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I: Basic Theory of MagnetismSpin: Origin of MagnetismIntroductionMagnetic 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 lowtemperature 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 ц_{в} the Bohr magneton defined by Physics of Magnetic Thin Films: Theory and Simulation Hung T. Diep Copyright © 2021 Jenny Stanford Publishing Pte. Ltd. ISBN 9789814877428 (Hardcover), 9781003121107 (eBook) www.jennystanford.com The negative sign of /x_{s} in (1.1) is due to the negative charge of the electron (—e). We define a = 2S. The components of a are the wellknown Pauli matrices
We see that a_{z} is diagonal. The eigenvalues of S_{z} are thus 1/2 and 1/2. Using the above matrices, one can show that they obey the following commutation relations
where a^{±} = a_{x} ± ia_{y}. 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 socalled 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 /x_{N} = 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 /x_{s}. 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 freeelectron 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 FreeElectron GasUnder 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 /(£_{T}4) is the FermiDirac distribution function given by Eq. (A.37). In the case where H is small, we can use the approximation p(F_{t}) ~ p[E_{t}) ~ p[E) because p{E) is a smooth function of E and p_{B}H is small. We can also replace /(E_{T}) 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 p_{t}(E_{F}) = 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 lowГ expansion shown in Problem 5 in Section 1.7. At high temperatures, у is proportional to 1 /Т (Curie's law) as seen in that exercise. Experimental data for normal metals confirm Eq. (119), 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 freeelectron gas model. Paramagnetism of a System of Free AtomsWe 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 singleatom 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 C_{v} per atom versus k_{B}T for дц_{в}В = 1.5. The peak separates two regions. In the lowГ region the number of moments parallel to В is larger than that antiparallel to B, so that Figure 1.1 Specific heat C_{v} as a function of k_{B}T forдц_{в}В = 1.5. there is a nonzero magnetic moment. In the highГ region, they are equal, there is thus no induced magnetic moment. For a higher B, the lowГ region is extended so that the peak of C_{v} 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 ManyElectron AtomsWe 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 N_{e} 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: A_{x} = yB/2, A_{y} = xB/2, A_{z} = 0. From Eq. (1.35), one obtains where Ho is the Hamiltonian in zero field, namely L_{z} and S_{z} are the z components of the total orbital moment L and the total spin S defined by The groundstate energy is where E_{0}[B = 0) is the energy in zero field. For a system of N free identical atoms, the total energy is equal to NE_{0}(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 SolidsExchange Interaction: Origin of MagnetismIn 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 HartreeFock 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 (n_{b} mi, a, ■ ■ ■ , n_{4}, m_{4}, ct_{4}). If ni = n_{2} = лз = n_{4}, 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 = m_{2} = m_{3} = m_{4}, 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 n_{2} = «4, the Coulomb term is given by
If ni = n_{4} and n_{2} = Л3 (by consequence, =
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: (faK_{t}1b_{nii}b+_{2}ib„_{21}$) = {фК^Ь„_{л}Ь^Ь„_{А}ф) = 0. We define next the following spin operators As we suppose one electron per site, we have
The righthand 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, n_{2}). We use now the notation _{n2}j instead of  _{Пг }where (n_{1(} n_{2}) indicates the pair (nin_{2}) 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„_{x}„_{2} 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. J_{ni}„_{2} 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 MaterialsIn magnetic materials, depending on the nature of the spins and the interaction between them, one can use several spin models as described below. Heisenberg ModelThe 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 onehalf 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:
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 socalled zeropoint quantum fluctuations make the spins contracted from its full length. The zeropoint 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 ModelsBesides 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 twocomponent 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 "easyplane 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 "KosterlitzThouless" transition discovered in the 1970s [10, 58,191, 380]. A discussion on this phase transition is given in Chapter 5. 3. Potts model: The qstate Potts model is defined by
where
where Q_{n} 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 qstate 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]. ConclusionThis 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 socalled Pauli paramagnetism where the susceptibility is a constant at the firstorder 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. ProblemsProblem 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:
Numerical application: Calculate AE and Дг for the following fields ц_{в}Н = 0.5,1, and 2 Tesla. Problem 3. FermiDirac distribution for freeelectron gas: Electrons are fermions which obey Pauli's exclusion principle. Microscopic states follow the FermiDirac statistics. The FermiDirac 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 М^{П})(Е)£__{Л} is the nth derivative of h[E) at E = д. Problem 5. Pauli paramagnetism: Calculate the susceptibility of a threedimensional 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.
the orbital plane, gives rise to the following variation of its magnetic moment Am = — (m_{e}: electron mass). Comment on the negative sign.
Numerical application: p = 2220 kg/m^{3}, 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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