Magnetism
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Magnetism. How to describe the physics: Spin model In terms of electrons. Spin model: Each site has a spin S i. There is one spin at each site. The magnetization is proportional to the sum of all the spins.

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Magnetism

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Magnetism

How to describe the physics:

Spin model

In terms of electrons


Spin model: Each site has a spin Si

  • There is one spin at each site.

  • The magnetization is proportional to the sum of all the spins.

  • The total energy is the sum of the exchange energy Eexch, the anisotropy energy Eaniso, the dipolar energy Edipo and the interaction with the external field Eext.


Dipolar interaction

  • The dipolar interaction is the long range magnetostatic interaction between the magnetic moments (spins).

  • Edipo=(1/40)i,j MiaMjbiajb(1/|Ri-Rj|).

  • Edipo=(1/40)i,j MiaMjb[a,b/R3-3Rij,aRij,b/Rij5]

  • 0=4£ 10-7 henrys/m

  • For cgs units the first factor is absent.


Interaction with the external field

  • Eext=-gB H S=-HM

  • We have set M=B S.

  • H is the external field, B =e~/2mc is the Bohr magneton (9.27£ 10-21 erg/Gauss).

  • g is the g factor, it depends on the material.

  • 1 A/m=4 times 10-3Oe (B is in units of G); units of H

  • 1 Wb/m=(1/4) 1010 G cm3 ; units of M (emu)


Anisotropy energy

  • The anisotropy energy favors the spins pointing in some particular crystallographic direction. The magnitude is usually determined by some anisotropy constant K.

  • Simplest example: uniaxial anisotropy

  • Eaniso=-Ki Siz2


Orders of magnitude

  • For Fe, between atomic spins

  • J¼ 522 K

  • K¼ 0.038 K

  • Dipolar interaction =(gB)2/a3¼ 0.254 K

  • gB¼ 1.45£ 10-4 K/Gauss


Last lecture we talk about J a little bit. We discuss the other contribution next

First: Hext


Hext g factor

  • We give two examples of the calculation of the g factor,the case of a single atom and the case in semiconductors.


Atoms

  • In an atom, the electrons have a orbital angular momentum L, a spin angular momentum S and a total angular momentum J=L+S.

  • The energy in an external field is given by Eext=-gB<Jz> by the Wigner-Eckert theorem.


Derivation of the orbital contribution: gL=1

  • E=-H¢ M.

  • The orbital magnetic moment ML= area x current/c; area= R2; current=e/(2) where  is the angular velocity. Now L=m R2=l~. Thus ML=  emR2 /(cm2 )= -0 I e/(2mc). Recall B=e/2mc

  • M=B l.

  • The spin contribution is MS=2B S

  • Here S does not contain the factor of ~

R


Summary

  • E=-M¢ H

  • M=B( gL L+gs S) where gL=1, gS=2; the spin g factor comes from Dirac’s equation.

  • We want <j,m|Jz|j,m>.

  • One can show that <j,m|M|j,m>=g <j,m|J|j,m> for some constant g (W-E theorem). We derive below that g=1+[j(j+1)+s(s+1)-l(l+1)]/[2j(j+1)].


Calculation of g

  • M=L+2S=J+S

  • <j,m|J¢M|j,m>=j’,m’<j,m|J|j’m’><j’,m’|M|j,m> = gj’,m’<j,m|J|j’m’>  <j’,m’|J|j,m>= g <j,m|J2|j,m> =g j (j+1).

  • gj(j+1)=<j,m|J¢ M|j,m>=<j,m|J2+J¢ S|j,m>=j(j+1)+<j,m|J¢ S|j,m>.

  • g=1+<j,m|J¢ S|j,m>/j(j+1).


Calculation of g in atoms

  • L=(J-S); L2=(J-S)2=J2+S2-2J¢ S.

  • <J¢ S>=<(J2+S2-L2)>/2= [j(j+1)+s(s+1)-l(l+1)]/2.

  • Thus g=1+ [j(j+1)+s(s+1)-l(l+1)]/2j(j+1)


Another examples: in semiconductors, k¢ p perturbation theory

  • The wave function at a small wave vector k is given by = exp(ik¢ r)uk(r) where u is a periodic function in space.

  • The Hamiltonian H=-~2r2/2m+V(r). The equation for u becomes [-~2r2/2m+V-~ k¢ p/2]u=Eu where the k2 term is neglected.


G factor in semiconductors

  • The extra term can be treated as perturbation from the k=0 state, the energy correction is

  •  Dijkikj= <|kipi|><|kjpj|>/[E-E]

  • In a magnetic field, k is replaced p-eA/c.

  • The equation for u becomes H’u=Eu;

  • H’= Dij(pi-eAi/c)(pj-eAj/c)-B¢ B). Since A=r£ B/2, the Dij term also contains a contribution proportional to B.


Calculation of g

  • H’=H1+…; H1= (e/c)p  D A+A D p.

  • Since A=r£ B/2, H1= (e/2c)p  D (r  B)+(r B) D p.

  • A B C=A B C, for any A, B, C; so H1= (e/2c)(pDrB -BrDp )=gBB

  • g= m(pDr - rDp)/.

  • Note pirj=ij/im+rjpi

  • gj= /i Diljli+O(p) where ijk= 1 depending on whether ijk is an even or odd permutation of 123; otherwise it is 0; repeated index means summation.


g=DA /i

  • g_z=(D_{xy}-D_{yx})/i, the antisymmetric D.

  • g is inversely proportional to the energy gap.

  • For hole states, g can be large


Effect of the dipolar interaction: Shape anisotropy

  • Example: Consider a line of parallel spins along the z axis. The lattice constant is a. The orientation of the spins is described by S=(sin, 0, cos ). The dipolar enegy /spin is M02 [1/i3-3 cos2 /i3]/40 a3=A-B cos2 .

  •  1/i3=(3)¼ 1.2

  • E=-Keff cos2(), Keff=1.2 M02/40.


Paramagnetism: J=0

  • Magnetic susceptibility: =M/B (0)

  • We want to know  at different temperatures T as a function of the magnetic field B for a collection of classical magnetic dipoles.

  • Real life examples are insulating salts with magnetic ions such as Mn2+, etc, or a gas of atoms.


Magnetic susceptibility of different non ferromagnets

Free spin paramagnetism

Van Vleck

Pauli (metal)

T

Diamagnetism (filled shell)


Boltzmann distribution

  • Probability P/ exp(-U/kB T)

  • U=-gB B ¢ J

  • P(m)/ exp(-gB B m/kBT)

  • <M>=NB gm P(m) m/m P(m)

  • To illustrate, consider the simple case of J=1/2. Then the possible values of m are -1/2 and 1/2.


<M> and 

  • We get <M>=NgB[ exp(-x)-exp(x)]/2[exp(-x)+exp(x)] where x=gB B/(2kBT).

  • Consider the high temperature limit with x<<1, <M>¼ N gB x/2.

  • We get =N(gB )2/2kT

  • At low T, x>>1, <M>=NgB/2, as expected.


More general J

  • Consider the function Z= m=-jm=j exp(-mx)

  • For a general geometric series 1+y+y2+…yn=(1-yn+1)/(1-y)

  • We get Z=sinh[(j+1/2)x]/sinh(x/2).

  • <M>=-d ln Z/dx=NgB[(j+1/2) coth[(j+1/2)x]-coth(x/2)/2].


Diamagnetism of atoms

  •  in CGS for He, Ne, Ar, Kr and Xe are -1.9, -7.2,-19.4, -28, -43 times 10-6 cm3/mole.

  •  is negative, this behaviour is called diamagnetic.


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