Electron-phonon mediated spin-ﬂip as driving mechanism for ultrafast magnetization dynamics in 3 dferromagnets Theodor Griepe1 2and Unai Atxitia1 2

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Electron-phonon mediated spin-ﬂip as driving mechanism for ultrafast magnetization

dynamics in 3dferromagnets

Theodor Griepe1, 2 and Unai Atxitia1, 2, ∗

1Dahlem Center for Complex Quantum Systems and Fachbereich Physik,

Freie Universität Berlin, 14195 Berlin, Germany

2Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain

(Dated: October 28, 2022)

Despite intense experimental eﬀort, theoretical proposals and modeling approaches, a lack of

consensus exists about the intrinsic mechanisms driving ultrafast magnetization dynamics in 3d

ferromagnets. In this work, we ﬁnd evidence of electron-phonon mediated spin-ﬂip as the driving

mechanism for the ultrafast magnetization dynamics in all three 3dferromagnets; nickel, iron and

cobalt. We use a microscopic three temperature model with parameters calculated from ﬁrst-

principles, which has been validated by direct comparison to the electron and lattice dynamics

extracted from previous experiments. By direct comparison to the experimentally measured

magnetization dynamics for diﬀerent laser ﬂuence, we determine the spin-ﬂip probability of each

material. In contrast to previous ﬁndings but in agreement to ab-initio predictions, we ﬁnd that

relatively small values of the spin-ﬂip probability enable ultrafast demagnetization in all three 3d

ferromagnets.

The discovery of femtosecond spin dynamics in the 3d

ferromagnet nickel opened the door for magnetic non-

volatile data processing on ultrafast time scales[1]. Since

then, ultrafast demagnetization has been demonstrated

in other transition metals [2,3], rare-earth metals [4–

7] and their alloys [8], semiconductors [9] and insulators

[10]. The fundamental mechanisms behind laser-induced

ultrafast magnetization dynamics, such as the nature of

spin-ﬂip scattering or the role of spin-polarized transport,

are still object of debate [11–16]. In the ﬁrst works,

Koopmans et al. [11,12] proposed electron-phonon

scattering mediated spin-ﬂip as the main mechanism

driving the ultrafast magnetization dynamics in 3d

ferromagnets. This idea was supported by ﬁtting the

so-called microscopic three temperature model (M3TM)

to ultrafast demagnetization traces in nickel and cobalt

[12]. Later on, it was argued that other mechanisms

should signiﬁcantly contribute to the demagnetization

since the ab-initio estimated values for the spin-ﬂip rates

were too small to produce the observed demagnetization

[14,17]. The main drawback of the M3TM as

originally used by Koopmans et al. [12] is the large

number of ﬁtting parameters. This has been partially

solved in subsequent works in nickel [15,18] where

the temperature dependence of the electron, lattice and

spin speciﬁc heats at equilibrium were directly taken

from experiments[19]. By doing so, the spin-ﬂip rates

necessary to ﬁt the experimental results reduced and

approached to ab-initio estimations. Besides electron-

phonon spin-ﬂip scattering, other alternative mechanisms

have been put forward, such as Coulomb scattering

of Elliot-Yafet type [3] and electron-magnon scattering

[2,20]. The latter aims to describe angular momentum

conservation due to an increase of the orbital momenta

upon creation of a magnon. However, experimental

studies suggest that angular momentum dissipation

to orbital momenta cannot explain the magnetization

quenching in iron and cobalt on this timescale and

support the claim of the lattice to absorb excess angular

momentum[21–24]. Superdiﬀusive spin currents have

been also proposed as dominant mechanism driving the

ultrafast demagnetization, however this process is only

eﬃcient when magnets are interfaced by nonmagnetic

metals [13,25]. Whereas in thin ﬁlms on insulating

substrates, superdiﬀusive transport plays no signiﬁcant

role in the demagnetization process of 3dferromagnets

[15]. We note that recent ab-initio time-dependent

density functional theory have shown to be accurate in

the early stages of the optically induced demagnetization

process before lattice vibrations set in [26]. However,

experimental evidence suggests that angular momentum

from the magnetic system is transferred to the lattice

[23,24]. These processes are not within reach of ab-initio

methods, and, therefore, their investigation needs to rely

on semi-phenomenological models.

Femtosecond-resolved experimental techniques that

measure the magnetization dynamics as well as the

electronic and lattice dynamics are nowadays accessible.

The electron energy distribution has been measured

in nickel using time- and angle-resolved photo-emission

spectroscopy (tr-ARPES) [27,28]. It has been

demonstrated that only a few tens of femtoseconds after

the laser hits the sample, the electron system thermalizes

into a Fermi-Dirac distribution with a well-deﬁned

electron temperature. The lattice energy dynamics has

been measured using time-resolved electron diﬀraction

in the three 3dferromagnets [24,29,30]. It has

been demonstrated that the so-called two- temperature

model (2TM) describes well the measured electron and

lattice temperature dynamics for parameters calculated

from ﬁrst principles, and by including the energy

exchange between the spin and electron systems within

arXiv:2210.15269v1 [cond-mat.mtrl-sci] 27 Oct 2022

an atomistic spin model framework [29,30]. However, the

lack of a theoretical model describing the magnetization

dynamics questions the full validity of the atomistic spin

model.

In this work, we demonstrate that the M3TM is

able to quantitatively describe the ultrafast dynamics

in 3dferromagnets, including electron, lattice and spin

degrees of freedom. To do so, we replicate the dynamics

of the experimentally measured electron and lattice

temperatures as well as magnetization for a range of laser

ﬂuences in all three 3dferromagnetic transition metals

iron, nickel and cobalt. Notably, we are able to reduce the

number of ﬁt parameters to only the spin-ﬂip probability.

We achieve so by using system parameters that have been

calculated from ab-initio methods and experimentally

validated. By contrast to previous works, we ﬁnd that the

spin-ﬂip probabilities agree with to those calculated via

ab-initio [14,17]. We conclude that our work evidences

electron-phonon spin-ﬂips as intrinsic mechanism driving

ultrafast magnetization dynamics in 3dferromagnets.

Our model is based on an extension to the M3TM [12,

31]. The energy ﬂow dynamics are described by the two-

temperature model (see Fig. 1),

dTe

dt =gep(Tp−Te) + S(t) + ˙

Qe−s(1)

dTp

dt =−gep(Tp−Te).(2)

When a metallic thin ﬁlm is subjected to an optical

laser pulse, only the electrons are excited by the photon

electric ﬁeld. Initially, the absorbed energy is barely

transferred to the lattice and consequently the electron

system heats up. On the timescale of 100 fs, the

electron temperature will rise far above the critical

temperature, Tc. The magnetic system responds to this

fast temperature rise by reducing its magnetic order on

similar time scales. The electron and lattice systems are

assumed to be thermalized so that their energy can be

described by a temperature, Tefor the electrons and Tp

for the phonons. In Eq. (1), the absorbed laser pulse

power is represented by a Gaussian function, S(t) =

S0∗G(τp), where τpis the pulse duration. The electron

heat capacity is Ce. The electron-phonon coupling allows

for temperature equilibration of hot electrons and the

lattice on the time scale determined by the ratio gep/Ce.

Since gep accounts only for spin-conserving scattering

events, in Eq. (1) we include a term that accounts for

the ﬁnite energy cost (gain) of a spin-ﬂip and couple

it to the electron dynamics: ˙

Qe−s=J0m˙m/Vat, with

J0/3=[S/(S+ 1)]kBTcin the MFA, mis the reduced

magnetization, J0is the exchange energy, Vat the atomic

volume and eﬀective spin S. In phase I (dm/dt < 0)

the energy cost of an electron-phonon mediated spin-ﬂip

of probability asf in minority direction is deducted from

the electronic energy, while in phase II (dm/dt > 0)

the direction of energy ﬂow is reverted (Fig. 1). In

FIG. 1. Energy absorbed by the spinless electrons

S(t)is distributed to the lattice (gep) and localized spins

(˙

Qe−s). Energy is transferred from electron to spin

during demagnetization (I), and spin to electrons during

remagnetization (II). The angular momentum from electron-

phonon mediated spin-ﬂips, with probability asf , is implicitly

exchanged with lattice.

other works the energy ﬂow between the spin and the

electron systems has been taken into account by adding

it to the electron speciﬁc heat, Ce→Ce+Cs, where Cs

is the equilibrium spin speciﬁc heat [27,29]. However,

recent works suggest that this energy ﬂow needs to

be calculated through the spin Hamiltonian for non-

equilibrium spin conﬁgurations [29,30]. An important

aspect of the two-temperature model in Eqs. (1) and

(2) is the exact value of Ce,Cp, and gep. In this work,

we use the ab-initio calculated parameters, which were

already used before in Ref. [29,30]. The parameters

are temperature dependent and were already validated

by direct comparison to the ultrafast lattice dynamics

in iron, nickel and cobalt [29,30]. The electronic and

lattice heat capacities agree very well with experimental

data [19,32]. The magnetization dynamics is calculated

using the M3TM for ﬁnite spin values, for details of

the M3TM see Supplemental Material and Ref. [31].

Within this model, the rate parameter deﬁning the

magnetization dynamics scales linearly with the so-called

spin-ﬂip probability asf . Here, we use S= 1/2for nickel,

S= 2 for iron and S= 3/2for cobalt [33]. For these

values, within the MFA, the equilibrium magnetization

as function of temperature is well reproduced for all

three 3dferromagnets. Figure 2shows an example

of the comparison of our model to experimental data

for the magnetization [Fig. 2(a)], electron [Fig. 2(b)]

and lattice [Fig. 2(c)] temperature dynamics in nickel.

Vertical orange line denotes the transition from phase I

(dm/dt < 0) to phase II (dm/dt > 0). The electron

temperature is retrieved from tr-ARPES measurements

on a 400 nm nickel ﬁlm, where the electronic energy

distribution around the Fermi edge is measured and ﬁtted

to the Fermi-Dirac distribution, yielding the electronic

temperature [27]. The lattice dynamics are retrieved

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Electron-phononmediatedspin-ipasdrivingmechanismforultrafastmagnetizationdynamicsin3dferromagnetsTheodorGriepe1,2andUnaiAtxitia1,2,1DahlemCenterforComplexQuantumSystemsandFachbereichPhysik,FreieUniversitätBerlin,14195Berlin,Germany2InstitutodeCienciadeMaterialesdeMadrid,CSIC,Cantoblanco,28049Madri...

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Electron-phonon mediated spin-ﬂip as driving mechanism for ultrafast magnetization dynamics in 3 dferromagnets Theodor Griepe1 2and Unai Atxitia1 2

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