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• Text: G. D. Mahan, Many Particle Physics• Topics:
– Magnetism: Simple basics, advanced topics include micromagnetics, spin polarized transport and itinerant magnetism (Hubbard model)
– Superconductivity: BCS theory, advanced topics include RVB (resonanting valence bond)
– Linear Response theory: advanced topics include the quantized Hall effect and the Berry phase.
– Bose-Einstein condensation, superfluidity and atomic traps
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.
Exchange energy
• Eexch=-Ji,d Si Si+
• The exchange constant J aligns the spins on neighboring sites .
• If J>0 (<0), the energy of neighboring spins will be lowered if they are parallel (antiparallel). One has a ferromagnet (antiferromagnet)
Alternative form of exchange energy
• Eexch=-J (Si-Si+)2 +2JSi2.
• Si2 is a constant, so the last term is just a
constant.
• When Si is slowly changing Si-Si+ r Si.
• Hence Eexch=-J2 /V dr r S|2.
Magnitude of J
• kBTc/zJ¼ 0.3
• Sometimes the exchange term is written as A s d3 r |r M(r)|2.
• A is in units of erg/cm. For example, for permalloy, A= 1.3 £ 10-6 erg/cm
Interaction with the external field
• Eext=-gB 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)
Dipolar interaction
• The dipolar interaction is the long range magnetostatic interaction between the magnetic moments (spins).
• Edipo=(1/40)i,j MiaMjbiajb(1/|Ri-Rj|).
• Edipo=(1/40)i,j MiaMjb[a,b/R3-3Rij,aRij,b/Rij5]
• 0=4 10-7 henrys/m
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=-Ki Siz2
Relationship between electrons and the spin description
Local moments: what is the connection between the description in terms of the spins and that of the
wave function of electrons？
• Itinerant magnetism:
Illustration in terms of two atomic sites:
• There is a hopping Hamiltonian between the sites on the left |L> and that on the right |R>: Ht=t(|L><R|+|R><L|).
• For non-interacting electrons, only Ht is present, the eigenstates are |+> (|->) =[|L>+ (-) |R>]/20.5 with energies +(-)t.
Non magnetic electrons
• For two electrons labelled by 1 and 2, the eigenstate of the total system is |G0>=|1,-up 〉 |2,- down 〉 -|1,-down 〉 |2,-up 〉 by Pauli’s exclusion principle. Note that <G0|Si|G0>=0.
• There are no local moments, the system is non-magnetic.
Additional interaction: Hund’s rule energy
• In an atom, because of the Coulomb interaction, the electrons repel each other. A simple rule that captures this says that the energy of the atom is lowered if the total angular momentum is largest.
Some examples:
First: single local moment
Single local moment
• H=k nk +Ed(nd++nd-)+Und+nd-- +k,(ck
+d+c.c.) .
• Mean field approximation: Hd=k nk +Ed
(nd++ nd-)+Und+ <nd- > + k,(ck+
d+c.c.).
Nonmagnetic vs Magnetic case
Illustration of Hund’s rule
• Consider two spin half electrons on two sites. If the two electrons occupy the same site, the states must be |1, up>|2,down>-|1,down>|2,up>. This corresponds to a total angular momentum 0 and thus is higher in energy.
• This effect is summarized by the additional Hamiltonian HU=Ui ni,upni,down.
Formation of local moments
• The ground state is determined by the sum HU+Ht. This sum is called the Hubbard model.
• For the non-interacting state <G0|HU+Ht|G0>=U-2t.
• Consider alternative ferromagnetic states |F,up>=|L,up>|R,up> etc and antiferromagnetic states, |AF>=(|L,up>|R,down>-|L,down>|R,up>)/20.5, etc. Their average energy is zero. If U>2t, they are lower in energy. These states have local moments.
Moments are partly localized
• Neutron scattering results for Ni: – 3d spin= 0.656– 3d orbital=0.055– 4s=-0.105
An example of the exchange interaction
• For our particular example, the interaction is antiferromagnetic. There is a second order correction in energy to the antiferromagnetic state given by J=|<L,up ； L,down|Ht|L,up ； R, down>|2/ E. This energy correction is not present for the state |F>. In the limit of U>>t, J=-t2/U.
• In general, the exchange depends on the concentarion of the electrons and the magnitude of U and t.
Local Moment Details:
PWA, Phys. Rev. 124, 41 (61)
