magnetic excitations
DESCRIPTION
Review on theoretical models of magnetic excitations in localized and itinerant solids.TRANSCRIPT
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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
• Building a candidate first excited state: flipping a spin at lattice site 𝑅 starting from the fully
oriented ground state.
𝑅 =1
2𝑆𝑆−(𝑅) 0
• Not a proper eigenstate: the x/y spin components in the Hamiltonian can be expressed in
terms of raising and lowering operators as −1
2
𝑅,𝑅′𝐽 𝑅 − 𝑅′ 𝑆−(𝑅′) 𝑆+(𝑅)
𝐻 𝑅 involves a sum over other states with a single flipped spin.
• Lattice periodicity suggests a state akin to Bloch wavefunctions
𝑘 =1
𝑁
𝑅𝑒𝑖𝑘∙𝑅 𝑅
lowering is distributed equally along the chain.
• Schrödinger equation yields a dispersion relation. Subtracting the ground state energy:
휀 𝑘 = 2𝑆
𝑅
𝐽(𝑅)𝑠𝑖𝑛2(𝑘 ∙ 𝑅
2)
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• Evaluation of the expectation value of the spin correlation function:
𝑘 S⊥(𝑅) ∙ S⊥(𝑅′) 𝑘 =2𝑆
𝑁cos(𝑘 ∙ (𝑅 − 𝑅′))
• Einstein-Bose statistics can be used to evaluate the mean occupation number as a function of
temperature (the state differs from 0 by integer 𝑆 = 1). Each excited magnon reduces the
magnetization from its saturation value by 1.
• Low temperature limit: 휀 𝑘 ≅𝑆
2
𝑅𝐽(𝑅) (𝑘 ∙ 𝑅)2. By summing the spin contribution over a 3d grid of
allowed wavevectors with EB average we get𝑀 0 −𝑀(𝑇)
𝑀(0)∝ 𝑇
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 (𝑇
𝑇𝑐)
3
2 in ferromagnetic
Ge and metallic compounds. The law
holds well up to 0.6Tc. From [6]
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• The energy density of the magnon spectrum at low temperature is 𝑈 ≅ 0
∞ 𝜀𝑔 𝜀
𝑒
𝜀𝑘𝐵𝑇−1
𝑑휀 ∝ 𝑇5
2, so
that the specific heat of the ferromagnet should follow 𝑐 =𝜕𝑈
𝜕𝑇∝ 𝑇
3
2
• In lesser-dimensional systems the density of states 𝑔(휀) is less regular; the number of excited magnons and the internal energy diverge: excitations compromise the ground state stability.
• Consistency with Mermin-Wagner theorem: no continuous symmetry can be spontaneously broken at 𝑇 ≠ 0 in systems with short-range interactons, if 𝑑 ≤ 2; long wavelengths excitations are not split by any energy gap from the ground state.
• 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
• Definition of fermionic creation and annihilation operators: Ψ𝜎 𝑟 = 𝑎𝑗𝑓𝑗 𝑟 , where 𝑓𝑗( 𝑟)’s are a
complete basis or often a set of eigenfunctions.
• Fourier expansion of the e-e repulsion Hamiltonian: 𝑒2
𝑟𝑖𝑗=
1
𝑉 𝑞
4𝜋𝑒2
𝑞2 𝑒𝑖𝑞∙(𝑟𝑖−𝑟𝑗)
• We choose plane waves1
𝑉𝑒𝑖𝑘∙ 𝑟 , approximate eigenfunctions in a generic conduction band, to
define creation and annihilation operators.
• 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:
𝑉𝑒𝑥𝑐ℎ = −1
𝑉
𝑘12𝜎
2𝜋𝑒2
𝑘1 − 𝑘2
2 𝑛𝑘1𝜎𝑛𝑘2𝜎
as in the diagonal term all contributions vanish except those with 𝑘2 − 𝑘1 = 𝑞, 𝜎1 = 𝜎2 = 𝜎
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Stoner Model
• The molecular field Stoner theory starts with the approximate Hamiltonian
𝐻 = 𝑘𝜎
휀𝑘𝑎𝑘𝜎
ϯ𝑎𝑘𝜎 +
𝑈
2𝑁
𝑘12𝑞𝜎𝜎′𝑎
𝑘1+𝑞,𝜎
ϯ𝑎
𝑘2−𝑞,𝜎′
ϯ𝑎𝑘2,𝜎′𝑎𝑘1,𝜎
One-body band spectrum,
diagonal in the base of its
eigenfunctions
Intra-atomic Coulomb
integral 2nd quantized expression
of 2-bodies interactions
Where the 𝑞 = 0 mean value is screened by the distributed lattice positive charge and the remaining 𝑞dependence is omitted in favour of the constant Coulomb integral.
• 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 𝜎𝑞𝑘− = 𝑎𝑘+𝑞,↓
ϯ𝑎𝑘,↑
and ψ𝑒 = 𝑘
𝑓𝑘𝜎𝑞𝑘−ψ𝑔 (suitable linear combination of states with a single promoted electron to throw in
the Schrödinger equation)
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Stoner Excitations
𝐻
𝑘
𝑓𝑘𝜎𝑞𝑘−ψ𝑔 = (𝐸𝑔 + 휀)
𝑘
𝑓𝑘𝜎𝑞𝑘−ψ𝑔
• The Schrödinger equation must be developed by use of the canonical fermionic
(anti)commutation relations: 𝑎𝑗ϯ, 𝑎𝑘 = 𝛿𝑗𝑘 , 𝑎𝑗 , 𝑎𝑘 = 0, 𝑎𝑗
ϯ, 𝑎𝑘
ϯ= 0 (implying Pauli’s
principle!)
e.g., 𝐻𝐶𝑜𝑢𝑙 , 𝜎𝑞𝑘− =𝑈
𝑁
𝑘1𝑞′𝜎(𝑎
𝑘+𝑞+𝑞′↓
ϯ𝑎
𝑘1−𝑞′𝜎
ϯ𝑎𝑘1𝜎𝑎𝑘↑ − 𝑎
𝑘 +𝑞↓
ϯ𝑎
𝑘1−𝑞′𝜎
ϯ𝑎𝑘1𝜎𝑎
𝑘−𝑞′↑)
• An extension to the Hartree-Fock approximation, called Random Phase Approximation, allows us to keep only diagonal elements (𝑘1 = 𝑘 + 𝑞′) when evaluating the effect of the operator on the tentative wavefunction; the other terms average out, being the phase rapidlyspatially-varying.
• The secular equation can now be expressed in terms of simple number operators in the separate spin-bands.
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• With the definition of a degree of spin polarization ξ =𝑁↑−𝑁↓
𝑁𝑒,
• Introduction of a Zeeman energy term for the interaction with an external field,
• A further correction that is necessary because the wavefunction doesn’t vanish anymorewith the application of the destruction operator 𝑎𝑘↑ (in fact the chosen ψ𝑔 is the ground state of the Hartree-Fock hamiltonian, not of the Stoner approximant),
The eigenvalue equation can finally be derived:
𝑁
𝑈=
𝑘
𝑛𝑘↑(1 − 𝑛𝑘+𝑞↓)
휀𝑘+𝑞 − 휀𝑘 + 2𝜇𝐵𝐻 +𝑁𝑒𝑈𝑁 ξ − ħω
• The numerator doesn’t vanish if the spin-flip happens from an occupied to an empty state.
• In the limit 𝑁 → ∞, the dispersion relation is ħω = 휀𝑘+𝑞 − 휀𝑘 + ∆
• The energy depends on 𝑘, so that Stoner excitations form a continuum with looseboundaries.
• Δ is the offset between the spin-split bands.
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Domain of the allowed spin-flip
transitions for fixed 𝑞, represented by
the shaded, non-overlapping region of
majority and minority Fermi spheres.
∆ +ħ2
2𝑚(𝑞2 ± 2𝑞𝑘𝐹)
𝒌𝑭↓
Incomplete overlap of the
Fermi spheres;
gapless excitation branch
In the low temperature limit,
these states reduce the
spontaneous magnetization
by 𝑀 0 −𝑀(𝑇)
𝑀(0)∝ 𝑇2
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Spin-Wave Branch
• 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 𝑞𝑚𝑎𝑥. In fact, such states existeven in the itinerant model.
• The long-wavelength dispersion relation can be obtained from the secular equation by Taylor expansion,
as long as ħ𝜔 − 2𝜇𝐵𝐻 ≪ ∆0=𝑁𝑒𝑈
𝑁ξ
ħ𝜔 ≅ 2𝜇𝐵𝐻 +ħ2𝑞2
2𝑚(1 − 1.3
𝑁휀𝐹
𝑁𝑒)
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 ħ𝜔𝑚𝑎𝑥 are
strongly damped by interaction with the electrons-
holes continuum and decay after short lifetimes.
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Inelastic neutron scattering
𝐸2 − 𝐸1 = ±ħ𝜔(±𝑝2 − 𝑝1
ħ+ 𝐺)
Conservation of energy, conservation of
crystal momentum (discrete symmetry) :
Measurement of dispersion relation
Triple-axis spectrometer is needed to
perform a scan in fixed- 𝑞 or fixed-휀 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
Ewald’s sphere
∆λ = −2𝑑 sin 𝜗 ∆𝜗
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
• Polarizing crystals (𝐶𝑢2𝑀𝑛𝐴𝑙)
• Polarizing mirrors
• Polarizing filters ( 3𝐻𝑒)
Spin-dependant nuclear cross section
for absorption, close to resonance
𝜎 = 𝜎0 ± 𝜎𝑃
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 𝑔𝐼𝜇𝑁 𝐼(𝐼 + 1) )
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Detection of Bragg peaks
with fixed- 𝑞 scan
Anisotropic dispersion of spin waves in the
antiferromagnet 𝑀𝑛𝐹2, 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 4.2°K." Journal of Applied Physics 35.3 (1964): 998-99. Web.