17 Domains in thin films

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 Contents

 

1.     Ferromagnetism in materials 

2.     Domain formation in ferromagnetic thin films

3.     Domain walls

(a) Bloch walls

(b) Neel Walls

4.     Ferroelectric domains

5.     Domain formation in ferroelectric thin films

6.     Dielectric hysteresis in perovskites

 

Ferromagnetism in materials

 

Ferromagnetic materials are an important class of solids which have played an indispensable role in data storage technologies of the digital age. Their utility for technological applications stems from the basic physical property of ferromagnets to exhibit spatially ordered magnetization patterns – magnetic domains – under a variety of conditions. The existence of ferromagnetism and ferromagnetic domains was postulated by Weiss (in 1906) where he suggested that atoms in a ferromagnetic material possessed permanent magnetic moments which could be aligned parallel to one another over an extensive region. In 1926, his postulates were converted into a well-defined theory of ‘domains’ of parallel moments (as shown in Figure 1). The total magnetisation (i.e. magnetic moment per unit volume) of a material is defined as the vector sum of the domain magnetisations. In a completely demagnetised state, this sum total is zero. Upon application of an external field, changes are observed in the domain configuration, like relative widths of domains change, thus producing a net magnetisation in the direction of applied field. The hypothesis made by Weiss was confirmed later by direct observations on ferromagnetic materials (Bitter, 1931).

 

Domain formation in ferromagnetic thin films

 

Ferromagnetic thin films have widespread applications in miniaturizing the photovoltaic devices and for the fabrication of compact memory structures. The extent of magnetisation in ferromagnetic thin films can be altered by material engineering and by optimizing the material dimensions. In ferromagnetic thin films, if there is an infinite lateral extension, such that the film is uniformly magnetized across a plane, the magnitude of magnetostatic energy is zero. But, if the film has a finite size, the effect of surface charging should also be considered, which gives rise to demagnetizing field. The demagnetizing field is responsible for separating the film into domains with dissimilar orientations of magnetization to reduce the magnetostatic energy (i.e. long range magnetic interaction). The boundaries between these domains are called magnetic domain walls. Inside the domain walls, the spins rotate gradually, leading to a certain width of the wall. The width of the domain wall is primarily determined by the competition between the exchange energy and the anisotropy energy. The exchange energy between neighbouring spins tends to increase the wall width. A larger rotation of spins between two neighbours causes a higher exchange energy. However, a wider wall leads to a higher value of anisotropy energy, because inside the domain wall, the direction of spins is away from the easy axis of magnetization. The various energy contributions which are responsible for domain formation in materials and are summarised as follows:

 

(a)   Magnetostatic energy

 

The magnetostatic (or dipole) energy is due to the magnetization M, the magnetic-dipole moment per volume, which arises from the alignment of atomic magnetic dipoles. In a solid, the dipoles arise primarily from electron spins. Although the orbital motion of electrons usually contributes less to the dipole strength, it plays a significant role for the magnetic anisotropy.

 

In the case of an infinitely extended magnetic film magnetized uniformly normal to the surface, for which the magnetostatic energy can be easily derived. The magnetic dipoles in the film, ?⃗, create the stray field,?????? . However, due to the surface charging, inside the film, there exists the field which has the same amplitude as Hstray but directs opposite to ?⃗ , the so called demagnetizing field, ??. So, the magnetostatic energy, which has the same amplitude to the demagnetizing energy in this case, will be described with ?⃗ , and Hd as

(b)  Exchange energy

 

The cooperative magnetic ordering which exists amongst the neighbouring spins in a material is the exchange interaction. The exchange interaction energy,??? , between two spins, ??⃗ and ??⃗ , scales scales with the exchange integral, J and is expressed as

 

Where??⃗ and ??⃗ are the unit vectors of interacting spins on two atoms. The exchange interaction stems out of the Coulomb interaction between the electron charges and the Pauli principle. To estimate the exchange energy in a magnetic domain wall, it is convenient to take all the spins together, i.e., a continuous model of spin rotation in a one-dimensional domain wall, the total ??? inside the wall is,

where ? = ?2??2??? /2 represents the exchange stiffness constant (J/m), and is a temperature dependent quantity, s is the spin quantum number (= 1/2), and refers to the lattice constant, indicates the number of nearest neighbour atoms per unit volume, whereas θ is the angle with respect to the easy axis of magnetization.

 

For simple cubic (SC) and body centred cubic (BCC) , J will be

? = 0.54????                                            ??? ??
? = 0.34????                                           ??? ???

 

where ?? and ?? are  the Boltzmann constant and the Curie temperature, respectively. It should be noted that the exchange energy tends to make a magnetic domain wall as wide as possible. Since the exchange energy declines upon reducing the angle between spins on neighbouring atoms inside the wall, the spins rotate gradually, leading to a certain width of the magnetic domain wall.

 

(c) Anisotropy energy

 

The anisotropy energy majorly depends on the orientation of the magnetization with respect to the crystallographic axes of the material. It is a manifestation of spin-orbit interaction. Many kinds of ferromagnetic films possess a uniaxial anisotropy, irrespective of their crystalline state i.e. this anisotropy can exist in polycrystalline, single crystal, or even alloyed films. In undisturbed crystals, the anisotropy energy will be minimized along certain crystal axes. However, anisotropy can be induced by symmetry breaking of the crystal structure at the interface and surface, by anisotropic modulation of atoms, or by alignment of surface/interface defects.

 

(d)  Zeeman energy

 

Zeeman energy is the interaction energy of the magnetization vector field M~ with an external magnetic field ????. Then the Zeeman energy is

 

 

To reduce the total magnetic energy ε, FM films create domains of a certain size. i.e.

The size of domains is defined by minimization of the sum of all the above energy terms, exchange, anisotropy, Zeeman, and demagnetizing energies.

 

Domain Walls

 

The transition region between domains magnetised in different directions was first studied by Bloch (1932). The change from one direction to the other is not discontinuous but occurs over a width determined by a balance between exchange and anisotropy energy. There are two main types of spin structures inside the domain walls, Bloch and Neel types. Besides many other kinds of domain walls exist. One of them is called the cross-tie wall, which is an intermediate state between Bloch and Neel walls, and it is composed of a mixture of Bloch and Neel walls. In Figure 2, the spin structures of Bloch and Neel walls are shown. The Bloch wall is usually preferable in bulk materials. Spins rotate in the plane parallel to the wall plane. The wall width of a 180◦ Bloch wall is most commonly defined by π p A/K, where A and K are the exchange constant and anisotropy energy, respectively. Then the wall profile basically follows a sine law. In thin films, however, a Bloch wall induces surface charges by its stray field. Then the Neel wall become more favourable when the film thickness becomes smaller than the wall width. In a Neel wall, spins rotate in the film plane (Figure 2). The width and profile of the Neel wall are difficult to define. The Neel wall has a narrow core and µm-long tails on both sides. The core width is of the order of the exchange length [= pA/Kd (nm scale)], where is the demagnetizing energy.

Figure 2: Rotation of magnetization vector in the (a) Bloch wall and (b) Neel Wall

 

Ferroelectric domains

 

Originally, the term “ferroelectricity” was coined in analogy with the similar behaviour of ferromagnets, which possess a finite switchable magnetization, such as that of iron (“ferro”), the canonical ferromagnetic metal. However, this similarity is only manifest from a purely macroscopic, thermodynamic point of view, as the microscopic mechanisms that give rise to a spontaneous electric or magnetic polarization are radically different

 

Ferroelectric materials are those which exhibit finite electric polarization in the absence of any external electric field. This polarization must have at least two stable states such that there is a possibility to reversibly switch between these two states upon application of an external electric field. Therefore, within a ferroelectric sample, different regions with localized orientation of polarization vector in any said direction may coexist. These regions of aligned electric dipoles are known as ferroelectric domains, which are separated by transition layers known as domain walls.

 

Domain formation in Ferroelectric thin films

 

The growth of epitaxial ferroelectric thin films is of prime concern for researchers working in the field of Photovoltaics, memory devices etc. Films are generally grown using Pulsed Laser Deposition or Sputtering technique to maintain the desired stoichiometry, which in turn controls the ferroelectric behaviour of the material.

 

In principle, the bound charges which arise at the surface of a polarized dielectric material are described by the equation

?? = −∇.P

 

Where, ?? refers to the bound charge density and P is the polarization vector. Specifically, in ferroelectric materials, the bound charges give rise to an internal electric field, which is opposite to the direction of spontaneous electric field. If the dimensions of the sample are reduced to a certain extent, this depolarization field overcomes the polarization and completely supresses it. Thus, if the depolarization field in a sample is not screened, the ferroelectricity of the material tends to disappear. However, in some cases, the presence of some screening charges at the sample surface can effectively reduce the extent of depolarizing field. In cases where sufficient screening charges are not available on the surface, the accumulation of bound charges can be overcome by employing ferroelectric domains of alternate polarization in a periodic fashion. As a result, the bound surface charges disappear on an average and subsequently, the extent of depolarization field into the material is greatly reduced. On the other hand, stray fields which originate due to the existence of domain walls are also accountable for the total energy of the system.

 

Dielectric hysteresis in perovskites

 

Owing to the complex set of elastic and electric boundary conditions at each grain, the ferroelectric grains in polycrystalline materials are always split into many domains. If the direction of the spontaneous polarization through the material is random or distributed in a such way as to lead to zero net macroscopic polarization, the piezoelectric effects of individual domains will cancel out and such materials will not exhibit piezoelectric effect, which requires at least non-centrosymmetric symmetry of the material.

 

Lead Titanate (PbTiO3) is a very well-known perovskite material that changes its phase from a non-ferroelectric cubic to ferroelectric tetragonal at a temperature of 490˚C Perovskite materials are known to have a general formula ABO3 where, valency of cation A can take values varying from +1 to +3 and of cation B can take values from +3 to +6. The structure may be viewed as consisting of BO6 octahedra surrounded by cations A. Amongst the known ferroelectric materials, the materials of practical interest exist in have a perovskite structure and many, such as lead Zirconate Titanate, Pb(Zr,Ti)O3, are actually solid solutions of PbTiO3. It has been observed that the spontaneous polarization in PbTiO3 lies along the cT-axis of the tetragonal unit cell and the crystal distortion is usually described in terms of the shifts of O and Ti ions relative to Pb.

 

A defining, and the most important, characteristic of ferroelectric materials is polarization reversal (or switching) by an electric field. We have seen that the natural state of a ferroelectric material is a multidomain state. Application of an electric field will reduce (in ceramics) or completely remove (in crystals) domain walls. One consequence of the domain-wall switching in ferroelectric materials is the occurrence of the ferroelectric hysteresis loop, as illustrated in Figure 3.

Beginning from the origin, where P = 0, E = 0, possible macroscopically if the thin film sample consists of multiple domains compensating each other and increasing the field (1), the total polarization of the material starts increasing towards a saturation value (2) which corresponds to a state where all unit cells are coherently oriented with respect to the field. Upon further ramping up the applied field, the polarization in the material is further enhanced due to dielectric charging (3). As the field is reduced back to zero, the total polarization decreases to some extent but remains finite, and the polarization value obtained at E = 0 is known as the remanent polarization +Pr. When the applied field is increased in the opposite direction, the polarization state starts switching suddenly for a specific field intensity called the coercive field−Ec. Upon further increase in applied field, the polarization saturates and a simple dielectric response is reached again (6). Generally, the same behaviour is observed when the electric field is reversed, and repeated cycling results in a symmetric hysteresis loop, with switching events at ±Ec. In the presence of internal fields, e.g. due to charge accumulation in a ferroelectric transistor, the hysteresis loop may however be offset with respect to E = 0, a phenomenon known as imprint. The spontaneous polarization ±Ps is usually defined as the extrapolation at zero field of the polarization value at high fields, where the slight decrease is due to charging. Ideally, Pr and Ps should be identical. In reality, Ps is often higher in polycrystralline materials (due for instance to the formation of opposite domains during the ramping of the field to zero), but can be very close in single crystals.

 

An ideal hysteresis loop is symmetrical, so the positive and negative coercive fields and positive and negative remanent polarizations are equal. The coercive field, spontaneous and remanent polarization, and shape of the loop may be affected by many factors including the thickness of the sample, presence of charged defects, mechanical stresses, preparation conditions, and thermal treatment.

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