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Module 2A: Theory of paramagnetism

Debanjan Bhowmik

Department of Electrical Engineering

Indian Institute of Technology Delhi

Abstract

In our rst module, we discussed the problems associated with the current proces- sor and memory technology in the industry, which are transistor based, and brie y 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 rst 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=0(~H+~M) (1)

M={m~H(2)

As a result,~B=0(~H+{m~H) =0r~H(3)

Parameters:r= relative permeability,{m= magnetic susceptibility

Fields:~H= applied magnetic eld,~B= magnetic

ux density and~M= magnetization

Dierence between

~Hand~Bis that~His what is applied, and~Bis the actual eld which is the combination of~Hand the response of the material to~H, which is given by the magnetization~M={m~H. 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~Bis much higher than magnitude of~Hbecause of the large magnetic moment of iron, in response to the~H, or the high{mof 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{mthat a material exhibits, we have dierent types of magnetic materials: When{mis small and negative, it is known as diamagnetic. When{mis small and positive, it is known as paramagnetic. When{mis 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{mis 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 know that the angular part of the wavefunctionYmllis an eigen function of the magnitude of orbital angular momentum (L2) and orbital angular momentum in one specic direction (say z) (Lz) : L

2Ymll=~2l(l+ 1)Ymll;LzYmll=~mlYmll(4)

wherel6ml6l. Magnitude of orbital angular momentum of the electron and its value in the z direction are given as follows: jmj= ~(l(l+ 1))0:5;mjz= ~ml(5) In addition to orbital angular momentum, electron has an intrinsic spin angular mo- mentum. Magnetic moment from the spin is given by: mjs= ~ms(6) where electron being a spin 12 particle hasmsequal to12 or12 Thus a state of an electron in an hydrogen atom can be given by the four quantum num- bers:n;l;mlandms. Energy only depends on then. 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, rst 18 electrons form the Ar core, after which 2 electrons go to 4s. Of the 6 electrons which ll up the 3d shell, which electron has what values of quantum numbers (mlandms) are described by the

Hund's rules, as follows:

1.S= msis 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 ll up same orbital states and hence electron-electron Coulombic repulsion is mini- mized.

2.L= mlis 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~L:~S. If >0 which is the case for less than half lled shell, L and S are anti-parallel andJ=jLSj. If <0 which is the case for less than half lled shell, L and S are anti-parallel and J=L+S. The four quantum numbers now aren;J;mj;ms.J6mj6J. 3 Now keeping the Hund's rules in mind, let us look at the quantum numbers of the

6 electrons of 3d shell of Fe atom. The rst ve electrons havems=12

and the last electron hasms=12 following the rst Hund's rule, to maximizeS= ms. The rst ve electrons have to go to dierent orbitals since theirmsis the same. Each of them go toml= 2;1;0;1;2 orbitals. The 6th electron goes toml= 2 orbital to maximize L= ml, following rule 2, and making L parallel to S, following rule 3, since the 3d shell of Fe atom is more than half lled. J= L+ S = 4. Net magnetic moment of the atom () is the combination of orbital and spin angular momentum of the electrons in the atom. It is given by: =gBmj(7) whereJ6mj6J.,Bis 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 nd out the average magnetization of a system of non-interacting atoms, which will give us the susceptibility{mof 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. We know that probability of a microstate the energy of which isE=eEbk

BT(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 eld; m is the moment of the atom:

State 1 :m=gBJ;E1=g0BJh

State 2 :m=gB(J1);E2=g0B(J1)h

4

State n-1 :m=gB(J1);En1=g0B(J1)h

State n:m=gBJ;En=g0BJh

Probability ofm1atoms in state 1,m2atoms in state 2, Probability ofm1atoms in state 1,m2atoms in state 2, .......mnatoms in state n= No. of possible microstates x Probability of each microstate = 1Z N!m

1!m2!:::::::mn!eE1m1k

BTeE2m2k

BT::::::eEnmnk

BT= 1Z N!m

1!m2!:::::::mn!eg0BJhm1k

BTeg0B(J1)hm1k

BT:::::eg

0BJhmnk

BT

Sum of all the probabilities = 1

X m

1;m2;:::::::mn1Z

N!m

1!m2!:::::::mn!eg0BJhm1k

BTeg0B(J1)hm2k

BT:::::eg

0BJhmnk

BT= 1 (8)

,or, the partition function Z=X m

1;m2;:::::::mnN!m

1!m2!:::::::mn!eg0BJhm1k

BTeg0B(J1)hm2k

BT:::::eg

0BJhmnk

BT= (eg0BJhk

BT+eg0B(J1)hk

BT+::::::::::eg

0BJhk

BT)N(9)

sincem1+m2:::::::+mn=N.

Average value of magnetization

< m >=1Z X m

1;m2;:::::::mn(gBJm1) + (gB(J1)m2) +::::::(gBJmn)N!m

1!m2!:::::::mn!

e g0BJhm1k

BTeg0B(J1)hm2k

BT:::::eg

0BJhmnk

BT kBT

0d(lnZ)dh

=NkBT

0d(eg0BJhk

BT+eg0B(J1)hk

BT+::::::::::eg

0BJhk BT)dh NkBT

0d(sinh(J+12

)g0Bhk

BT))dh

=gBJNBJ(gBhk

BT))) (10)

whereBJ(x) =1J [(J+12 )coth((J+12 )x)12 coth(12 )x)] is called the Brillouin function. When applied magnetic eld h is very large such that g0Bhk

BT>>1 ,BJ(g0Bhk

BT) = 1

and hence m=gBJN(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 eld is increased.

When applied magnetic eld h is small such that

g0Bhk

BT<<1 ,BJ(g0Bhk

BT) =g0Bhk

BTJ+13

and hence m=g202BJ(J+ 1)3kBTNh(12) Thus, when eld is small,mvshplot 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 elds is given by: m=g202BJ(J+ 1)3kBTNV (13) (Division by volume is needed since magnetization is expressed as magnetic moment per unit volume and permeability is dened as the change of magnetization with magnetic eld. It is unit-less in S.I. If we calculate the susceptibility in order of magnitude it turns out to be: m=g202BJ(J+ 1)3kBTNV

4107(91024)2131:381023T1030102

(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 ({m;Curie), and susceptibility mea- sured experimentally:

1. Susceptibility of metals measured experimentally is orders of magnitude lower than

m;Curie. For example, susceptibilities of sodium and aluminum metals are105(Figure

3.5, Modern Magnetic Materials by Robert O'Handley)

2. Also the plot of susceptibility of metals versus temperature, at small temperature, is

at, 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 solid. As a result, the metal needs to be treated as free electron gas instead of an assortment of atoms which we did above. Susceptibility, calculated using the free electron gas model of atoms, is called Pauli susceptibility ({m;Pauli) 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: =eikxxeikyyeikzz;E=~22m((kx)2) + (ky)2) + (kz)2)) =~22m(k2) (15)

From periodic boundary condition:

k x=2L xnx;ky=2L yny;kz=2L znz(16) whereLx,LyandLzare dimensions of the solid in x,y and z; andnx,nyandnzare 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=

N(E) =43

(k3)( 2L x)(2L y)(2L z)=V62(k3) =V62(2mE~ 2)32 (17)

Density of states is given by:

D(E) =dN(E)dE

=V42(2mE~ 2)12 2m~ 2(18) 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 eld, density of states of up spin and down spin electrons is dierent, which we will calculate next. 7 When a magnetic eld is applied say in the upward direction (") potential energy of the electrons is0Bhfor spin up (") electrons and0Bhfor spin down (#) electrons.

Hence for spin up (") electrons,

22m(k2") =E(0Bh) =E+0Bh;D"(E) =D(E+0Bh) (19)

and for spin down (#) electrons,

22m(k2#) =E(0Bh) =E0Bh;D#(E) =D(E0Bh) (20)

If we plotD"(E) andD#(E) as a function of energy (E), we see that they are parabolas shifted downward and upward vertically by0Bhcompared to the parabola D(E) vs. E.

No. of spin up (") and spin down (#) electrons =

N "=Z 1

0BhD(E+0Bh)f(E)dE;N#=Z

1

0BhD(E0Bh)f(E)dE(21)

For low temperatures, we can approximate that for energy < Fermi energy (EF) f(E)=1 and for energy higher thanEFf(E)=0. Hence, N "=Z EF

0BhD(E+0Bh)dE=Z

EF+0Bh

0

D(E0)dE0

Z EF

0D(E0)dE0+Z

EF+0Bh

E

FD(E0)dE0Z

EF

0D(E)dE+D(EF)0Bh(22)

Similarly,

N #Z EF

0D(E)dED(EF)0Bh(23)

Net magnetization:

< m >= (N"N#)B1V =202BhD(EF)V (24)

From equations (17) and (18),

D(EF) =V42(62NV

)13 2m~ 2(25)

Magnetic susceptibility is given by:

m= (62NV )13 2m~

212202Bm02B~

2(NV )13

4107(91024)291021(1:051034)21010104(26)

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~korkx;ky;kzcorresponds to a specic 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 eld say in up direction, energy of the spin up states is lower than the spin down states. So rst the k-states corresponding to spin up states ll up and then the same k-states corresponding to spin down. As a result after the electrons ll up all the available states up to Fermi energy, there is an imbalance due to some higher k-states, lled up only by spin up electrons. Now if a magnetic eld is applied in either direction, only the spin of electrons with energy close to the the Fermi energy can ip (high k-values) can ip 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 ipping 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 is equal to kBTk BTF.TFis the Fermi temperature corresponding to the Fermi energy (a constant). Since Curie susceptibility varies as 1T

Pauli susceptibility is independent of

temperature T. 9quotesdbs_dbs17.pdfusesText_23

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