Theory of Spin Torque Assisted Thermal Switching of Single Free Layer
Abstract
The spin torque assisted thermal switching of the single free layer was studied theoretically. Based on the rate equation, we derived the theoretical formulas of the most likely and mean switching currents of the sweep current assisted magnetization switching, and found that the value of the exponent in the switching rate formula significantly affects the estimation of the retention time of magnetic random access memory. Based on the Fokker-Planck approach, we also showed that the value of should be two, not unity as argued in the previous works.
I Introduction
Magnetic random access memory (MRAM) using tunneling magnetoresistance (TMR) effect [1],[2] and spin torque switching [3],[4] has attracted much attention for spintronics device applications due to its non-volatility and fast writing time with a low switching current. A high thermal stability () (more than 60) of magnetic tunnel junctions (MTJs) is also important to keep the information in MRAM more than ten years. Recently, Hayakawa et al. [5] and Yakata et al. [6],[7] respectively reported that the anti-ferromagnetically (AF) and ferromagnetically (F) coupled synthetic free layers show high thermal stabilities ( for AF coupled layer and for F coupled layer) compared to a single free layer.
The thermal stability has been determined by measuring the spin torque assisted thermal switching of the free layer and analyzing the time evolution of the switching probability by Brown’s formula [8] with the spin torque term. The theoretical formula of the switching probability is generally given by , where . Here, , , and are the attempt frequency, current magnitude, and critical current of the spin torque switching at zero temperature, respectively. is the exponent of the current term in the switching rate , and was argued to be unity by Koch et al. in 2004 [9]. On the other hand, recently, Suzuki et al. [10] and we [11],[12] independently studied the spin torque assisted thermal switching theoretically, and showed that the exponent should be two. Since the estimation of the thermal stability strongly depends on the value of , as discussed in this paper, the determination of is important for the spintronics applications.
In this paper, we study the spin torque assisted thermal switching of the single free layer theoretically. In Sec. II, we derive the theoretical formulas of the most likely and mean switching currents of the sweep current assisted magnetization switching, and study the effect of the value of the exponent on the estimation of the retention time of the MRAM. In Sec. III, the differences of the theories in Refs. [9],[10],[11] are discussed by analyzing the solution of the Fokker-Planck equation. Section IV is devoted to the conclusions.
Ii Theory of Magnetization Switching due to Sweep Current
In this section, we consider the spin torque assisted thermal switching of the uniaxially anisotropic free layer, which has two minima of its magnetic energy. At the initial time , the system stays one minimum. From , the electric current is applied to the free layer which exerts the spin torque on the magnetization and assists its switching. In this section, the current is assumed to increase linearly in time with the sweep rate , as done in the experiments [7],[13],[14]. The magnitude of the current should be less than because we are interested in the thermally activated region. The time evolution of the survival probability of the initial state, , is described by the rate equation,
(1) |
where the switching rate is given by
(2) |
We assume that the attempt frequency is constant. is the exponent of the current term, . The switching probability is given by . Also, we define the probability density by . Equation (1) describes the escape from one equilibrium to the others in many physical systems, and the value of reflects their energy landscape: for the Bell’s approximation [15], for the linear-cubic potential [16], and for the parabolic potential [17],[18]. The determination of the value of has been discussed not only in spintronics but also the other fields of physics [19]. The form of Eq. (2) is the special case of the model of Garg ( in Ref. [20] corresponds to ).
The solution of Eq. (1) with the initial condition is given by
(3) |
where is the lower incomplete function. Figure 1 the time evolutions of (a) the switching probability and (b) its density . The values of the parameters are taken to be GHz, mA, mA/s, and , respectively, which are typical values found in the experiments [6],[7],[13],[14]. As shown, suddenly changes from 0 to 1 at a certain time at which takes its maximum. We call the switching time. The switching time is determined by the condition , i.e., , and is given by
(4) |
for , and
(5) |
for . Here is the product logarithm which satisfies . For a large , , and () can be approximated to
(6) |
The current at , , is the most likely switching current for the thermal switching. Since we are interested in the switching after the injection of the current at , should be larger than zero. Thus, the above formula is valid in the sweep rate range , where the critical sweep rate is given by
(7) |
The value of estimated by using the above parameter values is on the order of mA/s, which is much smaller than the experimental values ( mA/s in Ref. [14]). Thus, the above analysis is applicable to the conventional experiments.
We also define the mean switching current by
(8) |
Since takes its maximum at , we approximate that
(9) |
where . Then, can be approximated to
(10) |
where . Thus, is given by
(11) |
where is the exponential integral. It should be noted that is expanded as [21]
(12) |
where is the Euler constant. In general, the moment is given by
(13) |
Then, the standard deviation of the current, , is given by
(14) |
Since the thermal stability can be estimated by evaluating the parameter , as shown below, let us derive the relations between and experimentally measurable variables. The difference between the most likely switching current and mean switching current is given by
(15) |
For , and are, respectively, given by
(16) |
(17) |
As shown in Refs. [22],[23] is around in the experimentally reasonable temperature and sweep rate regions (so called fast pulling regime or Garg’s limit [24],[25]). Thus, we can approximate that and for . Then, for is given by
(18) |
Similarly, for , by using the approximation , and are, respectively, given by
(19) |
(20) |
Then, for is given by
(21) |
is approximately zero for a sufficiently high thermal stability () which means a narrow width of the probability density. We also find
(22) |
(23) |
for arbitrary and . We numerically verify Eqs. (22) and (23) among the temperature region K, where the values of the parameters are same with those in Fig. 1 ( is taken to be 60 for K). Equation (22) or (23) can be used to determine the value of experimentally. Otherwise, can be estimated by using the relation
(24) |
Let us discuss the effect of the value of on the estimation of the retention time of MRAM. We assume that the value of is experimentally determined by some other experiments [5]. Then, the unknown parameter in Eq. (16) or (19) is only the thermal stability. As mentioned above, can be experimentally determined by using Eq. (22), (23), or (24). By setting , we found that the estimated values of the thermal stability with () and () satisfy the relation . Let us define the retention time of MRAM by . Then, the ratio of the estimated values of the retention time by () and () is given by , which is on the order of for and increases with increasing . Thus, the determination of the value of is important for the accurate estimation of the retention time of MRAM.
Iii Comparison with Theory of Koch et al.
In this section, we investigate the difference of the value of between Koch et al. [9] and Refs. [10],[11],[12] by comparing the solutions of the Fokker-Planck equation, and show that should be two. For simplicity, in this section, the current magnitude is assumed to be constant in time [9],[10],[11],[12].
First of all, it should be mentioned that the analytical solution of the switching probability can be obtained only for the two special cases. The first one is the uniaxially anisotropic system [10]. The second one is the in-plane magnetized thin film in which the switching path in the thermally activated region is completely limited to the film plane, and thus, the effect of the demagnetization field normal to film plane is neglected [11]. In these systems, the magnetization dynamics can be described by one variable (the angle from the easy axis, ), although, in general, the magnetization dynamics is described by two angles (the zenith angle and azimuth angle ). Then, the thermal switching of the magnetization can be regarded as the one dimensional Brownian motion of a point particle. Although the effect of the demagnetization field of an in-plane magnetized system is taken into account in the definition of the critical current of Ref. [9], the model of Ref. [9] should be regarded as the identical with the models in Refs. [10],[11] because the assumption in Ref. [9] is valid for the two special cases mentioned above, where and are the total magnetic field acting on the free layer and magnetization direction of the pinned layer, respectively.
The difficulty to calculate the spin torque assisted thermal switching probability arises from the fact that the spin torque cannot be expressed as the torque due to the conserved energy. Mathematically, it means that we cannot find any function whose two gradients, and , simultaneously give the spin torque terms of the Landau-Lifshitz-Gilbert equation in coordinate. Then, the steady state solution of the Fokker-Planck equation deviates from the Boltzmann distribution. However, in the two special cases mentioned above, since the magnetization dynamics depends on only , can be obtained by integrating the spin torque term with respect to . Then, the Fokker-Planck equation,
(25) |
has a steady state solution of the Boltzmann distribution form, . Here , , , , , , and are the magnetization, volume of the free layer, applied field, uniaxial anisotropy field, strength of the spin torque in the unit of the magnetic field, gyromagnetic ratio, and the Gilbert damping constant, respectively. is the magnetic energy, and is the effective magnetic energy given by
(26) |
The term in Eq. (26) corresponds to mentioned above. By using the steady state solution of the Fokker-Planck equation, we can calculate the switching probability, according to Refs. [8],[10],[11].
Koch et al. argued that Brown’s formula with the magnetic energy is applicable to the spin torque switching problem by replacing and with . These replacements arise from the assumption that the directions of the spin torque ( , where and
(27) |
The origin of the problem in Ref. [9] is that is not a steady state solution of the Fokker-Planck equation (25): the steady state solution is . Since the effect of the spin torque can be regarded as an additional term to the applied field, as shown in Eq. (26), the potential barrier height of the spin torque assisted thermal switching is, similar to Brown’s formula [8], given by
(28) |
where the thermal stability is defined by . By using the relation
(29) |
and defining the critical current by , we find that [11]
(30) |
Thus, the exponent of the current term should be two.
Iv Conclusion
In conclusion, we studied the spin torque assisted thermal switching of the single free layer theoretically. We derived the theoretical formulas of the most likely and averaged switching currents of the sweep current assisted magnetization reversal, and showed that the value of the exponent in the switching rate significantly affects the estimation of the retention time of MRAM. We also discussed the difference between the theories in Ref. [9] and Refs. [10],[11] from the Fokker-Planck approach, and showed that the exponent should be two.
Acknowledgment
The authors would like to acknowledge H. Kubota and S. Yuasa for the valuable discussions they had with us.
References
- [1] S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki, and K. Ando, ”Giant room-temperature magnetoresistnace in single-crystal Fe/MgO/Fe magnetic tunnel junctions,” Nat. Mater., vol.3, pp.868-871, 2004.
- [2] S. S. P. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B. Hughes, M. Samant, and S. H. Yang, ”Giant tunnelling magnetoresistance at room temperature with MgO (100) tunnel barriers,” Nat. Mater., vol.3, pp.862-867, 2004.
- [3] J. C. Slonczewski, ”Current-driven excitation of magnetic multilayers,” J. Magn. Magn. Mater., vol.159, pp.L1-L7, 1996.
- [4] L. Berger, ”Emission of spin waves by a magnetic multilayer traversed by a current,” Phys. Rev. B, vol.54, pp.9353-9358, 1996.
- [5] J. Hayakawa, S. Ikeda, K. Miura, M. Yamanouchi, Y. M. Lee, R. Sasaki, M. Ichimura, K. Ito, T. Kawahara, R. Takemura, T. Meguro, F. Matsukura, H. Takahashi, H. Matsuoka, and H. Ohno, ”Current-Induced Magnetization Switching in MgO Barrier Magnetic Tunnel Junctions With CoFeB-Based Synthetic Ferrimagnetic Free Layers,” IEEE Trans. Magn., vol.44, pp.1962-1967, 2008.
- [6] S. Yakata H. Kubota, T. Sugano, T. Seki, K. Yakushiji, A. Fukushima, S. Yuasa, and K. Ando, ”Thermal stability and spin-transfer switching in MgO-based magnetic tunnel junctions with ferromagnetically and antiferromagnetically coupled synthetic free layers,” Appl. Phys. Lett., vol.95, pp.242504, 2009.
- [7] S. Yakata, H. Kubota, T. Seki, K. Yakushiji, A. Fukushima, S. Yuasa, and K. Ando, ”Enhancement of Thermal Stability Using Ferromagnetically Coupled Synthetic Free Layers in MgO-Based Magnetic Tunnel Junctions,” IEEE. Trans. Magn., vol.46, pp.2232-2235, 2010.
- [8] W. F. Brown Jr., ”Thermal Fluctuations of a Single-Domain Particle,” Phys. Rev., vol.130, pp.1677-1685, 1963.
- [9] R. H. Koch, J. A. Katine and J. Z. Sun, ”Time-Resolved Reversal of Spin-Transfer Switching in a Nanomagnet,” Phys. Rev. Lett., vol.92, pp.088302, 2004.
- [10] Y. Suzuki, A. A. Tulapurkar, and C. Chappert, ”Nanomagnetism and Spintronics,” Elsevier, Chapter 3, 2009.
- [11] T. Taniguchi and H. Imamura, ”Thermally assised spin transfer torque switching in synthetic free layers,” Phys. Rev. B, vol.83, pp.054432, 2011.
- [12] T. Taniguchi and H. Imamura, ”Minimization of the Switching Time of a Synthetic Free Layer in Thermally Assisted Spin Torque Switching,” Appl. Phys. Express, vol.4, pp.103001, 2011.
- [13] E. B. Myers, F. J. Albert, J. C. Sankey, E. Bonet, R. A. Buhrman, and D. C. Ralph, ”Thermally Activated Magnetic Reversal Induced by a Spin-Polarized Current,” Phys. Rev. Lett., vol.89, pp.196801, 2002.
- [14] F. J. Albert, N. C. Emley, E. B. Myers, D. C. Ralph, and R. A. Buhrman, ”Quantitative Study of Magnetization Reversal by Spin-Polarized Current in Magnetic Multilayer Nanopillars,” Phys. Rev. Lett., vol.89, pp.226802, 2002.
- [15] G. I. Bell, ”A theoretical framework for adhesion mediated by reversible bonds between cell surface molecules,” Science, vol.200, pp.618-627, 1978.
- [16] Y. Sang, M. Dubé, and M. Grant, ”Thermal Effects on Atomic Friction,” Phys. Rev. Lett., vol.87, pp.174301, 2001.
- [17] G. Hummer and A. Szabo, ”Kinetics from Nonequilibrium Single-Molecule Pulling Experiments,” Biophys. J., vol.85, pp.5-15, 2003.
- [18] J. Husson and F. Pincet, ”Analyzing single-bond experiments: Influence of the shape of the energy landscape and universal law between the width, depth, and force spectrum of the bond,” Phys. Rev. E, vol.77, pp.026108, 2008.
- [19] S. Getfert and P. Reimann, ”Optimal evaluation of single-molecule force spectroscopy experiments,” Phys. Rev. E, vol.76, pp.052901, 2007.
- [20] A. Garg, ”Escape-field distribution for escape from a metastable potential well subject to a steadily increasing bias field,” Phys. Rev. B, vol.51, pp.15592, 1995.
- [21] N. N. Lebedev, ”Special Functions & Their Applications,” Dover, Chapter 3, 1972.
- [22] T. Taniguchi and H. Imamura, ”Dependence of spin torque switching probability on electric current,” J. Nanosci. Nanotechnol., accepted.
- [23] T. Taniguchi and H. Imamura, ”Theoretical study on dependence of thermal switching time of synthetic free layer on coupling field,” J. Appl. Phys., vo.111, pp.07C901, 2012.
- [24] Y. J. Sheng, S. Jiang and H. K. Tsao, ”Forced Kramers escape in single-molecule pulling experiments,” J. Chem. Phys., vol.123, pp.091102, 2005.
- [25] H. J. Lin, H. Y. Chen, Y. J. Sheng, and H. K. Tsao, ”Bell’s Expression and the Generalized Garg Form for Forced Dissociation of a Biomolecular Complex,” Phys. Rev. Lett., vol.98, 088304, 2007.