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Condensed Matter Physics

- Sharp 251
- 8115
- chui_at_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) - 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
- Spin model
- 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 Jgt0 (lt0), 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 S2.

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/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)

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

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

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 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.

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.

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.

Some examples

- First single local moment

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.).

Nonmagnetic vs Magnetic case

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.

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.

Moments are partly localized

- Neutron scattering results for Ni
- 3d spin 0.656
- 3d orbital0.055
- 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 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.

Local Moment Details

- PWA, Phys. Rev. 124, 41 (61)

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