Hot spin spots in the laser-induced demagnetization

Laser-induced femtosecond magnetism or femtomagnetism simultaneously relies on two distinctive contributions: (a) the optical dipole interaction (ODI) between a laser field and a magnetic system and (b) the spin expectation value change (SEC) between two transition states. Surprisingly, up to now, no study has taken both contributions into account simultaneously. Here we do so by introducing a new concept of the optical spin generator, a product of SEC and ODI between transition states. In ferromagnetic nickel, our first-principles calculation demonstrates that the larger the value of optical spin generator is, the larger the dynamic spin moment change is. This simple generator directly links the time-dependent spin moment change {\Delta}Mk z (t) at every crystal- momentum k point to its intrinsic electronic structure and magnetic properties. Those hot spin spots are a direct manifestation of the optical spin generator, and should be the focus of future research.

Femtomagnetism represents an emerging frontier in magnetic recording technology [1, 2], where a femtosecond laser pulse is used to read/write the magnetic information in the storage media within a few hundred femtoseconds. However, how such an ultrafast magnetization process occurs is not well understood. The true underlying microscopic mechanism is still under debate. Two major mechanims are proposed. One is the Elliot-Yafet (EY) mechanism proposed by Koopmans et al. [3,4], in which the spin relaxation occurs via electron scattering at impurities or defects and phonons. Further, Steiauf et al. [5] invoked a strong spin-mixing parameter to support the EY mechanism. However, the EY mechanism is challenged in rare earth compounds as Vahaplar et al. [6] demonstrated a nonthermal writing on a time scale of 10 ps. On such a long time scale, the phonon should play an important role by smearing out any initial polarization dependence on the laser light. In a group of the lanthanide-doped permalloys, Radu et al. [7] particularly reported an opposite behavior to the EY's prediction [3]. Nonthermal nature of femtomagnetism was clearly demonstrated in GdFeCo [8], which also invalidated stimulated Raman scattering mechanism. Another mechanism attributes the demagnetization to the coherent interaction of the laser beam with the electron of the system proposed by Bigot et al. [9]. In systems with spin-orbit coupling in general most of the optical electric dipole transitions of the stimulated electrons conserve the dominant spin character of the electrons, but a few transitions are spin-flip transitions. The number of spin-flip transitions increases with increasing spin-orbit coupling. The coherent interaction mechanism is strongly supported by the theoretical findings independently developed by Zhang et al. [10,11], and they found that a crucial cooperation between laser field and spin-orbit coupling is indispensable to ultrafast demagnetization. In 2009, Lefkidis et al. [12] proposed a spin-switch mechanism in the coherent temporal regime, which is based on the angular-momentum exchange between the light and the irradiated antiferromagnets. Motivated by those latest investigations, here we provide a new view on the coherent interaction mechanism in femtosecond magnetism.
To this end, the existing experimental results are not conclusive enough to develop a simple method to characterize the spin moment change in femtomagnetism. In particular, little attention has been paid to two most important contributions simultaneously: Optical dipole interaction (ODI) and spin expectation value change (SEC). SEC is defined as the difference between the spin expectation values of two transition states. ODI allows the laser pulse to influence the system, while SEC allows the spin moment change during laser excitation. However, those two contributions are intrinsically disconnected. For instance, the dipole interaction does not guarantee spin moment change. In optical dipole transitions, without spin-orbit coupling, the spin is conserved. The separation of ODI and SEC is unique to femtomagnetism, and is very different from the magnetization process driven by a thermal or magnetic field, where the spin moment change is the only quantity that should be considered. Therefore, to fully understand femtomagnetism, it is a must to take both contributions into account simultaneously.
In this Letter we take into account both contributions by introducing a new conceptoptical spin generator or SD. This generator is defined as a product of optical dipole transition moment and SEC for a particular transition. If multiple transitions are involved, a summation over those transitions should be carried out. We test this concept in ferromagnetic nickel. Our first-principles calculation shows that SD has the capability to single out optically hot spin spots. In nickel, two major hot spots are identified. Their structures in the Brillouin zone are almost identical to those constructed from the optical spin generator. Therefore, our finding not only greatly simplifies the interpretation of femtomagnetism, but also points out a new direction for future experimental investigations.
We start with a transition from state |a to state |b . The expectation value of optical spin generator SD is defined as (1) where D a→b is a dipole transition moment, ∆S a→b = S b − S a is SEC and S a refers to the spin expectation value in state |a , and c.c. refers to the complex conjugation. The direction indices of SD are omitted for brevity. To see an analytic example, we resort to two states characterized by total angular momentum quantum number j and its magnetic quantum number m j . First, we form the eigenstates for each j and m j . Then, we compute the dipole transition matrix element and the spin matrix element among those states. Finally, we compute the product of these two matrix elements. A lengthy but straightforward calculation [13] shows that for a transition with ∆m j = 0 and ∆j = 1, the expectation value of (SD) ab generator for this transition is where the radial contribution and its unit are not included. One sees explicitly that the expectation value of optical spin generator nonlinearly depends on j and m j ; such a nonlinear dependence results from their distinct dependence of spin and dipole expectation values on j and m j though both spin and dipole changes monotonically with j and m j , a finding which is verified in an entirely different system [14].
In the following, we are going to demonstrate the power of optical spin generator SD in ferromagnetic nickel. While the details of our theoretical formalism have been presented before [11], here in brief, we start our calculation by solving the Kohn-Sham equation selfconsistently. The spin-orbit coupling (SOC) is included explicitly, without using a spin mixing parameter [15]. Once we obtain eigenstates (ψ nk , E nk ), we then compute the spin and optical properties. We find that to converge our spin change, the number of required k- The interaction between the fs laser pulse field and the system is described by Here D = er is the electric-dipole operator with e being the electron charge and r the position operator. The dipole matrix elements are directly computed from the first-principles method, without using the constant matrix element approximation as others [15], which has a serious consequence (see below). The laser field E(t) peaks at 0 fs and has a Gaussian shape with duration of 12 fs and photon energy of 2 eV. We numerically solve the Liouville equation for the electron density matrices ρ k at each k point [18], where H = H 0 + H I . Due to the huge number of k points, all the calculations are run in parallel [18]. The spin moment change at each k point is computed from ∆M k , and S z k is the z-component of spin matrix. Since the spin moment change oscillates with time, we time-average it from 90 to 150 fs; averaging the changes from 120 to 150 fs yields a similar result.
We first show that those conventional high symmetry lines and planes in the Brillouin zone are not a good guide for spin moment change. Figures 1(a)-(c) show the crystal-momentum-resolved major magnetic spin moment changes ∆M k z (t) on those traditional high symmetry lines and planes, where the filled circles denote the moment increase and the empty circles the moment reduction, with quantitative changes shown below each figure. In the Γ-X-W-K plane, the major spin reductions occur close to the Fermi surface (see Fig. 1(a)), but in the Γ-K-L-U-X plane, the major change is away from the Fermi surface. The maximum reduction reaches −0.22µ B . The hexagonal L-K-W-U-W ′ plane, which does not intersect with the Fermi surface, has a major change around the L-U line. Note that k points with the spin moment increase also appear in these planes. The net spin change comes from the superposition of the two types hot spots.
Next we demonstrate the power of optical spin generator to characterize the spin moment change in two steps: (1) we thoroughly examine the spin moment change ∆M k z (t) at each k in the entire Brillouin zone and (2) directly compute the SD and compare it with the real spin moment change. Figure 2(a) illustrates the first comprehensive dispersion of ∆M k z (t), which is constructed out of 73763 irreducible k points. To reduce the huge volume of the data, we only plot 750 k points with spin moment change ∆M k z smaller than −0.1µ B (more negative). This comprehensive dispersion finally narrows our attention to only two major spin hot spots and some small islands. Our findings are insightful. First, structurally the hot spin spots consist of stacks of layers of spin hot points in the crystal momentum space. These hot spots with large spin moment reduction are big enough to be detected experimentally. A small portion of spot B is displayed at the bottom left of the figure, with one representative layer magnified on the bottom right. Then we choose a triangle, highlighted by dark red balls, for a close examination. Choosing such a triangle is purely arbitrary, and is mainly in the consideration of the continuation of the coordinates of the k-points; the conclusion is same when we choose a different triangle in different layers. To define the above triangle, we choose three k-points, k 1 , k 2 , and k 3 , whose coordinates are given in the caption of Fig.   2.
Second, different from the single-channel excitation process as seen in our above analytic example, the laser photon pre-selects multiple channels. To see this clearly, we present the energy band dispersion along three sides of the triangle (see Figure 2(d)). There are two transitions with their transition energies close to the laser photon energy, see two vertical arrows. According to our previous investigation [19], this leads to a possible resonant excitation. Therefore, the generator is constructed by summing over all the major transitions as (SD) k ≡ ij ∆S z k;ij D k;ij , with i and j being the initial and final state band indices. In our present case, there are two such transitions. The results are illustrated in Fig. 2(b).
The hot spin spots are nicely reproduced by SD except the small islands. This explicitly shows that the SD is able to characterize the spin moment change of femtomagnetism in the crystal-momentum space. Figure 2(e) compares the dispersion of spin moment change with that of SD. We see that from k 1 to k 2 , SD drops initially (more negative) and then increases sharply (less negative) when close to k 2 . These changes are a joint effect of the SEC and dipole moment. As seen in the above analytic example, the change is highly nonlinear. This is partly due to the fact that ∆S and D generally depend on quantum numbers and k points differently. Going from k 2 to k 3 , we see a monotonous decrease (more negative).
On the k 3 -k 1 side, SD shows a peak. These changes will manifest themselves in the spin moment change at each k point.
Figure 2(e) shows the spin moment change ∆M k z along three sides of the same triangle. It is astonishing that ∆M k z closely follows SD, with all main features reproduced. The difference is that SD is computed solely from the SEC and dipole matrix elements while ∆M k z is computed dynamically. Our additional extensive calculations (not included in the paper) yield the same conclusion. This proves unambiguously that SD largely governs the nature of laser-induced spin dynamics. If we compare those two figures more closely, we do see some differences between ∆M k z and SD. For instance, along the k 1 -k 2 side, SD has a smaller decrease than that in ∆M k z ; along the k 2 -k 3 side there is a kink in ∆M k z . Along the k 3 -k 1 side, the spin moment change is also much more pronounced than that in SD. These small differences are directly associated with the laser pulse shape and the photon energy.
Notwithstanding these small difference, the optical spin generator provides an easy way to evaluate the spin moment change, without complicated real time simulations.
Conceptually, the optical spin generator is very useful for future experimental and theoretical investigations, not only in ferromagnets but also in magnetic semiconductors, since SD is essentially a system quantity, and the only external input to construct SD is the energy window set by laser photon energy. The physics meaning of SD is very clear: It acts as a source term for the laser to influence spin. We emphasize that had we used the constant dipole matrix, SD would be a simple spin matrix, and the optical dipole selection rule would be ignored. The hot spin spots are a manifestation of SD. Future experimental research should be able to test our results.
In conclusion, we have introduced a new concept of the optical spin generator to simultaneously take into account both the dipole transition and spin expectation value change between transition states, each of which is indispensable to laser-induced femtosecond magnetism. The generator directly links to the spin moment change at each k point. This helps to unfold the hidden kernel of femtomagnetism in crystal-momentum space. In ferromagnetic nickel, we find that the hot spin spots consist of two major spin hot spots and a few smaller islands. These hot spots are not along any high symmetry lines or plane nor Fermi surfaces.