Precession frequency and switching time of the magnetization vector of the spin-valve free layer with longitudinal anisotropy

In this paper, a magnetization vector dynamics of a spin-valve free layer with the planar anisotropy based on various materials was simulated. Two dynamics types were identified, being of practical interest for MRAM (switching) and STNO (stable precession). The range of current and field values, corresponding to these spin-valve operation modes, were obtained. The numerical calculations of the switching time shown that Co80Gd20 is the best material for a spin valve as a part of MRAM. As a result of calculating the precession frequency, it was concluded that the most suitable for fabrication of STNO ferromagnetic layers is the alloy Fe60Co20B20.


Introduction
In the classical theory, the magnetic states of ferromagnets are controlled by an applied magnetic field. In 1996, J. Slonchevsky [1] predicted another way to change the magnetic configuration of ferromagnetic thin films using a spin-polarized current. The angular momentum transferred by the spin-polarized current transmits the torque to the magnetization vector, which leads to its switching or constant precession. A change of the magnetization vector projection onto the anisotropy axis causes a magnetoresistance variation of the structure, which entails voltage fluctuations in the external circuit. Magnetization switching is the main mode of the magnetoresistive random access memory (MRAM) operation [2,3], and the magnetization vector precession is used in the work of spin-transfer nanooscillators (STNO) [4].  The simplest configuration of the MRAM cell and the spin-transfer oscillator consists of a relatively thick "pinned" ferromagnetic layer, which serves as a current polarizer, a non-magnetic layer, and a relatively thin "free" layer. An antiferromagnetic layer is needed to fix the magnetization of the pinned layer (figure 1). A non-magnetic metal (for example, copper, platinum) or a thin dielectric (for example, MgO) is used as a nonmagnetic interlayer. The similar structure is usually called a spin valve or magnetic tunnel junction (MTJ), respectively.
The main objective of this work is to calculate the switching time and oscillation frequency of the spin valve placed in the magnetic fields of various directions, as well as the selection of the most suitable ferromagnetic materials and magnetic field configurations, which provide the best switching and frequency characteristics for MRAM and STNO, respectively.

Basic equations
The object of this study is a spin valve with planar anisotropy of the layers. The side of the valve square cross section was equal 11 nm a . The anisotropy axis was directed along one of the sides of the square. The free layer thickness was  1 2 nm d , the pinned layer thickness was  2 5 nm d , and the thickness of the copper nonmagnetic interlayer was The axis OX of the coordinate system associated with the structure was directed along the anisotropy axis. The structure was placed in a magnetic field H, which can be directed along one of the coordinate system axes.
, , H H H are the projection of the vector H on the corresponding axis. An electric current of density J is passed perpendicular to the plane of the layers. M is the magnetization vector of the spin-valve free layer, and s is the unit vector that coincides with the direction of the pinned layer magnetization (figure 1).
For the ferromagnetic layers, such materials as cobalt Co and iron Fe (single-crystal films which are easier and cheaper to obtain), alloys 60 20 20 Fe Co B and 70 30 Fe Co (they have a high spin polarization parameter  0.5 P ), and alloys 93 7 Co Gd and 80 20 Co Gd (they have the best magnetic properties to reduce the switching magnetic field) are considered. In [5], the magnetic parameters of these materials are presented in more detail. Defects in the microstructure of materials are not taken into account in our model.
The magnetization vector dynamics of the spin-valve free layer M is described by the Landau-Lifshits-Gilbert equation where  is the gyromagnetic ratio,  is the dimensionless dissipation coefficient, s M is the saturation magnetization.
The effective magnetic field eff H is included the magnetic anisotropy field, the demagnetization field, the effective field created by the spin-polarized injection current, and the external magnetic fie98ccld H. The details of the bifurcation analysis of the dynamic system (1) were given in [2,5,6].
The projection variation of the magnetization vector M on the OX anisotropy axis due to the giant magnetoresistance effect leads to the change in the output signal U [7]. Its value is determined by the following expression:      where X M is the vector M projection on the axis OX, P R and AP R is the resistance of the spin valve in the parallel and anti-parallel states (table 1), respectively. According to the resistor model of the giant magnetoresistance for the structure of the selected geometry, "the current perpendicular to plane", the equations for the resistances P R and AP R can be represented as [8] VII Euro-Asian Symposium "  For more detailed description of the calculation methodology of the two-component alloys resistivity see [11]. However, for the practical applications, the films resistivity must be measured experimentally.

Magnetization vector dynamics
The dynamics of the vector M was calculated by the Runge-Kutta method. The perturbation regarding the equilibrium position was taken 0.0001.
The switching mode and the stable precession mode are the main types of dynamics that are of practical importance for MRAM and STNO. From the point of view of the qualitative theory of dynamical systems, the precession mode represents a limit cycle.

Magnetization Vector Switching
The process of the logical "1" writing in the MRAM cell corresponds to the change of the direction of the magnetization vector M in the free layer to the antiparallel direction of the pinned-layer magnetization vector s. The parallel direction of the vector M corresponds to the equilibrium point 1 T , and the antiparallel direction -to the equilibrium position 2 T . Switching of the spin valve occurs when exposed to current, the magnetic field, or the combination of both. We agree that the logical "1" writing corresponds to the direction of the current opposite to the axis OZ . Then, the logical "0" recording corresponds to the direction of current along the OZ axis. Similarly, to switch the vector M from point Co Gd -based spin valve when  0 H and  62 2.41 10 A/cm J . Figure 2b shows the corresponding volt-second characteristic obtained by formula (2). The switching time  12 t of the spin valve, in this case, will be equal to 184 ns.
In figure 3a the numerical calculation result of the reciprocal of the recording time  12 t versus current density J for various materials were shown.    Fe Co B , at the equal values of the magnetic field.

Magnetization Vector Precession
For STNO, the regimes with precession, in which the magnetization vector projection on the anisotropy axis varies periodically, are important. This leads to a periodic significant change in the output signal U . In figure     Fe Co B , which is placed in the magnetic field parallel to the axis OY (     5 0, 5 10 A/m , The value of the current density J is  82 3.7 10 A/cm .  It should be noted that, in the case, when the magnetic field is parallel to the axis OY or OZ , the trajectory of the magnetization vector wounds on the limit cycle around the axis, along which the magnetic field is directed ( figure 5a, figure 6a), and the volt-second characteristics approach to harmonic one ( figure 5b, figure 6b). At the same time, if the field is directed along the axis OX , two , when placed in the magnetic field, directed along the axis OY . The spin valve with ferromagnetic layers made of cobalt, has the maximum frequency of the oscillations in the magnetic field parallel to the axis OZ . In a magnetic field directed along the axis OX , the 80 20 Co Gd -based spin valve has the lowest power consumption, however, the amplitude of oscillations, in this case, is very small.

Conclusion
In this work, the dependences of the spin-valve switching time to the antiparallel state on the spinpolarized current  12 () tJ and the reverse switching time on the magnetic field parallel to the anisotropy axis  21 () X tH were calculated numerically. It has been established that the most suitable material among those considered for implementing magnetoresistive random access memory is 80 20 Co Gd , because the spin valve based on this material has the smallest critical switching current and the highest switching speed among the considered materials. The 60 20 20 Fe Co B -based spin valve has the shortest switching time by a magnetic field. However, the minimum switching magnetic field, in this case, power consumption is 10,000 times greater than in switching by the spin-polarized current for the 80 20 Co Gd -based spin valve. The obtained data are consistent with the results in [5]. The intervals of current and magnetic field, at which stable precession takes place, were calculated. For this case, three orthogonal magnetic field directions were considered. It is shown that to obtain a signal close to saw-tooth one, it is necessary to use a magnetic field parallel to the ferromagnetic layers anisotropy axis. At the same time, the directions perpendicular to the anisotropy axis correspond to the harmonic signal. It was found that the alloy 60 20 20 Fe Co B is the most suitable among those considered for the STNO, since the spin valve based on it has the maximum amplitude of the output signal in a magnetic field parallel to the axis OY . In this case, the current and field ranges are 1.5 times lower than for the cobalt-based spin valve, which has the maximum frequency of the output signal.
The calculated data presented in the paper have general recommendatory nature and illustrate the use of the theoretical model of the spin valve.