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Magnetic Excitations
A conceptual overview in localized and itinerant paradigms
Alberto Beccari 24/09/2015
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Spin waves in the Heisenberg ferromagnet
β’ Building a candidate first excited state: flipping a spin at lattice site π starting from the fully
oriented ground state.
π =1
2ππβ(π ) 0
β’ Not a proper eigenstate: the x/y spin components in the Hamiltonian can be expressed in
terms of raising and lowering operators as β1
2
π ,π β²π½ π β π β² πβ(π β²) π+(π )
π» π involves a sum over other states with a single flipped spin.
β’ Lattice periodicity suggests a state akin to Bloch wavefunctions
π =1
π
π πππβπ π
lowering is distributed equally along the chain.
β’ SchrΓΆdinger equation yields a dispersion relation. Subtracting the ground state energy:
ν π = 2π
π
π½(π )π ππ2(π β π
2)
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β’ Evaluation of the expectation value of the spin correlation function:
π Sβ₯(π ) β Sβ₯(π β²) π =2π
πcos(π β (π β π β²))
β’ Einstein-Bose statistics can be used to evaluate the mean occupation number as a function of
temperature (the state differs from 0 by integer π = 1). Each excited magnon reduces the
magnetization from its saturation value by 1.
β’ Low temperature limit: ν π β π
2
π π½(π ) (π β π )2. By summing the spin contribution over a 3d grid of
allowed wavevectors with EB average we getπ 0 βπ(π)
π(0)β π
3
2, which better fits the experiments than
the exponential decay in molecular field theory.
Ratio of the magnetization to its saturation
value as function of (π
ππ)
3
2 in ferromagnetic
Ge and metallic compounds. The law
holds well up to 0.6Tc. From [6]
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β’ The energy density of the magnon spectrum at low temperature is π β 0
β ππ π
π
πππ΅πβ1
πν β π5
2, so
that the specific heat of the ferromagnet should follow π =ππ
ππβ π
3
2
β’ In lesser-dimensional systems the density of states π(ν) is less regular; the number of excited magnons and the internal energy diverge: excitations compromise the ground state stability.
β’ Consistency with Mermin-Wagner theorem: no continuous symmetry can be spontaneously broken at π β 0 in systems with short-range interactons, if π β€ 2; long wavelengths excitations are not split by any energy gap from the ground state.
β’ In 2D, any kind of anisotropy (shape, dipolar, spin-orbit) can invalidate the theorem and restore ferromagetism.
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Formalism of second quantization: exchange interaction
β’ Definition of fermionic creation and annihilation operators: Ξ¨π π = ππππ π , where ππ( π)βs are a
complete basis or often a set of eigenfunctions.
β’ Fourier expansion of the e-e repulsion Hamiltonian: π2
πππ=
1
π π
4ππ2
π2 πππβ(ππβππ)
β’ We choose plane waves1
ππππβ π , approximate eigenfunctions in a generic conduction band, to
define creation and annihilation operators.
β’ Evaluating the first-order correction from the set of two-bodies Coulomb interactions on the ground
state of the electron gas, we get an expression for the exchange term:
πππ₯πβ = β1
π
π12π
2ππ2
π1 β π2
2 ππ1πππ2π
as in the diagonal term all contributions vanish except those with π2 β π1 = π, π1 = π2 = π
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Stoner Model
β’ The molecular field Stoner theory starts with the approximate Hamiltonian
π» = ππ
νππππ
Ο―πππ +
π
2π
π12πππβ²π
π1+π,π
Ο―π
π2βπ,πβ²
Ο―ππ2,πβ²ππ1,π
One-body band spectrum,
diagonal in the base of its
eigenfunctions
Intra-atomic Coulomb
integral 2nd quantized expression
of 2-bodies interactions
Where the π = 0 mean value is screened by the distributed lattice positive charge and the remaining πdependence is omitted in favour of the constant Coulomb integral.
β’ Excited states are built from the ground, ferromagnetic state by promoting a majority-spin electron to
the minority spin-split band: in the language of second quantization πππβ = ππ+π,β
Ο―ππ,β
and Οπ = π
πππππβΟπ (suitable linear combination of states with a single promoted electron to throw in
the SchrΓΆdinger equation)
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Stoner Excitations
π»
π
πππππβΟπ = (πΈπ + ν)
π
πππππβΟπ
β’ The SchrΓΆdinger equation must be developed by use of the canonical fermionic
(anti)commutation relations: ππΟ―, ππ = πΏππ , ππ , ππ = 0, ππ
Ο―, ππ
Ο―= 0 (implying Pauliβs
principle!)
e.g., π»πΆππ’π , πππβ =π
π
π1πβ²π(π
π+π+πβ²β
Ο―π
π1βπβ²π
Ο―ππ1πππβ β π
π +πβ
Ο―π
π1βπβ²π
Ο―ππ1ππ
πβπβ²β)
β’ An extension to the Hartree-Fock approximation, called Random Phase Approximation, allows us to keep only diagonal elements (π1 = π + πβ²) when evaluating the effect of the operator on the tentative wavefunction; the other terms average out, being the phase rapidlyspatially-varying.
β’ The secular equation can now be expressed in terms of simple number operators in the separate spin-bands.
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β’ With the definition of a degree of spin polarization ΞΎ =πββπβ
ππ,
β’ Introduction of a Zeeman energy term for the interaction with an external field,
β’ A further correction that is necessary because the wavefunction doesnβt vanish anymorewith the application of the destruction operator ππβ (in fact the chosen Οπ is the ground state of the Hartree-Fock hamiltonian, not of the Stoner approximant),
The eigenvalue equation can finally be derived:
π
π=
π
ππβ(1 β ππ+πβ)
νπ+π β νπ + 2ππ΅π» +ππππ ΞΎ β Δ§Ο
β’ The numerator doesnβt vanish if the spin-flip happens from an occupied to an empty state.
β’ In the limit π β β, the dispersion relation is Δ§Ο = νπ+π β νπ + β
β’ The energy depends on π, so that Stoner excitations form a continuum with looseboundaries.
β’ Ξ is the offset between the spin-split bands.
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Domain of the allowed spin-flip
transitions for fixed π, represented by
the shaded, non-overlapping region of
majority and minority Fermi spheres.
β +Δ§2
2π(π2 Β± 2πππΉ)
ππβ
Incomplete overlap of the
Fermi spheres;
gapless excitation branch
In the low temperature limit,
these states reduce the
spontaneous magnetization
by π 0 βπ(π)
π(0)β π2
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Spin-Wave Branch
β’ The previous slide took for granted a collective excitation analogous to the excited states of the Heisenberg ferromagnetic chain, that dips into the Stoner continuum at ππππ₯. In fact, such states existeven in the itinerant model.
β’ The long-wavelength dispersion relation can be obtained from the secular equation by Taylor expansion,
as long as Δ§π β 2ππ΅π» βͺ β0=πππ
πΞΎ
Δ§π β 2ππ΅π» +Δ§2π2
2π(1 β 1.3
πνπΉ
ππ)
Spin-waves excited in an
inelastic neutron scattering
experiment on hcp Co and
measured with a TOF
spectrometer.
which is parabolic as in the Heisenberg model.
Excitations with energy greater than Δ§ππππ₯ are
strongly damped by interaction with the electrons-
holes continuum and decay after short lifetimes.
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Inelastic neutron scattering
πΈ2 β πΈ1 = Β±Δ§π(Β±π2 β π1
Δ§+ πΊ)
Conservation of energy, conservation of
crystal momentum (discrete symmetry) :
Measurement of dispersion relation
Triple-axis spectrometer is needed to
perform a scan in fixed- π or fixed-ν mode
(through angular degrees of freedom)
Moving parts are mounted over compressed-air
pads to ease constraint and low-friction
displacement
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Monochromators
Ewaldβs sphere
βΞ» = β2π sin π βπ
Some focusing can be
provided by curved cuts
Different crystals are chosen for
different wavelengths to
improve the neutron flux
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Polarization of neutron beams
Necessity to distinguish magnon
scattering from nonmagnetic, low-
energy excitations
β’ Polarizing crystals (πΆπ’2πππ΄π)
β’ Polarizing mirrors
β’ Polarizing filters ( 3π»π)
Spin-dependant nuclear cross section
for absorption, close to resonance
π = π0 Β± ππ
Magnetic particle optics, total external
reflection for one spin state happens
between two critical angles
Supermirror bender array
Nuclear and magnetic structure factors for
Bragg scattering can compensate for a
given diffraction peak and spin
polarization(Stern-Gerlach splitting would require huge
magnetic fields due to low ππΌππ πΌ(πΌ + 1) )
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Detection of Bragg peaks
with fixed- π scan
Anisotropic dispersion of spin waves in the
antiferromagnet πππΉ2, measured with TOF
spectrometer. Note the linear fit valid at low
wavelength.
From [7]
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Recap
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REFERENCES1. Ashcroft, Neil W., and N. David Mermin. Solid State Physics.
2014. Print.
2. Blundell, Stephen. Magnetism in Condensed Matter. Oxford UP, 2003. Print.
3. Yosida, Kei. Theory of Magnetism. Heidelberg: Springer-Verlag, 1996. Print.
4. Crangle, John. Solid State Magnetism. New York: Van Nostrand Reinhold, 1991. Print.
5. Stewart, Ross. Polarized Neutrons. Rep. Science & Technology Facilities Council, n.d. Web.
6. Holtzberg, F., T. R. McGuire, S. Methfessel, and J. C. Suits. "Ferromagnetism in Rare-Earth Group VA and VIA Compounds with Th3P4 Structure." Journal of Applied Physics 35.3 (1964): 1033-038. Web.
7. Low, G. G., and A. Okazaki. "A Measurement of Spin-Wave Dispersion in MnF2 at 4.2Β°K." Journal of Applied Physics 35.3 (1964): 998-99. Web.