Module 2A: Theory of paramagnetism Debanjan Bhowmik Department of Electrical Engineering Indian Institute of Technology Delhi Abstract In our first module, we discussed the problems associated with the current proces- sor and memory technology in the industry, which are transistor based, and briefly mentioned why magnetic devices can replace transistors, non-volatility of magnets be- ing cited as the main motivation. In the second module, we will discuss the theory of magnetism from the basics, keeping in mind the metallic spintronics systems relevant for technology. The metal used to store information in these systems is essentially ferromagnetic, so we need to understand ferromagnetism in this course but to do that first we need to understand paramagnetism, which we do in this part of the module (2A). 1 1 Types of magnetism We start with a familiar equation discussed in Electromagnetics courses: B~ = µ0(H~ + M~ ) (1) ~ ~ M = {mH (2) As a result, ~ ~ ~ ~ B = µ0(H + {mH) = µ0µrH (3) Parameters: µr= relative permeability, {m= magnetic susceptibility Fields: H~ = applied magnetic field, B~ = magnetic flux density and M~ = magnetization Difference between H~ and B~ is that H~ is what is applied, and B~ is the actual field which is the combination of H~ and the response of the material to H~ , which is given ~ ~ by the magnetization M = {mH. For example, in the case of a solenoid coil with an iron bar inserted inside it, H~ = NI (N= number of turns per unit length, I= current), but magnitude of B~ is much higher than magnitude of H~ because of the large magnetic ~ moment of iron, in response to the H, or the high {m of iron. The purpose of this set of lectures, or study of magnetism in general, is to look into the microscopic physics that goes into the determination of {m, which we don't cover in a standard Electromagnetics course. Based on the value of {m that a material exhibits, we have different types of magnetic materials: When {m is small and negative, it is known as diamagnetic. When {m is small and positive, it is known as paramagnetic. When {m is very large and positive, it is known as ferromagnetic. We do not cover diamagnetism in these lectures. Instead, we next discuss the theory of paramagnetism. Here we have to remember that the {m is a bulk parameter of the material, while the origin of the magnetism is microscopic. Hence we have to use quantum mechanics to understand the origin of magnetic moment in an atom, and then use statistical mechanics to understand how magnetic moments of individual atoms act in unison in the bulk, which is an aggregate of a large number of such atoms, to yield a bulk parameter like {m. 2 2 Quantum mechanical origin of magnetism in an isolated atom Let us look at the solution of Schrodinger's equation of an electron in an hydrogen atom. We ml know that the angular part of the wavefunction Yl is an eigen function of the magnitude of orbital angular momentum (L2) and orbital angular momentum in one specific direction (say z) (Lz): 2 ml 2 ml ml ml L Yl = ~ l(l + 1)Yl ; LzYl = ~mlYl (4) where −l 6 ml 6 l. Magnitude of orbital angular momentum of the electron and its value in the z direction are given as follows: 0:5 jµmj = γ~(l(l + 1)) ; µmjz = γ~ml (5) In addition to orbital angular momentum, electron has an intrinsic spin angular mo- mentum. Magnetic moment from the spin is given by: µmjs = γ~ms (6) 1 1 1 where electron being a spin 2 particle has ms equal to 2 or − 2 . Thus a state of an electron in an hydrogen atom can be given by the four quantum num- bers: n; l; ml and ms. Energy only depends on the n. However when we try to calculate the net magnetic moment in other atoms we have to keep in mind that the energy degeneracy is lifted because of electron- electron repulsion in a multi electron system which is not the case with hydrogen atom. Say in Fe atom which has 16 electrons, first 18 electrons form the Ar core, after which 2 electrons go to 4s. Of the 6 electrons which fill up the 3d shell, which electron has what values of quantum numbers (ml and ms) are described by the Hund's rules, as follows: 1. S = Σms is maximized. This is because if electrons have same spin quantum number, from Pauli exclusion principle, their other quantum numbers cannot be identical. So they do not fill up same orbital states and hence electron-electron Coulombic repulsion is mini- mized. 2. L = Σml is maximized. It has been found if orbits have same direction, electron-electron repulsion is lower since electrons have less chance of collision. 3. Due to spin-orbit coupling J becomes a new quantum number because L and S are dependent on each other. Spin orbit coupling energy is given by ξL:~ S~. If ξ > 0 which is the case for less than half filled shell, L and S are anti-parallel and J = jL − Sj. If ξ < 0 which is the case for less than half filled shell, L and S are anti-parallel and J=L+S. The four quantum numbers now are n; J; mj; ms. −J 6 mj 6 J. 3 Now keeping the Hund's rules in mind, let us look at the quantum numbers of the 1 6 electrons of 3d shell of Fe atom. The first five electrons have ms = 2 and the last 1 electron has ms = − 2 following the first Hund's rule, to maximize S = Σms. The first five electrons have to go to different orbitals since their ms is the same. Each of them go to ml = 2; 1; 0; −1; −2 orbitals. The 6th electron goes to ml = 2 orbital to maximize L = Σml, following rule 2, and making L parallel to S, following rule 3, since the 3d shell of Fe atom is more than half filled. J= L+ S = 4. Net magnetic moment of the atom (µ) is the combination of orbital and spin angular momentum of the electrons in the atom. It is given by: µ = gµBmj (7) where −J 6 mj 6 J., µB is the Bohr magnetron and g is the Lande g-factor. 3 Calculation of magnetization of a system of non-interacting atoms using partition function (Curie paramagnetism) So far we have derived the magnetic moment of an isolated atom using quantum me- chanics. Now we need statistical mechanics to find out the average magnetization of a system of non-interacting atoms, which will give us the susceptibility {m of a paramagnet. To calculate a macroscopic quantity like magnetization, we need to identify the number of microstates corresponding to a single value of magnetization. Then we take a ratio of that number of microstates to the total number of microstates, which gives the probability of the system exhibiting magnetization of that value. We do this for every value of magneti- zation. The sum of the product of all possible values of magnetization and the probability of the system to exhibit that value gives the expectation value of the magnetization, or the magnetization of the bulk. E − b We know that probability of a microstate the energy of which is E = e kB T (Refer to "Fundamentals of Statistical and Thermal Physics" by F. Reif for the derivation. Essen- tially we have to count the number of microstates of the bath, with which the system is in equilibrium.) Let there be n states for each atom; h is the applied magnetic field; m is the moment of the atom: State 1 : m = −gµBJ; E1 = gµ0µBJh State 2 : m = −gµB(J − 1); E2 = gµ0µB(J − 1)h ............ 4 State n-1 : m = gµB(J − 1); En−1 = −gµ0µB(J − 1)h State n: m = gµBJ; En = −gµ0µBJh Probability of m1 atoms in state 1, m2 atoms in state 2, Probability of m1 atoms in state 1, m2 atoms in state 2, .......mn atoms in state n= No. of possible microstates x Probability of each microstate = E m E m E m 1 N! − 1 1 − 2 2 − n n e kB T e kB T ::::::e kB T = Z m1!m2!:::::::mn! gµ0µ Jhm1 gµ0µ (J−1)hm1 gµ0µ Jhmn 1 N! − B − B B e kB T e kB T :::::e kB T Z m1!m2!:::::::mn! Sum of all the probabilities = 1 X 1 N! − gµ0µB Jhm1 − gµ0µB (J−1)hm2 gµ0µB Jhmn e kB T e kB T :::::e kB T = 1 (8) Z m1!m2!:::::::mn! m1;m2;:::::::mn ,or, the partition function X N! − gµ0µB Jhm1 − gµ0µB (J−1)hm2 gµ0µB Jhmn Z = e kB T e kB T :::::e kB T = m1!m2!:::::::mn! m1;m2;:::::::mn gµ µ Jh gµ µ (J−1)h gµ µ Jh − 0 B − 0 B 0 B N (e kB T + e kB T + ::::::::::e kB T ) (9) since m1 + m2::::::: + mn = N. Average value of magnetization 1 X N! < m >= (−gµBJm1) + (−gµB(J − 1)m2) + ::::::(gµBJmn) Z m1!m2!:::::::mn! m1;m2;:::::::mn − gµ0µB Jhm1 − gµ0µB (J−1)hm2 gµ0µB Jhmn e kB T e kB T :::::e kB T − gµ0µB Jh − gµ0µB (J−1)h gµ0µB Jh k T d(lnZ) Nk T d(e kB T + e kB T + ::::::::::e kB T ) = B = B µ0 dh µ0 dh d(sinh(J + 1 ) gµ0µB h )) NkBT 2 kB T gµBh = = gµBJNBJ ( ))) (10) µ0 dh kBT 1 1 1 1 1 where BJ (x) = J [(J + 2 )coth((J + 2 )x) − 2 coth( 2 )x)] is called the Brillouin function.