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.