CHAPTER 2 MAGNETISM. 2.1 Magnetic materials

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1 CHAPTER 2 MAGNETISM Magnetism plays a crucial role in the development of memories for mass storage, and in sensors to name a few. Spintronics is an integration of the magnetic material with semiconductor technology, to realize nano sized devices with better features like non-volatility, scaling, etc. We begin with a discussion on magnetism and the classification of various materials based on their magnetic properties. In section 2.2, we progress from the classical picture of magnetism to a quantum description, which better describes the origins of magnetism. 2.1 Magnetic materials The magnetic induction ( B) is the response of a material to the applied magnetic field ( H). The relationship between B and H depends on the material. The magnetization ( M) is defined as the magnetic moment per unit volume such that B = µ 0 ( H + M), (2.1) where, µ 0 is the permeability of free space. Apart from these, susceptibility ( χ) which indicates the response of a material to a magnetic field and the permeability ( µ), are defined from the relations M = χ. H and B = µ. H. (2.2) A material s magnetic property can be understood from the M or B versus H curves, called the magnetization curves. Based on the behaviour of the materials for an applied magnetic field H, materials can be classified as:

2 Diamagnetic Paramagnetic Antiferromagnetic Ferromagnetic Ferrimagnetic From Figure 2.1a, we observe that the net magnetization of diamagnetic, paramagnetic and antiferromagnetic materials is very low even at a very high applied magnetic field and the relationship between M and H is linear. But, in ferromagnetic and ferrimagnetic materials, even at a low applied magnetic field, the value of magnetization is very high. Moreover, the M H curve is non-linear, and the magnetization saturates at higher values of H as shown in Figure 2.1b. M emu/cm 3 M emu/cm Ferromagnetic Paramagnetic or or Ferrimagnetic Antiferromagnetic H A/m Diamagnetic H A/m -0.5 (a) (b) Figure 2.1: M-H curves for (a) diamagnetic, paramagnetic and antiferromagnetic (b) Ferromagnetic and Ferrimagnetic materials [N. A. Spaldin 2003] Diamagnetism Diamagnetism is a very weak property that exists in all materials. Diamagnetic substances are composed of atoms that have all the orbitals filled and hence the net magnetic 14

3 moment is zero in the absence of an applied field. However, when exposed to a field, a negative magnetization is produced, and hence the susceptibility is negative (χ < 0) as shown in Figure 2.1a. Paramagnetism Paramagnetic materials have a net magnetic moment due to the presence of unpaired electrons in the outermost orbitals. When a magnetic field is applied, there is a partial alignment of the atomic magnetic moments in the direction of the field. However, the individual magnetic moments do not interact and the net magnetic moment is zero when the field is removed. This results in a net positive magnetization and positive susceptibility as shown in Figure 2.1a. The ordering of magnetic moments in a paramagnetic material in the absence of H is shown in Figure 2.2a. (a) Paramagnet (b) Antiferromagnet (c) Ferromagnet (d) Ferrimagnet Figure 2.2: Magnetic ordering in different materials 15

4 Antiferromagnetism If, in a magnetic structure, the moments of two nearby sub-lattices are exactly equal but opposite, the net moment of the material is zero. This type of magnetic ordering is called antiferromagnetism (AFM), shown in Figure 2.2b. In some materials, the atomic magnetic moments exhibit very strong interactions, which are produced by electronic exchange forces that have quantum origins. This exchange force results in the parallel or antiparallel alignment of atomic moments. The exchange field is very large of the order of 1000 Tesla. In antiferromagnets, the exchange force between the neighbouring moments is antiparallel, unlike in ferromagnets, in which exchange force is parallel. Above a certain temperature, called the Néel temperature, thermal energy overcomes the exchange energy and randomizes the moments. The exchange coupling constant depends on the ratio of inter-atomic distance to the atomic diameter as shown in Figure 2.3. We observe that, for smaller values of this ratio, the exchange constant is negative for AFM materials like Cr and Mn indicating an antiparallel coupling of moments. Ferromagnetism Ferromagnetic materials exhibit parallel alignment of moments as shown in Figure 2.2c, which results in a large net magnetization at room temperature, even in the absence of a magnetic field. The transition elements Fe, Ni, Co and many of their alloys are typical examples of ferromagnetic (FM) materials. From Figure 2.3, we observe that the exchange coupling is positive for FM materials causing parallel alignment of adjacent moments. Moreover, FM materials retain the magnetization even after the applied field is removed, and this behavior is called hysteresis. 16

5 Ferro + Fe Co Ni Exchange Anti-ferro - Mn Cr Inter atomic distance Atomic diameter Figure 2.3: Exchange coupling in transition metals [N. A. Spaldin 2003] Ferrimagnetism More complex forms of magnetic ordering as shown in Figure 2.2d can occur in ferrimagnets where, the magnetic moments of two nearby sub-lattices are opposite but unequal, unlike in AFM. This results in a net magnetic moment at room temperature. Even though ferrimagnetism exhibits similar characteristics as ferromagnetism, the ferrimagnetic materials are not good conductors like FM materials. All ferromagnetic and ferrimagnetic materials exhibit the phenomenon of hysteresis as shown in the four quadrant hysteresis loop in Figure 2.4. The performance of any magnetic material can be defined with the help of this hysteresis loop. M s is the saturation magnetization and is a measure of how strongly the material can be magnetized. M r is the remanent magnetization which is the residual, permanent magnetization left after the removal of the applied field. In order to demagnetize the specimen from its remanent state, a reverse field H c, the coercive field is required to reduce the residual magnetization to zero. Higher the value of H c, larger is the retaining capacity of the material. This is the reason that, the PL in an MTJ is with a larger coercivity, so that it requires a larger field to change its magnetization than the FL. Even ferromagnetic materials become paramagnetic, with random orientation of moments, above a certain temperature called the Curie temperature. 17

6 M M s M r - H c H c H - M r Figure 2.4: Hysteresis loop for a ferromagnetic material 2.2 Origins of magnetism Just as a current circulating in a coil gives rise to Ampere s magnetic field in bulk materials, the electrons circulating in atomic orbitals give rise to a magnetic field. The three major sources of a magnetic field, considered to be the origins of magnetism in a free atom are: spin angular momentum, due to the precession of the individual electron, orbital angular momenta due to circulation of electrons around the nucleus, and change in orbital angular momentum due to an external field Quantum numbers The differing magnetic properties in materials are explained with the help of the four quantum numbers. 18

7 Principal quantum number (n) The principal quantum number n, determines the energy E n of a shell, but is not involved directly in contributing to the magnetic property. The shells for n = 1, 2, 3, 4... are referred to as K, L, M, N... As the value of n decreases, the energy of the atom decreases. There are n 2 orbitals in each shell, and each orbital can have a maximum of 2 electrons. Orbital quantum number (l): The orbital angular momentum of the electron due to the electron orbiting around the nucleus as shown in Figure 2.5a, is decided by the orbital quantum number l. The orbital angular momentum of a single electron is given as L = l(l + 1). l can take values ranging from 0 to (n 1) and 0, 1, 2, 3... refers to the s, p, d, f,... atomic orbitals. The magnitude of orbital angular momentum L of an electron in p orbital with l = 1 is L = 2 and for an electron in the d orbital with l = 2, is L = 5. Even though the value of principal quantum number n does not have a direct impact on the magnetic moment of an atom, the value of n decides the presence of s, p, d, f,... orbitals, and hence the magnetic moment. Magnetic quantum number (m l ) With the application of a field, the orientation of the orbital angular momentum changes. These changes are also quantized and the quantized m l values range from l to +l. The component of the angular momentum along the direction of field is given by Component of orbital angular momentum in direction of H app = m l. (2.3) For a d orbital with l = 2, m l takes values from -2 to +2, which indicates that the d orbital can have 5 orientations with the application of a magnetic field. 19

8 m l m s (a) (b) Figure 2.5: Origin of magnetic moment (a) Orbital moment (b) Spin moment Spin quantum number (s) Apart from orbiting around the nucleus, electrons spin about their own axis as shown in Figure 2.5b, which is the major contributor to the magnetic moment of an atom. The spin quantum number s, always has a value of 1. Analogous to orbital angular momentum, the 2 magnitude of the spin angular momentum S is defined as S = s(s + 1) = 3. (2.4) 2 Magnetic spin quantum number (m s ) Similar to m l for an orbiting electron in a field, the change in the spin angular momentum to an external field is also quantized as m s. It has only two values ± 1 2. Component of spin angular momentum in direction ofh app = m s = ± 2 (2.5) Since, m s is smaller than S, the spin angular momentum vector cannot point along the applied field, but precesses around the field. From Figure 2.6, we see that, Fe has 4 unpaired electrons, Co has three unpaired electrons and Ni has 2 unpaired electrons, in the 3d orbital. Even though Mn has 5 unpaired electrons, 20

9 Atom Mn Unpaired electrons 5 Fe 4 Co 3 Ni 2 Cu 0 Figure 2.6: Spin occupancy in 3d orbital the exchange coupling between adjacent atoms is negative making it antiferromagnetic, as shown in Figure Magnetic moments By convention, the magnetic field ( H) originating from the magnet, is taken to have fields of force from the positive of dipole (north pole) to the negative of dipole (south pole) as shown in Figure 2.7a. In the 1820s, it was discovered that an electric current can generate a magnetic field. The integral form of Ampere s statement, H.d l = I enclosed, (2.6) helps us find the magnetic field, which depends on the shape and the amount of current flowing through the circuit. A current carrying loop of area A, carrying a current I as shown in Figure 2.7b, is equivalent to a magnet with a moment m = I A.ˆn, (2.7) 21

10 H N H S I (a) (b) Figure 2.7: Magnetic fields of lines (a) from North pole to South pole in a dipole, (b) upward for a current in the anti-clock direction in a coil given by Ampere s right hand rule where ˆn is the unit vector normal to the plane of the loop. The energy of the magnetic dipole moment in an external field is E = µ 0 m. H = µ 0 mh cos θ. (2.8) An electron orbiting around the nucleus of an atom can be treated as a current carrying loop. We find that, if the electron is at a distance a from the nucleus and is moving with a velocity v, then the current due to this electron of charge e is given by I = ev 2πa. (2.9) The negative sign is due to the fact that the direction of electron flow is opposite to the conventional current direction. The magnetic dipole moment due to the orbiting electron from Equations 2.7 and 2.9 is m = I A = e v a. (2.10) 2 We know that the orbiting electron with a mass m e has an angular momentum L = m e v a. From Equation 2.3, we learn that the orbital angular momentum can take values m l. Equat- 22

11 ing these two expressions for angular momentum, we get, v = m l m e a. (2.11) Hence, the magnetic moment of an electron from Equations 2.10 and 2.11 is m = e 2m e m l = µ B m l, (2.12) where µ B = e /(2m e ) = J/T is the Bohr magneton. From Equation 2.12, we observe that due to the negative charge of electron, the dipole moment points in the opposite direction to that of the angular momentum. The energy of the magnetic moment is obtained from Equation 2.8 as E = µ 0 m. H = µ 0 e 2m e m l H = µ 0 µ B m l H. (2.13) We thus understand that the energy of the particle changes with the change in the applied field and the angular momentum of the orbital. This phenomenon is called the normal Zeeman effect. We also know that the spin of the electron precesses about its own axis at a precession frequency where γ, the gyromagnetic ratio of the electron, is γ = ω = γh, (2.14) magnetic moment angular momentum = e g = µ B g, (2.15) 2m e and, g is the Landé g-factor whose value for a free electron is approximately 2. Hence γ has a magnitude of s 1 T 1. Spin is unrelated to orbital moment and fully survives in a crystal field. Each spin corresponds to a moment m = µ B and is denoted by (spin-up) 23

12 and (spin-down). By sign convention, spin-up electrons in a thin film have a higher energy than spin-down electrons, because the moments of the latter are parallel to the quantization direction determined by the internal magnetic field, as described further in Section 3.1. The spontaneous magnetization (M) of a material is defined in terms of the density of states of spin-up (N (E)) and spin-down (N (E)) electrons and the Fermi function f(e) as M = µ B [N (E) N (E)]f(E)dE. (2.16) These topics are dealt in further detail in the next chapter. 0 24