Добавил:

Mendeleev Опубликованный материал нарушает ваши авторские права? Сообщите нам.

Вуз:

Российский химико-технологический университет им. Д.И. Менделеева

Предмет:

Химия

Файл:

Fundamentals of the Physics of Solids / 14-Magnetically Ordered Systems

.pdf

Скачиваний:

Добавлен:

08.01.2014

Размер:

943.46 Кб

Скачать

☆

<<< < Предыдущая 1 2 3 4 5 67 / 77

14.7 Magnetic Anisotropy, Domains

509

downward orientation of the moments is preferred, there are two characteristic types of domain walls. In most cases the rotation of the magnetic moment is such that it remains in the plane of the wall everywhere. Such a domain wall is called a Bloch wall.13 When the rotation of the moment is in a plane perpendicular to the wall, we speak of a Néel wall.14 These situations are shown in Fig. 14.19; the wall is perpendicular to the x-axis.

(a)

(b)

Fig. 14.19. Rotation of the magnetic moment for domain walls in the (y, z) plane: (a) Bloch wall; (b) Néel wall

In magnetically uniaxial crystals the energy density is the sum of (14.7.10) and (14.7.15). The derivative term comes from the exchange between neighboring spins, and J1 and J2 are related to the exchange integral. For simplicity, neglecting anisotropy in this term and the homogeneous isotropic part in the other,

E =

∂rμ

− 2

Mz2

dr ,

(14.7.17)

∂M

where K = K −K . Assuming that K > 0, magnetization is along the z-axis inside the domains, pointing either upward or downward. If the domain wall is in the (y, z) plane, and all the spatial variations occur in the x-direction,

E =

∞

∂x

− 2

Mz2 dx . (14.7.18)

∂Mx

∂My

∂Mz

−∞

Using the position-dependent polar angles θ and ϕ to describe the rotation of the magnetic moment,

Mx = M sin θ cos ϕ ,

My = M sin θ sin ϕ ,

Mz = M cos θ

(14.7.19)

in general. In terms of the variables θ(x) and ϕ(x),

E = M 2

∞

θ dx

−

2 cos2 θ dx .

(14.7.20)

+ sin2

dθ

dϕ

−∞

13F. Bloch, 1932.

14L. Néel, 1955.

510 14 Magnetically Ordered Systems

Considering the spatial variations of ϕ ﬁrst: the energy has its minimum when ϕ is constant. Two special cases are customarily distinguished: ϕ = π/2 corresponds to the Bloch wall, and ϕ = 0 to the Néel wall. In the Bloch wall the magnetic moment stays in the (y, z) plane, that is

Mx = 0 , My = M sin θ ,							Mz = M cos θ .		(14.7.21)
while in the Néel wall
Mx = M sin θ ,		My = 0 ,					Mz = M cos θ .		(14.7.22)
In both cases
E = M 2	∞		2	dx		2	− 2 cos2 θ dx .		(14.7.23)
E = M 2			2	dx			− 2 cos2 θ dx .		(14.7.23)
			J		dθ			K

−∞

The spatial variations of θ(x) have to be determined from the energy minimum. The Euler equation of the variational problem,

∂L

−

= 0 ,

(14.7.24)

∂θ

leads to the formula

d2θ

− K sin θ cos θ = 0 .

(14.7.25)

dx2

Integration gives

dθ

− K sin2 θ = constant .

(14.7.26)

The spins are fully aligned far from the domain wall, so the following boundary condition can be imposed at inﬁnity:

x =

−∞

θ = 0 ,

(14.7.27)

θ = π ,

θ = 0 .

x = +

∞

The value of the constant is therefore zero, and so

dθ

sin2 θ .

(14.7.28)

The solution of this equation that satisﬁes the boundary condition is

cos θ(x) = − tanh

K/J

x ,

(14.7.29)

since

dθ

− sin θ

= − K/J

1 − tanh2

K/Jx = − K/J sin2 θ .

(14.7.30)

		14.7 Magnetic Anisotropy, Domains							511
Solving (14.7.29) for ex/δ , where δ =				,
Solving (14.7.29) for ex/δ , where δ =			J/K	,
						θ(x)
ex/δ =	1	− cos θ(x) = tan			1	θ(x)	,
7								(14.7.31)
7	1	+ cos θ(x)			2			(14.7.31)
and hence
θ(x) = 2 arctan(ex/δ ) .								(14.7.32)

It can be immediately seen that δ is the domain-wall thickness. For obvious physical reasons, the exchange term prefers a slow rotation of the magnetic moment; therefore in itself it would lead to an inﬁnitely thick domain wall. On the other hand, the anisotropy energy is minimal when the reversal of the moment is abrupt. The competition of these two terms leads to a ﬁnite wall thickness. Choosing values that are typical in ferromagnets for the exchange constant and the anisotropy constant, δ 10−6 cm is obtained. The speciﬁc thickness values for iron, cobalt, and nickel are 40, 15, and 100 nm, respectively. Therefore the number of atoms across the domain wall is on the order of 100.

The energy of the wall can also be calculated. The energy di erence per unit surface area relative to the case of homogeneous magnetization along the z-axis is

∞ M 2

ΔE =

K sin2 θ(x) − K cos2 θ(x) + K

−∞

∞

−∞

(14.7.33)

K/Jx

= M 2K

1 − tanh2

= M 2√

∞

1 − tanh2 x dx = 2M 2√

−∞

This calculation gives the same energy for the Bloch wall and the Néel wall. This is because the energy contribution of magnetic dipoles has been neglected. When it is taken into account, the energy of the Bloch wall is found to be lower – except for very thin samples, since in thin magnetic ﬁlms it is energetically more favorable to have the magnetization aligned with the surface of the sample everywhere, as illustrated in Fig. 14.20.

In the case of very strong uniaxial anisotropy the orientation of the moments in the interior of the domain extends all the way to the surface. In such arrangements there is always a substantial fringing ﬁeld. When the anisotropy is weaker or cubic, the formation of closure domains at the surfaces is energetically more favorable, as illustrated in Fig. 14.21.

In this arrangement the magnetization component perpendicular to the surface is everywhere continuous across the boundaries of the closure domains,

512 14 Magnetically Ordered Systems

Fig. 14.20. Néel wall between two domains in a thin layer

Fig. 14.21. Formation of perpendicularly magnetized closure domains close to the surface

and no fringing ﬁeld appears. Using energy considerations, the dimensions of such closure domains – and through them, the width of the domains, i.e., the distance between the domain walls – can be determined. Denoting the width of the domains by d, there are Lx/d closure domains on the surface of a sample of linear size Lx. Since the volume of the closure domain is (d2/4)Ly, and the anisotropy energy per unit volume is 12 KM 2, the total anisotropy energy of all closure domains is

Eaniso =	Lx	·	d2		K	M 2 =	1	KdLxLyM 2 .	(14.7.34)
				Ly ·
	d		4		2		8

Neglecting the surface energy of the closure domains in comparison with the surface energy of the Lx/d large domain walls of surface area √Ly Lz separated by regular distances d, and making use of the formula 2M 2 JK for the surface energy density, the wall energy is found to be

Ewall = 2M

2√

LyLz = 2M

2√

LxLy Lz

Minimizing the full energy with respect to d,

− 2M

2√

LxLy Lz

= 0 ,

KLxLy M

from which

1/4

d = 2 Lz (J/K) .

(14.7.35)

(14.7.36)

(14.7.37)

The typical value of this distance is 4d 5	1–10 μm, that is, the linear ex-
tension of domains corresponds to 10 –10	atoms. By increasing d, further

14.7 Magnetic Anisotropy, Domains					513

Fig. 14.22. Spike domains close to the sample surface

spike-shaped domains may appear at the surface of the sample, as shown in Fig. 14.22.

For technical magnetization curves the displacement of the domain walls and the rotation of the magnetization direction within the domains are of the utmost importance. For weak ﬁelds wall motion dominates, while for stronger ﬁelds the rotation of magnetization.

Domains can be directly observed using, for example, the powder method, in which magnetic particles trace out the pattern on the surface, or magnetooptic methods, such as those based on the Kerr e ect15 or the Faraday e ect.16 The former method is rooted in the observation that the polarization of light becomes rotated upon reﬂection from the surface of the sample because of the interaction with magnetic material, so the domains become visible in a polarization microscope. The rotation of the polarization plane of light is observed with methods based on the Faraday e ect, too, however this time a beam penetrating through a thin magnetic layer is used. Owing to the relatively large size of domains, scanning and transmission electron microscopes may also be used for mapping the domain structure. A recent development in this ﬁeld is the magnetic force microscope (MFM), a new version of the atomic force microscope with a magnetic stylus, designed speciﬁcally for probing magnetic materials. It has been used successfully for measuring the spatial variations of magnetization at the sample surface.