Phase and spin dynamics in a superconductor/ferromagnet/superconductor junction

We study a phase dynamics induced by a spin dynamics in a ferromagnetic Josephson junction, in which two superconductors (SC’s) are separated by a ferromagnetic layer. A new phenomenological model for the phase variable is proposed by including the spin dynamics excited by the ferromagnetic resonance (FMR) in the gauge invariant phase of s-wave SC’s. We found that the current-voltage characteristics show step structures by tuning the microwave frequency to FMR. The result originats from the coupling between the spin and the phase dynamics, and provides a new route to observe the spin wave excitation by using the Josephson effect. The Josephson effect is a macroscopic quantum phenomenon regarding phase coherence of superconductor (SC). The dc Josephson effect is characterized by a current through a thin barrier without voltage drop [1]. When a finite voltage-drop (V ) appears in the junction, the phase difference, θ, evolves with time, t, and V as ∝ V t, which induces an alternating current, i.e. ac Josephson effect. When an alternating electric field generated by microwave irradiation is applied to the junction, the current-voltage (I-V ) characteristics show step structures called Shapiro step [2]. The phase dynamics providing the I-V characteristics is described by the resistively shunted junction (RSJ) model, which is a equivalent circuit composed of resistance, capacitor and Josephson current[3, 4]. A ferromagnetic Josephson junction, in which two SC’s are separated by a ferromagnetic metal (FM), is of considerable interest in recent years [5, 6, 7, 8]. One of the unique phenomena is the sign reversal of critical current with thickness of FM and temperature. The origin of this sign reversal is the exchange splitting of the conduction band in FM. On the other hand, ferromagnetic materials possess dynamic properties such as spin waves, which can be excited by ferromagnetic resonance (FMR). Hence, the interaction between Cooper pairs and spin waves in the FM is expected to create a new effect in transport properties. In this paper, we study the effect of coupling between phase and spin dynamics in a ferromagnetic Josephson junction. The RSJ model is extended to include the effect of spin wave excitations taking account of the gauge invariance. In a ferromagnetic Josephson junction composed of s-wave SC’s separated by FM, we choose a coordinate such that the cross section of the junction is parallel to the yz plane and the current flows along x. The Josephson junction is characterized by the phase difference θ(y, z, t) between 25th International Conference on Low Temperature Physics (LT25) IOP Publishing Journal of Physics: Conference Series 150 (2009) 052069 doi:10.1088/1742-6596/150/5/052069 c © 2009 IOP Publishing Ltd 1 SCs, which is described by the RSJ model given by, i = ic sin θ (y, z, t) + 1 R Φ0 2π dθ (y, z, t) dt + C Φ0 2π d2θ (y, z, t) dt2 , (1) where i is an applied current density and ic is the critical current density. The flux quantum is denoted by Φ0. The resistance, R, and the capacitance, C, in the Josephson junction are normalized by the junction area, Syz, as R = R0/Syz, C = C0/Syz. R0 and C0 are the resistance and the capacitance in the Josephson junction, respectively We consider the situation, in which the FM is exposed to a circularly polarized microwave. The microwave induces the precessional motion of the magnetization, which corresponds to a uniform mode of spin wave. The motion of magnetization due to the microwave radiation is given by the Landau-Lifshitz-Gilbert (LLG) equation [9], dM dt = −γM × Heff + α M [ M × dM dt ] , (2) where M (t) is the magnetization in the FM, γ is the gyromagnetic ratio, and α is the Gilbert damping. Below, α is set to 0.01[9]. The effective field acting on M (t) is given by Heff = H0 +hac (t), where H0 is an uniaxial magnetic anisotropic field, which is parallel to the z axis and hac = (hac cosΩt, hac sinΩt, 0) is the microwave driving field with frequency Ω. The linearized solution of Eq. (2) is given by, M± (t) = γMzhac Ω− Ω0 ∓ iαΩ ±iΩt, (3) which describes the precession of M± (t) = Mx(t) ± iMy(t) around H0 with frequency Ω and shows a resonance at Ω0 = γH0. The rotating magnetic field is induced in the FM through the magnetic flux density B (t) = 4πM (t). Here, it is noted that the superconducting phase difference is not gauge invariant in the magnetic field. The gauge invariant phase difference must satisfy the following differential equation [10], ∇y,zθ (y, z, t) = − Φ0 M (t)× n, (4) where y,z = (0, ∂/∂y, ∂/∂z), n is the unit vector perpendicular to the yz plane. Equation (4) is satisfied by the following solution, θ (y, z, t) = θ (t)− 4πMzd Φ0 y + 4πMy (t) d Φ0 z. (5) We adopt the gauge such that the superconducting phase difference is equal to θ (t) without magnetic field. Introducing Eq. (5) in Eq. (1) and integrating over the junction area, we obtain the extended RSJ model that includes the spin dynamics in the FM, I/Ic = sin [ ΦsM̃y (τ) ] ΦsM̃y (τ) sin [θ (τ)] + dθ (τ) dτ + βc d2θ (τ) dτ2 , (6)