where q is the smallest solution of the Bessel functions and is related to the aspect ratio m of the prolate ellipsoid. q = 1.8412 + 0.48694/m – 0.11381/m2. The red dashed line in Fig 2-3 is the critical length versus the aspect ratio for a permalloy ellipsoid by Eq. (2-14). The saturated magnetization Ms =830emu/cm3 and the exchange constant C = 2.1x10-6erg/cm were used. These two parameters are suitable for a confined structure and will be discussed in the section 3-3. The magnetization prefers to align uniformly for a wire with a high aspect ratio. While at a fixed aspect ratio, the uniform state would be observed in a shorter wire.
0 20 40 60 80
Fig 2-3: The theoretically domain diagram of a prolate ellipsoid as a function of the aspect ratio and the critical length.
Edge induced domain
For a rectangular thin film element, there are straight edges and sharp corners resulting in different domain configuration from an ellipsoid. The calculation of magnetostatic energy is complex and the numerical evaluation is usually used. The induced domain appears at the corner, due to the magnetic pole avoidance. Here, the edge induced domain in a wire is discussed based on a two-dimensional approximation which is suitable for a thin film of a soft magnetic material.
If the dimension of a rectangular thin film element with zero crystalline and induced anisotropy is much larger than the single domain limit, a flux closure magnetization configuration with vanishing magnetostatic energy may be favored. In such a pattern, the magnetization lies parallel to the film surface and is diverging free in the interior and at the edges. Therefore, domains and domain walls can be obtained under the consideration of the boundary conditions and the principle of pole avoidance. Van den Berg developed a comprehensive analysis on the prediction of the possible range of these domains and the position of domain walls for such thin film elements with arbitrary shape [9]. He proved that a stray field free magnetization pattern can be constructed when following the below conditions.
z Take circles that touch the edges at two (or more) points and lie otherwise completely within the figure. The centers of all such circles form the domain walls.
z If a circle touches the edges in more than two points, its centre forms a domain wall junction.
z In every circle the magnetization direction must be perpendicular to each touching radius.
For a rectangle, the domain pattern can be illustrated as a flux closure structure which is the well known Laudau-Lifshitz pattern. In Fig. 2-4, the circles marked 1, 2, and 3 touch at least two points at the edge and lie completely within the rectangle.
Their centers 1’, 2’, and 3’, respectively, are located at the domain walls. Moreover, circle 2 is touching the edge at three points implying that its center 2’ coincides with a domain-wall junction. As for the circles 4 and 5, they are touching two edge points;
however, they are lying only partly within the object, so that centers 4’ and 5’ are not at domain walls. Following the third condition, the magnetization direction must to be perpendicular to each touching radius resulting in the closure pattern.
Fig 2-4: A formation of a flux closure pattern in a rectangle.
Subsequently, we try to find the energy gain by the flux closure domain relative to the single domain state, and below which sample size the flux closure domain become unstable, taking into account the domain wall energy. For a uniformly magnetized rectangular thin film element, the total energy is simply the Emag. Although a thorough calculation of Emag is necessary to be approached by the finite element calculation [10], we regard the rectangle as a prolate ellipsoid and use Eq.
(2-11) to make calculations much easier. A flux closure domain may carry a zero net magnetization and hence, the total energy is the only domain wall energy and can be written as:EFC =[(2 2−1)w+l]⋅t⋅γ where w, l, and t are the width, length, and
thickness of a rectangle, respectively. γis the domain wall energy per unit area. The single domain limit Lcritical is obtained by equating the domain wall energy with Eq.
(2-11), as following:
The single domain limit is thus determined by the interplay between the domain wall energy and the magnetostatic energy.
2-2 Magnetization reversal in wires
Magnetization reversal is a process which refers to the variation of magnetic configuration under magnetic fields, pulse currents, and pulse fields, etc. The process in thin films and bulks has been widely investigated. Recent advantages in the nanofabrication methods have made the possibility of studying the magnetism at small length scale.
In our study, we focus on the reversal process of a wire. For a wide wire, these typical reversal mechanisms of a thin film, such as the nucleation and annihilation of domains and propagation of domain walls still play an important role in the reversal process. When a wire width is less than the critical length, the single domain is an energetically favorable state. The question of magnetization reversal mechanisms for such wire is whether the magnetic moment rotation is always in unison. Brown provided a set of differential equations from minimizing the additional energy contributed from the magnetization perturbation and claimed that the eigenfunctions of these equations are the state of system during reversal and the eigenvalues are the
switching fields (Hsw) at which field the magnetization has a significant change[11].
The physical system will choose the one yielding the least Hsw since the mode is more achievable. For a prolate ellipsoid, three eigenmodes are proposed such as the following: coherent rotation, magnetization curling, and magnetization buckling.
Their magnetization arrangements and the theoretical anticipation of the relationship between the Hsw and the wire width are presented in Fig. 2-5. In the coherent rotation mode, all magnetic moments remain parallel to each other during the reversal process. By contrast, the buckling mode and the curling mode have a zigzag configuration in the plane and a vortex structure in the cross-section, respectively. The switching field appears to be uncorrelated with dimensions for coherent rotation. The switching field decreases with increasing wire width as a reciprocal relation when the reversal is completed via the magnetization curling. A threshold between both is the exchange length lex,2 C /Ms. For a semi-wire width larger than lex, the reversal occurs through curling. While a semi-wire width is smaller than lex, the coherent rotation is expected by the Stoner–Wohlfarth model [12]. Because the buckling mode occurs only in a narrow-sized interval, these two modes are usually seen as the only switching modes that can occur.
Later, several reports using the micro-magnetic simulation method show that the complete magnetization moment rotation during the reversal is not always uniform.
The magnetization reversal can occur through a creation of domain wall pairs and then sweep across a wire [13-15]. Furthermore, a number of studies show that thermal fluctuation can activate magnetization reversal resulting in a non-uniformly spatial magnetization distribution [16,17].
Fig 2-5: (Left) Theoretical plot of the reduced switching field as a function of the reduced semi-wire width for the coherent rotation, the magnetization curling, and the magnetization buckling. (Right) Illustration of magnetization reversal for the corresponding mechanism [18].
2-2-1 Magnetization reversal by Uniform rotation
Coherent rotation – Stoner Wohlfarth model
The magnetic moment rotates in the same angle everywhere and it is therefore known as the coherent rotation mode. It can be expressed in terms of the classic Stoner-Wohlfarth model [12]. In this model, a particle is assumed that exchange energy holds all spins tightly parallel to each other and there is a uniaxial anisotropy.
Then, during the magnetization reversal the total energy consists of only Eani and EZ.
For a simply second-order uniaxial anisotropy, the total energy can be written as:
where φ and q are the angles of magnetization and applied field with respect to the easy axis of magnetization, respectively. Ku is the uniaxial anisotropic constant. When the applied field is zero, the Etotal exhibits a periodicity of π . There are minima and maxima at φ =nπ with n=0,1,2… and 0.5,1.5,2.5…., respectively. The magnetization always lies along the easy axis at remanence. As illustrated in Fig. 2-6, the Etotal is plotted as a function of q, H, and φ according to Eq. (2-16). The magnetization reversal is determined by the local minimum and derivative of Etotal. Here, we consider three different orientations of magnetic field relative to the easy axis. Let us start with the simple case that magnetic field is along the easy axis (q=0o), as shown in Fig. 2-6(b). At remanence, magnetic moment is along the easy axis and E is a local minimum at φ =0o. As the magnetic field is reversed, although Etotal
increases, the local minimum of Etotal remains at φ =0o before reaching the Hsw. Until the magnetic field is equal to Hsw, φ =0o becomes a local maximum and magnetic moment rotate suddenly by 180o. For a new local minimum of Etotal at
180o
φ = with further increasing magnetic field, the moment stays at φ =180o. In another extreme case that magnetic field is perpendicular to the easy axis (q=90o) as shown in Fig. 2-6(c). The magnetization is along the direction of the applied field when the applied field is large enough to saturate the magnetization, implying that
90o
φ = is the local minimum of the Etotal. With reducing the applied field, φ =90o is no longer the local minimum, resulting in smooth rotation of magnetization until along the easy axis (φ =0o or 180 ). There are two possible paths toward o φ =0o or 180 , due to both paths are equivalent. The last case is that magnetic field is o
neither along nor perpendicular to the easy axis. As shown in Fig. 2-6(a), the magnetic field makes an angle of 30o with respect to the easy axis. At remanence, magnetic moment is along the easy axis and E is a local minimum at φ =0o. As the magnetic field is reversed, the local minimum of E moves slightly from φ =0o to φ =30o with increasing magnetic field resulting from the competition of the shape anisotropy and magnetic field. When the magnetic field is equal to Hsw, the magnetic moment switches to an angle φ =150o, where has the lowest energy. Fig. 2-6(a) shows this switching behaviors indicated by the dashed line.
The switching field can be calculated by the analysis of the stability of the total energy. For given values of q and H, the magnetization lies at an angle φ where the energy is a minimum locally. The magnetization direction must fulfill that the first derivative of Etotal with respect to φ is zero. Using the reduced fieldh=H (2Ku /Ms)and the reduced magnetizationm=M Ms =cos(θ −ϕ), this
The magnetization hysteresis loop can be calculated from this equation by solving for m as a function of h. Then, Hsw as a function of q can be obtained
A polar plot of reduced Hsw(q) shown in Fig. 2-7 by the solid line is the famous Stoner-Wohlfarth astroid. There is four-fold symmetry of Hsw(q). Hsw is a maximum when the applied field is parallel and perpendicular to the easy axis and is a minimum at q=45o±90o, implying the magnitude of Hsw can be significantly reduced as
operating at this point. This concept has been used in the writing process of MRAM.
Fig 2-6: The total energy as a function of the magnetization angle for several magnetic field values based on Stoner-Wohlfarth model. (a) q = 150o (b) q = 0o (c) q = 90o.
2-2-2 Nonuniform magnetization reversal
As seen in Fig. 2-5, magnetization should reverse by coherent rotation for small samples. For larger samples, nonuniform reversal modes are more likely. Two principal nonuniform reversal modes of an ellipsoid are curling mode and buckling mode. The latter was identified only in a narrow-sized interval. We ignore it here because of its minor importance. The former is discussed in the following.
Magnetization Curling
Curling mode assumes that the magnetization direction rotates in a plane perpendicular to the anisotropy axis of the wire, effectively reducing the stray field.
The feature of this mode has been presented in Fig. 2-5. Although analytical solutions of the magnetization hysteresis loop by curling mode are not available. Aharoni assumed that magnetic moments rotate uniformly before switching and solved the Brown’s differential equations based on the curling mode to obtain the angular dependence of the switching field for a prolate ellipsoid [19]. The relation is given by
θ ellipsoid along the major and minor axes, respectively, which are presented in section 2-1-1.The parameter S is the reduced radius: S = 2r/lex, r is the minor semi-width. k = q2/p where q is the same geometrical factor used in Eq. (2-14). The switching field for several values of S as a function of q is plotted as dashed lines in Fig 2-7. In the case of S=2, switching field of magnetization curling is a minimum when the applied
field is parallel to the easy axis and has an increasing trend when the magnetic field is applied approaching the hard axis. As we can see, the switching field of magnetization curling is less than coherent rotation when q<45o±90o, implying the reversal occurs via magnetization curling at small angle but via coherent rotation at large angle. The amount of the angle with the reversal completed via coherent rotation increases with decreasing S. For S<1, the reversal is completely through the coherent rotation and the switching field is given by the Stoner-Wohlfarth model.
Fig 2-7: Angular dependence of the switching field of a prolate ellipsoid for several reduced radii S. For S<1, the switching field is given by the Stoner-Wohlfarth model and is presented by the solid line.
Propagation of domain walls
In section 2-1-2, we mentioned that the edge domain of a thin film element exists based on the principle of pole avoidance. Even for a rectangle with a high aspect ratio of a near single domain state, the edge domain is still observed. The behavior of these domain structures in applied fields have been theoretically investigated by Bryant and Suhl [20]. Their initial ideal was that the magnetic charges induced in an applied field should be distributed as in the analogous electrostatic problem. The magnetic charges are expected at the edges of the element like the electric charges reside on the surfaces of a conductor. Once the distribution of these magnetic charges is known, the domain pattern can be calculated numerically for any field. As for a rectangle with a uniform magnetization for almost all volume except both ends, the propagation and the annihilation of domain walls play a major role during the magnetization reversal. The angular dependence of the switching field may be appropriately expected by 1/cosq, according to the projection of the magnetic field on the wire axis [21].
2-3 Magnetoresistance of ferromagnetic wires
The definition of magnetoresistance (MR) is the change in the resistance induced by an applied magnetic field. There are a number of different kinds of MR, such as Lorentz magnetoresistance (LMR), anisotropic magnetoresistance (AMR), giant magnetoresistance (GMR), colossal magnetoresistance (CMR), and domain wall magnetoresistance (DWMR), each with a different origin. Among them, AMR and DWMR have a major contribution for a single layer ferromagnetic specimen.
Focusing on these two kinds of MR, their origin and behavior are discussed as follows.
2-3-1 Anisotropic magnetoresistance
The resistance of a material depends on its state of magnetization. This phenomenon is called magnetoresistive effect. The origin of the magnetoresistive effect in semiconductors and normal metals is the influence of the Lorentz force on the current path, with a result that the resistance is proportional to the square of the magnetic field. Apart from the ordinary magnetoresistive effect in semiconductors and normal metals, there is another effect in transition metals due to the spin-orbital interaction. In this effect the resistance depends on the orientation of the magnetization with respect to the direction of the electric current. The effect is often called the anisotropic magnetoresistance (AMR), which was first discovered by Kelvin [22].
It is convenient to explain AMR effect by taking into account the existence of the spin-orbital coupling and the anisotropic scattering mechanism of s and d electrons. In the case of the magnetic field direction with an angle relative to the current direction, the spin orbital interaction causes the resultant perturbed d-electron wave function to have a complex dependence on the angle. The transition probability of s electron to d orbital is large for electrons traveling parallel to the magnetic field. In the transition metal, the band structure is split into two different sub-bands of different orientations of the electron spins. When the 3d band is not fully filled, the transition of 4s electrons to 3d band is probable. The current of 4s electrons with smaller effective mass predominates the transport. The 3d electrons with large effective mass have low mobility. If there is a large transition probability of 4s electrons to 3d band, the increase of 3d electrons in the conduction current results in the decrease of conductance. Thus, when the current is parallel to the magnetic field direction, the
large transition probability leads to large resistance.
For a single crystalline metal with saturated magnetization, the Ohm’s law is described by the following expressionEri ij Mr Jrj
) ρ (
= , where Er
is the electric field,
Jr
is the current density, and ρis the tensor of the resistivity. Taking into account the crystal symmetry, Thomas obtained the resistivity tensor for a simple case that the magnetic field is in the film plane (x-y plane) [23]:
⎥⎥
whereρtandρlare the resistivity at the current direction perpendicular and parallel to the magnetization direction, respectively.Δρ =ρl −ρt. ρ is the Hall coefficient. H φis an angle between the orientation of the magnetization and the current direction.
When the current is in the x-axis, the resistivity in the x-axis is described by the expression.
It is the general formula of AMR effect. The resistivity in the y-axis is written as φ
ρ θ
ρxy( )= 21Δ sin2 (2-22)
It is the so-called Planar Hall effect. The angular dependences of resistivity for both effects are presented in Fig.2-8. Both cases show an periodicity of π . AMR and PHE have maxima at φ =nπ with n=0,1,2… and 0.25,1.25,2.25…, respectively, and minima with n=0.5,1.5,2.5… and 0.75,1.75,2.75…, respectively.
Fig 2-8: Schematic plot of the angular dependences of the resistivities, AMR (red) and PHE (blue), respectively.
2-3-2 Domain wall resistance
Domain wall resistance arises from the scattering when electric current passes through a domain wall. In general cases, the contribution of DW scattering to the MR is concealed by conventional sources of low temperature MR such as AMR and LMR in a ferromagnetic system. In order to isolate DWR, it is necessary to create artificial domain walls using a unique pattern such as adding a neck to the wires [24], designing zigzag structures [25], forming a striped domain by thickness modulation [26] or exchange biases [27], and implementing an elaborate magnetic field history process [28]. Although DWR was observed in various systems, both positive[24-28]
and negative[29] values were reported with their theoretical justifications [30-32]. Up to now, the sign and the magnitude of the DWR and the fundamental mechanisms of DW scattering are still controversial. In this section, we introduce the historical
theories about the positive DWR.
Spin mistracking effect
The intrinsic DWR arising from the spin polarized current passing through a domain wall has been developed by two methods, semi-classical and fully quantum mechanical, to predict the magnitude of this effect. When electrons pass through a domain wall, the electron spin will precess around the rotating exchange field and lags behind in orientation with respect to the local magnetization, as seen in Fig. 2-9. Viret estimated the average of this angle via the exchange field rotating through in half of a Larmor precession. He treated the spin vector classically and found that the mean free path depends on the spin-dependent scattering rate and the cosine of the behind angle
The intrinsic DWR arising from the spin polarized current passing through a domain wall has been developed by two methods, semi-classical and fully quantum mechanical, to predict the magnitude of this effect. When electrons pass through a domain wall, the electron spin will precess around the rotating exchange field and lags behind in orientation with respect to the local magnetization, as seen in Fig. 2-9. Viret estimated the average of this angle via the exchange field rotating through in half of a Larmor precession. He treated the spin vector classically and found that the mean free path depends on the spin-dependent scattering rate and the cosine of the behind angle