Outline
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Para-, ferro-, antiferro-, ferrimagnets, ...
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Classical theory
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Langevine theory of paramagnetism, Curie law
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Weiss molecular field theory, TC, Curie-Weiss law
Quantum theory of dia- and paramagnetism
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Larmor, van Vleck, Brillouin function, Pauli
Interaction of moments
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Heitler-London model of H2 and exchange interaction
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Heisenberg model hamiltonian
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mean field approximation, Curie-Weiss law, TC
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quantum Heisneberg model, magnons, Bloch's law
Stoner model
Types of magnetic structures
Classical theory of paramagnetism
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Model of Langevin – magnetic moment as a vector in magnetic field
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Energy of moment m in field H=(0,0,H):
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Mean value of magnetization on atom
where Langevin function is defined as
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Susceptibility (for large T)
Curie law (inverse proportionality to T)
Weiss molecular field theory
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Attempt to explain magnetic order
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Introduction of an internal field caused by neighboring atomic moments
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Huge fields ~1000T! No physical interpretation by Weiss
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In zero external magnetic field, using Langevin theory
we get
→ graphical solution:
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For T>Tc there is only a solution M=0
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Critical temperature
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Susceptibility
→ Curie-Weiss law
Towards quantum theory
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Langevin's and especially Weiss theories describe a wide range of magnetic
phenomena relatively well
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paramagnetism, ferromagnetism, antiferromagnetism (Néel – Nobel prize 1970),
ferrimagnetism – both latter cases can be qualitatively described as two nonequivalent sublattices generating the internal magnetic field on each other
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lack of mechanism for the internal magnetic field
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the Langevin's function does not fit too well for many systems
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no explanation of magnetocrystalline anisotropy, metamagnetic transitions, etc.
Quantum theory
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improves over the Langevin's function and provides explanation of the internal
magnetic field
insight into MAE, crystal field excitations, ...
complete ab initio theory of magnetism consistently treating localized and band
character of electrons is still lacking
Macroscopic definitions
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Thermodynamics: free energy
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Magnetization
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Susceptibility
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Force in inhomogeneous magnetic field
Electrons in magnetic field
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change of canonical momentum operator
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the kinetic energy operator becomes
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add interaction of spins with magnetic field
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summary of new terms in Hamiltonian
Perturbation theory
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the new terms are small at fields which we can produce in laboratories,
therefore we treat it as a perturbation
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susceptibility is 2nd order in H
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changes in energy levels up to 2nd order in H:
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the linear term is of the order of ~1 meV in fields ~10 T, the other terms are
3-4 orders of magnitude smaller
individual terms lead to a dia- or paramagnetic behavior of the system in
magnetic field
Larmor diamagnetism
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insulators, closed shells in ground state (L=S=J=0), i.e., only last term
contributes
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ground-state energy change due to mag. Field:
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if all excited states are high in energy, then
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This is Larmor diamagnetic susceptibility (negative), typically ~10-5
Hund's rules
(1) Maximize the total spin S.
A.k.a. “the bus seat rule”.
Maximizing spin reduces screening and
allows electrons to get closer to cores.
(2) While fulfilling (1), maximize the
total orbital momentum L.
Classically: orbiting in the same direction reduces probability
that electrons meet, i.e., reduces repulsion.
(3) For less than half-filled shells J = |L-S| and for more
than half-filled shells J = L + S.
This is a consequence of spin-orbit coupling
Van-Vleck paramagnetism
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if J=0, but not L or S, the first term vanishes
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we get two non-zero terms in susceptibility
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first term is Larmor diamagnetism
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second term is Van-Vleck paramagnetism (positive)
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if the first excited state is close in energy to the ground state, more
complicated formulas apply
Paramagnetism
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For atoms with unfilled shells with nonzero S, L and J, the first term is nonzero
and dominates
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We have (2J+1) degenerate state in zero field
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In nonzero field we need to diagonalize (2J+1)x(2J+1) matrix with elements
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Wigner-Eckart theorem states
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with Landé factor
Paramagnetism
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Thanks to Wigner-Eckart theorem we see that the matrix is actually diagonal
we can interpret (within the lowest J-multiplet)
as a magnetic moment of the ion
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To get susceptibility we need to consider all these (2J+1) states split by
magnetic field
Free energy
Brillouin function
From free energy we get magnetization
where Brillouin function is defined by
At low temperatures BJ=1
At high temperatures
Curie's law
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At high temperatures we get for susceptibility
i.e., susceptibility is inverse-proportional to temperature – Curie law
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This paramagnetic susceptibility is at room temperature of the order of 10 -210-3 and thus dominates the diamagnetic contribution
Comparing to the Curie-law derived in the classical case, we can define an
effective moment / effective Bohr magneton number p:
Example: rare-earth paramagnets
Good agreement theoryexperiment,
except for Sm & Eu
Sm & Eu have low-lying excited
states, which we neglected
Example: 3d transition metals
For 3d transition metals Curie's
law works if we assume L=0
Quenching of orbital momentum
due to crystal field splitting
Modification of the third Hund's
rule
Magnetism of conduction electrons
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Delocalized conduction electrons
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Magnetic field shifts energy levels by
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Since
we can expand density of states
and obtain for number of occupied states
i.e., the magnetization is
Pauli paramagnetism
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From magnetization
we get a susceptibility
which is independent of temperature
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This contribution is called Pauli paramagnetism
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It is of order 10-6, i.e., comparable to Larmor diamagnetic contribution
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Note: conduction electrons also have Landau diamagnetic contribution to
susc., see A&M
Sources (29.4. & 6.5.)
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Main source:
Ashcroft & Mermin: Solid State Physics
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Chapter 31: Diamagnetism and Paramagnetism (29.4.)
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Chapter 32: Electron Interactions and Magnetic Structure
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Chapter 33: Magnetic Ordering
See also:
Mohn: Magnetism in the Solid State
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Chapter 7: Heisenberg Hamiltonian
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Section 16.1: Heitler-London Theory for the Exchange Field
Interaction between moments
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We developed a quantum theory of magnetic susceptibility originating from
various terms (Larmor, van Vleck, nonzero-J paramagnetism, Pauli)
So far we included no interaction between moments → no mechanism for a
spontaneous magnetization in zero external magnetic field
Possible sources of such interaction:
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Dipolar – too weak, ~0.1meV
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Spin-orbital – stronger in heavy elements, up to ~1eV for actinides
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Electro-static + Pauli principle – the strongest, ~1eV and more
Heitler-London model of hydrogen molecule as a starting point for
constructing model Hamiltonians for interactions of localized moments
Hydrogen molecule (summary)
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Two electron system, four possible spin arrangements
We can classify them as
spin singlet (S=0) and
spin triplet (S=1):
The ground state is singlet
If we neglect all the higher excited states and restrict ourselves to singlet &
triplet, we can reproduce the energy levels by the following model
Hamiltonian expressed in spin-space only:
Heitler-London model of H2
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2 hydrogen atoms, 2 electrons, assumption that there is always one electron
close to every proton → symmetric and antisymmetric wavefunctions
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the complete Hamiltonian can be written as
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energies of the spatially symm./antisymm. states are
where
Heitler-London model of H2
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Let's add spins → possible spin configurations for two electrons:
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S=0, MS=0 – spin singlet – antisymmetric in spins
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S=1, MS=-1,0,1 – spin triplet – symmetric
Total 2-electron wave-function is always antisymmetric → lower lying state
that is symmetric in orbital space must have antisymmetric spin part, i.e.,
spin singlet & higher lying state, which is antisymmetric in orbital space will
have be a spin triplet
Using relations
the same energy levels can be directly obtained by
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we introduced “exchange constant” → generalization gives Heisenberg
model hamiltonian
Heisenberg model
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Generalization of the situation with hydrogen molecule:
Extracting Jij is not a trivial problem and to some extent it is still not
completely solved
Solving the quantum model itself without approximations is computationally
unsolvable except for smallest systems
Mechanisms/sources of Jij (Olle's lecture next week):
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Direct exchange
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Super-exchange
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Indirect exchange (RKKY)
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Itinerant exchange
Magnetic structures
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J>0 for nearest neighbors → ferromagnet
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J<0 for nearest neighbors → antiferromagnet
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non-negligible J's for more distant neighbors or in geometries leading to
magnetic frustrations → more complicated magnetic structures, e.g., spin
spirals, non-collinear structures, etc.
Mean-field theory – Weiss field
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rewrite the Heisenberg model, including field:
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this looks like a set of spins in effective field
which does not depend on i due to periodicity
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yet, the effective (Weiss) field is an operator → the mean-field theory
replaces it with its thermodynamic mean value
Critical temperature
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Taking mean-field approximation and zero external field we obtain equation
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When M(T) goes to zero,
and we get
the mean-field approximation of the magnetic transition temperature
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Weak points: over-estimation of TC, wrong low-temperature behavior, also
around TC
More advanced methods
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Random-phase approximation
Monte-Carlo simulations – exact answers within the classical Heisenberg
model, though demanding calculations
Using numerical methods we can also get M(T) within all three methods
Doing Monte-Carlo Heisenberg model quantum-mechanically is still an
active field of research
Ground state of ferromagnet
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At T=0, all moments aligned parallel, i.e., total moment NS → |NS,NS>
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Rewrite Heisenberg Hamiltonian using raising and lowering operators:
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Applying the HH on the |NS,NS> gives
i.e., it is an eigenstate of the HH
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No lower energy-state of ferromagnetic HH exists → it is a ground state |0>
Hint:
Low-T excitations of ferromagnet
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Lowering spin projection by one at one site is not an eigenstate of HH:
Because
can construct linear combinations
, i.e., translational invariance, we
which are eigenstates of Heisenberg Hamiltonian:
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One can show that
Low temperature magnetization
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Superposition of magnons (like for phonons) is only an approximation here,
but “quite OK” for low-lying excited states
The magnetization is reduced by one per magnon, i.e.
At low temperatures only the lowest
energy excitations happen, and for these
i.e.,
Bloch's 3/2 law
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Mermin-Wagner theorem → no magnetization in 2D or 1D
Ground state of antiferromagnet
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Intuitively: arrangement of alternating up/down moments
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It is not an eigenstate of Heisenberg Hamiltonian!
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Assume S=1/2 chain and apply
i
j
i j
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In classical case (spins as vectors) this is a ground state with lowest energy
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For nearest-neighbor (NN) interaction the following bounds are valid:
which actually coincide in the classical case, where