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# I: Basic Theory of Magnetism

## Spin: Origin of Magnetism

### Introduction

Magnetic properties of a material are due to spins of atoms, molecules or nucleus which compose the material. The spins can be independent of each other or they interact with each other leading to various collective phenomena. There are many spin models and various kinds of interaction. This chapter treats some properties of systems of independent spins and explains the origin of the exchange interaction in materials. Exchange interactions, or magnetic interactions, between spins of neighboring atoms can give rise to a magnetic ordering which is responsible for the principal low-temperature properties of the system. We will consider various spin models and several kinds of magnetic interaction and their consequences in the following chapters.

We consider an electron of charge —e (e > 0), of mass m and of spin S. The magnetic moment associated with the spin is written as

where g = 2.0023 is the Lande factor and цв the Bohr magneton defined by

Physics of Magnetic Thin Films: Theory and Simulation Hung T. Diep

Copyright © 2021 Jenny Stanford Publishing Pte. Ltd.

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The negative sign of /xs in (1.1) is due to the negative charge of the electron (—e).

We define a = 2S. The components of a are the well-known Pauli matrices

We see that az is diagonal. The eigenvalues of Sz are thus -1/2 and 1/2. Using the above matrices, one can show that they obey the following commutation relations

where a± = ax ± iay.

In addition to the magnetic moment defined from its spin in (1.1), the electron has also an orbital moment due to its motion

where r and v are the position and the velocity of the electron. The kinetic orbital moment 1 is given by

This moment yields, as seen below, the so-called diamagnetic phenomenon observed under the application of a magnetic field.

The nuclear magnetic moment д/ associated to the nuclear spin / is defined by a relation similar to (1.1) with /х в replaced by /xN = where M is the mass of the nucleus (proton or neutron) which is equal to 1836m. The heavy mass of a nucleus makes the effect of ixi veiy small with respect to that of /xs. Therefore, when these two kinds of moment exist in the system, the effect of the nuclear moment is often neglected.

We consider now a system of independent spins with random orientations ±1/2, such as a free-electron gas. Since the spins are independent, their states are affected only by a variation of an external parameter such as the temperature and/or an applied magnetic field. In the absence of an applied magnetic field, the random distribution of spin magnetic moments results in a zero total moment. Under an applied magnetic field B, a number of spins will turn themselves into the field direction giving rise to a nonzero total moment M. The susceptibility / defined as x = jff is then positive. The system is paramagnetic.

We consider the case of a system of atoms where each atom has initially zero magnetic moment. Under an applied field, the electrons of each atom may modify their states so as to create an induced moment to resist the field effect: The induced moment M is in the direction opposite to that of the field. This gives rise to a negative susceptibility. The system is diamagnetic.

We study the paramagnetism and the diamagnetism in the following.

### Paramagnetism of a Free-Electron Gas

Under an applied magnetic field H, the energy of an electron of spin parallel to H decreases by an amount мвЯ, and that of an electron of spin antiparallel to H increases by the same amount. This effect is called "Zeeman effect." We write

For a system of free electrons, the total magnetic moment induced by the field is

where and Nj are, respectively, the numbers of t and i spins.

Using the density of states for each type of spin given by Eq. (A.44) we have

where /(£T4) is the Fermi-Dirac distribution function given by Eq. (A.37). In the case where H is small, we can use the approximation p(Ft) ~ p[Et) ~ p[E) because p{E) is a smooth function of E and pBH is small. We can also replace /(ET) and /(£t) by their first- order Taylor expansion around E:

Replacing these into Eq. (1.14) we have

At low temperatures, ^ is not zero only near Ef (see Problem 3 in Section 1.7). We have therefore

The susceptibility is thus

where pt(EF) = 2p(Ef)is the "total” density of states with the spin degeneracy [Eq. (A.45)].

Equation (1.19) is known as "Pauli paramagnetism” susceptibility which is independent of T. To calculate M at higher orders of T, we can use a low-Г expansion shown in Problem 5 in Section 1.7. At high temperatures, у is proportional to 1 (Curie's law) as seen in that exercise. Experimental data for normal metals confirm Eq. (1-19), but strong variations of у with temperature have been observed in some transition metals (Pd, Ti,...) [186]. To explain these variations, it is necessary to take into account various interactions neglected in the free-electron gas model.

### Paramagnetism of a System of Free Atoms

We consider a system of N independent atoms each of which has a total moment J = S + L. The maximum modulus J of J is the sum of the amplitudes S and L: J = S + L. The magnetic moment of the /-th atom is

where the Lande factor is given by

Under an applied magnetic field В along the z direction, the Zeeman energy is

where J? = J, J — 1 ,■■■,—) (2J +1 values) and E, is the energy of the /-th atom. The partition function of N atoms is written as the product of single-atom partition function z [see Eq. (A.9)]

where

The last equality was obtained by using the formula for the geometric series of 2] +1 terms, of ratio e_/3a"sB. For J = 1/2 one has

The free energy F is (see Appendix A)

The energy of the microscopic state / is

where ц. is the magnetic moment of atom i and Mi the total magnetic moment, in the state /. The average magnetic moment is calculated by

Replacing Eq. (1.25) in Eq. (1.26) to obtain F, then using F in the last equality, one obtains for / = 1/2

The magnetization m, defined as magnetic moment per unit volume, is given by

At high T, tanh (\$£) -* one has The susceptibility is thus

This is the Curie’s law, similar to that of a gas of free electrons at high T (see Problem 5 in Section 1.7).

At low T, one has tanh (1, so that the magnetization is

maximum, i.e., m = in the case / =1/2. This result is different from that of Pauli paramagnetism of free electrons at low T given by Eq. (1.19).

The average energy of the system is (see Appendix A)

The paramagnetic heat capacity for ] = 1/2 is thus

Figure 1.1 shows Cv per atom versus kBT for дцвВ = 1.5. The peak separates two regions. In the low-Г region the number of moments parallel to В is larger than that antiparallel to B, so that

Figure 1.1 Specific heat Cv as a function of kBT forдцвВ = 1.5.

there is a non-zero magnetic moment. In the high-Г region, they are equal, there is thus no induced magnetic moment. For a higher B, the low-Г region is extended so that the peak of Cv moves to a higher temperature. Beyond the peak position, the magnetic moment induced by the magnetic field at low T is destroyed by the temperature.

### Diamagnetism of Many-Electron Atoms

We consider the case where the valence orbital of an atom is completely occupied, i.e., the atom has no permanent magnetic moment from its electrons. As seen below, the applied magnetic field will result in a diamagnetic effect. However, when the valence orbital is only partially occupied, the applied magnetic field will give rise to both paramagnetism and diamagnetism. The paramagnetism is studied in the previous section. In the following, we study the diamagnetism.

We consider here an atom which has Ne electrons in its valence orbital. We suppose that the valence orbital is full: the orbital moment is L = 0 and the total spin is S = 0. The Hamiltonian of the valence electrons under the applied magnetic field В is written

as

where A(r,) is the vector potential associated with B, namely rotA(r,) = B, and U (r,) represents the interaction between electron / with the remaining electrons of the orbital. For В || Oz, one can choose A(r,) as follows: Ax = -yB/2, Ay = xB/2, Az = 0. From Eq. (1.35), one obtains

where Ho is the Hamiltonian in zero field, namely

Lz and Sz are the z components of the total orbital moment L and the total spin S defined by

The ground-state energy is

where E0[B = 0) is the energy in zero field. For a system of N free identical atoms, the total energy is equal to NE0(B). At T = 0, one has F = E- TS=E. The magnetization is

The susceptibility is thus negative (diamagnetic) and given by

### Magnetic Interactions in Solids

#### Exchange Interaction: Origin of Magnetism

In this paragraph, we show that the magnetic interaction between neighboring atoms leads to the Heisenberg spin model. We suppose that the reader has some knowledge of the Hartree-Fock approximation and is familiar with the second quantization method. If not, he/she can skip the following demonstration and go directly to Eq. (1.57).

We consider the Coulomb interaction between two electrons written in the second quantization (see Appendix B)

where j/„ and are field operators defined by

where y>nm(r) and m at the site n of the crystal, b and b+ fermion annihilation and creation operators. The wave functions nm constitute an orthogonal set. Equation (1.43) becomes

where the sum runs over (nb mi, a, ■ ■ ■ , n4, m4, ct4).

If ni = n2 = лз = n4, the interactions are between electrons of the same site. Equation (1.46) is the origin of Hund’s atomic empirical rules. In addition, if mi = m2 = m3 = m4, this equation is the Coulomb term in the Hubbard Hamiltonian which will be shown below.

For simplicity, we suppose one electron per site and one orbital per electron in the following.

If rii = П3 and n2 = «4, the Coulomb term is given by

If ni = n4 and n2 = Л3 (by consequence, = 2), the exchange term becomes

where

where the last two terms have been added. Note that these terms do not affect the result because their averages are zero in the diagonal representation: (faKt1bniib+2ib21|\$) = {фК^Ь„лЬ^Ь„Аф) = 0.

We define next the following spin operators

As we suppose one electron per site, we have

The right-hand side of (1.50) becomes Using

we rewrite (1.48) as

where we added a factor | to remove the double counting of each pair (ni, n2). We use now the notation n2j instead of | Пг where (n1( n2) indicates the pair (nin2) counted only once. Finally, one has

The first term does not depend on spins. The second term is the Heisenberg model which shall be used later throughout this book.

Hamiltonian (1.57) is thus the origin of ferromagnetism observed in ferromagnetic materials. If the interaction is a Coulomb interaction as we suppose here, then J„x2 is positive as shown below. We write

where the wave functions y>„(r) have been supposed to be orthogonal. They are Wannier wave functions constructed from linear combinations of Bloch wave functions. Making use of

we obtain where

The above two integrals are identical because the indices and variables are dummy. Jni2 is thus positive. From (1.57), we see that if S()I and S„2 are parallel, then the energy is lowest. This state of spin ordering is called "ferromagnetic."

#### Spin Models: Magnetic Materials

In magnetic materials, depending on the nature of the spins and the interaction between them, one can use several spin models as described below.

#### Heisenberg Model

The Heisenberg model for the interaction between two spins localized at the lattice sites / and j is given by

where /,■; is the exchange integral resulting from the Coulomb interaction between two electrons of spins S, and S; , localized at r, and Гр We have demonstrated (1.61) in the previous section [see Eq. (1.57)]. In general, the value of Jjj depends on the distance between the spins and on the orientation of r, - r, with respect to the crystalline axes.

In the quantum model, S, is a quantum spin whose components obey the spin commutation relations. For example, in the case of spin one-half its components are the Pauli matrices (1.3)—(1.5). In the classical model, S, is considered as a vector.

We consider hereafter the simplest case where the exchange interaction is limited between nearest neighbors and this interaction is identical and equal to J for all pairs of nearest neighbors. In this case, the Hamiltonian reads

where the sum is performed over all pairs of nearest neighbors. We see that if/ > 0, V. is minimum when all spins are parallel. This spin configuration corresponds to the ferromagnetic ground state.

In the case where / <, 0, we have to distinguish classical and quantum spin models:

• (i) In the classical model, the spins are vectors. Except for lattices composed of equilateral triangular faces in their elementary cell such as the two-dimensional triangular lattice, the face- centered cubic lattice and the hexagonal-close-pack lattice, H is minimum in the other lattices when all nearest neighbors are antiparallel throughout the crystal. This spin configuration is called "classical antiferromagnetic ground state" or "Neel state”.
• (ii) In the quantum model, the spins can be decomposed into spin operators [see (1.51)—(1.53)]: (Sf, S* = Sf + iSf, S~ = S? — iSf). These spin operators obey the spin commutation relations

It is known that the Neel state is not the quantum ground state for antiferromagnets. As will be shown in Chapter 3, at zero temperature the so-called zero-point quantum fluctuations make the spins contracted from its full length. The zero-point spin contraction is of the order of a few percents depending on the lattice structure. The real quantum ground state of antiferromagnets is not known, though we know that it is not far from the classical Neel state, except in low dimensions where strong quantum fluctuations can destroy the Neel order. Note that in the case of ferromagnets, the perfect parallel spin configuration is the real ground state in both classical and quantum spin models.

When the geometry of the lattice does not allow us to satisfy all the interactions between a spin with its neighbors, the system is said "frustrated." This happens when the elementary lattice cell is composed of equilateral triangles such as in the triangular lattice. There are many striking effects due to the frustration. The reader is referred to the book "Frustrated Spin Systems" for reviews [85]. We will outline some remarkable properties of frustrated systems in Section 5.7.3.

#### Ising, XY, Potts Models

Besides the Heisenberg model, there are three other spin models which are very popular in magnetism and in statistical physics:

1. Ising model: The Ising model is defined by

where

2. XY model: The XY model is defined by the interaction between two-component spins. It is sometimes called “model of plane rotators." The Hamiltonian is given by

The XY model is used to study spin systems with a strong planar anisotropy called "easy-plane anisotropy" which exists in some magnetic materials. It is also used in statistical physics. A very interesting ferromagnetic XY model in two dimensions has been extensively studied because it gives rise to a very special phase transition known as the "Kosterlitz-Thouless" transition discovered in the 1970s [10, 58,191, 380]. A discussion on this phase transition is given in Chapter 5.

3. Potts model: The q-state Potts model is defined by

where q values, cr, = 1, 2, • • • , q for example, and Sa.ia. denotes the Kronecker symbol. The sum is performed over pairs of nearest neighbors. If interaction/,y = J > 0 for nearest neighbors (/, ;'), then in the ground state there is only one value of q: It is ferromagnetic. Note that if q = 2, the model is equivalent to the Ising model. We define the Potts order parameter <2 by

where Qn is the spatial average defined by

where n = 1,..., q, the sum runs over all lattice sites, and N is the total site number. From this definition we see that the ground state containing only one kind of spin has Q = 1, while in the disordered state q kinds of spin are equally present in the system, namely Qi = Qq = l/<7- so that <2 = 0.

The q-state Potts model is used to study systems of interacting particles where each particle has q individual states. Exact methods to treat the Potts models in two dimensions are shown in a book by Baxter [25].

### Conclusion

This chapter introduces the spin and shows some principal behaviors of a system of independent spins under the application of a magnetic field. We have examined three cases. The first case concerns free electrons at low temperatures. We have obtained the so-called Pauli paramagnetism where the susceptibility is a constant at the first-order approximation. The second case is the system of free atoms where each atom has a permanent magnetic moment. Under the application of a magnetic field, the susceptibility is positive (paramagnetic) and proportional to 1/T (Curie’s law). The third case is a diamagnetic case: The reaction of the electrons in an atom to an applied magnetic field gives rise to a negative susceptibility. This phenomenon is called "atomic diamagnetism."

We have also demonstrated the Heisenberg model describing the magnetic interaction between two spins which leads to a magnetic ordering in solids. We have also presented several spin models such as the Ising, XY and Potts models. These models are used in the following chapters to study theoretically behaviors of bulk materials and thin films.

### Problems

Problem 1. Orbital and spin moments of an electron:

Using the theory of angular momentum, calculate the orbital and spin moments of an electron. Determine the total magnetic moment.

Problem 2. Zeeman effect:

• (a) Calculate the magnetic moment per atom for Fe, provided the saturated magnetization under an applied magnetic field equal to 1.7 x 106 A/m, the mass density of Fe p = 7970 kg/m3 and the atomic mass of Fe M = 56.
• (b) Calculate AE the separation of the energy levels due to the Zeeman effect on the atomic level corresponding to the wavelength A = 643.8 nm of a cadmium atom. Calculate the variation of frequency Ди of the initial level.

Numerical application: Calculate AE and Дг for the following fields цвН = 0.5,1, and 2 Tesla.

Problem 3. Fermi-Dirac distribution for free-electron gas:

Electrons are fermions which obey Pauli's exclusion principle. Microscopic states follow the Fermi-Dirac statistics.

The Fermi-Dirac distribution is given by (see Appendix A)

where д is the chemical potential, fi = кв the Boltzmann constant and T the temperature. The function f[E, T, д) is the number of electrons of the microscopic state of energy E at temperature T.

Give the properties of /(£, T, д) at T = 0. Plot f[E, T, д) as a function of E for an arbitrary д(> 0), at T = 0 and at low T.

Problem 4. Sommerfeld’s expansion :

Consider the function

where h(E) is a function differentiable at any order with respect to E.

Show that h[E) can be expanded in powers of T at low Г as follows:

where МП)(Е)|£_Л is the n-th derivative of h[E) at E = д. Problem 5. Pauli paramagnetism:

Calculate the susceptibility of a three-dimensional electron gas in an applied magnetic field B, at low and high temperatures. One supposes that В is small.

Problem 6. Paramagnetism of free atoms for arbitrary J:

Consider a gas of N atoms of moment J in a volume V. Show that the average of the total magnetic moment per volume unit of the gas is

where By (x) is the Brillouin function given by

Show that at high temperature one has

Find the limit of m at T = 0.

Problem 7. Langevin's theory of diamagnetism:

Consider an electron in an atom. In the theory of diamagnetism by Langevin, the motion of the electron around the nucleus is equivalent to the motion of a magnetic moment m generated by a current / which circulates in a closed loop of surface A.

• (a) Write a relation between i, m and A.
• (b) Show that the magnetic moment of the electron is written as m = evr/2 where e is the charge of the electron, v its velocity and r its orbital radius.
• (c) Show that an applied magnetic field H, perpendicular to

the orbital plane, gives rise to the following variation of its magnetic moment Am = — (me: electron mass).

Comment on the negative sign.

• (d) What will be the result if H makes an angle в with the surface normal ?
• (e) Calculate the susceptibility of a material of mass density p made of atoms of Z electrons, of mass M.

Numerical application: p = 2220 kg/m3, e = 1.6 x 10-19 C, Z = 6,r = 0.7 x 10-10 m.

Problem 8. Langevin's theory of paramagnetism:

Consider an atom of permanent magnetic moment m (atom having an odd number of electrons). Using the Maxwell- Boltzmann statistics, show that the magnetic moment resulting from the application of a magnetic field H on a material of N atoms per volume unit, in an arbitrary direction is given by

where £(x) = coth(x) — £ (Langevin function). Calculate the susceptibility in the case of a weak field.

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