Condensed Matter Physics

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Condensed Matter Physics

• Sharp 251
• 8115
• chui_at_udel.edu

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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

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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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Exchange energy

• Eexch-J?i,d Si ? Si?
• The exchange constant J aligns the spins on
  neighboring sites ?.
• If Jgt0 (lt0), the energy of neighboring spins will
  be lowered if they are parallel (antiparallel).
  One has a ferromagnet (antiferromagnet)

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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-J?2 /V ? dr r S2.

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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

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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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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

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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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Relationship between electrons and the spin
description
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Local moments what is the connection between the
description in terms of the spins and that of the
wave function of electrons?

• Itinerant magnetism

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Illustration in terms of two atomic sites

• There is a hopping Hamiltonian between the sites
  on the left Lgt and that on the right Rgt
  Htt(LgtltRRgtltL).
• For non-interacting electrons, only Ht is
  present, the eigenstates are gt (-gt) Lgt (-)
  Rgt/20.5 with energies (-)t.

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Non magnetic electrons

• For two electrons labelled by 1 and 2, the
  eigenstate of the total system is
  G0gt1,-upgt2,- downgt-1,-downgt2,-upgtby Paulis
  exclusion principle. Note that ltG0SiG0gt0.
• There are no local moments, the system is
  non-magnetic.

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Additional interaction Hunds 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.

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Some examples

• First single local moment

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Single local moment

• H?k nk? Ed(ndnd-)Undnd-- ??k,?(ck?d?c.c.)
  .
• Mean field approximation Hd??k nk? Ed (nd
  nd-)Und? ltnd-? gt ??k,?(ck?d?c.c.).

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Nonmagnetic vs Magnetic case
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Illustration of Hunds rule

• Consider two spin half electrons on two sites. If
  the two electrons occupy the same site, the
  states must be 1, upgt2,downgt-1,downgt2,upgt.
  This corresponds to a total angular momentum 0
  and thus is higher in energy.
• This effect is summarized by the additional
  Hamiltonian HUU?i ni,upni,down.

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Formation of local moments

• The ground state is determined by the sum HUHt.
  This sum is called the Hubbard model.
• For the non-interacting state ltG0HUHtG0gtU-2t.
• Consider alternative ferromagnetic states
  F,upgtL,upgtR,upgt etc and antiferromagnetic
  states, AFgt(L,upgtR,downgt-L,downgtR,upgt)/20.5,
  etc. Their average energy is zero. If Ugt2t, they
  are lower in energy. These states have local
  moments.

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Moments are partly localized

• Neutron scattering results for Ni
• 3d spin 0.656
• 3d orbital0.055
• 4s-0.105

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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 JltL,upL,downHtL,upR,
  downgt2/? E. This energy correction is not
  present for the state Fgt. In the limit of Ugtgtt,
  J-t2/U.
• In general, the exchange depends on the
  concentarion of the electrons and the magnitude
  of U and t.

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Local Moment Details

• PWA, Phys. Rev. 124, 41 (61)

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