Spin-flip processes

Spin-flip processes are very popular to explain the ultrafast demagneti-zation. The most important ones are electron-electron, electron-magnon and electron-phonon spin-flip scattering which are described in the fol-lowing. In principle other scattering mechanisms are imaginable, e.g.,

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electron-defect or electron-interface scattering, but these are assumed to be of minor importance [46].

1. Spin-flip scattering is often discussed in the simplified version of Elliott [63] and Yafet [64] where the matrix-elements and rate equa-tions are not calculated directly. The spin-flip relaxation time T₁ is related with the non-spin-flip (Drude) relaxation time τ. The Elliott-Yafet relation reads

1 T1

=pb²1

τ (4.2)

wherepis a material-dependent proportionality factor andb² is the spin-mixing factor explained in subsection 2.2.3. It was shown by Beuneu and Monod [65] thatpis between 1 and 10 for most metals.

The spin-mixing factorb² is calculated for several elements in refs.

[22, 23]. It is 0.025 for Ni and 0.024 for Fe using a thermal smearing of 25 meV. Koopmans et al. showed that an Elliott-Yafet parameter pb² well below 1 can be sufficient to explain the demagnetization experiments [66]. The Elliott-Yafet parameter for Ni and Fe is ap-proximately in the range between 0.02 and 0.3 (if 1 < p < 10).

Therefore it was concluded in ref. [23] that the demagnetization ex-periments could well be explained with the Elliott-Yafet mechanism.

Though, the Elliott-Yafet relation does not say anything about the underlying scattering mechanism.

2. Electron-electron scattering was theoretically investigated by Krauß et al. [67]. A simple band structure and simplifying assumptions for the Coulomb and the dipole matrix elements were used. Within this model the demagnetization in Ni and Co could be explained by electron-electron spin-flip processes. However, some of the same authors admitted an overestimation of the energy deposited by the excitation pulse [68] and hence, the result is not on a firm footing.

I want to make an important remark: If electron-electron spin-flip scattering is responsible for the demagnetization, one would have to explain where the spin angular momentum goes to. It can only stay in the electronic system, i.e., go to the orbital angular momentum which is a contradiction to Stamm’s experiments [47]. The authors of ref. [67] implicitly assume that the lattice is a perfect sink of

angular momentum which would require electron-phonon scatterings in addition.

3. Electron-magnon scattering was first suggested by Carpene et al.

[69]. Without spin-orbit coupling a magnon absorption or emission cannot change the magnetization because of the angular momentum conservation required for the single-scattering process. In the au-thors’ mind an electron scatters with a magnon decreasing the spin angular momentum in the following way: Because of the spin-orbit couplingLbbS=LbzSbz+ 1/2·(bL⁺Sb⁻+Lb⁻Sb⁺) a decrease of the spin angular momentum Sb⁻ creates an increase of the orbital angular momentumLb⁺ which is immediately quenched by the crystal field.

It is claimed that the crystal field quenching (also called orbital quenching) is a very fast process. I want to note that the crys-tal field quenching is probably also mediated via electron-phonon scattering processes. Because of ref. [47] an increase of the orbital angular momentum can only occur on a very short timescale (smaller than the experimental resolution of ref. [47]), and therefore one had to assume that the orbital angular momentum is transferred to the lattice by electron-phonon scatterings in that extremely short time.

Another publication showed experimentally [70] that magnon emis-sion by hot electrons indeed occurs on the fs timescale in Fe.

4. Most promising is certainly the electron-phonon spin-flip scatter-ing because the lattice is a possible sink for angular momentum whereas in electron-electron or electron-magnon scattering one has problems to argue where the spin angular momentum goes to. The microscopic three-temperature model showed that electron-phonon scattering combined with atomic spin-flips can explain the demag-netization experiments in principle [18]. However, several approxi-mations enter the microscopic three-temperature model. Ab-initio calculations by Carva et al. [71, 72] and Essert et al. [68, 73] came to the conclusion that electron-phonon scattering within a fixed band structure cannot explain the experimentally observed demagnetiza-tion. Only few approximations entered the ab-initio calculations. It was suggested in ref. [68] that a dynamic band structure which re-spects a varying exchange splitting should be used in the calculation instead. A change in the band structure and especially in the

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change splitting was observed in experiments [48, 50, 74] and could in principle change the results. In ref. [74] Stamm et al. observed a reduction of the spin-orbit coupling of about 6%. Carva et al. [71]

found out that a non-equilibrium electron distribution results in a higher demagnetization rate than a thermal electron distribution but still not enough to explain the experiments. This is in contradiction to Koopmans et al. [18] who claimed that the thermal electrons are responsible for the main demagnetization.

5. A very recent publication by Schellekens and Koopmans [75] sug-gests that single-electron flips in combination with atomic spin-flips (using the microscopic three-temperature model) can explain the demagnetization whereas single-electron spin-flips in a rigid-band structure alone can never reproduce the experimental demag-netization rates because of a lack of driving force of demagdemag-netization.

The authors showed that the microscopic three-temperature model naturally explains the demagnetization by a thermal disorder of the atomic magnetic moments. So, they claim that one should not only look at the longitudinal reduction of the atomic magnetic moment (length reduction) but also on the transverse reduction (disorder)³. This is only possible if both magnons and phonons are taken into account. One can imagine several processes where phonons and magnons are involved:

• Three-particle scattering processes between electron, magnon and phonon: Three-particle scattering processes are in general very unlikely. To the best of my knowledge it has never been discussed before in the literature of ultrafast demagnetization.

• Phonon-magnon scattering processes: Phonon-magnon scatter-ing is a very slow process in the order of 100 ps [76] (whereas electron-phonon scattering is in the order of 1 ps) and usually seen as irrelevant for Ni and Fe on a fs timescale [1, 18, 61].

• Depletion/diffusion of magnons: Magnons could diffuse into the substrate so that they deplete in the sample. In princi-ple this could be very similar to the dynamics suggested by Battiato et al. [77] discussed in subsection 4.2.6.

3It was always assumed that a disorder (transverse reduction) is too slow to explain a dynamics on the fs timescale since phonon-magnon scattering (which is necessary to explain Bloch’sT^3/2-law) is a very slow process on the 100 ps timescale [76].

• Combined electron-phonon and electron-magnon scattering pro-cesses: Electron-phonon scattering and electron-magnon scat-tering could happen one after another. Spin-up and spin-down electrons can both absorb or emit phonons and flip their spin.

However, only spin-down electrons can emit a magnon and flip their spin (↓→↑+magnon) and only spin-up electrons can ab-sorb magnons and flip their spin (↑+magnon→↓) due to an-gular momentum conservation which is required in electron-magnon single-scattering processes. A combined electron-pho-non and electron-magelectron-pho-non spin-flip scattering process could be:

↑+phonon→ ↓

↓ → ↑+magnon (4.3)

or effectively

↑+phonon→ ↑+magnon. (4.4)

The electron-phonon scattering transports the angular momen-tum to the lattice and the electron-magnon scattering changes the directions of the atomic magnetic moments which leads naturally to a demagnetization. It could well be that these combined electron-phonon and electron-magnon scattering pro-cesses are important, though, one has to check carefully how realistic this model is. Especially, one has to compare the electron-phonon and electron-magnon scattering rates (investi-gations on this line are under way in our group, see PhD thesis of Michael Haag and ref. [78]). They could be in the same order of magnitude or have a totally different order of magnitude. If the rates are different, the slower process determines the com-bined electron-phonon/electron-magnon scattering dynamics.

This list is certainly not complete and there are definitely even more possibilities involving phonons and magnons.

The ultrafast demagnetization process can also be described by a Landau-Lifshitz-Bloch equation [79] which is a generalized Gilbert equation including temperature and a longitudinal and transverse damping parameter. While the model includes nicely both longitu-dinal and transverse damping, the underlying microscopic mecha-nisms are not specified.

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In summary, the electron-phonon spin-flip scattering is more promising to explain the ultrafast demagnetization than the electron-electron and electron-magnon scattering alone since the sink for angular momentum (the lattice) is clear for electron-phonon scattering. However, ab-initio results including the very accurate results of the present thesis (see chap-ter 7) give hints to a minor role of electron-phonon scatchap-tering when calcu-lating with a fixed band structure. It has to be tested whether the results change for a dynamical change of the band structure (see chapter 7) and whether combined single-electron spin-flip and atomic spin-flip scattering processes suggested by Schellekens and Koopmans [75] are realistic.