Theoretical Investigation of All Optical Switching by Intersystem Crossing

The dynamics of spin flips induced by the femtosecond laser are theoretically investigated in this article. The spin flips in this scenario are attributed to the intersystem crossing (ISC) described within the frame of the Rabi model. This new explanation is a step attempting to explain the mechanism of the all-optical magnetic orientation switching in the perspective of the conservation of the angular momentum and breaks of the selection rule, which is ignored in the Raman scattering related explanations. The final spin states discussed herein are closely related to the intensity of the incident laser and the ISC decay rate. The quantitative analysis of the relation between decay rate, temperature and the intensity of the laser is discussed.


Introduction
Occurring on numerous ferri [1][2][3][4][5][6] and ferromagnetic [7][8][9][10] materials, all-optical switching (AOS) is considered as a universal phenomenon. The sub-picosecond time scale within which this magnetization switching is accomplished shows the potential of the ultrafast magnetic data storage. However, the evolution of the magnetization reversal in AOS scenario remains an open question. Two-temperature model [11,12] and the inverse Faraday Effect (IFE) [13] are the two most accepted models describing the dynamics of AOS.
Taking the ultrafast heating as a dominant factor, the two-temperature model works well in the magnetic orientation reversal of anti-ferromagnetic coupled materials [14,15]. However, it fails to explain the transient ferromagnetic state in which the magnetic orientation of the transient metal sublattice reverses and gets parallel to that of the rare earth after demagnetization of the transient metal sublattice.
Proposed by Ziel et al. [16], IFE is interpreted as a thermodynamic potential describing the internal energy of a system. In their later work [17], Pershan et al. deduced the effective Hamiltonian to quantify IFE, separating the effective Hamiltonian into two parts: diagonal matrix elements and off-diagonal ones. The former is responsible for the eigenstates of the unperturbed Hamiltonian, and the latter describes the process of the stimulated Raman scattering (SRS). Popova et al. [18,19] further developed SRS for AOS, and quantified SRS induced magnetization. However, the selection rule requiring the Appl. Sci. 2020, 10 conservation of electrons' angular momentum, suggests that electron transition during SRS should keep the orientation of spins unchanged. Even though results, Popova proposed, show the imbalance of spin-up and spin-down in a few hundred femtoseconds, it should be a temporary state that electrons stagnate at excited states, and finally spins will return to their original states. Though the two mentioned models ignore the conservation of the angular momentum of the system, some researchers argued that spins could be reversed by spin-orbit coupling (SOC) [20][21][22][23][24][25]. The SOC model assumes that the variation of the angular momentum of spins are balanced by their orbital counterparts. Electron transitions from singlet state to triplet state are used to explain the process of the ultrafast demagnetization [26]. However, experimental evidences [27,28] show that the angular momentum of spins and orbits both quench to zero during demagnetization, meaning that orbits cannot be the reservoir of the spin angular momentum. Therefore, the angular momentum of spins should flow to lattice [29,30].
Intersystem crossing (ISC) is a non-radiative process, which can be described by vibrational SOC. Because of following reasons, ISC can be a candidate to explain the mechanism of a specific AOS that is induced by one linear polarized laser shot. (1) ISC solves the deficiency in Popova's work that spins can be reserved with no need to meet selection rule; (2) ISC could be a channel to transfer angular momentum from spins to lattices by vibrational SOC [31]; (3) the time scale of AOS is comparable with ISC. (Recent researches [32,33] on the ultrafast ISC show that the transition time spans from 100 fs to a few picoseconds.) Our model is based on a four-level system, but some of its parameters are "borrowed" from Fe. The model is in the formulation of Rabi oscillation. The calculation of the model shows that ISC could be a plausible approach leading AOS. The impact of temperature and the intensity of the incident laser pulse to the final state of AOS are also discussed.

Dynamic Description of Electron in Laser Field
We construct our Hamiltonian for AOS as: The right terms in Equation (1) represent an unperturbed Hamiltonian, interaction between laser and material and SOC. In our model, laser interaction and SOC are considered as perturbations that induce electron transitions. H int describes electron oscillation between ground and excited state; H SOC describes electron transitions from singlet state to triplet state, but in our paper, SOC is associated with vibration.
The time-dependent Schrodinger equation with the Hamiltonian in Equation (2) has the following solution: where c n (t) describes electron transitions between N eigenstates. The probability of finding an electron on state n at time t is denoted as P n (t), which can be calculated by P n (t) = c n (t) 2 . The possibility that the final state will take can then be calculated by P n (∞). In our study, the state vector is a 4-dimensional one; state vectors Ψ n represent singlet ground state 0 S, singlet excited state 1 S, triplet state 1 T and ground state G, respectively. As shown in Figure 1, we decompose the spin flip process into three stages as excitation, ISC, and decay. Firstly, electrons are excited from 0 S to 1 S and oscillate between 0 S and 1 S; then some electrons in a singlet state 1 S transfer to triplet state 1 T, obeying the ISC rate, which is mainly determined by SOC and the vibration of nucleus; finally, electrons in 1 T state decay to state G but maintaining the same spin state as 1 T.

Figure 1.
A schematic four-level system for our model. Electrons non-radioactively decay from singlet state 1 S to triplet state 1 T. The displacement ΔQ corresponds to different equilibrium geometries in 1 S and 1 T. ΔE is the energy gap between the lowest vibrational levels and λ is the electron-phonon selfenergy difference. These two parameters are associated with the Franck-Condon factor.
Rabi model [34] corresponds to Hint that describes electron oscillation. The dynamics of ISC can be incorporated into the framework of the Rabi model by treating it as a decay rate k. We use Equation (3) to describe the dynamic process electrons experiencing in AOS. The first term on the right hand side describes electron oscillation between 0 S and 1 S, which has a fundamental impact on the decay speed from 1 S to 1 T. The second term describes the effect of ISC. Hamiltonian. Its off-diagonal elements describe the perturbation responsible for the change of the probability of electrons, and its diagonal elements are detuning elements that determine the coupling efficiency between energy bands and the incident laser. In Equation (4), ( ) t Δ dominated by the energy difference between the incident laser pulse and the energy gap, determines the amplitude of the Rabi oscillation. For clearly presenting electron transition between 0 S and 1 S, 0 S in Figure 1 is not pictured as a vibrational state. The vibration of both 0 S and 1 S may induce the mismatch between laser and energy gap between 0 S and 1 S, but we assume that ( ) t Δ is detuned to zero at the very beginning of our tentative calculation.
Ω is the scaled projection of the electric dipole moment d onto the electric field ( ) E t , as shown in Equation (5).
In our study, the incident laser pulse is chosen to have a Gaussian envelope with 100 fs pulse width. Comparing with the frequency of the incident laser and the Rabi oscillation, the rotating wave approximation [35] is valid when the intensity of the laser is not high. Therefore, ( ) E t in Equation (5) has the envelope of the Gaussian distribution. ℏ is the reduced Planck constant. These four states are combined into two systems. The first two-level system is constituted by 0 S and 1 S; the second two-level system is constituted by 1 T and G. The spin orientation in these two systems are opposite of each other. ISC links these two separated systems by breaking the selection rule with a non-radiative radiation. Therefore, the difference of probabilities between electrons being on two systems determines the final orientation of spins.
Rabi model [34] corresponds to H int that describes electron oscillation. The dynamics of ISC can be incorporated into the framework of the Rabi model by treating it as a decay rate k. We use Equation (3) to describe the dynamic process electrons experiencing in AOS. The first term on the right hand side describes electron oscillation between 0 S and 1 S, which has a fundamental impact on the decay speed from 1 S to 1 T. The second term describes the effect of ISC.
c 0 (t) and c 1 (t) in Equation (3) have the same meaning as c n (t); W(t) is a 2 × 2 matrix of the Hamiltonian. Its off-diagonal elements describe the perturbation responsible for the change of the probability of electrons, and its diagonal elements are detuning elements that determine the coupling efficiency between energy bands and the incident laser. In Equation (4), ∆(t) dominated by the energy difference between the incident laser pulse and the energy gap, determines the amplitude of the Rabi oscillation. For clearly presenting electron transition between 0 S and 1 S, 0 S in Figure 1 is not pictured as a vibrational state. The vibration of both 0 S and 1 S may induce the mismatch between laser and energy gap between 0 S and 1 S, but we assume that ∆(t) is detuned to zero at the very beginning of our tentative calculation.
Ω is the scaled projection of the electric dipole moment d onto the electric field E(t), as shown in Equation (5).
In our study, the incident laser pulse is chosen to have a Gaussian envelope with 100 fs pulse width. Comparing with the frequency of the incident laser and the Rabi oscillation, the rotating wave approximation [35] is valid when the intensity of the laser is not high. Therefore, E(t) in Equation (5) has the envelope of the Gaussian distribution.h is the reduced Planck constant.
The ISC rate can be described by El-Sayed's rules [36]. In his work, El-Sayed highlighted the importance of SOC and predicted that the transition from 1(ππ*) to (nπ*) would be faster than 1(ππ*) to 3(ππ*), as the latter involves the same orbitals. The electron transition rate is calculated by Fermi's golden rule. If we consider SOC with adiabatic states (following El-Sayed's rule), the ISC rate from 1 S to 1 T can be written as: S and T are the singlet and triplet states; H SO is the Hamiltonian for SOC; E 1 S and E 1 T are the energy of 1 S and 1 T. However, if vibrations of states are considered, El-Sayed's rules will be broken and the speed of ISC will be accelerated. ISC can be rewritten by considering a thermal contribution and assuming that SOC has no dependency on vibration [37]: k B is Boltzmann constant; T is temperature; j, k represents vibrational states; v Tk and v Sj are vibration states; Z is a canonical partition function for vibrational motion in the singlet state. Comparing Equation (7) with Equation (6), except the thermal related term Z, we can find that vibrational terms contribute to the ISC rate, which ensures ISC within the same orbit. Hence, this formulation is consistent with experimental results that spin angular momentum transfers to lattice but not orbit. Equation (8) can be simplified to the following formulation [38]: V is the intensity of interaction between levels where spin-obit coupling drives the transition from singlet state to triplet state. To figure out the contribution of vibration to ISC, SOC is treated as a constant V. F is the Franck-Condon (FC) integrals corresponding to the vibration part in Equation (8).
Jortner theory [39] can be safely applied to present FC integrals [40]: where the footnote p (m) is the integer with its value closest to (∆E − m ω s )/ ω s ; I n (z) in Equation (9) represents the modified Bessel function; the electron-phonon coupling strength S and S s are dimensionless Huang-Rhys parameters [41]; T l is the temperature of the lattice. The maximum of m = ∆E/ ω and S =λ/ ω are rounded down as natural numbers. The low frequency mode of exterior medium specified in terms of the mean frequency ω s is approximated to ω s 10-100 cm −1 in solids, and the high frequency mode is approximated to ω 200-3000 cm −1 [40]. The limitation of the FC integralfor a room temperature, ω s k B T, is applied for the low frequency part of Equation (9), which results in Equation (11): Appl. Sci. 2020, 10, 128 In work [40], the exp(−S(2( _ v + 1)) term in Equation (11) is simplified to exp(−S), we found this simplification deteriorates the impact of temperature on the ISC rate leading an unreasonable upshifting minimum ISC rate. Moreover, this limitation even gives a wrong tendency of the ISC rate, when ω ≈< k B T. Therefore, in this paper, we take the limiting process in Equation (11) to the high frequency vibration and get the reasonable results.

Discussion and Results
We start out our calculation from Equation (11) to get the decay rate k needed in Equation (3). As Fe is the core element in the AOS scenario, our four-level system uses some parameters that refer to Fe. According to the typical displacement ∆Q of a few tenths of Angstrom [32,33,42], the values of λ and |∆E| are taken as 0.2 eV. In classical description, ∆E is interpreted as Gibbs free energy, ∆G. If ∆G < 0, the process involved is spontaneous and accompanied with energy releasing. In the case of spin flip, the energy of electrons at triplet states is smaller than that at singlet states, so ISC is an energy releasing process, in which phonons are released, so ∆E = −0.2 eV. For SOC, we take a typical atomic Fe value [32], V = 0.05 eV, ω = 0.03 eV.
Evaluating Equation (11) with all parameters above, we get Figure 2a which shows the relation between temperature and the ISC rate. The initial temperature starts at room temperature, 300 K. The ISC rate dramatically declines with temperature increasing from 300 K to 600 K. In work [40], the _ exp( (2( 1)) S v − + term in Equation (11) is simplified to exp( ) S − , we found this simplification deteriorates the impact of temperature on the ISC rate leading an unreasonable upshifting minimum ISC rate. Moreover, this limitation even gives a wrong tendency of the ISC rate, when . Therefore, in this paper, we take the limiting process in Equation (11) to the high frequency vibration and get the reasonable results.

Discussion and Results
We start out our calculation from Equation (11) to get the decay rate k needed in Equation (3). As Fe is the core element in the AOS scenario, our four-level system uses some parameters that refer to Fe. According to the typical displacement ΔQ of a few tenths of Angstrom [32,33,42], the values of λ and |ΔE| are taken as 0.2 eV. In classical description, ΔE is interpreted as Gibbs free energy, ΔG. If ΔG < 0, the process involved is spontaneous and accompanied with energy releasing. In the case of spin flip, the energy of electrons at triplet states is smaller than that at singlet states, so ISC is an energy releasing process, in which phonons are released, so ΔE = −0.2 eV. For SOC, we take a typical atomic Fe value [32], V = 0.05 eV, Evaluating Equation (11) with all parameters above, we get Figure 2a which shows the relation between temperature and the ISC rate. The initial temperature starts at room temperature, 300 K. The ISC rate dramatically declines with temperature increasing from 300 K to 600 K.
Because of the sensitivity of the ISC rate to the temperature, the temperature fluctuation heated by the incident light influences the ISC rate. Another well-developed theory, the two-temperature model, successfully describes ultrafast heating process. The details of the two-temperature model, where the 'two-temperature' refers to the temperature of electrons and lattices, are discussed by Mathijs, etc. [43] The lattice temperature is calculated using this model, and inserted into Equation (11) to represent the temperature of the sample. The two-temperature model is used to reflect the temperature fluctuation and we ignore the influence of the effect on magnetization. Even though we do not present the details of the two-temperature model, but equations and values can be found in Reference [6]. Taking the time Tr when the electric field of the incident pulse gets its maximum as reference, our simulation starts 200 fs earlier than Tr. After substituting the electric field amplitude in Equation (5) and the pump intensity P in two-temperature model into Equation (3)  , we then plot the temperature fluctuation over time in Figure 2b. Figure 2b shows the electron temperature increases Because of the sensitivity of the ISC rate to the temperature, the temperature fluctuation heated by the incident light influences the ISC rate. Another well-developed theory, the two-temperature model, successfully describes ultrafast heating process. The details of the two-temperature model, where the 'two-temperature' refers to the temperature of electrons and lattices, are discussed by Mathijs, etc. [43] The lattice temperature is calculated using this model, and inserted into Equation (11) to represent the temperature of the sample. The two-temperature model is used to reflect the temperature fluctuation and we ignore the influence of the effect on magnetization. Even though we do not present the details of the two-temperature model, but equations and values can be found in Reference [6].
Taking the time T r when the electric field of the incident pulse gets its maximum as reference, our simulation starts 200 fs earlier than T r . After substituting the electric field amplitude in Equation (5) and the pump intensity P in two-temperature model into Equation (3) by P ∝ E 2 , we then plot the temperature fluctuation over time in Figure 2b. Figure 2b shows the electron temperature increases dramatically at the beginning, while the lattice temperature lags much later, that it cannot reach 600 K within the first 0.5 ps. The temperature in Equation (11) corresponds to the lattice temperature because the ISC rate is associated with vibration of nucleus. Constrained by the pulse width of the incident laser, the Rabi oscillation and ISC mainly occur within 0.5 ps. Hence, the ISC rate keeps decreasing within the first 0.5 ps.
In the simulation, we assume that all electrons stay at ground state initially. The amplitude of electron field E 0 equals to 10 9 V/m. Since we have the ISC rate, W(t) already, we then can calculate c(t) and P(t). The probability of electrons staying at 0 S, P( 0 S), shown as the blue curve in Figure 3a, declines from 1 dramatically and starts oscillating. This ultrafast laser pulse induced oscillation lasts about 300 fs, during which some electrons secede to triplet state. After the oscillation, there are still electrons staying at 1 S with the probability P( 1 S) equals 0.2. The remaining electrons decay to 1 T corresponding to the smooth decline of the orange curve, because the decay from 1 S to 0 S is neglected, comparing with the two orders lager decay rate.
decreasing within the first 0.5 ps.
In the simulation, we assume that all electrons stay at ground state initially. The amplitude of electron field E0 equals to 10 9 V/m. Since we have the ISC rate, ( ) W t already, we then can calculate ( ) c t and ( ) P t .The probability of electrons staying at 0 S, P( 0 S), shown as the blue curve in Figure 3a, declines from 1 dramatically and starts oscillating. This ultrafast laser pulse induced oscillation lasts about 300 fs, during which some electrons secede to triplet state. After the oscillation, there are still electrons staying at 1 S with the probability P( 1 S) equals 0.2. The remaining electrons decay to 1 T corresponding to the smooth decline of the orange curve, because the decay from 1 S to 0 S is neglected, comparing with the two orders lager decay rate. We plot the probability that electrons decay to the triplet state in blue, and the probability of the final spin state in orange in Figure 3b. The probability of electrons transferring to 1 T, P( 1 T), can be presented as , which is the equivalence of the probability of electrons at the second two-level state. Electrons decaying to triplet state will be trapped even for a few seconds, sothe transition from 1 T back to 1 S in our system is assumed to be zero and the direct transition between 0 S and G is forbidden. The final spin state is determined by the difference between two subsystems. Dynamics of electrons in the second two-level system also can be described by Rabi model, but the oscillations do not affect the final state of spins. The final spin state, m P can be calculated by Equation (12).
For about 300 fs, spin states of the system get the balance between spin up and down. The time scale of the simulation is approximately consistent with the sub-lattice of Fe [14]. Then, the orientation of spins reverses slowly from the initial state. The electron transitions cease at about 1 ps, and the normalized difference between the spin up and down reaches −0.2 from 1 which agrees with experimental results [44,45].  We plot the probability that electrons decay to the triplet state in blue, and the probability of the final spin state in orange in Figure 3b. The probability of electrons transferring to 1 T, P( 1 T), can be presented as P( 1 T) = 1 − [P( 0 S) + P( 1 S)], which is the equivalence of the probability of electrons at the second two-level state. Electrons decaying to triplet state will be trapped even for a few seconds, sothe transition from 1 T back to 1 S in our system is assumed to be zero and the direct transition between 0 S and G is forbidden. The final spin state is determined by the difference between two sub-systems. Dynamics of electrons in the second two-level system also can be described by Rabi model, but the oscillations do not affect the final state of spins. The final spin state, P m can be calculated by Equation (12). P m = P 0 S) + P( 1 S) − P( 1 T = 2[P( 0 S) + P( 1 S)] − 1 (12) For about 300 fs, spin states of the system get the balance between spin up and down. The time scale of the simulation is approximately consistent with the sub-lattice of Fe [14]. Then, the orientation of spins reverses slowly from the initial state. The electron transitions cease at about 1 ps, and the normalized difference between the spin up and down reaches −0.2 from 1 which agrees with experimental results [44,45]. Now we focus on the impact of amplitude of the electric field E 0 of the ultrafast laser pulse on final system states. The impact is shown in Figure 4: the blue curve is the final probability of electrons at the first two-level system, the orange one is the final spin state of whole system over E 0 . The most impressive element in Figure 4 is the oscillating pattern of blue and orange curves. In the Rabi model, when laser pulse areas equal to 2π and the pulses traverse samples without photon absorption during oscillations, the self-induced transparency is achieved and electrons oscillate back to initial state, 0 S. Illustrated in Figure 4, when P( 0 S) + P( 1 S) gets its peaks, the first two-level system appears a self-induced transparency. On the contrary, when laser pulses excite electrons to their excited state with an abrupt ending, most electrons at 1 S decay to triplet state, which realizes the near 100% spin flip.
Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 10 Now we focus on the impact of amplitude of the electric field E0 of the ultrafast laser pulse on final system states. The impact is shown in Figure 4: the blue curve is the final probability of electrons at the first two-level system, the orange one is the final spin state of whole system over E0. The most impressive element in Figure 4 is the oscillating pattern of blue and orange curves. In the Rabi model, when laser pulse areas equal to 2π and the pulses traverse samples without photon absorption during oscillations, the self-induced transparency is achieved and electrons oscillate back to initial state, 0 S. Illustrated in Figure 4, when P( 0 S) + P( 1 S) gets its peaks, the first two-level system appears a selfinduced transparency. On the contrary, when laser pulses excite electrons to their excited state with an abrupt ending, most electrons at 1 S decay to triplet state, which realizes the near 100% spin flip.
The dynamics of the spin state, shown as the orange curve in Figure 4, undergo two phases: a dramatic decrease before E0 reaches 9 0 0.25 10 V/m E = × followed by oscillations. The first phase corresponds to the situation that the laser with low intensity excites electrons to 1 S; at this time, the ISC rate is faster than the speed of the excited radiation, which in turn lowers the probability from 1 with the increasing intensity of the laser. During the oscillating phase, the final spin state oscillates with up-shifting valleys because of the temperature. As shown in Figure 2, the high intensity of the laser apparently causes an increase in temperature and the high temperature declines the decay rate of the transition from 1 S to 1 T. Some non-trivial issues need to be addressed. Firstly, the intensity of the laser causes AOS to range within a relatively narrow window 3 , which is similar to the results shown in Figure 4 before E0 reaches studies. This result is induced by the ultra-high intensity of laser which cannot be treated as perturbation in the Hamiltonian. Hence, our model should have a cut-off intensity at about 0.5 × 10 3 V/m. Secondly, in our investigation, the final saturated magnetization we have shown here has not been observed in AOS induced magnetization reversal. The saturated magnetization may result from the mismatch between the incident laser wavelength and the energy gap, Δ. When the mismatch ℏΔ equals to 1 eV, the amplitude of Rabi oscillation is half of Δ = 0. However, the femtosecond laser has a relative broad frequency spectrum. The mismatch will not dramatically affect our result but slightly reduce the reversed magnetization.

Conclusion
The model of the electron transition coupling with the spin flip is presented to explain the process of AOS. The dynamic of the spin flip is described on the base of Rabi model, and the main cause for AOS is attributed to ISC. Though the spin flip is not direct evidence of magnetization reversal, magnetization originates from the spin states of electrons, especially for transition metal. The dynamics of the spin state, shown as the orange curve in Figure 4, undergo two phases: a dramatic decrease before E 0 reaches E 0 = 0.25 × 10 9 V/m followed by oscillations. The first phase corresponds to the situation that the laser with low intensity excites electrons to 1 S; at this time, the ISC rate is faster than the speed of the excited radiation, which in turn lowers the probability from 1 with the increasing intensity of the laser. During the oscillating phase, the final spin state oscillates with up-shifting valleys because of the temperature. As shown in Figure 2, the high intensity of the laser apparently causes an increase in temperature and the high temperature declines the decay rate of the transition from 1 S to 1 T. Some non-trivial issues need to be addressed. Firstly, the intensity of the laser causes AOS to range within a relatively narrow window 3 , which is similar to the results shown in Figure 4 before E 0 reaches E 0 = 0.5 × 10 9 V/m. However, the periodic appearances of the window have not been confirmed by other studies. This result is induced by the ultra-high intensity of laser which cannot be treated as perturbation in the Hamiltonian. Hence, our model should have a cut-off intensity at about 0.5 × 10 3 V/m. Secondly, in our investigation, the final saturated magnetization we have shown here has not been observed in AOS induced magnetization reversal. The saturated magnetization may result from the mismatch between the incident laser wavelength and the energy gap, ∆. When the mismatchh∆ equals to 1 eV, the amplitude of Rabi oscillation is half of ∆ = 0. However, the femtosecond laser has a relative broad frequency spectrum. The mismatch will not dramatically affect our result but slightly reduce the reversed magnetization.

Conclusions
The model of the electron transition coupling with the spin flip is presented to explain the process of AOS. The dynamic of the spin flip is described on the base of Rabi model, and the main cause for AOS is attributed to ISC. Though the spin flip is not direct evidence of magnetization reversal, magnetization originates from the spin states of electrons, especially for transition metal. The spin state is a result of the electron transition between energy levels. However, to maintain the spin state as the final magnetization, dynamics of spins should be coupled to a process with longer time scale such as magnetic interaction. Meanwhile, the final spin state depends on the laser intensity, allowing the manipulation of magnetization, as well as the extension of the application of AOS.