All optical control of magnetization in quantum confined ultrathin magnetic metals

All-optical control dynamics of magnetization in sub-10 nm metallic thin films are investigated, as these films with quantum confinement undergo unique interactions with femtosecond laser pulses. Our theoretical analysis based on the free electron model shows that the density of states at Fermi level (DOSF) and electron–phonon coupling coefficients (Gep) in ultrathin metals have very high sensitivity to film thickness within a few angstroms. We show that completely different magnetization dynamics characteristics emerge if DOSF and Gep depend on thickness compared with bulk metals. Our model suggests highly efficient energy transfer from femtosecond laser photons to spin waves due to minimal energy absorption by phonons. This sensitivity to the thickness and efficient energy transfer offers an opportunity to obtain ultrafast on-chip magnetization dynamics.

This section is organized to present our theoretical derivation of the microscopic three temperature model. A ferromagnetic ultrathin metal illuminated by an ultrashort (femtosecond) laser pulse can be microscopically described by a generic Hamiltonian for interacting electron and phonon baths as the following: H = He + Hp + Hep (S1) where He and Hp are the Hamiltonians of free electrons and phonons, respectively. Hep models the electron-scattering from the lattice, ignoring the spin flips. We remark that the baths are, in fact, not exactly free. The interactions in the same subsystems Hee and Hpp lead to fast equilibration and hence, they are dropped after instant thermalization assumptions. Their effects are implicit in the dynamics; we will not express them here explicitly. Ignoring the spin degree of freedom of the electrons, non-interacting gas of them has the Hamiltonian S-2 Here ck (c † k) annihilates (generates) an electron with excitation energy E(k) in Bloch state |k⟩. Initially, the gas in metal is in equilibrium at an ambient temperature T0. After the ultrashort pulse excitation, this gas can quickly reach a new temperature due to (screened) Coulomb interaction Hee.
An ensemble of non-interacting quantum harmonic oscillators describes the bath of lattice vibrations, phonons, with the Hamiltonian H p = ∑ ħω q ( † + 1/2) q (S3) Here the operators a † q and aq generate and annihilate phonons with quasi-momentum q, respectively. Phonons obey the Bose-Einstein statistics. Einstein model is the simplest choice to describe them, while Debye model is also possible.
Fröchlich interaction describes the scattering of spinless electrons from the lattice as where we denote the matrix element of the scattering process with Vkk′q.
The calculation of electron-phonon coupling parameter Gep in the two-temperature model proceeds applying of the Fermi's golden rule using Hep. We will present a brief review of the derivation in Ref. 1 , which is based upon pioneering works 2 , and allows for including beyond free electron theory effects and arbitrary DOS. Accordingly, we write the energy transfer rate from electrons to the lattice Ee as Here, is the thermal factor, that depends on the Fermi-Dirac (fk) and Bose-Einstein (nq) distribution functions of the electron and phonon baths in their "local" thermal equilibrium states 1 . Explicitly they are defined as fk = 1/(1 + exp (E(k) − µ)/kBTe) and nq = 1/( exp (ħωq/kBTp − 1). Near room temperature, after the conversion of the k summation to continuum energy integrals, we can write with gF denoting the electronic DOS at Fermi level. The limitation of the DOS to the Fermi level value assumes that only those electrons near the Fermi energy could contribute to the scattering process from the lattice vibrations around room temperature. The term α 2 F(Ω) is the Eliashberg spectral function 3 . If we assume temperatures are below Debye temperatures but higher than phonon mode energy ħΩ≪ kBTp, kBTe, then we can replace the population distributions with a temperature gradient between the baths such that where we used dEe = CedTe. Electron-phonon coupling factor Gep is given by where kB is the Boltzmann constant, λ is the electron-phonon mass enhancement parameter 4 , and ⟨ω 2 ⟩ is the second moment of the phonon 3 . At low temperatures, we can take Ce = γTe, where γ = π 2 kB 2 gF/3, according to the Sommerfeld expansion 5 . Numerical examination of Ce for different metals at higher temperatures, which is beyond the scope of the present contribution, can be found in the literature 6 . Depth of the metallic confinement potential is taken to be Vz = 10 eV. Electronic density is used as n = 3/4πrs 3 with rs = 4aB, and aB being the Bohr radius.

Dependence of DOSF on temperature
In this section, we describe a theoretical method with which the dependence of DOS at Fermi level changes with temperature, to be able to take into account this size effect for Gep.
Our numerical method starts with taking kz for nz = 1 in Eq. (S4) as a guess set of Kz to evaluate kµ by solving Eq. (S8) in the main manuscript. We then compare kz with kµ; if kz < kµ, then we increase the size of Kz by increasing nz one more. The iteration repeats till the largest kz in Kz is more than calculated kµ. The results for corresponding chemical potential µ (or temperaturedependent Fermi energy) and DOS at Fermi energy are plotted in figure S2. Both µ and gF are dimensionless, normalized with their corresponding bulk values. Generalizing the methods in Ref. 7 to this temperature-dependent Fermi energy case, we found how the Lz dependence of DOS at Fermi level changes with temperature, to be able to take into account this size effect for Gep.
In figure S2, the chemical potential change with electron temperature, and Fermi level density of state, are shown. In addition, the electron phonon coupling coefficient is shown in figure S2(c). Both DOSF and Gep are independent of the electron temperature at the temperature ranges applicable in our calculations (where maximum electron temperature does not exceed 2000 K).
S-4 The effect of substrate (especially if it is non-magnetic) might be negligible in changing of the magnetization order. However, it might influence the contributing electron and phonon density and so Gep. Nevertheless, assuming an isolated thin film is reasonable considering the previous studies 8,9 . In figure 4 of the main manuscript, we showed the magnetization dynamics of the Ni ultrathin film for different film thicknesses. A zoomed in version of the magnetization dynamics at a smaller time ranges are shown in figure S3.
S-5 Figure S3. Magnetization dynamics of the Ni ultrathin film in 300 fs after interaction with the laser pulse.

Sensitivity of quantum confinement effect of magnetic ultrathin films to the potential well depth (Vz)
In the main manuscript, we reported our calculation results of quantum confinement effect in figure, 2 and 4, using the potential well depth of Vz= 10 eV. In this section of SI document, we present the sensitivity of our results to the size of the potential well. Practically, the Vz is an infinite value, however, in the FEM calculations, a finite value is given to it. We consider three extra cases of Vz=5 eV, 15 eV, and 20 eV.   The calculations results for the various potential well depths (Vz=5, 10,15, and 20 eV) shows that only in case of sub-20 Å film thicknesses, Vz size might result in a few percent difference in the calculations. However, it does not influence the qualitative predictions of our model on decrease and oscillations in the electron density of states and electron-phonon coupling as the result of quantum confinement effect. In other words μ increases slightly (~ 3%), in sub-20 Å range, from Vz=5 eV to Vz=10 eV, but the oscillations are not considerable when Vz changes from 10 eV to 15 eV or 20 eV. Accordingly, gF, Gep, and γ oscillations decrease (~ 4%) changing from Vz=5 eV to Vz=10 eV, while remains unchanged for other Vz values. We also investigated the sensitivity of the figure 4 results to the size of Vz. Despite similar negligible variations for Vz= 5 eV in case of low film thicknesses, changing the depth of the potential well did not influence the conclusions of the quantum confinement effect on the oscillations of the magnetization dynamics as well as electron and phonon temperatures (figure S6-S7). Increasing the Vz from 5 eV, a small increase (~ 3%) in the maximum electron temperature and decrease in demagnetization dip was observed for sub-20 Å thin films. However, for other Vz values, the results remained unchanged. Since the decrease in the chemical potential and consequently the gF, and Gep, is averagely monotonic 7 (the amplitude of the oscillations does not exceed a few percent), the qualitative results of our calculations are consistent using different potential well depths.

M3TM model considering spin-phonon and spin-electron scattering
In this section, we calculated the effect of film thickness including the spin-phonon, and spinelectron scattering in the M3TM models shown in figure S8 (a). We used the following equations: where Cs is the spin heat capacity, Ges, and Gps are the electron-spin and phonon-spin coupling coefficients, respectively. Ts represents the spin temperature. Note that both Gep and R change with the film thickness, as mentioned in the manuscript. if we heuristically assume that Ges and Gps have similar size dependence as with Gep, through their dependence to DOSF, we find that our results with constant Gep, and Ges do not change significantly. We therefore intuitively expect that the size dependence of Ges and Gps are negligible on Te and Tp dynamics, which determine the magnetization evolution.
According to the figure S8(a), due to the energy loss to the spin-phonon scattering phenomena, the laser fluence needed to recover the magnetization is larger compared to the condition where the spin-lattice scattering is neglected (I0=35 mJ·m -2 ). In addition, the magnetization drop time is slightly longer, while the recovered magnetization ratio is slightly lower as the result of spin scattering. Figure S8 (b) shows the thickness dependence of the lattice temperature. According to this figure, the maximum lattice temperature increases due to the energy exchange between spin and phonon baths. However, maximum Tp does not exceed 305.976 K, which is still well below the Curie temperature of Ni. In the previous studies, the easy axis of the Ni changes from perpendicular to in-plane when the film thickness changes from around 12-14 Å to 24 Å 10,11 . The size-dependent change in the magnetic anisotropy is due to shape anisotropy (dominant anisotropy in the thicker films that dictate the in-plane easy axis), which is completely geometry dependent. In our magnetization dynamics model, the "m" stands for the normalized magnetization vector magnitude (|Mz/Ms|). After interaction with the low-fluence fs laser pulse, the magnetization vector orientation might change but not switch completely. The motion is described as precession, canting, or maybe toggle from an initial stable orientation, which we assume to be m0=1, regardless of the direction. The magnetization magnitude, which changes due to spin reorientation, does not influence the temperature dynamics in the 2TM or M3TM. The easy axis and magnetic anisotropy effects are S-9 studied using LLG equations where the magnetization is a vector, which is beyond our paper's scope [12][13][14][15] .

Effect of the femtosecond laser pulse width on the magnetization dynamics of the quantum confined ultrathin Ni film
According to the laser pulse power distribution, P(t) = , and I0 and t0 are the laser pulse fluence in J·m -2 , pulse width (fs), respectively). It leads to increase in the time of the demagnetization and also electron temperature equilibration. Moreover, for wider pulses such as 200 fs and 500 fs, the injected energy is small such that the quenched magnetization does not recover back and stays constant, and the demagnetization ratio is smaller compared to the lower pulse widths. Electron temperature rises to the lower amounts in case of wide laser pulses, due to the lower laser energy injection and similar to the magnetization quenching time, the electron equilibration time increases by increasing the pulse width. This effect is also reflected in the maximum electron temperature, which is considerably smaller for 200 fs and 500 fs pulse width, compared to 50 fs. In addition, figures S9-S11 show that for each specific pulse width, the behavior of the magnetization dip oscillation and maximum Te and Tp are almost similar. In conclusion, increasing the pulse width decreases the laser power injected to the thin film which leads to increase in the demagnetization time as well as electron equilibration time. Choosing very long pulse durations, increases the laser fluence needed for recovery of the magnetization after quenching, which is not favorable for the scope of our manuscript.

Effect of laser pulse fluence on magnetization dynamics, and temporal electron and phonon temperatures
In the laser-driven all-optical switching experiments, not all the incoming laser energy is absorbed by the ultrathin Ni film; we investigate the case of Ni thin films with a few nm thickness, in which S-11 only 20% of the incoming laser fluence is absorbed by the thin film and most of it transmits through the Ni sample. We have redone the calculations using incoming laser fluences of 2, 5, 7, and 10 mJ·cm -2 (ranges used in the experiments [16][17][18] ) for both nm-thick Ni film using M3TM model of Koopmans, and quantum confined Ni thin film. The results are shown in Figure S12. The results of Figure S12 show that using the same experimental parameters for solving M3TM, results in magnetization drop in sub-200 fs and recovery to near 65%-95% of its initial stage in 1-2 ps (depending on the laser fluence). The plots on the right side of Figure S12 shows that, despite the nm-thick Ni film, using similar laser fluences, the lattice temperature exceeds the Curie temperature in the quantum confined 20 Å Ni film and results in complete thermal demagnetization, even before the electron-phonon equilibration (in case the material is not evaporated).

Effect of laser pulse width on the magnetization dynamics, and temporal electron and phonon temperatures
The effect of the laser pulse duration (width) appears in the incident laser power to the thin film.
According to the P(t) = ) formula, by increasing the pulse duration, t0, the injected laser energy to the thin film decreases which leads to a lower rise in Te, when the pulse width is higher.
We have shown and compared the results of M3TM both for nm-thick (experimental parameters 8,9 ) and quantum confined Ni film for different pulse widths in fs and ps range, shown in Figures S13 and S14 of this supporting information, respectively. We show that using the pulse width in sub 500 fs timescales, the sudden rise in the electron temperature is inevitable due to the excess energy concentration and low heat capacity of the electron. The conclusions from Figures S13 and S14 are as follows: 1) In the case of illumination with the pulse widths in the picosecond regime, the rise in the electron temperature happens more smoothly.
2) Due to the lower energy injection in the case of wider pulses, the magnetization recovery is suppressed, and it stabilizes after magnetization quenching.
3) The timescales of the magnetization quenching and recovery increase in the case of wider pulses, which is undesirable for ultrafast manipulation of magnetization. 4) Electron temperature equilibration time increases with increasing the pulse width/duration.