sign of 𝑀��⃑ but also by the way in which 𝑀��⃑ varies with 𝐻��⃑.

The scalar ratio of these two quantities is called the susceptibility

χ

:

𝜒 =𝑀

𝐻 (2.6)

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Now they can be roughly classified into three main groups in accordance with their

χ

values[3, 16, 17].

(1) Diamagnetic, if

χ

is a very small negative number.

Electrons which constitute a close shell in an atom usually have their spin and orbital moments oriented so that the atom as a whole has no net moment.

Thus the monoatomic rare gases He, Ne, Ar, etc., which have closed-shell electronic structures, are all diamagnetic. The macroscopic effect of this is equivalent to that of a negative magnetization that can be described by a negative magnetic susceptibility. The effect is usually very small, and

χ

for most known diamagnetic materials is in the order of -10^-5

(2) Paramagnetic, if

χ

is a very small positive number.

Arises mainly from the magnetic dipole moments of the spinning electrons.

The alignment forces, acting upon molecular dipoles by the applied field, are counteracted by the deranging effects of thermal agitation. Unlike diamagnetism, which is essentially independent of temperature, the paramagnetic effect is temperature dependent, being stronger at lower temperatures where there is less thermal collision. (i.e., Na, Al)

. (i.e., Cu, Hg, Ag)

(3) Ferromagnetic, if

χ

is a large positive number.

The magnetization of ferromagnetic materials can be many orders of magnitude larger than that of paramagnetic substances. Ferromagnetism can be explained in terms of magnetized domains. I will show more detail in the next section.

13

Besides, engineers are usually concerned only with ferromagnetic materials and need to know the total flux density 𝐵�⃑ produced by a given field, then engineers become the definition in mks system. In addition, engineers are usually only concerned with ferromagnetic materials and the total flux density 𝐵�⃑

produced by a given field. The mks system is generally used as the unit system in engineering application. They often find the 𝐵�⃑, 𝐻��⃑ curve, also called a magnetization curve, more useful than the 𝑀��⃑, 𝐻��⃑ curve. The ratio of B to H is called the permeability

µ

:

𝜇 = 𝐵

𝐻 ( 𝐻

𝑚 , 𝑖𝑛 𝑚𝑘𝑠 𝑠𝑦𝑠𝑡𝑒𝑚) (2.7)

When the magnetic properties of the medium are linear and isotropic, the magnetization is directly proportional to the magnetic field intensity:

𝐵�⃑ = 𝜇𝐻��⃑ is the permeability of free space is chosen to be

Where the proportionality constant k is equal to 4π/µ0 in the mks system in

14

equation (2.1)

Fig. 2.1 Bar magnet in a uniform field[3].

15

2.2 Ferromagnet

According to the models of magnetized domains, which have been experimentally confirmed, a ferromagnetic material (such as Co, Ni, and Fe) is composed of many small domains, their linear dimensions ranging from a few microns to about 1 mm. These domains, each contain about 1015 or 1016 atoms, are fully magnetized in the sense that they contain aligned magnetic dipoles resulting from spinning electrons even in the absence of an applied magnetic field. Quantum theory asserts that strong coupling forces exist between the magnetic dipole moments of the atoms in a domain, holding the dipole moments in parallel. Between adjacent domains there is a transition region about 100 atoms thick called a domain wall. In a demagnetized state the magnetic moments of the adjacent domains in a ferromagnetic material have different directions, as exemplified in Fig. 2.2 by the polycrystalline specimen model shown[2, 3, 16, 17]. There were two different real examples shown in Fig. 2.3 and Fig. 2.4, which were observed domain structure by two techniques involved. In overall term overall, the random nature of the orientations in the various domains results in no net magnetization.

When an external magnetic field is applied to a ferromagnetic material, the walls of those domains having magnetic moments aligned with the applied field and which move in such a way as to make the volumes of those domains grow at the expense of other domains. As a result, magnetic flux density is increased.

For weak applied fields, domain-wall movements are longer/ long acting reversible, and domain rotation toward the direction of the applied field will occur. For example, the M-H plane for Fe86V14 film is shown in Fig. 2.5, if an

16

applied field is reduced to zero at point P1, the M-H relationship will not follow the red curve P2P1O, but will go down from P2 to P2’, along the lines of the broken curve in the figure. This phenomenon of magnetization lagging behind the field producing it is called hysteresis. As the applied field becomes even much stronger (past P1 to P2), domain-wall motion and domain rotation will cause essentially a total alignment of the microscopic magnetic moments with the applied field, at which point the magnetic material is said to have reached saturation Ms. The curve OP1P2

If the applied magnetic field is reduced to zero from the value at P

on the M-H plane is called the virgin magnetization curve.

2, the magnetic magnetization does not reduce to zero but assumes the value at Mr

To make the magnetic magnetization of a specimen zero, it is necessary to apply magnetic field intensity H

. This value is called the residual or remanent magnetization (10 kOe =1 T) and is dependent on maximum applied field intensity. The existence of a remanent magnetization in a ferromagnetic material makes permanent magnets possible.

c in the opposite direction. This required Hc; the coercive force; but a more appropriate name is coercive field intensity (in Oe). Like Mr, Hc also depends on the maximum value of the applied intensity.

17

Fig. 2.2 Domain structure of a polycrystalline specimen model[17].

Fig. 2.3 The Bitter method image, which was taken in a zero field and at room temperature, of the Fe81Ni19 array films in a completely demagnetized state.

18

Fig. 2.4 The Magnetic Force Microscopy (MFM) image, which was taken in a zero field and at room temperature, of the

La

_0.7

Sr

_0.3

MnO

₃

(LSMO)

films in a completely demagnetized state.

Fig. 2.5 The virgin magnetization (in red circles) and the major hysteresis (in black squares) curves of the Fe86V14 film. 4πM is the magnetization of the film[18].

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2.2.1 Soft magnetic materials and Hard magnetic materials

Ferromagnetic materials for use in electric generators, motors, and transformers should have a large magnetization for a very small applied field.

As the applied magnetic field intensity varies periodically between + Hmax

Good permanent magnets, on the other hand, should show a high resistance to demagnetization. This requires that they are made with materials that have large coercive field intensities H

, the hysteresis loop is traced once per cycle. The area of the hysteresis loop corresponds to energy loss (hysteresis loss) per unit volume per cycle.

Hysteresis loss is the energy lost in the form of heat in overcoming the friction encountered during domain-wall motion and domain rotation. Ferromagnetic materials, which have tall narrow hysteresis loops with small loop areas, are referred to as “soft” materials, there is shown in Fig. 2.6 red curve; they are usually well-annealed materials with very few dislocations and impurities so that the domain walls can move easily.

c and hence wider hysteresis curve, like the blue curve in Fig. 2.6. These materials are referred to as “hard” ferromagnetic materials. The coercive field intensity of hard ferromagnetic materials can be 500 Oe or more, whereas that for soft materials is usually 50 Oe or less[2, 3, 16].

20

Fig. 2.6 This figure is the soft ferromagnetic material hysteresis (in red circles) and the hard ferromagnetic material hysteresis (in blue squares) curves.

2.2.2 Curie temperature

As previously indicated, ferromagnetism is the result of strong coupling effects between the magnetic dipole moments of the atoms in a domain. When the temperature of a ferromagnetic material is raised to such an extent that the thermal energy exceeds the coupling energy, the magnetized domains become disorganized. Above this critical temperature, known as the Curie temperature (Tc); a ferromagnetic material behaves like a paramagnetic substance. Hence, when a permanent magnet is heated above its curie temperature, it loses its magnetization. The Curie temperature of most ferromagnetic materials lies between a few hundred to a thousand degrees Celsius, that of iron being 770°C[2, 3].

21

2.3 Magnetic anisotropy

The magnetization changes from zero to the saturation value, which the value of Ms

One factor which may strongly affect the shape of the M, H curve, or the shape of the hysteresis loop, is magnetic anisotropy. This term simply means that the magnetic properties depend on the direction in which they are measured.

This general subject is of considerable practical interest, because anisotropy is exploited in the design of most magnetic materials of commercial importance.

A thorough knowledge of anisotropy is thus important for an understanding of these magnetic materials[2, 3].

itself will be regarded simply as a constant of the material. If we understand the several factors that affect the shape of the M, H curve, we will then understand why some materials are magnetically soft and others are magnetically hard.

There are several kinds of anisotropy such as crystal anisotropy, shape anisotropy, stress anisotropy, and induced anisotropy.

2.3.1 Crystal anisotropy

The existence of crystalline anisotropy may be demonstrated by the magnetization curves of single-crystal specimens. By magnetization curve we mean the component of magnetization in direction of applied field M, plotted as a function of the applied field[16]. Magnetization curves for single crystals of iron, nickel, and cobalt for various orientations of the applied field with respect to the crystal axis for room temperature are shown in Fig. 2.7, Fig. 2.8, and Fig.

2.9. It is clear that much smaller fields are required to magnetize the crystals to

22

saturation along which the magnetization tends to lie are called easy axis (EA);

the axis along which it is most difficult to produce saturation are called hard axis (HA).

Fig. 2.7 Magnetization curves of single-crystal of iron[16].

Fig. 2.8 Magnetization curves of single-crystal of nickel[16].

23

Fig. 2.9 Magnetization curves of single-crystal of cobalt[16].

2.3.2 Shape anisotropy

Consider a polycrystalline specimen having no preferred orientation of its grains, and therefore no net crystal anisotropy. If it is spherical in shape, the same applied field will magnetize it to the same extent in any direction. But if it is nonspherical, it will be easier to magnetize it along a long axis than along axis. The reason for this is the demagnetizing field along a short axis is stronger than along a long axis. The applied field along a short axis then has to be stronger to produce the same true field inside the specimen. Thus shape alone can be a source of magnetic anisotropy.

24

2.3.3 Stress anisotropy

The main reason for stress anisotropy is the inverse magnetostrictive effect, which will be discussed in section 2.4. Simply put, there exists an inverse effect which causes such properties as permeability and the size and shape of the hysteresis loop to be strongly dependent on stress in many materials.

Magnetostriction therefore has many practical consequences, and a great deal of research has accordingly been devoted to it[2, 3, 16].

2.3.4 Induced anisotropy

Various other anisotropies may be induced in certain materials, chiefly solid solutions, by appropriate treatments. These induced anisotropies are of interest both to the physicist, for the light they throw on basic magnetic phenomena, and to the technologist, who may exploit them in the design of magnetic materials for specific applications[2, 3, 16].

The following treatments can induce magnetic anisotropy:

(1) Magnetic annealing:

This mean heat treatment in a magnetic field, sometimes called a thermomagnetic treatment. This treatment can induce anisotropy in certain alloys.

(2) Stress annealing:

This means heat treatment of a material that is simultaneously subjected to an applied stress.

(3) Plastic deformation:

This can cause anisotropy both in solid solutions and in pure metals, but by quite different mechanisms.

(4) Magnetic irradiation:

25

This means irradiation with high-energy particles of a sample in a magnetic field.

26

2.4 Magnetostriction[2, 3]

When a ferromagnetic substance is exposed to a magnetic field, its dimensions change. This effect is called magnetostriction. It was discovered in 1842 by James Joule, who showed that an iron rod increased in length when it was magnetized lengthwise by a weak field. The fractional change in length

∆l/l is simply a strain, and, to distinguish it from the ε caused by an applied stress, we give the magnetically induced strain a special symbol λ[2, 3]:

λ =Δ𝑙

𝑙 (2.10)

The value of λ measured at magnetic saturation is called the saturation magnetostriction λs, and, when the word “magnetostriction” is used without qualification, λs is usually meant. Magnetostriction occurs in all pure substances. However, even in strongly magnetic substances, the effect is usually small: λs is typically of the order of 10^-5

The longitudinal, sometimes called Joule, magnetostriction just described is not the only magnetostrictive effect. Others include the magnetically induced torsion or bending of a rod. These effects, which are really only special cases of the longitudinal effect, will not be described here.

[3].

The value of the saturation longitudinal magnetostriction λs can be positive, negative, or, in some alloys at some temperature, zero. The value of λ depends on the extent of magnetization and hence on the applied field, and Fig. 2.10 shows how λ typically varies with 𝐻��⃑ for a substance with positive

27

magnetostriction. As mentioned in the preceding, the process of magnetization occurs by two mechanisms, domain-wall motion and domain rotation. For example, the magnetostriction of an iron crystal dependence on magnetic field in the [100] direction is shown in Fig. 2.11.

Between the demagnetized state and saturation, the volume of a specimen remains very nearly constant. This means that there will be a transverse magnetostriction λt

λ_𝑡 = −1

2 λ (2.11)

very nearly equal to one-half the longitudinal magnetostriction and opposite in sign, or

When technical saturation is reached at any given temperature, in the sense that the specimen has been converted into a single domain magnetized in direction of field, further increase in field cause a small further strain. This causes a slow change in λ with H called forced magnetostriction, and the logarithmic scale of

H in Fig. 2.10 roughly indicates the fields required for this effect to become

appreciable. It is caused by the increase in the degree of spin order which very high fields can produce[3].

The longitudinal, forced-magnetostriction strain λ shown in Fig. 2.10 is a consequence of a small volume change, of the order of ∆V/V=10^-10 per Oe, occurring at fields beyond saturation and called volume magnetostriction. It causes an equal expansion or contraction in all directions. Forced magnetostriction is a very small effect and has no bearing on the behavior of practical magnetic materials in ordinary fields[2, 3].

28

Fig. 2.10 shows dependence of magnetostriction on magnetic field[2].

29

Fig. 2.11 indicates magnetostriction of an iron crystal in the [100] direction[3].

2.4.1 Measure λ on bulk or ribbon

The measurement of longitudinal magnetostriction is straightforward but not trivial, especially over a range of temperatures. While early investigators used mechanical and optical levers to magnify the magnetostrictive strain to an observable magnitude, today this measurement on bulk or ribbon samples is commonly made with an electrical-resistance strain gage cemented to the specimen. The gage is made from an alloy wire or foil grid, embedded in a

30

thin paper or polymer sheet, which is cemented to the sample. When the sample changes shape, so does the grid, and the change in shape also causes a change in the electrical resistance of the gage. With ordinary gages, the fractional change in resistance is about twice the elastic strain. This is typically a small resistance change, but one fairly easily measured with a bridge circuit, either ac or dc[3].

Fig. 2.12 Magnetostriction measurement on a sample (bulk or ribbon) using a strain gage[3].

31

2.4.2 Measure λ on thin film

Thin film samples present special challenges in the measurement of

Thin film samples present special challenges in the measurement of

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