Download Lecture Two

Magnetism
How to describe the physics:
(1) Spin model
(2) 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/R33Rij,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=AB 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.