Green Nanomaterials. Siddharth Patwardhan

Electron spin and nanomagnetism

      Nanomagnetism is another perfect illustration of a property on the boundary between bulk and atomic scale. Again we will see here how some nanoparticles behave like atoms (superparamagnets) while others behave more like the bulk (single-domain nanomagnets). Furthermore, we see how the quantum interplay between charge, electron spin and magnetic field can be harnessed in nanomaterials for spintronic applications. To understand the nanoscale properties we must first understand the interplay of electrons and magnetism at the atomic scale, appreciate how this builds to bulk scale, then interrogate the middle.

      In fundamental terms, a magnetic field is generated when there is movement of charge. This is seen by the magnetic field surrounding a wire conducting electricity, which is due to the charged electrons moving in the wire (figure 2.7(Ai)). Electrons in an atom have two sorts of motion (orbital and spin (figure 2.7(Aii))) and as such have a magnetic moment: generally it is the electrons in a material that give it its magnetic properties. The moment of a single electron can be calculated from first principles, and is found to be

      μB=πr2I=eℏ2me=9.274×10−24Am2,(2.6)

      where r is the radius of the orbital, I is the current, e is the charge of an electron, m_e is the mass of an electron and ℏ is the reduced Planck constant.

      Figure 2.7. Description of magnetism from the atomic to the bulk. (A) Two demonstrations of magnetic fields generated by electrons. (Ai) Macroscale example of magnetic field lines of magnetic field generated by a wire conducting electricity due to the flow of electrons (the current I) in the wire. (Aii) The motion of an electron within an atom (both orbital and spin) which generates an atomic magnetic field. (B) Any unpaired electrons contribute to a bulk magnetic property depending how they are arranged in the solid. Each arrow represents the direction and magnitude of a paramagnetic atom. (C) The nanomagnetic properties, demonstrating how the largest single domain magnetism have the highest coercivity due to no loss of energy through domain wall formation, while superparamagnetic nanoparticles have near zero coercivity. (D) Description of the magnetic properties of hysteresis, plotting the magnetisation with increasing field, then reversing the field and increasing again.

      Two electrons only exist in the same atomic orbital if they have opposite spin values (either + 1/2 or −1/2). The electrons are ‘paired’ in an orbital and these two opposite electronic spins cancel out each other’s magnetic moment, so there is said to be no net overall magnetic effect. However, this is not technically true: such orbits actually have a week repulsion to the magnetic field (in the order of a million times less than the unpaired magnetic effect). If an atom has only paired electrons it is diamagnetic. The presence of an unpaired electron in an atom gives the atom an overall magnetic moment and such atoms are described as paramagnetic. The application of a magnetic field causes these dipoles to align with the direction of an external magnetic field. The fact that unpaired electrons create a magnetic field and in turn are affected by an external field is fundamental to spintronics. This new field uses the quantum physical properties of ultrathin (single atomic thick layer) 2D nanomaterials, and harnesses the effect of fields on electron spin and spin–orbit coupling to convey electronic information rather than the transport of free elections down a wire. In purely paramagnetic materials the electron dipoles are only weakly coupled, and as a result thermal energy causes these moments to randomly align. Examples of purely paramagnetic materials include many transition metals and rare-earth salts. These materials have a net magnetic moment due to incomplete electron shells.

      Bulk magnetism is a cooperative effect of the alignment of many paramagnetic atoms, and as such it is a property reliant on structural order. There are three main categories of bulk magnetic ordering (figure 2.7(B)). Ferromagnetism is when all the paramagnetic atoms are aligned in the same direction even when a field is removed, resulting in the material having a magnetic moment. Ferromagnetism is rare, with iron, cobalt, nickel and some rare-earth metals being the only elements to be ferromagnetic. Ferromagnetism is rare as paramagnetic atoms/ions will tend to align in an anti-parallel way; this form of ordering is termed antiferromagnetic and will exhibit no net magnetisation as the opposing paramagnetic moments will cancel each other out. Examples include numerous transition metal compounds, such as iron manganese (FeMn). The final bulk magnetic coupling utilises the strong anti-parallel alignment, but results in a net magnetic moment simply by ensuring there is an uneven number of magnetic dipoles in each direction, so the dipoles are not fully cancelled out. A material with this magnetic ordering is called a ferrimagnet. Ferrimagnetic materials tend to be made up of different materials or ions, with examples including garnets such as YIG (Y₃Fe₂(FeO₄)₃) and ferrites such as the oldest known magnetic material: magnetite (Fe₃O₄).

      Magnetic order is dependent on temperature: when the temperature is increased above a certain threshold (blocking temperature (T_B)) (also called Curie temperature (T_c) or Néel temperature (T_N) for ferr(o/i)magnetic and antiferromagnetic materials respectively), specific to that material, the thermal energy destroys the ordering and the bulk magnetism is lost. Above this temperature the material becomes paramagnetic.

      It is well known that magnetic materials can become demagnetised over time. This is because a material will gradually form magnetic domains, which are small regions of local dipole alignment within the material, each separated by a domain wall. It is always worth remembering that maintaining order carries an energy cost, so long-range order will inevitably reduce to short-range order (domain), to eventual disorder at any elevated temperature if there is nothing to maintain the order (i.e. a field). Domains form to minimise the material’s energy by reducing magnetic poles at the surface to reduce the magnetostatic energy. However, the creation of a domain wall requires energy to overcome this stability, so an equilibrium is reached. If a demagnetised material is placed back in a magnetic field, the domains most closely aligned to the field direction grow as the walls recede at the expense of the misaligned domains, continuing until all the dipoles are aligned with the higher field and the material is said to be magnetically saturated (figure 2.7(D)).

      A material that will easily or reluctantly align with a changing magnetic field is termed magnetically soft or hard, respectively, a property that is measured by its magnetic coercivity (figure 2.7(D)). For a soft magnet, simply removing the field (field = 0) is enough to demagnetise the material. However, a hard magnet will retain its moment at zero field, and even in an opposing magnetic field (figure 2.7(D)). The electrons in such materials have a preferred (lower energy) direction of alignment, so more energy is required to realign them. Coercivity can be quantified from a hysteresis plot (figure 2.7(D)) where a material is driven to saturation by application of a cycling magnetic field through both opposing directions. Where there is a hysteresis loop, work is done to oppose the field and this energy is proportional to the area of the hysteresis curve and is dissipated in the form of heat.

      If the bulk material’s dimensions are reduced to the nanoscale, a critical volume is reached where it costs more energy to create a domain wall than the total magnetostatic energy of the material, and so a single domain magnetic particle is formed (figure 2.7(C)), which is also dependent on the anisotropy of the material. These are the smallest permanent magnets, and have the advantage that energy is not wasted on the formation of domain walls, so coercivity is the highest for this material (by volume). As the size is reduced further the magnetisation energy of the particles approaches the thermal energy (K_BT) and so the magnetic dipole of these particles will continually fluctuate with thermal fluctuations. These particles behave like paramagnetic atoms would (not a bulk magnet) and are thus termed superparamagnetic. Such particles do not display hysteresis as the particle itself flips in the field. Interestingly there is not a cut-off defined size at which a single domain magnetic nanoparticle starts to display superparamagnetic behaviour. It is entirely dependent on the temperature and the time-scale of an experiment. That is, the same single domain ferromagnetic nanoparticle could be a permanent static magnet in one experiment where the measurement was conducted over a short