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# Magnetism - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Magnetism' - lixue

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### Magnetism

How to describe the physics:

Spin model

In terms of electrons

• 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.

• 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.

• 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)

• 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

• 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

H other contribution nextext 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 other contribution next

• 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 other contribution nextL=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 other contribution next

• 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 other contribution next

• 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 other contribution next

• 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 other contribution next¢ 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 other contribution next

• 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 other contribution next

• 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 other contribution nextA /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 other contribution next

• 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 other contribution next

• 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 other contribution next

Free spin paramagnetism

Van Vleck

Pauli (metal)

T

Diamagnetism (filled shell)

Boltzmann distribution other contribution next

• 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 other contribution next

• 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 other contribution next

• 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 other contribution next

•  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.