Magnetism

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Magnetism

• How to describe the physics
• Spin model
• In terms of electrons

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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.

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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
  
• ?04? 10-7 henrys/m
• For cgs units the first factor is absent.

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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/m4? times 10-3Oe (B is in units of G) units
  of H
• 1 Wb/m(1/4?) 1010 G cm3 units of M (emu)

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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

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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

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

• First Hext

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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.

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Atoms

• In an atom, the electrons have a orbital angular
  momentum L, a spin angular momentum S and a total
  angular momentum JLS.
• The energy in an external field is given by
  Eext-g?BltJzgt by the Wigner-Eckert theorem.

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Derivation of the orbital contribution gL1

• E-H M.
• The orbital magnetic moment ML area x current/c
  area? R2 currente?/(2?) where ? is the angular
  velocity. Now Lm? R2l. Thus ML ? emR2 ?
  /(cm2? ) -?0 I? e/(2mc). Recall ?Be?/2mc
• M?B l.
• The spin contribution is MS2?B S
• Here S does not contain the factor of

R
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Summary

• E-M H
• M?B( gL Lgs S) where gL1, gS2 the spin g
  factor comes from Diracs equation.
• We want ltj,mJzj,mgt.
• One can show that ltj,mMj,mgtg ltj,mJj,mgt for
  some constant g (W-E theorem). We derive below
  that g1j(j1)s(s1)-l(l1)/2j(j1).

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Calculation of g

• ML2SJS
• ltj,mJMj,mgt?j,mltj,mJjmgt?ltj,mMj,mgt
  g?j,mltj,mJjmgt ? ltj,mJj,mgt g
  ltj,mJ2j,mgt g j (j1).
• gj(j1)ltj,mJ Mj,mgtltj,mJ2J
  Sj,mgtj(j1)ltj,mJ Sj,mgt.
• g1ltj,mJ Sj,mgt/j(j1).

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Calculation of g in atoms

• L(J-S) L2(J-S)2J2S2-2J S.
• ltJ Sgtlt(J2S2-L2)gt/2 j(j1)s(s1)-l(l1)/2.
• Thus g1 j(j1)s(s1)-l(l1)/2j(j1)

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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-2r2/2mV(r). The equation for
  u becomes -2r2/2mV- k p/2uEu where the k2
  term is neglected.

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G factor in semiconductors

• The extra term can be treated as perturbation
  from the k0 state, the energy correction is
• ? Dijkikj? lt?kipi?gtlt?kjpj?gt/E?-E?
• In a magnetic field, k is replaced p-eA/c.
• The equation for u becomes HuEu
• H? Dij(pi-eAi/c)(pj-eAj/c)-?B? B). Since Ar
  B/2, the Dij term also contains a contribution
  proportional to B.

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Calculation of g

• HH1 H1 (e/c)? p ? D? AA? D? p.
• Since Ar B/2, H1 (e/2c)? p ? D? (r ? B)(r?
  B)? D? p.
• A? B? CA? 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/imrjpi
• gj? /i? Dil?jliO(p) where ?ijk? 1 depending on
  whether ijk is an even or odd permutation of 123
  otherwise it is 0 repeated index means
  summation.

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gDA /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

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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 a3A-B cos2 ? .
• ? 1/i3?(3)¼ 1.2
• E-Keff cos2(?), Keff1.2 M02/4??0.

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Paramagnetism J0

• 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.

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Magnetic susceptibility of different non
ferromagnets
Free spin paramagnetism
?
Van Vleck
Pauli (metal)
T
Diamagnetism (filled shell)
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Boltzmann distribution

• Probability P/ exp(-U/kB T)
• U-g?B B J
• P(m)/ exp(-g?B B m/kBT)
• ltMgtN?B g?m P(m) m/?m P(m)
• To illustrate, consider the simple case of J1/2.
  Then the possible values of m are -1/2 and 1/2.

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ltMgt and ?

• We get ltMgtNg?B exp(-x)-exp(x)/2exp(-x)exp(x)
  where xg?B B/(2kBT).
• Consider the high temperature limit with xltlt1,
  ltMgt¼ N g?B x/2.
• We get ?N(g?B )2/2kT
• At low T, xgtgt1, ltMgtNg?B/2, as expected.

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More general J

• Consider the function Z ?m-jmj exp(-mx)
• For a general geometric series 1yy2yn(1-yn1)
  /(1-y)
• We get Zsinh(j1/2)x/sinh(x/2).
• ltMgt-d ln Z/dxNg?B(j1/2) coth(j1/2)x-coth(x/
  2)/2.

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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.