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Condensed Matter Physics

Condensed Matter Physics. Sharp 251 8115 chui@udel.edu. Text: G. D. Mahan, Many Particle Physics Topics: Magnetism: Simple basics, advanced topics include micromagnetics, spin polarized transport and itinerant magnetism (Hubbard model)

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Condensed Matter Physics

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  1. Condensed Matter Physics • Sharp 251 • 8115 • chui@udel.edu

  2. 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

  3. Magnetism How to describe the physics: Spin model In terms of electrons

  4. 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.

  5. Exchange energy • Eexch=-Ji,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)

  6. 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+rSi. • Hence Eexch=-J2 /V dr |rS|2.

  7. 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

  8. 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)

  9. 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

  10. 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

  11. Relationship between electrons and the spin description

  12. Local moments: what is the connection between the description in terms of the spins and that of the wave function of electrons？ • Itinerant magnetism:

  13. 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.

  14. 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.

  15. 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.

  16. Some examples: First: single local moment

  17. 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.).

  18. Nonmagnetic vs Magnetic case

  19. 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=Ui ni,upni,down.

  20. 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.

  21. Moments are partly localized • Neutron scattering results for Ni: • 3d spin= 0.656 • 3d orbital=0.055 • 4s=-0.105

  22. 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.

  23. Local Moment Details: PWA, Phys. Rev. 124, 41 (61)

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