Superparamagnetism

The collapse of domain walls

Superparamagnetism is a distinctive behavior of single-domain nanoparticles, originated from the fast flipping process of the total magnetic moment due to thermal energy. Below, some basic ideas leading to the concept of SPM.

Contents

· Atomic Origin of Magnetism

· Magnetic Energy: Domain Walls

· Single Domain Particles: Superparamagnetism

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Each electron can be viewed as a charged particle that rotates on its axis, similar to a toy top. This spinning movement produces another characteristic of electrons (and other atomic particles): the magnetic moment. (figure 1). 

These numbers are the 'fingerprints' that identifies the element and determine its physical and chemical behavior. To maintain the charge neutrality, each atom (in its neutral state) has thus a typical number of electrons orbiting around the nucleus. These electrons are responsible for (most of) the magnetic properties of each chemical element. 

Since the electron has an intrinsic electrical charge, also the orbits of electrons around the nucleus can give rise to the so-called orbital moment. (Fig. 2) 

These two types of magnetic moment, when observed in macroscopic scale, is what produces the force of attraction, for example, between two magnets.

Depending on the number of electrons of each atom, and its overall arrangement, the total magnetic moment of an atom can differ greatly, causing some atoms to be more suitable for the production of, for example, permanent magnets. Also, there is a force of quantum origin (called the exchange or superexchange) that tends to align the magnetic moments of two neighboring atoms in parallel (ferromagnetic) or antiparallel (antiferromagnetic) directions, as shown in Fig. 3. This interaction is stronger than any other between atoms in a material (eg, to destroy the order that exists inside of a piece of metallic iron , we should warm up (i.e., provide thermal energy) to a temperature of 800 ºC!. 

Thus, each atom within a material contributes individually to the total magnetic moment. Due to the huge number of atoms that exist, for example, 1 g of a given material (which is around N_A, the Avogadro's Number, which is 10²³ = 1,000,000,000,000,000,000,000,000), if all times atomic were actually pointed towards the same direction (that is, if all were aligned because of the strength of exchange interactions within a ferromagnet), the total value of the magnetic moment would be enormous and would be almost impossible to separate two pieces of, for example, metallic iron. In practice this does not occur because this bulky order needs a lot of energy to be maintained, and all physical systems obey the principle of minimum enrgy, namely to seek the state of lower energy that is compatible with the conditions of system in question.

Comment. 

Avogadro’s number is just that: a number. Was created to represent the huge number of atoms in a portion of macroscopic material. It has the same purpose as the number you call "a dozen". A dozen is exactly 12 (without units!) Represents a set of 12 things: eggs, oranges, planets or cows. The number of Avogadro N_A = 10²³ represents a set of 100,000,000,000,000,000,000,000 (one hundred trillion) of things: cows, oranges or atoms. 

Just as, for example, 1 mole of oranges from radio ~ 10cm (quite an ordinary orange!) will occupy a volume of V = (pi/6)x(10³)x(N_A)=3.1x10^26 cm³, a figure comparable with the volume of the Earth ( V_E ~ 1.1x10²⁷ cm³) !!!!.

The Minimum Energy Principle

The principle of minimum energy lays at the foundations of physics: it provides a basis for predicting the direction of events in the universe. Thus, intuitively we know that an object launched through a window at the 12^nd floor will move in the direction as to reduce its distance to the ground (height), for minimizing its gravitational potential energy (which is proportional to height). ANother example would be two balls loaded with equal electrical charge (lets say, both positively charged): these two charged bodies will move in the direction to fall apart to the maximum possible extent (ideally they'll go to the infinite), thereby reducing the electrostatic energy of the system (i.e. the pair of balls).

Magnetic Energy: Domain Walls

So a solid piece of magnetic material should have ALL its magnetic moments (spins) aligned by exchange forces and pointing to the very same direction, thus storing huge amounts of magnetic energy. 

The way by which such a situation is avoided and the energy minimized, is to break down this parallel alignment at several points of the material, and forming regions with parallel alignment inside, but each volume having random directions so that the sum of overall moments of the fields is essentially zero. This is shown in Fig 4 where the magnetic moment (big red arrow) in each figure is a measure of the energy stored in the material. Although two neighboring areas, pointing in different directions will 'dislike' to be close (two neighboring atoms of different areas, want to be aligned in the same direction because of the exchange interaction), the decrease of total energy due to the cancellation of the moments between areas favors this configuration.

Figure 4. Left: single-domain with all moments parallel. Rigth: Multi-domain configuration having interfaces (domain walls, black lines) with broken magnetic order.  

The single-domain state: superparamagnetism

On the other hand, if the dimensions of the material under consideration is reduced drastically, until a few nanometers (1 nanometer = 10^-9 = 0.000000001 meter), these domain walls will be forced to shrink and coexist in a small volume, thus increasing the "repulsion" among them. This proximity (density) of domains when the material reaches dimensions at the nanometer scale, yields the domains to merge into a single one, creating what is known as a single-domain particle (usually a nano-particle). The exact dimensions at which a nanostructured material passes from single-to multi-domain depends on each material, and is known as Critical Size. If the particle is spherical, it is called in critical diameter. (Figure 5)

Fig. 5: Multi-domain (left) and monodomain (right) configurations for a nanoparticle. The yellow regions indicate that at the particle surface the spins are usually misaligned.

In a nano-particle each atom is part of a magnetically ordered arrangement, with the magnetic moments aligned in one direction space, and therefore the total magnetic moment is the sum of all moments of the atomic particle. If we realize that the separation between atoms is of the order of tenths of a nanometer, one can conclude that in a sphere of radius of a few tens of nanometers, the total number of atoms is of the order of 1000-10000. Therefore, the total magnetic moment of a single-domain nanoparticle can be about 10,000 times greater than the atomic moments of the constituent atoms.

The behavior of a set of such a single-domain nanoparticles can be very complex due to the fact that the magnetic moment of each particle disturbs the behavior of neighbor particles. In general, this interaction between particles serves to align the magnetic moments in order to also reduce the energy of the whole. In some cases a preferred direction can exist for the magnetic moment of each particle, as for example the case of needle-shaped particles (Figure 6). In such a situation, the magnetic moment prefers to be oriented along the major axis of particle, either up or down (so to diminish another energy: the magnetostatic energy). The preferred direction or axis is called "easy axis", and may also occur in particles of any shape because of asymmetries of chemical bonds typical of many materials. We see that, for a particle with anisotropy axis of the type shown in Figure 6 (called uniaxial), the magnetic moment can turn from the position theta = 0 ° to 180 °, only if it has an energy equal to or greater than the energy of the 'anisotropy barrier' E_A.

Figure 6: Example of an 'anisotropy axis' originated from shape factors. The larger energy of the perpendicular direction of the magnetic moment (up right) compared to the parallel to the particle axis (up left) is known as anisotropy energy barrier E_A.

There are different frameworks to make a theory of Superparamagnetism:

 

1) The starting point could be the magnetic response of a magnetic momento to an external magnetic field H, at T = 0 (no thermal energy). Then the effect of Temperature is added to the model. This approach has been developed by Stoner and Wolfhart and followers since 1948.

2) We can start from the relaxation of the magnetic momento due to thermal fluctuations, to describe the time evolution of M(t), and from there, to develop the equations of superparamagnetic behavior. This thermal relaxation, or Néel-Arrhenius relaxation, is the basis of many approaches describing the heating mechanisms of single domain magnetic nanoparticles.

 

The magnetic relaxation of single domain MNPs and heating models are discussed HERE.