Quantum mechanism of nonlocal Gilbert damping in magnetic trilayers

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(1)PHYSICAL REVIEW B 91, 214435 (2015). Quantum mechanism of nonlocal Gilbert damping in magnetic trilayers Ehsan Barati and Marek Cinal Institute of Physical Chemistry of the Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland (Received 16 April 2015; published 30 June 2015) A fully quantum-mechanical calculation of the Gilbert damping constant α in magnetic trilayers is done by employing the torque-correlation formula within a realistic tight-binding model. A remarkable enhancement of α in Co/NM1 /NM2 trilayers is obtained due to adding the caps NM2 =Pd, Pt, and it decays with the thickness of the spacers NM1 =Cu, Ag, Au in agreement with experiment. Nonlocal origin of the Gilbert damping is visualized with its atomic layer contributions. It is shown that magnetization in Co is damped remotely by strong spin-orbit coupling in NM2 via quantum states with large amplitude in both Co and NM2 . DOI: 10.1103/PhysRevB.91.214435. PACS number(s): 75.78.−n, 75.40.Gb, 75.70.Tj. I. INTRODUCTION. Employing magnetic layered structures in spintronic devices such as hard-disk read heads and magnetic random access memories is the key ingredient in data storage technology and its ongoing developments. This followed the discoveries of interlayer exchange coupling [1,2], giant [3] and tunneling [4,5] magnetoresistance, and spin-transfer torque [6–8] in trilayers built of ferromagnetic layers separated by nonmagnetic spacers. Metallic trilayers are also commonly used to investigate magnetization dynamics in view of potential spintronic applications like racetrack memories [9] and spin torque nano-oscillators [10]. The dynamical processes in magnetic nanodevices and, in particular, magnetization switching are profoundly affected by magnetic damping due to spin-flip scattering and transfer of spin angular momentum. Magnetic relaxation in ferromagnetic metals is governed by the Gilbert damping, which enters the phenomenological Landau-Lifshitz-Gilbert (LLG) equation [11,12]. The Gilbert damping plays a crucial role in magnetization dynamics of magnetic layered systems. In particular, it affects the threshold spin current required for magnetization switching [13] and the domain wall velocity in current-carrying domain wall structures [9]. In the last two decades, extensive research activities have been devoted to magnetization dynamics in magnetic films [14–25]. The Gilbert damping constant α in ferromagnet/nonmagnet (FM/NM) metallic bilayers is found to be appreciably enhanced in comparison with its bulk value [21,22]. The damping is also enhanced in FM/NM1 /NM2 trilayer structures with spacer layers of NM1 =Cu due to adding the NM2 =Pd, Pt, Ru, and Ta caps [16–18,21,23–25]. This experimental evidence clearly shows that the enhancement of the Gilbert damping in magnetic layered systems is of nonlocal origin. An early theoretical paper on nonlocal magnetic damping is due to Berger [26]. He argued that the exchange coupling between itinerant sp electrons passing through the FM/NM interface and d electrons in the FM yields an enhanced Gilbert damping due to spin-flip electron transitions in which spin waves are emitted or absorbed near the interface. The enhanced Gilbert damping in FM/NM layered systems is explained in a semiphenomenological way in Refs. [27,28] by pumping spin angular momentum from the FM into the adjacent nonmagnetic (normal metal) layers. According to 1098-0121/2015/91(21)/214435(5). the spin pumping theory, the predicted damping enhancement in FM/NM1 /NM2 trilayers decays with the thickness of the NM1 spacer with low spin-flip rate. Although this theory gives a plausible general explanation of spin relaxation in magnetic layered systems, it does not provide a fully quantum-mechanical description of this phenomenon. Such a description can be achieved using Kamberský’s torque correlation model [29] on which the present calculations are based. Despite numerous experiments no quantum calculations of magnetic damping in magnetic trilayer systems have been reported, except a recent paper [30] which addresses the Gilbert damping only in NM/Py/NM symmetric trilayers within an ab initio scattering formalism. In our recent work [31] the damping constant in bulk ferromagnets, ferromagnetic films, and FM/NM bilayers was calculated with the torque correlation formula within a realistic tight-binding (TB) model. Therein, it has been shown that magnetic damping in Co/Pd and Co/Pt bilayers has large nonlocal contributions from their nonmagnetic parts adjacent to the ferromagnetic Co layer. This paper is devoted to FM/NM1 /NM2 trilayers in which a significant damping contribution comes from the second nonmagnetic part NM2 separated from the ferromagnetic layer by a magnetically inactive spacer. The aim of the present work is to establish the quantum mechanism of the nonlocal Gilbert damping in such trilayers. Calculations have been performed for Co/NM1 /NM2 trilayers with NM1 =Cu, Ag, and Au as the spacer and NM2 =Pd and Pt as the cap. The dependence of α on the spacer thickness and the electron scattering rate is investigated. The nonlocal origin of the Gilbert damping in such systems is visualized via atomic layer contributions to α. To better understand the mechanism of the nonlocal damping we investigate the spatial distribution of contributing electron states. II. THEORY. A phenomenological description of magnetization dynamics in magnetic systems is given by the LLG equation ∂m ∂m = −γ m × Heff + α m × (1) ∂t ∂t that represents the time evolution of the unit vector m pointing along magnetization M. The first term in Eq. (1), with the gyromagnetic ratio γ , describes the Larmor precession of. 214435-1. ©2015 American Physical Society.

(2) EHSAN BARATI AND MAREK CINAL. PHYSICAL REVIEW B 91, 214435 (2015). magnetization around the effective magnetic field H eff , applied externally and/or due to magnetic anisotropy. The second term, proportional to the Gilbert damping constant α, describes the relaxation of magnetization towards the direction of the field. A pioneering quantum-mechanical description of the Gilbert damping dates back to 1976 when Kamberský proposed his torque correlation model [29]. The expression for α within this model takes the following form for a magnetic layered system [31] 1 π |Ann (k)|2 Fnn (k). (2) α= dk NFM μs BZ n,n It includes the integration over the wave vector k in the two-dimensional (2D) Brillouin zone (BZ) of the volume BZ and the sum over band indices n,n . The parameters μs and NFM stand for the atomic magnetic moment (in units of the Bohr magneton μB ) and the number of atomic layers in the ferromagnetic part of the system, respectively. The matrix elements Ann (k) = nk|A− |n k are found for the torque A− = [S − ,HSO ] due to the spin-orbit (SO) interaction HSO where the spin operator S − = 12 (σx − iσy ) is given by the Pauli matrices σx , σy . The factor Fnn (k) is defined as the integral over energy ∞ d η()L( − n (k))L( − n (k)). (3) Fnn (k) = −∞. Here, η() = −dfFD /d is the negative derivative of the Fermi-Dirac function fFD (), and the two Lorentzians L depend on the energies n (k), n (k) of the electron states |nk,|n k, respectively. The width of the Lorentz function L(x) = ( /2π )/(x 2 + 2 /4) is the average electron scattering rate , treated here as an independent parameter. The present calculations are based on the TB model of the electronic structure in magnetic layered systems [31,32]. The TB Hamiltonian, including the SO interaction, is constructed within the Slater-Koster formalism [33,34]. The expression (2) is employed to calculate α in FM/NM1 /NM2 trilayers with out-of-plane magnetization; cf. Ref. [35] for a discussion on an arbitrary direction of M. The calculations are done for a wide range of scattering rates 0.001 eV 2.0 eV (expressed as /τ with the lifetime τ in Refs. [36,37]). The integral in Eq. (3) is evaluated efficiently by summing over the Matsubara frequencies and the poles of the two Lorentz functions [31]. Since the calculated α is weakly dependent on temperature T entering fFD () [31,37], finite T = 300 K is used to obtain a fast convergence of α with (60)2 k points in the 2D BZ. The calculations are further speeded up by limiting the integration to 1/8 of the 2D BZ. III. RESULTS. In this paper, we particularly concentrate on calculation of α in Co/NM1 /NM2 trilayers. The considered NM1 =Cu, Ag, and Au spacers are poor spin sinks as possessing long spindiffusion lengths λsd (Refs. [18,25,38,39]) and the NM2 =Pd and Pt caps with short λsd (Ref. [25]) are known as perfect spin sinks.. FIG. 1. (Color online) The Gilbert damping constant α in an Co(6 ML) film, Co(6 ML)/NM1 bilayers (NM1 =Cu, Ag and Au), and Co(6 ML)/NM1 /NM2 (4 ML) trilayers (NM2 =Pd, Pt) vs NM1 thickness; the scattering rate = 0.01 eV.. In Fig. 1 we depict the damping constant α versus the NM1 spacer thickness in Co(6 ML)/NM1 /NM2 (4 ML) trilayers for the scattering rate = 0.01 eV. For comparison, α for the corresponding Co(6 ML)/NM1 bilayers and the Co(6 ML) film (α 0.0026 for = 0.01 eV) are also shown. The calculated α in the Co/NM1 /NM2 trilayers declines almost monotonically, while slightly oscillating, with increasing the thickness N of the NM1 spacer layer; the oscillation periods are 5 ML for Cu and 5–7 ML for Ag. These oscillations are attributed to quantum well states with energies close to the Fermi level F . The damping constant α found for the Co/Cu/Pt trilayer is larger than that of the Co/Cu bilayer in accord with experiment [18,23]. The enhancement is over threefold at the Cu thickness of 3 and 5 ML and more than twofold for 5 ML < N 70 ML. Using Pd as the cap instead of Pt also results in significant damping enhancement though with much smaller values of α due to the weaker SO coupling in Pd. Almost the same results are obtained if Ag is used instead of Cu as the spacer, whereas the Au spacer leads to a higher damping due to its strong SO coupling. The presently obtained 1/NCo dependence of α on the Co thickness NCo in Co/NM1 /NM2 trilayers (not shown) is also in agreement with experiment on FM/Cu/NM2 heterostructures [16,17]. Other experimental reports on Py/Cu/Ta trilayers [21] and an Cu/Py/Cu/Pt system [18] have shown that the contribution from the second nonmagnetic layer (i.e., NM2 =Ta and Pt, respectively) vanishes for the spacer layer thicker than its λsd . However, such spacers are too thick for calculating α in the present model. For a spacer with thickness N much smaller than its spin-diffusion length, the analytical formula for α derived in the spin pumping theory [25,28] yields the following simple B dependence of α = A + N+C on N . Here A, B, and C are expressed with NCo , the spin mixing conductance of the Co/NM1 interface and the parameters of both nonmagnetic metals: λsd and the electrical conductance. Our results for. 214435-2.

(3) QUANTUM MECHANISM OF NONLOCAL GILBERT DAMPING . . .. FIG. 2. Gilbert damping constant α in Co(6 ML)/Cu(N ML)/ Pt(4 ML) trilayers against the scattering rate for different Cu spacer thicknesses N .. Co/NM1 /NM2 trilayers (Fig. 1) are perfectly fitted with this general formula within the considered range of the NM1 spacer thicknesses (up to 70 ML ≈ 12.5 nm) which are smaller by at least one order of magnitude than λsd of NM1 (200 ± 50 nm for Cu [18,25]). Figure 2 illustrates that the Gilbert damping in Co(6 ML)/Cu(N ML)/Pt(4 ML) trilayers alters with the scattering rate in a similar way for different thicknesses of the Cu spacer. The minimum of α occurs at ∈ [0.01 eV,0.1 eV] depending on N . Such a minimum occurs for bulk ferromagnets in the same range of [37,40]. As seen, the damping constant is almost independent of the spacer thickness for 0.05 eV. The experimentally observed decrease in α with increasing N is obtained for the range < 0.05 eV, including = 0.01 eV used in the present work. We attribute the obtained enhancement of the Gilbert damping in Co/NM1 /NM2 trilayers to the strong SO coupling in the NM2 cap as well as the high density of states at F in NM2 . The effect of the former has already been confirmed for Co/Pt bilayers by switching off the SO coupling in the Pt cap [31]. The composition of the quantum states contributing most to α is discussed in more detail below. A deeper understanding of the nonlocal enhancement of the Gilbert damping can be achieved by analyzing its spatial distribution. In our recent paper [31], an analytical expression −1 for the damping constant α = NFM l αl represented by a sum of contributions αl from individual atomic layers l has been derived and applied to ferromagnetic films and Co/NM bilayers. Therein, it has been shown how the Gilbert damping which stems from the ferromagnetic (Co) part is also damped nonlocally in the nonmagnetic part of the bilayers. Here, the analysis of layer contributions is utilized to investigate the nonlocal Gilbert damping in the Co/NM1 /NM2 trilayers. Figure 3 presents the layer contributions to the damping constant for Co(6 ML)/NM1 /Pt(4 ML) trilayers with different thicknesses of the NM1 spacer. It is seen that the distribution of the Gilbert damping within such trilayer structures is similar for different Cu spacer thicknesses. There are significant layer contributions in the Co part and almost no contributions from atomic layers inside the Cu spacer. Dominating contributions. PHYSICAL REVIEW B 91, 214435 (2015). FIG. 3. Layer contributions to the damping constant α in Co/NM1 /Pt trilayers (NM1 =Cu, Au) with different NM1 thicknesses; = 0.01 eV.. come from the Pt layers in a similar way as previously reported for Co/NM bilayers with NM=Pd and Pt [31]. As the Cu spacer gets thicker the contributions from the Pt layer get smaller, in accordance with experiment [18] and prediction of the spin pumping theory [28]. However, even for the thickest considered spacers (70 ML thick) the total contribution from the Pt cap is larger than the contribution from the Co film. Such spatial distribution of the Gilbert damping is due to the lack and presence of d bands with energies very close to F in Cu and Pt, respectively, as well as the strong SO coupling in Pt. Similar damping distributions (not shown) are obtained for Ag spacers and for the NM2 =Pd cap whose top of the d band lies above F as in Pt. Since the SO coupling in Pd is weaker than in Pt the layer contributions inside the Pd cap are smaller than in the Pt cap. This pattern is noticeably changed if Au is used as the spacer instead of Cu since there are nonzero contributions from the Au atomic layers at both the Co/Au and Au/Pt interfaces as well as the modified contributions from the Co and Pt interface atomic layers. The obtained results prove the nonlocal nature of the relaxation process in the investigated trilayers where magnetization precesses in the ferromagnetic Co film, but it is damped in the distant nonmagnetic cap separated from the Co film by a magnetically inactive spacer. To understand this mechanism on an even more fundamental level we examine the quantum states that contribute to the Gilbert damping. The Gilbert damping in the torque-correlation model stems from two kinds of electron transitions: intraband (n = n ) transitions within a single energy band and interband (n = n ) transitions between different energy bands [29,37]. The main source of the damping enhancement in Co(6 ML)/Cu(N ML)/Pt(4 ML) trilayers with = 0.01 eV is the intraband transitions though the interband transitions also give a significant contribution to the damping constant as shown in Fig. 4(a). The spatial composition of quantum states contributing to the intraband term of α is visualized in Figs. 4(b)–4(d). It is found that, while large contributions come from states almost entirely localized inside the Co film, the majority of states that significantly contribute to the Gilbert damping span throughout. 214435-3.

(4) EHSAN BARATI AND MAREK CINAL. PHYSICAL REVIEW B 91, 214435 (2015). damping enhancement due to the combination of their sizable amplitude in the Pt cap and the large SO coupling strength of Pt. The fraction of these states in Cu grows with increasing the Cu spacer thickness however the average probability per Cu atomic layer is similar for all investigated Cu thickness (3 ML, 9 ML, 30 ML) and it is around 0.02 ML−1 . Thus, the states leading to enhanced α in the Co/Cu/Pt trilayers with thick Cu spacer are composed of bulklike sp states in Cu and, attached to them, d states in Co and Pt with amplitudes up to a few times larger than in Cu. IV. CONCLUSIONS. FIG. 4. (Color online) (a) Intraband and interband terms of the damping constant α in the Co(4 ML)/Cu(N ML)/Pt(4 ML) trilayers with = 0.01 eV vs the Cu spacer thickness N ; (b)–(d) contributions to the intraband term of α in the trilayers with N = 3, 9, 30 ML from quantum states |nk with various fractions in the Co film and the Pt cap. The inclined yellow lines correspond to states with a fixed fraction in the Cu spacer.. the whole trilayer. Such states have a substantial fraction in each of its three constituent parts: Co, Cu, and Pt. In the trilayers with the Cu spacer a few ML thick [Fig. 4(b)] these fractions range from 0.2 to 1 in Co, from 0.0 to 0.8 in Pt, and up to 0.2 in Cu while summing up to 1 for each state. For thicker spacers the states giving predominant contributions to α have smaller fractions in Pt, and they tend to be split into two groups. At N = 30 ML the group of states with fractions between 0.1 and 0.5 in both Co and Pt gives a contribution of 0.010 to the total α = 0.022. These states are responsible for the. [1] P. Grünberg, R. Schreiber, Y. Pang, M. B. Brodsky, and H. Sowers, Phys. Rev. Lett. 57, 2442 (1986). [2] S. S. P. Parkin, N. More, and K. P. Roche, Phys. Rev. Lett. 64, 2304 (1990). [3] M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472 (1988). [4] T. Miyazaki and N. Tezuka, J. Magn. Magn. Mater. 139, L231 (1995). [5] J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meservey, Phys. Rev. Lett. 74, 3273 (1995). [6] L. Berger, J. Appl. Phys. 49, 2156 (1978). [7] P. P. Freitas and L. Berger, J. Appl. Phys. 57, 1266 (1985). [8] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). [9] S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008).. We present a quantum-mechanical calculation of the Gilbert damping constant α in Co/NM1 /NM2 trilayers within the torque-correlation model. The damping is found to be remarkably enhanced due to adding Pt as the second nonmagnetic layer NM2 , and it decreases with increasing the thickness of the NM1 =Cu, Ag, and Au spacers in agreement with experiment. The analysis of atomic layer contributions to α elucidates the nonlocal nature of the Gilbert damping in magnetic trilayers. The spatial decomposition of quantum states contributing to the damping shows that its enhancement is due to delocalized electrons whose wave functions are sizable in all parts of the trilayers. The spins of such electrons contribute to the magnetization in the ferromagnetic Co layer, but they also strongly interact, via the SO coupling, with heavy atoms in the Pt layer. Therefore, the precession of these spins is damped efficiently, and this leads to enhanced damping of the total magnetic moment, although it is almost entirely confined to the Co layer. This paper thus provides insight into quantum mechanisms of magnetic damping in metallic layered systems. ACKNOWLEDGMENTS. We acknowledge the financial support of the Foundation for Polish Science within the International PhD Projects Programme, cofinanced by the European Regional Development Fund within Innovative Economy Operational Programme “Grants for innovation”.. [10] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature (London) 425, 380 (2003). [11] T. L. Gilbert, Phys. Rev. 100, 1243 (1955). [12] T. L. Gilbert, IEEE Trans. Mag. 40, 3443 (2004). [13] J. Z. Sun, Phys. Rev. B 62, 570 (2000). [14] S. J. Yuan, L. Sun, H. Sang, J. Du, and S. M. Zhou, Phys. Rev. B 68, 134443 (2003). [15] R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87, 217204 (2001). [16] A. Ghosh, J. F. Sierra, S. Auffret, U. Ebels, and W. E. Bailey, Appl. Phys. Lett. 98, 052508 (2011). [17] A. Ghosh, S. Auffret, U. Ebels, and W. E. Bailey, Phys. Rev. Lett. 109, 127202 (2012). [18] S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66, 104413 (2002).. 214435-4.

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