Critical current density of a spin-torque oscillator with an in-plane magnetized free layer and an out-of-plane magnetized polarizer

Spin-torque induced magnetization dynamics in a spin-torque oscillator with an in-plane (IP) magnetized free layer and an out-of-plane (OP) magnetized polarizer under IP shape-anisotropy field ($H_{\rm k}$) and applied IP magnetic field ($H_{\rm a}$) was theoretically studied based on the macrospin model. The rigorous analytical expression of the critical current density ($J_{\rm c1}$) for the OP precession was obtained. The obtained expression successfully reproduces the experimentally obtained $H_{\rm a}$-dependence of $J_{\rm c1}$ reported in [D. Houssameddine $et$ $al$., Nat. Mater. 6, 447 (2007)].

A spin-torque oscillator (STO) [1][2][3][4][5][6] with an in-plane (IP) magnetized free layer and an out-of-plane (OP) magnetized polarizer [7][8][9][10][11][12][13][14][15] has been attracting a great deal of attention as microwave field generators 12,16-20 and high-speed field sensors [21][22][23][24][25][26] . The schematic of the STO is illustrated in Fig. 1(a). When the current density (J) of the applied dc current exceeds the critical value (J c1 ), the 360 • in-plane precession of the free layer magnetization, so-called OP precession, is induced by the spin torque. Thanks to the OP precession, a large-amplitude microwave field can be generated, 12,14,15 and a high microwave power can be obtained through the additional analyzer. 8 The critical current density, J c1 , for the OP precession of this type of STO has been extensively studied both experimentally 8,15 and theoretically. 7,[9][10][11]13,27  In this letter, we theoretically analyzed spin-torque induced magnetization dynamics in the STO with an IP magnetized free layer and an OP magnetized polarizer in the presence of H k and H a based on the macrospin model. We obtained the rigorous analytical expression of J c1 and showed that it successfully reproduces the experimentally obtained H a -dependence of the critical current reported by D. Houssameddine et al. 8 The system we consider is schematically illustrated in Figs. 1(a) and (b). The shape of the free layer is either a circular cylinder or an elliptic cylinder. The lateral size of the nanopillar is assumed to be so small that the magnetization dynamics can be described by the macrospin model. Directions of the magnetization in the free layer and in the polarizer are represented by the unit vectors m and p, respectively. The vector p is fixed to the positive z-direction. The negative current is defined as electrons flowing from the polarizer to the free layer. The applied IP magnetic field, H a , is assumed to be parallel to the magnetization easy axis of the free layer. The easy axis is parallel to x-axis.
The energy density of the free layer is given by 28 Here (m x , m y , m z ) = (sin θ cos φ, sin θ sin φ, cos θ), and θ and φ are the polar and azimuthal angles of m as shown in Fig. 1(b). The demagnetization coefficients, N x , N y , and N z are assumed to satisfy N z ≫ N y ≥ N x . K u1 is the first-order crystalline anisotropy constant, µ 0 is the vacuum permeability, M s is the saturation magnetization of the free layer, and H a is applied IP magnetic field.
Hereafter we conduct the analysis with dimensionless expressions. The dimensionless energy density of the free layer is given by Here, k u1 and h a are defined as k u1 = K u1 /(µ 0 M 2 s ) and h a = H a /M s . We discuss on the spin-torque induced magnetization dynamics at h a ≥ 0 in this letter, however, the dynamics at h a < 0 can be calculated in the similar way.
The spin-torque induced dynamics of m in the presence of applied current is described by the following Landau-Lifshitz-Gilbert equation, 28 where τ , χ, h θ , and h φ are the dimensionless quantities representing time, spin torque, and θ, φ components of effective magnetic field, h eff , respectively. h eff is given by h eff = −∇ǫ.
α is the Gilbert damping constant. The dimensionless time is defined as τ = γ 0 M s t, where γ 0 = 2.21 × 10 5 m/(A·s) is the gyromagnetic ratio and t is the time. h θ and h φ are given by Here h k is dimensionless IP shape-anisotropy field being expressed as is the effective first-order anisotropy constant where the demagnetization energy is subtracted. Since we are interested in the spin-torque induced magnetization dynamics of the IP magnetized free layer, we concentrate on k eff u1 < 0. The prefactor of the spin-torque term, χ, is expressed as where η(θ) = P/(1 + P 2 cos θ) is spin-torque efficiency, P is the spin polarization, J is the applied current density, d is the thickness of the free layer, e (> 0) is the elementary charge and is the Dirac constant. For convenience of discussion, the sign of Eq. (7) is taken to be opposite to that in Ref. 28.
In the absence of the current, i.e., J = 0, the angles of the equilibrium direction of m are obtained as θ eq = π/2 and φ eq = 0 by minimizing ǫ with respect to θ and φ. Application of J changes θ and φ from its equilibrium values. If the magnitude of J is smaller than the critical value, the magnetization converges to a certain fixed point. 29 The equations determining the polar and azimuthal angles of the fixed point (θ 0 , φ 0 ) are obtained by setting dθ/dτ = 0 and dφ/dτ = 0 as The fixed point around the equilibrium direction (θ eq = π/2, φ eq =0) are obtained as follows.
Assuming |φ 0 | ≤ π/2, i.e., cos φ 0 ≥ 0 and noting k eff u1 < 0, one can see that the quantity in the square bracket of Eq. (5) is positive and θ 0 = π/2 to satisfy h 0 θ = 0. Substituting θ 0 = π/2 to h 0 φ = −χ sin θ 0 , the equation determining φ 0 is obtained as where ξ = 2h a /h k . Since Eq. (10) does not contain the Gilbert damping constant, α, φ 0 is independent of α. In Fig. 2(a), the function, sin 2φ + ξ sin φ, is plotted against φ for various values of 0 ≤ ξ ≤ 4. One can clearly see that the azimuthal angle of the maximum (minimum) increases (decreases) towards π/2 (−π/2) with increase of ξ. The azimuthal angle of the fixed point is given by the intersection of this sinusoidal curve and a horizontal line at 2χ/h k , and it increases with increase of 2χ/h k as shown in Fig. 2(b). In Fig. 2 At the maximum, the derivative of the LHS of Eq. (10) with respect to φ 0 is zero, that is, Expressing cosine functions by tan φ 0 , one can easily obtain the solution of Eq. (11) as where the subscript "c1" stands for the critical value corresponding to J c1 . Fig. 3(a) shows ξ dependence of φ c1 given by Eq. (12). φ c1 = π/4 for ξ = 0, i.e., h a = 0. It monotonically increases with increase of ξ and reaches π/2 in the limit of ξ → ∞, i.e., h k → 0.
The maximum value, Λ, can be obtained by substituting φ = φ c1 into the LHS of Eq.
(10) as where X = ξ(ξ + ξ 2 + 32). Equating this maximum value with 2χ/h k and using Eq. (7), the critical current density is obtained as This is the main result of this letter. It should be noted J c1 is also independent of α. In the absence of the applied IP magnetic field, i.e., H a = 0, Eq. (14) becomes In the limit of H k → 0, it reduces to For small magnetic field such that H a ≪ H k , i.e., ξ ≪ 1, it can be approximated as by noting that the Taylor expansion of Λ around ξ = 0 is given by Once the current density, J, exceeds J c1 , the OP precession is excited and further increase of J moves the trajectory towards the south pole (θ = π). Around θ = 0 and π, there exist the fixed points other than θ 0 = π/2, which are determined by h k 2 sin θ 0 sin 2φ 0 + h a sin φ 0 = χ sin θ 0 .
After some algebra, the fixed point is obtained as where π/2 < |φ 0 | ≤ π. In the absence of the applied IP magnetic field, i.e., h a = 0, the polar angle of the fixed point is θ 0 = 0 or π. It is difficult to obtain the exact analytical expression for the critical current density, J c2 , above which the OP precession can not be maintained, and m stays at the fixed point given by Eqs. (20) and (21). Since the average polar angle of the trajectory of the OP precession is determined by the competition between the damping torque and spin torque, this critical current density should depend on α. The approximate expression was obtained by Ebels et al. 11 as (17), the critical current for small magnetic field can be approximated as I c ∝ H k + √ 2H a rather than I c ∝ H k + 2H a .
In summary, we theoretically studied spin-torque induced magnetization dynamics in an STO with an IP magnetized free layer and an OP magnetized polarizer. We obtained the rigorous analytical expressions of J c1 for the OP precession in the presence of IP shapeanisotropy field (H k ) and applied IP magnetic field (H a ). The expression reproduces the experimental results very well and revealed that the critical current is proportional to H k + √ 2H a for H a ≪ H k .