Spin dynamics in superconductor/ferromagnetic insulator hybrid structures with precessing magnetization

For the numerical calculations, we have considered niobium as a superconducting metal with the following parameters: Tc = 9.2 K, Δ(0) ≈ 1.76kB·Tc = 1.4 meV, D = 0.8·10−3 m2·s−1, and εF ≈ 5.32 eV. We approximate the DOS on the Fermi level with the free electron gas value N0 ≈ 4.9·1046 J−1·m−3. The coherence length has been estimated using [Graphic 9], where kB is the Boltzmann constant and ξ0 ≈ 11 nm. We numerically solve Equation 1 in mixed representation with the normalization condition. To obtain physical observables from the quasiclassical Green’s functions, one should find the harmonic coefficients in Equation 6 and Equation 7 and directly calculate observable values at the space-time points. In this work, we are interested in the calculation of spin current distributions along the thickness of the superconducting film, as well as the influence of induced magnetization dynamics on the electron perturbations in the S film. The dynamics of any observable will be periodic and can be characterized by its amplitude value. Thus, we only need to calculate the doubled absolute value of the coefficients in Equation 6 and Equation 7, which are exactly the amplitudes of the spin current and magnetization in the linear regime. Nonadiabatic processes are unlikely because the ratio Δ/ℏΩ ≫ 1 for the Nb/Y3Fe5O12 (YIG) hybrid structure. However, the superconducting order parameter may be partially reduced near the S/FI interface because of the inverse proximity effect. It gives rise to the spin distribution of quasiparticles with energies close to the spectrum gap near the interface.

Both spin current and induced magnetization in the superconductor originate from the singlet–triplet Cooper pair conversion mechanism, which is the main origin of the inverse proximity effect. The spin current can be induced only by the nonstationary flow of triplet Cooper pairs, just as in a conventional spin-pumping bilayer structure with a normal metal [33]. Thus, spin currents cannot emerge when the magnetization is stationary inside the ferromagnetic insulator layer. However, there is a possibility to induce stationary pure spin currents inside trilayer superconducting structures [1].

The distributions of spin current amplitudes into the S layer are depicted in Figure 2. The amplitudes are normalized by the factor js0 = (ℏ/2e)je0. The charge current density normalization factor is je0 = 2eN0DΔ(0)/ξ = 6.262·106 A·cm−2.

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Figure 2: Distributions of the spin current density inside the superconducting layer at different frequencies of the magnetization precession. The interface between the superconducting layer and ferromagnetic insulator is located at z = 3ξ

Distributions of spin current and induced magnetization for aluminum were calculated in our previous work [29]. One can see that the spin current amplitudes decay at a distance of the coherence length, similarly to the induced magnetization. However, the amplitude of the spin current strongly depends on the frequency of the magnetization precession. This effect is similar to ferromagnetic resonance spin pumping in normal metal/ferromagnetic insulator structures. In the last case (normal metal), the decay of the spin current is a consequence of spin relaxation processes, but we do not take into account any spin relaxation mechanisms within our model for a superconductor. We should mention that both spin pumping mechanisms in superconductors and normal metals are determined by the penetration of nonequillibrium spin density from the interface. In metals, such a penetration is limited by the spin flip scattering, while inside the superconductor, the spin relaxation time is usually much longer. Thus, induced magnetization and spin current in our problem are determined mainly by the competition between spin singlet and spin triplet orders [34] Therefore, we conclude that the main mechanism of the spin current decay is similar to that for the induced magnetization. It corresponds to the lowering of the triplet pair density away from the magnetic interface where the singlet–triplet conversion occurs. Moreover, we should point out that the decrease of spin current inside the superconducting layer completely agrees with the boundary condition of the zero matrix current at the interface between the free space and superconducting layer at z = 0.

Now let us consider the Fourier coefficients for the induced magnetization. Earlier, we have shown that the induced magnetization almost does not depend on the precession frequency [29]. This is because the absolute value of the projection of the magnetization vector to the interface plane does not change with a change of the precession frequency and may be given by the stationary component of the induced magnetization [35].

However, more precise results presented in Figure 3 show that the induced magnetization at the interface depends nonmonotonically on the precession frequency. Moreover, a maximum becomes obvious with increasing temperature, even if we do not take into account the thermal suppression of the superconducting order parameter. The competition between two different spin pumping mechanisms can explain this interesting behavior. The first mechanism is the adiabatic spin pumping of the superconducting condensate, and the second one is the spin pumping of the thermally generated quasiparticles, for example, unpaired electrons and holes. The competition of these two spin pumping mechanisms gives rise to the nonmonotonous frequency dependence of the induced magnetization, which is the sum of the quasiparticle spin density and the triplet correlations component.

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Figure 3: In-plane component of the induced magnetization at the S/FI interface as a function of the magnetic precession frequency at different temperatures.

The interplay between magnetization precession and proximity effect can suppress superconductivity at the interface causing an increasing number of quasiparticles. To explore the spin dynamics of the quasiparticles more deeply, let us investigate spin components of the electron block of the distribution function. Spin polarization of quasiparticles can be obtained by applying a spin polarization operator to the distribution function matrix. Due to the block-diagonal structure of the spin operator in electron–hole space, the spin distribution of quasiparticles can be represented as a superposition of electron-like and hole-like spin distributions [Graphic 10]. The first term in this expression corresponds to the spin polarization of electron-like quasiparticles and is mathematically equivalent to the trace of the product of Pauli matrix and the left upper block of the distribution matrix. Figure 4 illustrates the dynamics of the quasiparticle distribution function at magnetization precession frequencies of 1 and 8 GHz. The color maps for quasiparticles with x and y spin component evolution Sx,y(z, ε, t) = Tr[σx,yψel] are presented in Figure 4. The spin distribution function splits into two almost symmetric peaks around the spectrum gap value with increasing frequency (Figure 5). The asymmetry of the electron spin distribution is very small but visible in Figure 5, where two peaks emerge twice during one period of magnetization oscillation. This picture is similar for hole excitations due to the electron–hole symmetry. It should be noticed that a fraction of the spin distribution is lying inside the gap and should not be taken into account. But in the time-dependent case, there is always an energy shift equal to ±ℏΩ/2. This energy shift appears in every time convolution. The real consequences of these undergap states may be found if one takes into account also the density of states correction, which is beyond the scope of the current paper. Thus, the effect of spin distribution function splitting can be revealed in superconducting hybrid systems with nonequilibrium electron–hole distributions such as superconductor/normal metal contacts [36]. This is one more evidence of the significant role of quasiparticles in the spin dynamics of hybrid superconducting structures.

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Figure 4: Evolution of the spin-resolved distribution function at the S/FI interface (Sx component in the upper panels and Sy component in the lower panels) at magnetization precession frequencies of (a) 1 GHz and (b) 8 GHz. The normalized time is equal to t/Tτ, where Tτ = 2π/Ω is the period of magnetization precession.

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Figure 5: Snapshots of the spin distribution function at different moments of the precession period at a frequency of 8 GHz.

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