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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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How to describe the physics:

Spin model

In terms of electrons

Spin model each site has a spin s i

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

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

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

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

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

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

First: Hext

H ext g factor

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.



  • 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 g l 1

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 ~




  • 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

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

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

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

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 g1

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 d a i

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

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

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

Magnetic susceptibility of different non ferromagnets

Free spin paramagnetism

Van Vleck

Pauli (metal)


Diamagnetism (filled shell)

Boltzmann distribution

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

<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

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

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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