Quick viewing(Text Mode)

Module 2A: Theory of Paramagnetism

Module 2A: Theory of Paramagnetism

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 be- ing cited as the main motivation. In the second module, we will discuss the theory of 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 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 = κmH (2) As a result, ~ ~ ~ ~ B = µ0(H + κmH) = µ0µrH (3)

Parameters: µr= relative permeability, κm= Fields: H~ = applied magnetic field, B~ = magnetic flux density and M~ =

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 = κmH. For example, in the case of a solenoid coil with an 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 κm 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 κm, which we don’t cover in a standard Electromagnetics course.

Based on the value of κm that a material exhibits, we have different types of magnetic materials: When κm is small and negative, it is known as diamagnetic. When κm is small and positive, it is known as paramagnetic. When κm is very large and positive, it is known as ferromagnetic.

We do not cover in these lectures. Instead, we next discuss the theory of paramagnetism. Here we have to remember that the κm is a bulk parameter of the material, while the origin of the magnetism is microscopic. Hence we have to use to understand the origin of in an , and then use statistical mechanics to understand how magnetic moments of individual act in unison in the bulk, which is an aggregate of a large number of such atoms, to yield a bulk parameter like κm.

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 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 |µm| = γ~(l(l + 1)) ; µm|z = γ~ml (5) In addition to orbital angular momentum, electron has an intrinsic angular mo- mentum. Magnetic moment from the spin is given by:

µm|s = 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 = |L − S|. 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 κm 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.

gµ0µB h gµ0µB h When applied magnetic field h is very large such that >> 1 , BJ ( ) = 1 kB T kB T and hence m = gµBJN (11)

5 Moments of all individual atoms line up with each other and the net moment is the moment of individual atom multiplied by the number of atoms. It is not physically possible for the net moment to be higher than this value, so it saturates even if field is increased.

gµ0µB h gµ0µB h gµ0µB h J+1 When applied magnetic field h is small such that << 1 , BJ ( ) = kB T kB T kB T 3 and hence g2µ µ2 J(J + 1) m = 0 B Nh (12) 3kBT Thus, when field is small, m vs h plot follows a straight line. Comparing with equation (2), slope of the straight line determines the magnetic permeability. Thus we have derived what we set out as our goal in the beginning of this lecture, i.e. permeability of a paramagnet. The fact that the individual atoms form a system in this derivation without interacting with each other, i.e. ,moment of one atom is independent of the other (the way we have counted the microstates takes care of that) is applicable only for paramagnets and not ferromagnets.

Thus, magnetic susceptibility at small fields is given by:

2 2 g µ0µBJ(J + 1) N κm = (13) 3kBT V (Division by volume is needed since magnetization is expressed as magnetic moment per unit volume and permeability is defined as the change of magnetization with magnetic field. It is unit-less in S.I. If we calculate the susceptibility in order of magnitude it turns out to be: 2 2 g µ0µBJ(J + 1) N −7 −24 2 1 30 −2 κm = ≈ 4π × 10 × (9 × 10 ) × × 10 ≈ 10 3kBT V 3 × 1.38 × 10−23T (14) at room temperature. Each atom is considered to have a volume of (1 Angstrom)3.

The above calculation of paramagnetism assumes electrons are localized to their atoms. This assumption works for molecules and free atoms and ions, which have a net magnetic moment, but in metals there are two major discrepancies between susceptibility calculated from equation (13), also known as Curie susceptibility (κm,Curie), and susceptibility mea- sured experimentally: 1. Susceptibility of metals measured experimentally is orders of magnitude lower than −5 κm,Curie. For example, susceptibilities of and aluminum metals are ≈ 10 (Figure 3.5, Modern Magnetic Materials by Robert O’Handley) 2. Also the plot of susceptibility of metals versus temperature, at small temperature, is flat, showing that susceptibility is independent of temperature, which contradicts Curie’s

6 Law. This discrepancy happens because in metals magnetism often originates from conduction electrons which are not localized in individual atoms but are delocalized in the entire . As a result, the metal needs to be treated as free electron instead of an assortment of atoms which we did above. Susceptibility, calculated using the free electron gas model of atoms, is called Pauli susceptibility (κm,P auli) and the associated paramagnetism is called Pauli paramagnetism, which we discuss in the next section.

4 Calculation of paramagnetism of a free electron gas (Pauli paramagnetism)

In a metal, conduction electrons that contribute to magnetic moment, are all delocalized. As a result, ”free electron gas” model has to be used. In 3 dimensions (x,y,z) wave function of a free electron and the corresponding energy are written as:

2 2 ψ = eikxxeikyyeikzz; E = ~ ((k )2) + (k )2) + (k )2)) = ~ (k2) (15) 2m x y z 2m From periodic boundary condition: 2π 2π 2π kx = nx; ky = ny; kz = nz (16) Lx Ly Lz where Lx, Ly and Lz are dimensions of the solid in x,y and z; and nx,ny and nz are integers.Volume of the solid: V = LxLyLz Number of states with energy less than E = volume of the sphere with radius k such that corresponding energy is E/ volume occupied by each state in k space=

4 3 3 π(k ) V 3 V 2mE 3 N(E) = = (k ) = ( ) 2 (17) ( 2π )( 2π )( 2π ) 6π2 6π2 2 Lx Ly Lz ~ is given by:

dN(E) V 2mE 1 2m D(E) = = ( ) 2 (18) dE 4π2 ~2 ~2 We have deliberately not put the ”2” factor here because we want to calculate density of states for up-spins and down- spins separately. This is because under a magnetic field, density of states of up spin and down spin electrons is different, which we will calculate next.

7 When a magnetic field is applied say in the upward direction (↑) of the electrons is −µ0µBh for spin up (↑) electrons and µ0µBh for spin down (↓) electrons. Hence for spin up (↑) electrons,

2 ~ (k2) = E − (−µ µ h) = E + µ µ h; D (E) = D(E + µ µ h) (19) 2m ↑ 0 B 0 B ↑ 0 B and for spin down (↓) electrons,

2 ~ (k2) = E − (µ µ h) = E − µ µ h; D (E) = D(E − µ µ h) (20) 2m ↓ 0 B 0 B ↓ 0 B

If we plot D↑(E) and D↓(E) as a function of energy (E), we see that they are parabolas shifted downward and upward vertically by µ0µBh compared to the parabola D(E) vs. E. No. of spin up (↑) and spin down (↓) electrons = Z ∞ Z ∞ ↑ ↓ N = D(E + µ0µBh)f(E)dE; N = D(E − µ0µBh)f(E)dE (21) −µ0µB h µ0µB h

For low temperatures, we can approximate that for energy ¡ (EF ) f(E)=1 and for energy higher than EF f(E)=0. Hence,

Z EF Z EF +µ0µB h ↑ 0 0 N = D(E + µ0µBh)dE = D(E )dE −µ0µB h 0 Z EF Z EF +µ0µB h Z EF 0 0 0 0 = D(E )dE + D(E )dE ≈ D(E)dE + D(EF )µ0µBh (22) 0 EF 0 Similarly, Z EF ↓ N ≈ D(E)dE − D(EF )µ0µBh (23) 0 Net magnetization:

1 2µ µ2 hD(E ) < m >= (N ↑ − N ↓)µ = 0 B F (24) B V V From equations (17) and (18),

2 V 6π N 1 2m D(EF ) = ( ) 3 (25) 4π2 V ~2 Magnetic susceptibility is given by:

2 2 6π N 1 2m 1 2 mµ0µB N 1 κm = ( ) 3 µ0µB ≈ ( ) 3 V ~2 2π2 ~2 V 4π × 10−7 × (9 × 10−24)2 × 9 × 10−21 × 1010 ≈ 10−4 (26) (1.05 × 10−34)2

8 Orders of magnitude lower susceptibility in the case of Pauli paramagnetism compared to Curie paramagnetism can be explained as follows: Unlike Curie paramagnetism, here electrons are delocalized, i.e, all the electrons belong to the entire solid. A particular value of ~k or kx, ky, kz corresponds to a specific orbital state of the electron and for each such orbital state, there can be two possible spin states of the electron: spin up(↑) and spin down (↓), only one electron in each state following Pauli exclusion principle. Due to the presence of the magnetic field say in up direction, energy of the spin up states is lower than the spin down states. So first the k-states corresponding to spin up states fill up and then the same k-states corresponding to spin down. As a result after the electrons fill up all the available states up to Fermi energy, there is an imbalance due to some higher k-states, filled up only by spin up electrons. Now if a magnetic field is applied in either direction, only the spin of electrons with energy close to the the Fermi energy can flip (high k-values) can flip because for the low energy or low k- states, already there are 2 electrons one with spin up and one with spin down, and no additional electron can be allowed since it will violate the Pauli exclusion principle. As a result, the calculated value of Pauli susceptibility is orders of magnitude lower than of Curie Susceptibility.

Another thing to be noted is that unlike Curie susceptibility, Pauli susceptibility is not dependent on temperature. This happens because as explained before, only the electrons close to the Fermi surface take part in the spin flipping in Pauli paramagnetism unlike Curie paramagnetism. So Curie susceptibility has to be multiplied by the ratio of no. of electrons close to the Fermi surface to the total no. of electrons to get the Pauli susceptibility, which kB T is equal to . TF is the Fermi temperature corresponding to the Fermi energy (a kB TF 1 constant). Since Curie susceptibility varies as T Pauli susceptibility is independent of temperature T.

9

Top View