magnetic excitations
DESCRIPTION
Review on theoretical models of magnetic excitations in localized and itinerant solids.TRANSCRIPT
Magnetic Excitations
A conceptual overview in localized and itinerant paradigms
Alberto Beccari 24/09/2015
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)
• 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]
• 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.
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
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)
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.
• 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.
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
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.
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
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
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) )
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]
Recap
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.