1. Introduction

Interface-induced magnetic properties of materials attract much attention nowadays [1]. The most studied effects here are the interface-induced magnetic anisotropy [2] and Dzyaloshinskii-Moriya interaction [3,4]. As well, a number of exciting effects have been observed in ultrathin 2D magnetic heterostructures such as enhanced interlayer exchange [5], topological Hall effect in thin oxide films [6], interface dependent magnetization in rare-earth and transition metal layers [7], strong modification of antiferromagnetic behavior [8], specific magnetic states such as spin ice [9,10]. Magnetic interactions at interfaces can bring about enhanced or reduced magnetic moments and reorientation of the magnetic easy axis, which were treated as a source of surface-induced magnetooptical effect [11]. Magnetization-induced optical second harmonic generation (SHG) based on simultaneous breaking of space- and time-reversal symmetry is an efficient probe of surface and interface magnetism, as it combines intrinsic SHG interface sensitivity with high values of magnetooptical effects at the SHG wavelength [12,13]. Exciting capabilities of this technique were demonstrated when studying magnetic nanolayers [14], burried interfaces [15] and nanostructures [16,17]. At the same time, magnetic field induced SHG can appear due to inhomogeneous magnetization distribution, as was shown theoretically in [18] and confirmed by the experiments on visualization of magnetic domains via the MSHG probe [19].

Here we study specific magnetic field induced SHG (MSHG) effects in three-layer Pt/Co/W and Pt/Co/Pt films with the thickness of cobalt 2-10 nm. We show experimentally that besides common MSHG effects extra ones appear that are absent for homogeneously magnetized structures. Symmetry analysis of the considered MSHG effect supports the existence of the out-of-plane magnetization gradient.

2. Phenomenological approach

It is well known that SHG is symmetry forbidden in the electric dipole approximation in the bulk of centrosymmetric materials, thus for the case of metal layered nanostructures broken inversion symmetry at interfaces gives rise to interface SHG described by second-order nonlinear susceptibility $\hat {\chi }^{(2)} \equiv \hat {\chi }^{(0),cr}$ through $I_{2\omega }=\vert \textbf{P}^{(2)} \vert ^2 \propto \vert \hat {\chi }^{(2)} \vert ^2$. In the case of magnetic nanostructures, additional symmetry breaking happens due to their magnetization, which leads to the appearance of local magnetic field induced second-order susceptibility, $\hat {\chi }^{(2)M}$, which can be presented as a series in magnetization M as $\hat {\chi }^{M}(\textbf{M})$=$\hat {\chi }^{(1)M}\textbf{M}+\hat {\chi }^{(2)M}\textbf{M} \textbf{M}+\cdots$, the first term here describes magnetooptical Faraday and Kerr effects at the SHG wavelength. If considering nonlocality of M, one should take into account the following gradient in M terms of the SHG susceptibility [18]: $\hat {\chi }^{\nabla }(\textbf{M})$=$\hat {\chi }^{(3)\nabla }\boldsymbol{\nabla } \textbf{M}+\hat {\chi }^{(4)\nabla } \textbf{M} \boldsymbol{\nabla } \textbf{M} +\hat {\chi }^{(5)\nabla } \boldsymbol{\nabla } \textbf{M} \boldsymbol{\nabla } \textbf{M}$, where $\textbf{M}$ is the local magnetization of the sample. The terms with even numbers of polar vector associated subindices ( i.e. $\hat {\chi }^{(3)\nabla }$ and $\hat {\chi }^{(4)\nabla }$) are symmetry allowed in centrosymmetric materials and thus contribute to bulky SHG, while the term $\hat {\chi }^{(5)\nabla }$ can be neglected. For in-plane isotropic magnetic structures, one can find the MSHG experimental geometry when local $\hat {\chi }^{(1)M}$ term vanishes, while magnetic gradient terms of the SHG susceptibility $\hat {\chi }^{(3)\nabla }$ and $\hat {\chi }^{(4)\nabla }$ remain symmetry-allowed. This is the case of longitudinal magnetic field and p-polarized SHG, which was chosen for the experiments described below. The coordinate frame is shown in the inset of Fig. 1(b); longitudinal magnetic field H is oriented along the (OX) axis, normal to the structure corresponds to the Z-direction, p-polarization of the fundamental or SHG waves lies in the (XOZ) plane.

 

Fig. 1. (a) MOKE for Pt(2)Co(3)Pt(3) (dashed line, red symbols) and for Pt(3)Co(4)W(3) (open squares, solid line) films; (b) dependencies of the transmitted SHG intensity on the longitudinal magnetic field for Pt(3)Co(3)W(3) film for the angle of incidence $\theta = \pm 20^{\circ }$ and "p-in, mix-out" geometry, inset: scheme of the experimental setup with the used coordinate frame.

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In the experiment described below we studied in-plane ferromagnetic/heavy metals isotropic films (see Sec. 3.). In saturating magnetic fields, they are homogeneously magnetized in the (OX) direction, thus we can neglect the in-plane magnetic gradient terms in the directions along and perpendicular to H that are described by $\nabla _x M_{x,z}$ and $\nabla _y M_{x,z}$. At the same time, due to interface effects, the terms proportional to out-of plane gradients of magnetization, i.e. $\nabla _z M_{x,z}$, may exist.

Based on the full symmetry analysis of the MSHG in structures with local and nonlocal magnetization, one can find that odd in M effects in the p-polarized SHG in saturating longitudinal magnetic field involves only $\hat {\chi }^{(4)\nabla }$ components, while those of $\hat {\chi }^{(3)\nabla }$ are absent. Odd in H terms of the SHG polarization are given by the following components: $\chi _{xxxZzX}^{(4)}, \chi _{xxxXzZ}^{(4)}, \chi _{xzzZzX}^{(4)}, \chi _{xzzXzZ}^{(4)}, \chi _{zzxZzX}^{(4)}, \chi _{zzxXzZ}^{(4)}$, where capital letters in subindices denote the corresponding components of M, small letters correspond to polar vectors of the fundamental electric field and the gradient [18]. The SHG intensity measured in the experiment is thus given by the expression $I_{2\omega }=\vert \textbf{P}^{(2)cr}+\textbf{P}^{(4)\nabla } \vert ^2$ and thus may reveal odd in M magnetooptical SHG effect.

3. Results and discussion

Three-layer ferromagnetic/heavy metals films of the composition Pt(3)Co(x)W(3), where the layer thickness is indicated in nm and x=2-10 nm, were deposited on glass substrates using a AJA 2200 multichamber system at the basic pressure of 10$^{-5}$ Pa. In order to avoid the uniaxial magnetic anisotropy, the substrates were azimuthally rotated during the metals deposition. Figure 1(a) shows the magnetic hysteresis loops of the longitudinal magnetooptical Kerr effect (MOKE) in Pt(3)Co(4)W(3) and Pt(2)Co(3)Pt(3) films; the in-plane coercive field being less than 100 Oe.

SHG experiments were performed when using the radiation of the Ti-sapphire laser (800 nm wavelength, 50 fs pulse duration, 80 MHz repetition frequency) that passed through the sample at the angle of incidence $\theta$=$[+40^{\circ };-40^{\circ }]$, the laser spot on the sample surface being about 30 $\mu m$ in diameter. Transmitted SHG passed through an analyzer, 6mm BG39 (Schott) filters and detected by a PMT Hamamatsu R4220P. MSHG effects were studied for the longitudinal magnetic field applied along the (OX) direction as shown schematically in the inset of Fig. 1(b). The common longitudinal MSHG effect was studied for the mixed SHG polarization (the analyzer set at 45$^{\circ }$ between p- and s-polarizations) so that the SHG intensity variation was detected as the result of the second harmonic polarization plane rotation. Meanwhile, gradient in M effect was studied for the p-polarized SHG, as no modulation of the SHG intensity is expected in that case for homogeneous magnetic structures [20]. For brevity, the first one is denoted below as ”allowed” SHG effect, the latter one as ”forbidden” [21]. First we studied the ”allowed” longitudinal MSHG effect in transmission through the films of various compositions. Figure 1(b) shows the SHG magnetic hysteresis loops for Pt(3)Co(4)W(3) film for the angles of incidence $\theta = \pm 20^{\circ }$. One can see sharp SHG intensity peaks close to zero values of H, which are caused by the re-magnetization of the films through the in-plane rotation of the averaged magnetization [21]. It is worth noting that changing the sigh of the incident angle results in the inversion of the sign of the MSHG effect, indicating the change in the direction of the SHG polarization plane rotation associated with the SHG asymmetrical propagation through a magnetized medium. Similar MSHG effects were attained for all the studied films of both symmetrical (Pt/Co/Pt) and asymmetrical (Pt/Co/W) compositions.

Figure 2 shows the SHG magnetic field dependencies measured for the normal incidence of the laser beam polarized parallel to the applied magnetic field, as shown schematically on the insets. For in-plane isotropic films, only odd in M magnetization-induced components like $\chi ^{(2)}_{yxx}$ contribute to the $y$-polarized SHG (polarized orthogonal to the pump beam) [20], so the SHG intensity $I_{2\omega } \propto (\chi ^{(2)}_{yxx})^2$ is of the same value for positive and negative fields; SHG intensity minima situated symmetrically with respect to zero magnetic field reveal the remagnetization processes. At the same time, for symmetric Pt(3)Co(3)Pt(3) film (black squares) the SHG intensity is absent within the experimental accuracy, showing that the magnetization induced susceptibility is close to zero, as was expected for symmetrical layered structures [22].

 

Fig. 2. Dependencies of the transmitted SHG intensity on longitudinal magnetic field for normal incidence of the fundamental beam polarized parallel to H; SHG is polarized perpendicular to the magnetic field of the pump: Pt(3)Co(3)W(3) film (red and blue filled symbols) and Pt(2)Co(3)Pt(3) film (black circles); SHG polarization is parallel to H: Pt(3)Co(3)W(3) film (open symbols).

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On the contrary, as the polarizations of the incident and SHG waves are parallel to the external magnetic field (lower inset in Fig. 2), the first three terms of the second-order susceptibility $\chi ^{(0)cr}, \chi ^{(1)}, \chi ^{(2)}$ vanish, while small but still non-zero SHG signal that appears in this case can originate from the gradient terms $\chi _{xxxXzZ}^{(4)}$ and $\chi _{xxxZzX}^{(4)}$ associated with the fan-type magnetization distribution at the interfaces [23]. Being the only possible SHG source under the considered geometry of the nonlinear interaction, they produce equal SHG signal for positive and negative H values, which is consistent with the experimental data. One can also see the peaks in the SHG intensity close to zero magnetic field associated with the magnetization reversal.

In order to investigate MSHG effects induced by magnetic gradient terms, we studied the magnetic field dependence of the SHG intensity in the ”forbidden” geometry on the cobalt thickness $d_{Co}$ in asymmetric PtCoW films. Figure 3 shows the SHG hysteresis loops for $d_{Co}$=2,4 and 10 nm measured in transmission through the films at 20$^{\circ }$ angle of incidence. One can see that similarly to observed previously for uniaxial magnetic structures, changing the sign of the saturating longitudinal magnetic field leads to SHG intensity variation characterized by the asymmetry coefficient $A \equiv \frac {I^+ -I^-}{I^+ +I^-}$, where $I^+$ and $I^-$ denote the SHG intensity for the opposite directions of H; the highest $A$ values are achieved for the smallest Co thickness of 2 nm and decreases with $d_{Co}$ (inset in Fig. 3(c)). As a possible explanation we can suggest that the effect of structural asymmetry in the MSHG efficiency decreases as the heavy metals’ layers are spaced at larger distance.

 

Fig. 3. Dependencies of the SHG intensity (”p-in, p-out” polarization combination) on longitudinal magnetic field for the angle of incidence $\theta = 20^{\circ }$ for (a) Pt(3)Co(3)W(3) film, (b) Pt(3)Co(4)W(3) film, (c) Pt(3)Co(10)W(3) film.

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Figure 4 shows the magnetic hysteresis loops of the ”forbidden” effect for Pt(3)Co(3)W(3) sample for the angles of incidence $\theta = \pm$(10-40)$^{\circ }$. One can see that the asymmetry $A$ in saturating magnetic fields is an odd function of the angle of incidence and decreases with $\theta$ (Fig. 4(d)), contrary to the angular dependence of the non-magnetic SHG intensity determined as $\bar {I} = 0.5 \cdot \lbrace I^+ + I^-\rbrace$ (Fig. 4(c)). Strong angular dependence of $\bar {I}$ indicates that the most efficient crystallographic SHG contribution is induced by the $\chi ^{(2)cr}_{zzz}$ component, which grows as $sin^3 \theta$. When taking into account that only crystallographic and gradient in M contributions participate in SHG under these experimental conditions as $I_{2\omega }=\vert \textbf{P}^{(2)cr}+\textbf{P}^{(4)\nabla } \vert ^2$ and assuming that the phase shift between these two terms is constant, one gets $A(\theta ) \propto \chi ^{\nabla }(\theta )/\chi ^{cr}(\theta )$ as $\chi ^{\nabla } \ll \chi ^{cr}$. Further, as $\bar {I} \propto (\chi ^{cr})^2$, one can get the angular dependence of $\chi ^{\nabla }(\theta )$ shown in Fig. 4(d), right scale. One can see that it is a much more flat function of the angle of incidence, as compared to $\bar {I} (\theta )$, which is consistent with the angular dependencies of the SHG nonlinear polarization induced by $\chi _{xxxXzZ}^{(4)}$ and $\chi _{xxxZzX}^{(4)}$ components, where the first three subindexes give the $cos^3 \theta$ dependence. This also supports the assumption on the contribution of the magnetization gradient terms to the observed odd in M MSHG intensity effect in saturating longitudinal magnetic field.

 

Fig. 4. Dependencies of the SHG intensity on longitudinal magnetic field for Pt(3)Co(3)W(3) for (a) positive and (b) negative angles of incidence; angular dependencies of (c) the averaged intensity of p-polarized SHG component and (d) of the relative value of the SHG asymmetry and effective gradient term $\chi ^{\nabla }$.

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4. Conclusion

Summing up, we studied experimentally magnetization gradient contributions to the second-order nonlinear optical response of three-layer films composed of ferromagnetic and heavy metals. We show that for the longitudinal magnetic field and for the p-polarization of the second harmonic wave magnetic contrast in the SHG intensity is observed for the saturating magnetic fields of the opposite signs, which is symmetry forbidden for homogeneously magnetized structures. We suppose that this effect is induced by gradients of magnetization of the film in the out-of-plane direction, which is supported by the symmetry analysis.