The notion of a band structure can be applied to the development of magnetism in a material. Magnetism originates from the presence of electron magnetic dipole mo- ments (often referred to as spins) and their alignment within the material. There is a
2
-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24 E - E F [ e V ] X L W X L
Figure 2.5: Calculated electronic band structure of Si plotted along Γ→L in reciprocal space. The region shaded in grey is the band-gap, indicating that Si is a semiconducting material. Band structure calculated using the ELK DFT code.
wide range of different magnetic orderings which have been observed in physics, some of which are illustrated in figure 2.6. The magnetic ordering of a material is heavily dependent on the arrangement of the magnetic moments and how the system reacts to an applied magnetic field. Diamagnetism is the weakest form of magnetism and is found in all materials. In a diamagnetic material, the electron spins will oppose an applied magnetic field. Paramagnetism is stronger than diamagnetism and is characterised by the alignment of a material’s electron dipole moments with a sufficiently large magnetic field. When the applied magnetic field is removed, the spins lose their alignment. Below the so-called Curie Temperature TC, a material may develop a spontaneous alignment
of its spins. This is called Ferromagnetism. Ferromagnetism is a strong, long range ordering of the electron spins where the material will retain a net magnetisation even in the absence of an applied magnetic field. If a magnetic field is applied opposite to the alignment of the electron spins, they will only change their polarisation after a
(a) No Ordering (b) Ferromagnetism (Observed in Co2MnSi) (c) Antiferromagnetism (Observed in CeB6 and Ca3Co2O6) (d) Ferrimagnetism (Observed in Ca3Co2O6)
Figure 2.6: The types of magnetic ordering which were encountered within this thesis. The arrows represent the direction and magnitude of the magnetic moments in the material.
sufficiently large magnetic field is applied. This phenomenon is called hysteresis. For some materials, Antiferromagnetism will develop where the net magnetisation of the sublattices which comprise the material develop a spontaneous alignment anti-parallel to each other. Ferrimagnetism is a similar phenomenon to antiferromagnetism however the magnitudes of the sublattices’ magnetisations are different.
These types of ferroic ordering are very strong and do not develop solely from the electron magnetic dipole interaction. They develop from a type of interaction called the
Exchange interaction.
2.6.1 The Exchange Interaction
For many materials, magnetic ordering occurs as a result of the exchange interaction. The exchange interaction is a quantum mechanical phenomenon which occurs as a result
of the Pauli exclusion principle and Hund’s rules.21 The electrons of a system must obey exchange symmetry3 and cannot occupy the same quantum mechanical state. These
conditions restrict the electrons to certain states and force them to not occupy like- states (this is also often calledPauli repulsion). In a material such as Fe, the magnetic moments align parallel to each other because this is the state which minimises the exchange interaction. All other arrangements of the electrons require more energy than that of a parallel arrangement of magnetic moments.
For a system of localised magnetic moments which interact via the exchange interaction, the exchangeenergy over all electronsiand j of a many-electron system is defined as
Ex=− X ij Jij Si·Sj (2.19)
whereJij is the exchange interaction between spinsSi and Sj.22
Direct Exchange
Electrons which are close enough to have a large overlap of their electronic wavefunc- tions tend to interact via Direct exchange interactions. This is a strong, short range coupling which diminishes as the ionic separation increases. When the atoms are close together, the electrons occupy the intermediary space because this is where the Coulomb interaction is minimal. This spatial overlap forces the electrons to have opposing spins and as a result, direct exchange is responsible for causing antiferromagnetic ordering in materials.
Indirect Exchange
Over large inter-atomic separations, the Indirect exchange interaction contributes to the electronic coupling in a material. The more formal name of this interaction is the
Ruderman-Kittel-Kasuya-Yoshida (RKKY) interaction. This type of exchange interac- tion is often mediated by the conduction electrons of metals due to their long range,
3
de-localised nature.23
2.6.2 Spin-Orbit Coupling
An electron’s total magnetic moment is dependent not only on the spin angular momen- tum of the electron but also its orbital angular momentum. The interaction between the spin and orbital angular momenta cause shifts and splitting of the electron energy levels. This contributes an additional term to the Hamiltonian, HSO, defined as
HSO= e~2 2mec2r
dV(r)
dr S·L (2.20)
where e is the charge of the electron and S and L are the spin and orbital angular momenta of the electron, respectively. This phenomenon is calledSpin-Orbit Coupling. For materials which exhibit large orbital moments (such as 4f systems), the spin-orbit coupling will be strong and as a result, will need to be considered when attempting to accurately describe the band structure. For materials where the orbital moment is small (such as in most 3dalloys4), the spin-orbit coupling is negligible.23
2.6.3 Spin-split Bands
Very many materials have non-integer magnetic moments per atom. This value is not possible within the framework that magnetism originates solely from the unpaired elec- trons in a system. This type of magnetism is due to the spin-splitting of the bands of a material. Sometimes in a material, due to the interaction with the molecular potential, it is energetically favourable for the spin-up and spin-down bands to be unequal in terms of their populations. An example DOS has been sketched in figure 2.7 which shows the spin-split spin up and down bands of a ferromagnet. The energy change ∆E due to spin-down electrons moving into the spin-up band is
∆E= 1
2n(EF)(δE)
2(1−U
n(EF)) (2.21)
4
This is not the case for all 3d compounds. For example, in chapter 8, the calculations find a significant orbital moment in Ca3Co2O6.
D(E)
EF E
δE
Figure 2.7: Sketch of the DOS of a ferromagnetic material. The spin splitting of the spin- up (↑) and spin-down (↓) is what gives the compound its net, magnetic spin moment.
where n(EF) is the number of electrons atEF, δE is the increase in energy as a result
of moving electrons into the spin up band, and Un(EF) is the Stoner Parameter. For
materials which satisfy
Un(EF)≥1, (2.22)
Chapter 3
Theoretical Techniques
In this chapter the main theoretical methods for calculating the electronic structure of materials will be discussed. The work in this thesis will see applications of several different types of theoretical technique. These are the Hartree-Fock method, Density Functional Theory (DFT) through the use of the Full Potential Linearised Augmented Planewave (FP-LAPW), the Spin-Polarised Fully-Relativistic Korringa-Kohn-Rostoker
(SPR-KKR) methods, and theMolecular Orbital Wavefunction Method.