Role of the heat effect and field effect in single femto-second laser pulse induced helicity-dependent all-optical magnetic switching

Chudong Xu; Wanjie Xiong

doi:10.1364/OME.8.003133

1. Introduction

The femto-second laser pulse induced helicity-dependent all-optical magnetic switching (HD-AOS) [1] in ferrimagnetic and ferromagnetic materialshas received widespread attention [2–21] due to its potential application for magnetic data storage with ultrahigh data rates. Two different mechanisms for HD-AOS have been distinguished; a single-pulse HD-AOS for ferrimagnetic GdFeCo alloys and synthetic ferrimagnets [22], and a multishot HD-AOS for ferrimagnetic TbFe, TbCo alloys, ferromagnetic multilayers, ferromagnetic thin films, ferromagnetic granualar recording media, and so on. For multishot HD-AOS, a purely heat-driven thermal accumualtion mechanism in the presence of magnetic circular dichroism (MCD) was originally proposed [14], whereby the system begins in a demagnetised state and the laser pulse induces reversal in grains in one orientation as they absorb more (or less heat) due to the MCD. More recently, it was shown by Hadri and co-workers [17] that it is possible to induce switching using multiple pulses starting from a magnetized state and leading to a completely reversed state. However, so far multiple pulses are necessary to fully deterministically switch the magnetization for HD-AOS. The use of single-pulse switching would be interesting because it is ultra-fast and energy-effecient. As for the single-pulse HD-AOS, though numerous theoretical and experimental investigations have strived to reveal the underlying physical mechanism, the underlying mechanism is still a matter of some debate. So far there are three main effects [23] based on the interaction of ultrashort laser pulses with magnetic materials to describe the underlying microscopic mechanism. The first one is magnetic circular dichroism (MCD) [8] with different effective absorbed energy for different helicity light, which is attributed to the pure thermal effect, though this differential heating does not in itself give a mechanism for switching. The second one [2,3] proposes that the light helicity alone can drive the switching process, thus assigning it to nonthermal photomagnetic effects. The latter is accepted by the some groups [1,4,7], which suggest that the switching process is a combination of two effects, namely, heating of the sample in a strong nonequilibrium and action of the circular light as an effective magnetic-field pulse via the inverse Faraday effect (IFE), whose existence has been already demonstrated in dielectrics [24]. Accordingly, the single-pulse HD-AOS should be sorted into the thermal effects and the nonthermal optomagnetic effects. Based on this mechanism, Nieves et al [16] recently assumed and tuned the inverse Faraday field strength and duration to reproduce HD-AOS in ferromagnetic material, whose results showed that the field should be longer than the duration of laser pulse itself. They concluded that the inverse Faraday effect should be strong enough to serve as mechanism for the single-pulse HD-AOS once the decay time of the inverse Faraday field is sufficiently long. However, it is still unclear why the induced field pulse should decay sufficiently long to induce the single-pulse HD-AOS.

There are many unclear factors to impede the understanding for the single-pulse AO-HDS, while the main controversial issue of the above three mechanism can be ascribed to the role of the heat effect and the induced field effect in the single-pulse AO-HDS. As both of these two effects are generated by the circularly polarized laser pulse, it is difficult to say whether both of these effects are necessary, or not. Furthermore, it is difficult to distinguish the role of these two effects in the the single-pulse HD-AOS experiment, thus requiring detailed model calculations to gain insight into their roles. Using atomistic spin models or micromagnetic models one can achieve insight into these two effects via separating heat pulse and field pulse.

Besides, though GdFeCo has been the subject of many theoretical studies, TbFe has been so far mostly investigated experimentally [25] on AOS, motivating the need for a theoretical investigation of switching. At the same time, TbFe present larger anisotropy than GdFe, and thus potentially are more relevant for applications. Recently nanoscale magnetic recording with AOS technology using near-field optics has been reported on TbFeCo thin films [26]. Most of the experimental studies [9,10,12,13] on TbFe have been subject to switching through the use of circularly polarized laser pulses that confirmed the occurrence of AOS in TbFe, but the mechanism of switching is still unclear .

Based on the existing debates, and the discussions mentioned above, here we attempt to demonstrate the role of the heat effect and the effective magnetic field induced by circularly polarized laser pulse via IFE in the single-pulse HD-AOS, and try to make it clear why the induced field pulse should decay sufficiently long to induce HD-AOS. For that, we investigate the single-pulse HD-AOS in TbFe film based on the Landau-Lifshitz-Bloch (LLB) equation of motion, and especially, we use the laser heat pulse combined with the magnetic field pulse, whose amplitude is proportional to the pump fluence of the laser pulse, to simulate the effect of circularly polarized laser pulse considering its IFE.Firstly, we remark that our simulations can reproduce the single-pulse HD-AOS. Then the simulation of ultrafast magnetization dynamics under the different ratio of the duration of heat pulse to that of field pulse is performed, which shows whatever the duration of the filed pulse is, it should be longer than that of the heat pulse to induce the switching. To explain this result, we further simulate the magnetization dynamics by separating the heat pulse and the same duration of the field pulse for different time intervals. The results show that only the field pulse is after the moderate delay of the heat pulse, can the field have a strong effective action on the sample to induce switching, when there is no delay or at a short delay of field pulse to the heat pulse no switching can be obtained for the magnetic field has little effect on the sample at the process of ultrafast thermal demagnetization. Furthermore, our results demonstrate that the single-pulse HD-AOS should be the combined result of heat and field effects, which support its mechanism sorting into the thermal effect and the nonthermal optomagnetic effect.

2. Modeling method

2.1. Landau-Lifshitz-Bloch model

The LLB equation of motion describes the time evolution of a magnetic macrospin. The equation allows for longitudinal relaxation of the magnetization, and was derived by Garanin [27] within a Mean Field approximation from the classical Fokker-Planck equation for atomic spins interacting with a heat bath. In this sense the equation attempts to describe, in a spatially averaged way, the motion of an ensemble of magnetic moments. The thermal fluctuations induced by laser heating are taken into account as the stochastic field in LLB equation [28]. Models based on the resulting expressions have been shown to be consistent with atomistic spin dynamics simulations [29], as well as comparisons with experimental observations, for example, in laser induced demagnetization [30] and domain wall mobility measurements in Yittrium Iron Garnet crystals close to the Curie point [31] . The equation is similar to the Landau-Lifshitz-Gilbert (LLG) equation [32], with precessional and relaxation terms, but with an extra term that deals with changes in the length of the magnetization:

\begin{array}{l} m_{i} = & - γ (m_{i} \times H_{i}^{eff}) + ζ_{i, ‖} + \frac{γ α_{‖}}{m_{i}^{2}} (m_{i} \cdot H_{i}^{eff}) m_{i} \\ - \frac{γ α_{⊥}}{m_{i}^{2}} [m_{i} \times [m_{i} \times (H_{i}^{eff} + ζ_{i, ⊥})]], \end{array}

where m_i is a spin polarisation, M_i/M_s(0). The spin polarisation tends towards equilibrium, m_e, which is a temperature dependent quantity (discussed below). α_‖ and α_⊥ are dimensionless longitudinal and transverse damping parameters. γ is the gyromagnetic ratio taken to be the free electron value. The stochastic fields ζ_i,‖ and ζ_i,⊥, have zero mean and the variance [33]

〈 ζ_{i, ⊥}^{η} (0) ζ_{j, ‖}^{θ} (t) 〉 = \frac{2 k_{B} T (α_{⊥} - α_{‖})}{| γ | M_{s} V α_{⊥}^{2}} δ_{i j} δ_{η θ} δ (t),

〈 ζ_{i, ‖}^{η} (0) ζ_{j, ‖}^{θ} (t) 〉 = \frac{2 | γ | k_{B} T α_{‖}}{M_{s} V} δ_{i j} δ_{η θ} δ (t),

where is the additive noise, and η and θ represent the Cartesian components. T is the temperature of the heat bath to which the macrospin is coupled. The LLB equation is valid for finite temperatures and even above T_C, though the damping parameters and effective fields are different below and above T_C. For the transverse damping parameter:

α_{⊥} = {\begin{array}{l} λ (1 - \frac{T}{3 Tc}) & T < Tc \\ λ \frac{2 T}{3 Tc} & T \geq Tc \end{array}

and for the longitudinal:

α_{‖} = λ \frac{2 T}{3 Tc} for all T .

In the above equations, λ is a microscopic parameter which characterizes the coupling of the individual, atomistic spins with the heat bath.

The effective field $H_{i}^{eff}$ is given by [27]

H_{i}^{eff} = B + H_{A, i} + \frac{1}{2 {\tilde{χ}}_{i, ‖}} (1 - \frac{m_{i}^{2}}{m_{i, e}^{2}}) m_{i} + H_{e, i} + H_{demag, i}

where B represents an external magnetic field and

H_{A, i} = - (m_{i}^{x} e_{i}^{x} + m_{i}^{y} e_{i}^{y}) / {\tilde{χ}}_{⊥}

an anisotropy field. Ms(0) is the saturation magnetization (magnetization at 0K), V_i represents the volume of grain i and μ₀ is the permeability of free space. Here, the susceptibilities χ̃_l are defined by χ̃_l = ∂m_l/∂H_l and H_e,i is the exchange field:

H_{e, i} = \frac{A (T)}{m_{e}^{2}} \frac{2}{M_{s} (0) Δ^{2}} \sum_{j \in neigh (i)} (m_{j} - m_{i})

where A(T) represents exchange stiffness, which is depends on a power law to the equilibrium magnetization,

A (T) = A (0) m_{e}^{1.76}

. Δ is the size of modulation system.

The final term in equation (6), H_demag,i is the demagnetising field. This is calculated by writing the magnetostatic field in a (cubic) cell i as:

H_{demag, i} = - M_{s} (0) \sum_{j} N (r_{i} - r_{j}) \cdot m_{j}

where N is a 3×3 symmetric demagnetizing tensor. The sum runs over all cells at positions r_i,j and the demagnetising tensor is given by

N (r_{i} - r_{j}) = \frac{1}{4 π} \oint_{S_{i}} \oint_{S_{j}} \frac{d S_{i} \cdot d S_{j}}{| r - r^{'} |}

S_i(S_j) are the surface of cell i (j), respectively, r and r′ are the points on the surface i and j.

For application of the LLB equation one has to know a-priori the spontaneous equilibrium magnetization m_e(T), the exchange stiffness A(T), the perpendicular (χ̃_⊥(T)) and parallel (χ̃_‖(T)) susceptibilities beforehand. In this work, the input functions are calculated using a mean field approach. Table 1 shows a summary of the parameters that are used in the model.

Table 1. Physical parameters entering into the LLB model.

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2.2. Spatially dependent two-temperature model

To account for the laser induced ultrafast dynamics in this model, we utilize the semi-classical two-temperature model [34] of laser heating. This model defines a temperature associated with the electron and phonon heat baths through the simplified equations:

C_{e} \frac{\partial T_{e}}{\partial t} = - G (T_{e} - T_{l}) + P (r, t)

C_{l} \frac{\partial T_{l}}{\partial t} = G (T_{e} - T_{l})

where C_e (C_l) and T_e (T_l) are the electron (lattice) specific heats and temperatures respectively and G is the electron-lattice coupling constant. The time-and-spacial dependent laser power P(r, t) is assumed to be Gaussian in both time and space:

\begin{array}{l} P (r, t) & = ℱ exp (- {(\frac{t - t_{0}}{τ_{p}})}^{2}) \\ \times exp (- \frac{{(x - x_{0})}^{2}}{2 σ_{x}^{2}}) exp (- \frac{{(y - y_{0})}^{2}}{2 σ_{y}^{2}}) \end{array}

where t₀ is the pump delay, τ_p is the pump width, x₀ and y₀ are the pump centres in x and y respectively, and σ_x,y are the spatial widths in x and y. As well as implementing the spatial dependence of the pump fluence we have also added a spatial dependence of the induced field intensity arising from the IFE. Besides, we consider the IFE field to be linearly dependent on pump fluence, which is described by the following equation [35].

H_{IFE} \approx α E (ω) \times E^{*} (ω) \approx p α I_{0} Z .

where E(ω) is the electric field of the light, α is the magneto-optical susceptibility which also defines the Faraday rotation and I₀ is the pump intensity. The degree of ellipticity p spans from −1 to +1 when the polarization of the pump pulse is tuned from left- to right- handed circularly polarized. H_IFE is directed along the wavevector of the pump pulse close to the z-axis in our simulation.

3. Results and discussions

First, we simulate the ultrafast magnetization dynamics by choosing the opposite direction of the effective light-induced magnetic field pulse for the opposite helicity circularly polarized light. For example, we choose the positive direction of the magnetic field pulse (e.g., +H) for the right handed circularly polarized light (RCP) and the negative one (e.g., −H) for the left handed light (LCP). Besides, we consider the effective light-induced magnetic field pulse builds up with no delay with respect to the laser heat pulse. The decay of magnetic field is, in turn, assumed to be slower and is given by the decay time of the microscopical process responsible for the emerging of the induced magnetic field. So far to the best of our knowledge, no theoretical estimations for the decay time of the induced magnetic field are available. However, a few experiments performed on magnetic metals have confirmed that the laser-induced effective field can persist in a magnetic metal considerably longer than the duration of the laser pulse [24]. Here, we choose the decay time of field given in Ref [4], in which the ratio of the duration of the induced field pulse to that of the heat pulse is chosen as 2.5 times. Specially, we choose the duration of the heat pulse as 150 femtosecond (fs) and the duration of the field pulse as 375fs. Then the simulation is performed by varying the combinations of helicity (e.g., LCP or RCP) and magnetization(e.g., +M or −M), whose result is shown in Fig. 1. The normalized magnetization $M = \sum_{i = 1}^{N} m_{i z} / N$ , and N is the total number of macrospins.

Fig. 1 Simulation of ultrafast magnetization dynamics with the different initial magnetization state and the opposite helicity circularly polarized light. The laser fluence of heat pulse is 1.8GJ/cm³ns and the amplitude of field pulse is 18T. The duration of the heat pulse is 150fs and the duration of the field pulse is 375fs, whose duration is 2.5 times of heat pulse. The solid lines represent the results of using LCP and the dash lines represent that of using RCP.

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It’s obvious that the dynamics is strong dependent on both the helicity of laser pulse and the initial magnetization state. As for the positive initial magnetization state, using the left handed light can induce the switching while using the right handed light only induces an ultrafast demagnetization and then recovery rather than switching. However, this result is inverse under the negative initial magnetization state. Therefore, these results clearly show that our simulation based on the above construct model can reproduce the single-pulse HD-AOS.

So far we have shown that our model, which is based on the LLB equation of motion via using the heat pulse combined with the magnetic field pulse, can simulate the result of the single-pulse HD-AOS. Next it is important to make it clear whether both the heat effect and the field effects are necessary in the single-pulse HD-AOS. First, we present the simulation only using the heat pulse with the same duration as that one used in Fig. 1, whose result is shown in Fig. 2. When the same laser fluence as Fig. 1 is used, as shown with the black line in Fig. 2, there is no switching but only demagnetization and then magnetization recovery. Even the fluence is increased to induced the sample to demagnetize completely, as shown with the green line in Fig. 2, it still only shows a pure thermal demagnetization but no switching. Compared this result with Fig. 1, it is obvious that the switching is not the pure thermal origin and the field effect is necessary. Then we only use the field pulse with the same amplitude and the same duration with that one used in Fig. 1, whose result is also plotted in Fig. 2 with the blue line. Also, there is no switching. This result indicates that the heat effect could no be neglected in the switching. Therefore, we can exclude both a nonthermal origin and a pure thermal origin of the single-pulse HD-AOS, and the mechanism of the single-pulse HD-AOS should be the combined result of heat and field effects.

Fig. 2 Simulation of ultrafast magnetization dynamics induced by LCP at +M initial magnetization, while separate the heat effect by only using the heat pulse (shown as the black, red and green lines represent the different pump fluence) and the field effect by only using the field pulse (shown as the blue lines), respectively. The duration of the heat pulse is chosen as 150fs, which has the same duration with that one used in Fig. 1. The amplitude of field pulse is chosen as 18T, which has the same amplitude with that one used in Fig. 1.

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Since we have confirmed that both the heat effect and the field effect are required in the single-pulse HD-AOS, in the following, to further study how the field effect influence the switching, we present the simulation by varying the ratio of the duration of heat pulse (ranges from 50fs to 250fs at 50fs duration increment) to that of field pulse, whose results are shown in Table 2. As for the same duration of heat pulse (the same row in Table 2), the minimum threshold fluence for switching initially decreases with the increase of the ratio, which means that the duration of field pulse have a strong effect on the switching and the switching process is favoured by the longer action time of field in these ranges of pulse duration. And then when the duration of field pulse increase to about 400fs in our simulation, the minimum switching threshold will keep saturated. What’s more, the result also shows whatever the duration of the filed is, if the ratio of the duration of heat pulse to that of field pulse equals 1 (shown as the second column in Table 2), that is the field pulse has the same duration with the heat pulse, it could not induce any switching and only shows the pure thermal demagnetization even at very high fluence. On one hand, it means that in this situation the duration of field pulse has little impact on the switching, which seems to be contrary to the switching process favoured by the longer field pulse at the duration range below 400fs. On the other hand, it also indicates that the key of the induced magnetic filed via IFE to achieve HD-AOS is that the duration of the field pulse should be longer enough than that of the heat pulse rather than the absolute duration of field pulse. This agree with the recent report [16] that the induced field pulse should decay sufficiently long to induce the single-pulse HD-AOS.

Table 2. The dependence of the ratio of the duration pulse to the field pulse on the minimum threshold fluence for switching in the single-pulse HD-AOS.

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Now the question is why the duration of field pulse should be longer than that of heat pulse to induce the switching. To answer this question, we should make it clear the role of field on the switching process and when the magnetic field start to have a strong effective action on the switching. Accordingly, we further separate the heat pulse and the field pulse for different time interval to distinguish the role played by laser heating and field action in the switching. Since we focus on when the induced magnetic field have a strong effective action on the dynamics, we choose the same duration of the field pulse as the heat pulse for simplification, whose duration of pulse is 150fs. When there is a 100fs delay of field pulse to the heat pulse, as shown in Fig. 3(a), it only shows the pure thermal demagnetization but no switching even the fluence is further increased. This result is very similar to that only using the 150fs heat pulse without the field pulse(as shown in Fig. 2), which indicates that in the situation of the short delay between the field pulse and the heat pulse, the filed effect has little impact on this ultrafast magnetization dynamics. While when the delay time of field pulse to heat pulse is increased to 400fs, as shown in Fig. 3(b), it shows the switching occurs at the laser fluence above 1.8 GJ/cm³ ns. Compared with the result at 100fs delay, it is obvious that at this longer delay the field have a stronger effective action on the sample to induce the switching, which further support the opinion [16] that the induced field pulse should decay sufficiently long to induce the single-pulse HD-AOS.

Fig. 3 Simulation of ultrafast magnetization dynamics under the condition of separating the heat pulse and the field pulse for different time interval. The delay time of field pulse to the heat pulse is chosen as 100fs in Fig. 3(a), (b), (c), (d), respectively.

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To further study the dependence of the field effect to the switching on the delay time, we increase the delay time up to 1000fs and 5000fs, as shown in Fig. 3(c) and Fig. 3(d). Also there are switching at both these delay time. The only difference is at the longer delay time the minimum switching threshold fluence increases, which can be seen that the switching can be obtained with the fluence of 1.8 GJ/cm³ ns at 400fs delay in Fig. 3(b), 2.0 GJ/cm³ ns at 1000fs delay in Fig. 3(c) and 2.2 GJ/cm³ ns at the 5000fs delay in Fig. 3(d).

So far we have obtained the obviously different dynamics at the different delay time between the field pulse and the heat pulse, furthermore our results demonstrate that only at the appropriate delay of the field pulse to the heat pulse, can the switching be obtained. In the following, to clearly compare the difference of field effect on the ultrafast dynamics at the different delay of field pulse to the heat pulse, we plot the figure with the magnetization dynamics curves under the same heat pulse and field pulse while at different delays time between them, also the result without the field pulse is plotted here for comparison. As shown in Fig. 4, the delay time are chosen as no delay, 100fs, 400fs, 1000fs and 5000fs. It is obvious that the field show a gradually strengthening effect on the magnetization dynamics with the increase of the delay time at first (as shown in Fig. 4 of the dynamics curves at no delay, 100fs delay and 400fs delay), that is the field induce the more change of the magnetiztion. Then the field shows an almost invariant effect on the magnetization with the increase of the delay time via comparing the dynamics curves at 1000fs delay and 5000fs delay, which is shown that the field induced almost the same change of magnetization.

Fig. 4 Comparison of ultrafast magnetization dynamics with the same laser fluence (1.8 GJ/cm³ ns) while at the different delay time of field pulse to the heat pulse.

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Now we try to explain the dependence of the delay time on the switching via comparing the results of Fig. 3 and Fig. 4. First, as for the dynamics at 100fs delay time of field pulse to the heat pulse, as shown in Fig. 3a), the results of only showing pure thermal demagnetization even at high pump fluence, can be explained that the time of demagnetization is longer than the 100fs delay time (as shown in Fig. 4 with the yellow line) so that the field can not have a strong effective action on the magnetization at this ultrafast demagnetization process, which is clear by comparing the dynamics curves of “no-field” and “no-delay” (as shown in Fig. 4 with the grey line and red line, respectively)that all of these cureves show the similar dynamics. However, as for the dynamics at the 400fs delay time of field pulse to the heat pulse (as shown in Fig. 3b)), the effect of the field can induce the switching because after 400fs of the heat pulse exciting, the temperature of the sample has been below the Curie temperature and magnetization has start to recovery (as shown in Fig. 4 with the blue line) rather than to be in the completely demagnetization state, now the field can have a strong effective action on the sample to induce reversed domain nuclearation and domain wall motivation leading to switching. Besides, these explanation can answer the question proposed above why the duration of field pulse should be longer than that of heat pulse to induce switching. Because when the heat pulse excites the sample, it will induce ultrafast demagnetization after the first several hundred femtosecond. At this process the magnetic field have little impact on the sample so that it can not induce reversed domain nuclearation and domain wall motivation to induce the switching. Only the magnetization start to recover after the demagnetization can the magnetic field have a strong effective action on the sample to induce switching, which require that the duration of field pulse should be longer than heat pulse to have a long enough effective action time on the sample after the magnetization starting to recover.

However, when the delay time of the field pulse to the heat pulse is further increased, although the field have started to have an effective action on the magnetization, the recovery of the magnetization will lead to the increase of the net remnant magnetization to exceed to the low-remanence criterion for HD-AOS. In this situation no switching can be achieved, which is clearly shown in Fig. 4 of the dynamics curves at 1000fs delay and 5000fs delay. Or if we want to obtain the switching, we should accordingly increase the pump fluence with the increase of the delay time to ensure that the net remnant magnetization below the low-remanence criterion. This is exactly the case of the results in Fig. 3c) and Fig. 3d), at the longer delay time range, the minimum switching threshold fluence would increase with the delay time, which is also agree with the net remnant magnetization of sample influencing the minimum switching threshold [13]. The increase of the delay time means the later the magnetic field begin to act on sample, the more the magnetization has been recover and the larger the net remnant magnetization is. Therefore, to reach the same net remnant magnetization with the situation at the shorter delay time, we should increase the laser pump fluence, which lead to the minimum switching threshold increasing accordingly.

To study how long the delay of field pulse to heat pulse can still induce the switching, we have simulated this magnetization dynamics process by increasing the delay time to no switching can be obtained whatever the pump flunence is increased. The result (not shown here) demonstrate the delay window is wide up to at least over 10ps dependent on the pump fluence of the heat pulse, which indicates that after the heat induced ultrafast demagnetization, the field can induce the switching, even after a long delay after the heat pulse, as long as the magnetization have begun to recover and at the same time the net remnant magnetization is still smaller than the low-remanence criterion [13] for HD-AOS.

Based on the discussions above, there exists a window interval of the delay time of the field pulse to the heat pulse to induced the switching. The minimum delay time is decides by the time of the magnetic field start to have a strong effective action on the magnetization, while the maximum delay time is decided by the time of the recovery of the net remnant magnetization to exceed the low-remanence criterion for HD-AOS. What’s more, there exists an optimal delay time of the field pulse to the heat pulse. If the delay time is too short, no switching can not be obtained for the field have no strong effective action on the magnetization. While if the delay time is too long, either no switching can not be obtained for the or the minimum switching threshold will increase. The key point for the field favour to the switching is founding out the optimum start action time of the magnetic field, that is the field start to strongly effectively act on the sample just as the magnetization start to recover, because in this moment the net remnant magnetization is smaller so that the smaller minimum switching threshold is required. Furthermore, we can carefully suppose that the mechanism of the single-pulse HD-AOS should be the combined result of heat and field effects. The heat effect induce the sample to a small net remnant magnetization, and then it is reversed by the induced effective field for the small remnant magnetization is very sensitive to the effect of the field due to the highly susceptible state.

Furthermore, we compared our results with the HD-AOS induced by multiple pulses in the ferromagnetic materials [20,21] whose switching proceeds two stages [20]. First, the heat effect is dominant, and it shows the accumulating heat effect by multiple pulse will increase the degree of demagnetization. And only when the demagnetization reaches a certain value, the helicity effect (or field effect) will effectively act on magnetization and induce the switching. This requires the induced magnetic field to be beyond the pulse duration in the switching process [21]. By comparison, it is clear that our simulation of the switching process induced by a single pulse is similar to the switching induced by multiple pulses. They are in accordance in the following three factors: 1) the demagnetization should reach a certain value, 2)the induced field can effectively act on the magnetization after heat effect of laser pulse, 3) it requires the duration of the field to be longer than that of laser(heat) pulse.

Finally, it should be noted that our method by separating the heat pulse and the field pulse for different time interval is difficult to be realized in the experiment by using the circularly polarized laser pulse considering the induced magnetic field pulse via IFE, because the effective light-induced magnetic field pulse always builds up with no delay with respect to the laser heat pulse. However, Sabine et all [6] used two pulse excitations in HD-AOS experiments, one is the circularly polarized laser pulse and the other is linearly polarized laser pulse, try to study the helicity effect and the heat effect, respectively. We compared our theoretical calculations with the experimental results presented by Sabine et al [6], which shows that they are in accordance in that the HD-AOS is the combined results of heat and field effects, and the field effect could act effectively after the heat effect. Therefore, it should be acceptable that by separating the heat pulse and field pulse it is helpful to distinguish the role played by laser heating and field action in switching. Furthermore, it can help to make it clear when the field can show an strong effect on the sample after the laser heating, so that it can explained why the induced field pulse should decay sufficiently long to induce the single-pulse HD-AOS and help to understand the mechanism of the single-pulse HD-AOS. Also, it give a propect for optimizing the process of the single-pulse HD-AOS. For instance, as is well known that the single-pulse HD-AOS only work in a relatively narrow fluence window [4]. Exceeding the upper threshold fluence, the sample forms a multidomain structure due to the time of sample’s temperature above the Curie temperature leading to sample in an complete demagnetization state is longer than the duration of the induced field pulse. Actually, according to our results, the underlying reason is the demagnetization existing so long that the field can not have a strong effective action on the magnetization at this demagnetization process. Therefore, if we can increase the heat transfer coefficient to decrease the time of sample’s temperature above the Curie temperature and shorten the time of magnetization start to recovery, means increase the time of the field effective action, it should increase the upper threshold fluence to further enlarge the switching fluence window.

4. Conclusion

To conclude, to distinguish the role of the heat effect and the inverse Faraday field induced by circularly polarized laser pulse in the single-pulse HD-AOS, and to make it clear when the field start to have a strong effective action on the magnetization so that it help to explain why the induced field pulse should decay sufficiently long to induce the single-pulse HD-AOS, we investigate the single-pulse HD-AOS in TbFe film based on the LLB equation of motion for high-temperature magnetization dynamics. In our model, we present the heat pulse combined with the magnetic field pulse to simulate the effect of circularly polarized laser pulse considering its IFE. First, we present the simulation with the duration of the field pulse 2.5 times as that of the heat pulse, which shows it can reproduce the result of the single-pulse HD-AOS. Then we present the simulation only using the heat pulse and only using the field pulse, respectively. Both the results show no switching can be obtained, which exclude both a nonthermal origin and a pure thermal origin of the single-pulse HD-AOS. To further study how the field effect influence the switching, we vary the ratio of the duration of heat pulse (range from 50fs to 250fs) to that of field pulse, and the results show whatever the duration of the filed is, the duration of the magnetic field pulse should be longer than that of the heat pulse to induce the single-pulse HD-AOS, which agree with the recent report that the induced field pulse should decay sufficiently long to induce the single-pulse HD-AOS. To explain this result, we further separate the heat pulse and the field pulse for different time interval. The results show there exist a window interval of the delay time, only in this window, the single-pulse HD-AOS can be obtained. The reason of no switching at short delay time is the field pulse can not have a strong effective action on the magnetization, which also explains why the induced field pulse should decay sufficiently long to induce the single-pulse HD-AOS. However, the reason of no switching at long delay time is the magnetization has recovered more complete and the net remnant magnetization has exceed the minimum net magnetization that the magnetic field can induce switching. Furthermore, our results show at the delay time in the switching window, the minimum switching threshold fluence increase with the delay time due to the net remnant magnetization of sample influencing the minimum switching threshold.

Based on the above results, we can draw the conclusion that the single-pulse HD-AOS should be the combined result of heat and field effects. The detailed process of switching is the heat effect acting on the sample to a small net remnant magnetization at first, and then when the magnetization start to recover, during which the magnetic field is able to start to have a strong effective action on the sample to induce the switching. The reason for the field pulse should be decay long enough to induce the single-pulse HD-AOS, is the existing delay between the time of the induced field starting to strong effective action and the time of induced magnetic field being built up via IFE. Our result helps to understand the role of heat effect and field effects in HD-AOS so that it helps to understand the underlying mechanism of the single-pulse HD-AOS. Finally, it gives a prospect to optimize the single-pulse HD-AOS by increasing the heat transfer coefficient to increase the time of the field effective action to enlarge the switching fluence window.

Funding

Natural Science Foundation of Guangdong China (2015A030313400, 2017A030313009); Guangzhou Scientific Research Project (201707010347).

Acknowledgements

The authors would like to acknowledge support of the Natural Science Foundation of Guangdong China and the Guangzhou Scientific Research Project. Chudong Xu also gratefully acknowledges the kind help of T. A. Ostler.

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Quantity	Value	Units
λ	0.05
M_s(0)	2.5×10⁵	[emu/cc]
K(0)	3.8×10⁵	J/m³
γ	1.76×10⁻¹¹	[T⁻¹s⁻¹]
System size	2 × 2	[μm²]
Number of macrospins	500×500×1
Macrospin size	4×4×4	[nm³]
T_C	500	[K]

heat pulse [fs]	threshold at ratio=1:1 [GJ/cm³ ns]	threshold at ratio=1:1.5 [GJ/cm³ ns]	threshold at ratio=1:2 [GJ/cm³ ns]	threshold at ratio=1:2.5 [GJ/cm³ ns]	threshold at ratio=1:3 [GJ/cm³ ns]
50	No switching	No switching	4.7	4.2	3.9
100	No switching	2.7	2.3	2.2	2.1
150	No switching	1.7	1.6	1.5	1.5
200	No switching	1.3	1.2	1.2	1.2
250	No switching	1.1	1.0	1.0	1.0

Quantity	Value	Units
λ	0.05
M_s(0)	2.5×10⁵	[emu/cc]
K(0)	3.8×10⁵	J/m³
γ	1.76×10⁻¹¹	[T⁻¹s⁻¹]
System size	2 × 2	[μm²]
Number of macrospins	500×500×1
Macrospin size	4×4×4	[nm³]
T_C	500	[K]

heat pulse [fs]	threshold at ratio=1:1 [GJ/cm³ ns]	threshold at ratio=1:1.5 [GJ/cm³ ns]	threshold at ratio=1:2 [GJ/cm³ ns]	threshold at ratio=1:2.5 [GJ/cm³ ns]	threshold at ratio=1:3 [GJ/cm³ ns]
50	No switching	No switching	4.7	4.2	3.9
100	No switching	2.7	2.3	2.2	2.1
150	No switching	1.7	1.6	1.5	1.5
200	No switching	1.3	1.2	1.2	1.2
250	No switching	1.1	1.0	1.0	1.0

Role of the heat effect and field effect in single femto-second laser pulse induced helicity-dependent all-optical magnetic switching

Abstract

1. Introduction

2. Modeling method

2.1. Landau-Lifshitz-Bloch model

2.2. Spatially dependent two-temperature model

3. Results and discussions

4. Conclusion

Funding

Acknowledgements

References

Cited By

Figures (4)

Tables (2)

Equations (13)

Optical Materials Express