Luigi Paolasini
paolasini@esrf.fr
LECTURE 6:
“The dark side of magnetism: magnetic in metals”
- Free electron gas model.
- Magnetism in metals.
- Itinerant exchange interactions.
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
Localized electron model
- localized atomic magnetic moments
- described by the phenomenological Heisenberg Hamiltonian
- magnetism carried out by the partially filled shells
- saturated magnetic moment with integer number of µB
- valid for rare earths 4f magnetism (Hund rules)
Itinerant electron model
- metallic bonding
- competition between electronic correlation and kinetic energy
- electron delocalization and band description of electronic states
- non integral value of magnetic moment at saturation.
(ex: Fe 2.2 µB, Co 1.6 µB)
Hubbard Hamiltonian take into account both the kinetic and the Coulomb
interactions
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
- Electrons move freely in a volume V=L3
- We ignore the periodic potential due to the lattice periodicity
- Each state is doubly degenerate (spin-up and spin-down states)
Total number of electrons
Density of states
Double occupation
Fermi energy
k-space
Fermi sphere
Spin-up and
spin-down states
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
At T=0 : the electron momentum distribution fill the k-space up to wavevector kF.
Density of states at the Fermi energy EF
Fermi distribution
Density of states
At T>0 : the occupancy of states is governed
by the Fermi-Dirac distribution:
Chemical potential
Chemical potential µ = EF at all T, because the
degenerate limit valid for most of metals!!
The Fermi surface is the set of point in k-space whose µ = EF
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
- An applied magnetic field rises or lowers the electron sub-bands in n⇑ and n⇓
- The excess of electron in n⇑ si compensated by the deficit n⇓ - We neglet the smearing of the Fermi surface due to finite temperatures
- We neglet the orbital contribution (g=2)
Electron density with spin up (down)
In the degenerate case (T=0), the
magnetization is given by:
B
Pauli susceptibility
- Weak because concerns only
the electron on the Fermi surface
- Temperature independent
Because χP<<1
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
- Molecular field magnetization of electron gas (Pauli paramagnetism)
- Bootstrapping mechanism
-δn⇓ are flip in n⇑ sub-band close to EF+δE
δn⇑=1/2g(EF) δE
Kinetic energy cost =>unfavourable
>0
B=0
This energy cost can be compensated by the
molecular field λM produced by the band
splitting M=(n⇑-n⇓ )µB
Molecular field energy:
Notice that the molecular field is due to exchange interaction that, in turn, is a
consequence of Coulomb interaction
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
Antisymmetric wavefunction (ferromagnetic coupling and fermionicity)
Electron density
Probability density for two spin up (averaged over the Fermi sphere)
Exchange charge density
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
Averaging over a Fermi sphere we have :
(for spin-up band)
And considering also the spin-down density we have
the effective charge density:
The effective charge density is reduced because
the exchange correlations
This imply a renormalization of electron energy,
which is the basic starting point of Hartree-Fock
approximation
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
Stoner and Wohlfarth model for itinerant magnetism
The Stoner parameter I describes the energy reduction due to
electron correlation
Edmund Clifton Stoner
Magnetization
Energy renormalization of electrons sub-bands
where
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
Fermi distributions
~
evaluated at E=E±IR/2
and for µ=EF
Approximation:
f(x-Δx/2)-f(x+Δx/2) ~ -Δx f’(x) + (Δx)3 f’’’(x)/3! + ...
>0
<0
if the magnetization M>0
>0
Condition for a spontaneous ferromagnetism
We can evaluate this term at T=0
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
At T=0 we have:
Stoner criterion for ferromagnetism
where
Strong Coulomb interactions and/or large density of states at the Fermi
energy cause a spontaneous ferromagnetism!
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
If the Stoner criterion is not satisfied,the system do not have a spontaneous
magnetization.
However the Stoner term induces an enhancement of the Pauli’s susceptibility.
In an external applied magnetic field B
Zeeman term
Magnetization
B
Stoner enhancement of magnetic susceptibility
⇒ Coulomb interactions enhance the Pauli susceptibility.
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
We suppose that the electron density of the states is concentrated at EF
Spin-up bands
We set
Spin-down bands
and
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
Strong deviation in the critical behaviour near the Curie temperature TC
Experimental determined critical exponent : M~(1-T/TC)1/3 ≠(1-T/TC)1/2
Ni
TC= 631 K
The Stoner model does’nt take into account the excited states (magnetic excitations)!!
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
Pauli paramagnetism is associated to the spin polarization of
conduction bands. What about the orbital contribution?
A magnetic field applied to a metal produces cyclotronic orbits
which are distributed in quantized levels
Lev Davidovitch Landau
Energy eigenvalues
Landau tubes
Cyclotronic frequency
- The electron states are not uniformly spaced in k-space
- Eigenfunctions product of plane waves in z-direction
and a function of (kx2+ky2)1\2
- The Landau tubes are parallel to kz direction, resulting
in a cylindrical geometry.
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
A magnetic field applied on an electron gas break up the electron distribution,
and gives rise to the formation of Landau levels.
Landau diamagnetic susceptibility:
The diamagnetic Landau susceptibility is
weak compared to the Pauli susceptibility
All metals are paramagnetic, because χP>χL
Band structure effects
The effective mass m* of the electrons may be renormalized and the Fermi’s
density of states g(EF) will be enanced by a factor (me/m*).
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
Both the Pauli paramagnetism and the Landau diamagnetism will be
enhanced by (me/m*), and we can prove that:
The total susceptibility of a metal is then:
If m* < 1/√3 me ~ 0.58 me => χ < 0 => diamagnetic metal
Ex. Bismuth:
m*= 0.01 me => is a strong diamagnet
Notice also that the diamagnetism of core electrons in general is weaker with
respect to the contribution of conduction electrons
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
- We suppose to have a spatially modulated magnetic field:
- The electron gas will respond to this
perturbation and the magnetization will
be also spatially modulated:
Spatially varying magnetization
The q-dependent susceptibility is then
(Fermi surface)
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
- In analogy with the paramagnetic case, also the q-dependent
diamagnetic susceptibility can be calculated:
χL=-χP/3
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
- We consider the effect of a local perturbation of the magnetic field on
the magnetization of the free electron gas
- Magnetic field: delta function Fourier expanded
- The susceptibility of the electron gas have a spatial oscillatory
dependence of the form:
Real space susceptibility
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
- Provide a long-range exchange interaction between conduction
electrons and localized magnetic moments
- Effective in rare-earths and transition metals binary compounds where
5d conduction electrons interact with itinerant 3d electron of metals
- The resulting exchange interaction have an oscillatory behaviour:
At large distances (q>>1/kF)
- The exchange interaction can be positive (ferromagnetic) or negative
(antiferromagnetic) depending on the distances between the magnetic
moments
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
4f
- Hybridization between Fe 3d and 5d electrons
- Reduction of elemental Fe moment
- The net 3d and 5d moments are anti-parallel
- Local 4f-5d exchange interaction
L. Paolasini - LECTURES ON MAGNETISM- LECT.6
L. Paolasini - LECTURES ON MAGNETISM- LECT.6