arXiv:1604.08704v1 [cond-mat.supr-con] 29 Apr 2016
Current induced magnetization dynamics and magnetization switching in superconducting ferromagnet hybrid (F|S|F) structures Saumen Acharjee∗ and Umananda Dev Goswami† Department of Physics, Dibrugarh University, Dibrugarh 786 004, Assam, India, We investigate the current induced magnetization dynamics and magnetization switching in a superconducting ferromagnet sandwiched between two misaligned ferromagnetic layers by numerically solving Landau-LifshitzGilbert equation modified with current induced Slonczewski’s spin torque term. Ginzburg-Landau Gibb’s free energy functional in London’s limit has been used for this purpose. We demonstrated the possibility of current induced magnetization switching in ferromagnet-superconductor hybrid structures with strong easy axis anisotropy and the condition for magnetization reversal. The switching time for such arrangement is also calculated and is found to be highly dependent on the magnetic configuration. PACS numbers: 67.30.hj, 85.75.-d, 74.90.+n
I.
INTRODUCTION
During the last 15 years, a number of very interesting compounds have been discovered which reveal the coexistence of ferromagnetism and superconductivity in the same domain in bulk [1–6]. The interplay between ferromagnetic order and superconductivity thus gains lots of attention from varity of research communities [7]. Among those, some peoples were hunting for superconductivity in a ferromagnetic spin valve made up of two ferromagnetic substances separated by a superconducting element (F|S|F system). The discovery of spin triplet superconductivity in F|S|F spin valves [8] by Banerjee and others provides the unique opportunity to study the system. The major interest of the superconductor|ferromagnet (F|S) hybrid structures is due to the dissipation less flow of charge carriers offered by the superconducting environment. To completely understand this hybrid structure it is important to study the spin polarized transport. Moreover, the transport of spin is closely related to the phenomenon of current induced magnetization dynamics [9] and spin transfer torque [10, 11]. Spin transfer torque (STT) which is the building block of spintronics is based on the principle that when a spin polarized current is applied into the ferromagnetic layers, spin angular momentum is transferred into the magnetic order. It is observed that for a sufficiently large current, magnetization switching can occur [12, 13] in a magnetic layer. Thus the flow of electrons can be served to manipulate the configuration of the spin valves. Traditionally, a lot of works had been done on current induced magnetization dynamics and STT earlier on ferromagnetic layers. Soon after, a lot attention have been made on anti-ferromagnetic layers [13–16] also. Making a hybrid structure of a superconductor with a ferromagnet and the concept of current induced magnetization dynamics suggest a very interesting venue for combining two different fields namely superconductivity and spintronics. A few works had been earlier done on F|S hybrid structures [17–20]. In Ref. [12] supercurrent-induced magne-
∗
[email protected] †
[email protected]
tization dynamics in Josephson junction with two misaligned ferromagnetic layers had been studied and demonstrated the favourable condition for magnetization switching and reversal. Motivated by the earlier works, in this paper we studied the current induced magnetization dynamics of a superconducting ferromagnet in a hybrid structure of F|S based on Landau-Lifshitz-Gilbert (LLG) equation with Slonczewski’s torque (LLGS) using the Ginzburg-Landau-Gibb’s free energy functional. The proposed experimental setup is shown in Fig.1, in which two ferromagnets are separated by a thin superconducting ferromagnet. The coersive fields of the ferromagnets are such that the magnetization is hard in one layer while soft in the other and the orientation of magnetization of the soft ferromagnetic layer is supposed to be misaligned with the hard ferromagnetic layers by an angle θ. When the junction is current-biased, it gets spin polarized in the hard layer and thus transfer angular momentum to the magnetic order. This generates an induced magnetization contributing to the magnetic order. The dynamics of this induced magnetization has been studied by numerically solving the LLGS equation. The paper is organised as follows. In the Section II, a theoretical framework of the proposed setup is developed. The results of our work is discussed in the Section III by solving LLGS equation numerically . Finally we conclude our work in the Section IV. II. THEORY
To study the current induced magnetization dynamics of a ferromagnetic superconductor with easy axis anisotropy in F|S|F spin valve we utilized Landau-Lifshitz-Gilbert (LLG) equation with Slonczewski’s spin transfer torque (LLGS). The resulting LLGS equation takes the form ∂m ∂m = −γ(m × Hef f ) + α(m × )+T (1) ∂t ∂t where γ is the gyromagnetic ratio, α is the Gilbert’s damping constant and Hef f is the effective magnetic field of superconducting ferromagnet. T is the current induced spin transfer torque and can be read as [12]
2 quantum and θS represent the superconducting order parameter phase. The term B.H/4π accounts the contribution from the external magnetic field. K1 and K2 are analogous to quadratic and biquadratic constants accounting the magneto crystalline anisotropy. In general the U-based ferromagnetic superconductors have a very strong easy-axis magneto crystalline anisotropy [22]. To account this we consider the anisotropy axis to be directed along x-direction and contribution from the first anisotropy term is considered. In view of this the LLGS equation (1) takes the form
FIG. 1: The proposed experimental setup. A ferromagnetic superconductor is sandwiched in between two ferromagnetic layers. The magnetization orientation of the ferromagnetic layers are supposed to be misaligned by an angle θ. When a current I is injected it gets polarized and transfer spin torque to the magnetic order causing magnetization dynamics. Different colours of the ferromagnetic layers indicates the level of magnetization. Here, the bottom layer is hard in magnetization.
T = Iζ(m × [m × (mT − ǫmB ])
III.
ν~µ0 . 2em0 V
(3)
Here, e is the electronic charge, ν is the polarization efficiency, ~ is the Planck’s constant, µ0 is the magnetic permeability, m0 is the amplitude of magnetization and V is the volume of the system. I is the applied current bias. The effective magnetic field of the system can be obtained from the functional derivative of the free energy with respect to the magnetization: Hef f = −
δF δm
RESULTS AND DISCUSSIONS
(2)
where mT and mB respectively represents the normalized magnetization vector in the top and bottom magnetic layers of the spin valve and is taken as mT = (0, 1, 0) and mB = (0, cos θ, sin θ) such that for θ = 0, the configuration is parallel and is anti-parallel for θ = π. ǫ provides the factor of asymmetry in polarization in top and bottom ferromagnetic layers. The term ζ is given by ζ=
∂m ∂m = −γ[m×(Kexc ∇2 m+K1 m⊥ −B)]+α(m× )+T, ∂t ∂t (6) where m⊥ is the component of m perpendicular to the anisotropy axis which we direct along the y - direction with B = −B0 zˆ. The equation (6) is a non linear coupled differential equation in m and can be solved by numerical simulation. To model a realistic superconducting ferromagnet, the spin exchange coupling and the anisotropy field can be taken as [22] K1 ∼ 103 and Kexc ∼ 107 . Furthermore we set, ǫ = 0.1, B0 = 1 and ζ = 1.
(4)
Within the London limit, the Ginzburg-Landau Gibb’s free energy functional of a magnetic superconductor is given by [21] Z φ0 1 1 B2 1 (A+ ∇θS )2 + F = d3 r[ a|ψ|2 + b|ψ|4 + 2 2 4 8πλ 2π 8π 1 1 B.H 1 ], + K1 |m|2 + K2 |m|4 + Kexc |∇m|2 − B.m − 2 4 2 4π (5) where B = ∇ × A, ψ is the superconducting order parameter, Kexc is a coupling constant represents the exchange interaction, λ is the London penetration depth, φ0 is the flux
The magnetization dynamics and switching behaviour of our setup are investigated based on the above mentioned parameters initially for very weak damping (α ≪ 1) and then for strong damping (up to α = 0.5) with a very small angle of misalignment θ0 = 0.1π. Furthermore, to solve the equation (6) the time coordinate has been normalized to τ = γm0 t, where m0 is the magnitute of the magnetization. Few of the corresponding numerical solutions are shown in Fig. 2 for two current biasing. The inset figures show the parametric plot of the time evolution of the magnetization components. The figures in left panels show the weak damping regime with Gilbert’s damping parameter α = 0.05 for two different choices of current biasing 0.1 mA and 1.2 mA respectively from top to bottom. While the damping is considered to be strong with α = 0.5 in the figures of the right panels for the respective current biasings. It is seen that the magnetization components show quite different behaviours. The components mx and mz display oscillating decay until it vanishes completely, while on the other hand the component my saturates with the increasing value of τ . It is to be noted that the qualitative behaviour of the components of magnetization is similar for different damping parameters. But the quantitative difference is that in strong Gilbert’s damping regime, the oscillation of the magnetization components mx and my die out faster in time scale, while the component mz saturates too rapidly as seen from the right panels of Fig. 2. It is also our interest to see what happens when the baised current of the system is increased. To check that we increase the biased current keeping the damping parameter fixed and is found that when the current biasing goes to or goes beyond 1.2 mA, the y and z components of magnetization suddenly reverses for all values of damping parameter as seen from the bottom panels of the Fig. 2. This result indicates that it is
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FIG. 2: The time evolution of normalized components of magnetization with initial angle of misalignment θ = 0.1π. The figures in left panels depicted the weak damping α = 0.05, while the figures in the right panels are for strong damping α = 0.5 with a current biasing 0.1 mA and 1.2 mA respectively from top to bottom panels. The inset figures display a parametric plot representing the behaviour of the magnetization.
possible to generate current induced magnetization reversal of a superconducting ferromagnet in a F|S|F spin valve setup shown in Fig.1 by means of current biasing. Moreover, it is also seen from the figure that for the weak damping but with higher current biasing, oscillations and switching for reversal of respective components of magnetization are delayed by some factors. In view of these results it is also important to see explicitly what happens to the spin valve if configuration is changed and what influence the spin valve configuration has on the switching time τswitch ? To answer these two questions we have stuided our setup for different angles of misalingments as well as for different current biasings keeping the damping factor as low. One of the important results of this study is that for higher angles of misalingment (θ ≥ 0.4π) reversal of corresponding components of magnetization occurs at 1.0 mA current bias, whereas for other angles of misalingment this occurs at 1.2 mA current bias as shown in the representative plots in the Fig. 3. Also, we have observed that for all angles of misalingment except θ = 0.5π delying times for oscillation and reverval of respective components of magnetization at the reversal current bias are almost same. But this time is clearly different for the angle 0.5π and is maximum among all the angle as seen from the bottom right panel of the Fig. 3. From this study we have also plotted the switching time τswitch as a function of θ in Fig.4 for three different current bias. The switching time is defined as the time required by the magnetization component to attain numerically the 0.975 times of the saturated value [13]. From the first panel of this figure it is observed that the switching time gradually increases in a very monotonic fashion for a current biasing of 0.5 mA with different system configurations. This suggests
that the configuration near the parallel (θ = 0) offers rapid switching then the anti parallel (θ = π) for a current bias of 0.5 mA. However, for a current biasing of 1 mA the variation of switching time is non-monotonic with sharp peak and then suddenly it falls as seen from the middle panel of the Fig. 4. The peak of the switching time in this case supports our earlier observation of maximum time delay for ocillation and reversal of magnetization components at θ = 0.5π and current bias of 1.0 mA. The variation of switching time for this case is exactly opposite of the previous case, here anti parallel configuration offers much rapid switching then the parallel orientation which is quite obvious as in this condition the STT becomes stronger. A very similar observation is also made for a current bias of 1.5 mA seen from the right panel of the Fig. 4. Only difference is the peak position shifted towards the parallel orientation. So from Fig. 4 it can be concluded that the switching time has a strong dependence on the spin valve configuration and also on the current biasing. Our final interest is to study the influence of magnetic field on the component of magnetization. To analyse this we have plotted the magnetization components with τ for different values of the magnetic field in Fig. 5, keeping the biased currents fixed at 0.1mA. It is seen that for low value of B0 the components of magnetization shows quite similar behaviour as seen earlier in Fig. 2. However, on the the hand as the magnetic field increases the components of magnetization shows quite irregular behaviour. For example, for B0 > 10 as seen from the top right panel of the Fig.5, the my component suddenly reverses. As B0 becomes of the order of the anisotropy field, the components of magnetization behaves quite differently. In this condition, the component mz saturates while my shows an oscillating decay. So, here the magnetic field dominates
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FIG. 3: The time evolution of normalized components of magnetization with initial angles of misalignment are θ = 0.01π (left panels) and θ = 0.5π (right panels) for a weak damping α = 0.05 only. In the left panels current biasing are 0.1 mA and 1.2 mA respectively from top to bottom, while in the right panels they are 0.1mA and 1.0mA. The inset figures display a parametric plot representing the behaviour of the magnetization. 0.24
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FIG. 4: Switching time and its dependence on the spin valve configuration for a current bias of 0.5 mA, 1.0 mA and 1.5 mA respectively from left to right panels.
over the anisotropy field. It can be easily visualized from the parametric plot shown inset of bottom left of panel of the Fig. 5 where the motion takes place about the direction of magnetic field. The motion stabilize itself for more higher values of B0 . We have found the influence of magnetic field as mentioned above is exactly similar for all biasing currents and hence higher value of magnetic field (B0 ≥ 10) eliminate the effect of biasing current.
IV.
CONCLUSIONS
In summary, in this paper we have investigated the current induced magnetization dynamics and magnetization switching in a superconducting ferromagnetic sandwiched between two misaligned ferromagnetic layers with easy axis anisotropy by numerically solving Landau-Lifshitz-GilbertSlonczewski’s equation. For this purpose we have used the
Ginzburg-Landau-Gibbs free energy functional. We have demonstrated about the possibility of current induced magnetization switching for an experimentally realistic parameter set. It is observed that for the realization of magnetization switching sufficient biased current and moderate field are suitable for the case low Gilbert damping. Although, very rapid switching can be obtained for large damping, however such system can not be used because the system become highly unstable in such situation, which is unrealistic. Also switching is highly dependent on the magnetic configuration. It shows a monotonic increase for low current while it becomes non monotonic for higher values. Another conclusion can also be drawn that configuration near anti parallel offers more rapid switching than the parallel. Again, it can also be conclude that the dynamics is highly dependent and controlled by the magnetic field as it becomes of the order of the anisotropy field. As a concluding remark, the results indicates about the switching mechanism in F|S|F spin valve setup for an exper-
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FIG. 5: The time evolution of normalized components of magnetization with initial angle of misalignment θ = 0.1π and for current biasing I = 0.1 mA for different values of external magnetic field, viz., B0 = 10 (top left), B0 = 102 (top right), B0 = 103 (bottom left) and B0 = 2x103 (bottom right). The figures in insets display a parametric plot representing the behaviour of the magnetization.
imentally favourable parameter set which may be utilized to bind Superconductivity and Spintronics together for making
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
S.S. Saxena et al., Nature (London) 406, 587 (2005) D. Aoki et al., Nature (London) 413, 613 (2001) C. Pfleiderer et al., Nature (London) 412, 58 (2001) N.T. Huy et al., Phy. Rev. Lett. 99, 067006 (2007) J. Flouquet and A. Buzdin, Phys. World 15, 41 (2002) S. Nandi et al., Phys. Rev. B. 89, 014512 (2014) A.I. Buzdin, Rev. Mod. Phys. 77, 935 (2005) N. Banerjee, et al., Nature Communications (London) 5, 4048 (2014) I. Zutic, J. Fabian and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004) J. C. Slonczewski, J. Magn. Magn. Mater, 159, L1 (1996) L. Berger, Phys. Rev. B. 54, 9353 (1996) Jacob Linder, Takehito Yokoyama, Phys. Rev. B 83, 012501 (2011).
practical Superconducting Spintronic devices.
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
Jacob Linder, Phys. Rev. B. 84, 094404 (2011). A.S. Nunez et al., Phys. Rev. B. 73, 214426 (2006) X.L. Tang et al., Appl. Phys. Lett. 91, 122504 (2007) S. Urazhdin and N. Anthony, Phys. Rev. Lett. 99, 046602 (2007) X. Waintal and P. W. Brouwer, Phys. Rev. B. 65, 054407 (2002) S. Takahashi et al., Phys. Rev. Lett. 99, 057003 (2007) E. Zhao and J. A. Sauls, Phys. Rev. B. 78, 174511 (2008) V. Braude and Ya. M. Blanter, Phys. Rev. Lett. 100, 207001 (2008) T.K. Ng, C.M. Verma, Phys. Rev. B. 58, 11624 (1998). A.A. Bespalov, A.S. Mel’nikov and A.I. Buzdin, Phys. Rev. B. 89, 054516 (2014).
