Transcript Magnetism - Bartol Research Institute

Condensed Matter Physics
• Sharp 251
• 8115
• [email protected]
• 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=-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)
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 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=-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)
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/R33Rij,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=-Ki Siz2
Relationship between electrons
and the spin description
• Itinerant magnetism:
Local moments: what is the
connection between the description
in terms of the spins and that of the
wave function of electrons？
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=Ui 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>=U2t.
• 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)