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Composite excitation of Josephson phase and spin waves in Josephson junctions with ferromagnetic insulator

Coupling of Josephson-phase and spin-waves is theoretically studied in a superconductor/ferromagnetic insulator/superconductor (S/FI/S) junction. Electromagnetic (EM) field inside the junction and the Josephson current coupled with spin-waves in FI are calculated by combining Maxwell and Landau-Lifshitz-Gilbert equations. In the S/FI/S junction, it is found that the current-voltage (I-V) characteristic shows two resonant peaks. Voltages at the resonant peaks are obtained as a function of the normal modes of EM field, which indicates a composite excitation of the EM field and spin-waves in the S/FI/S junction. We also examine another type of junction, in which a nonmagnetic insulator (I) is located at one of interfaces between S and FI. In such a S/I/FI/S junction, three resonant peaks appear in the I-V curve, since the Josephson-phase couples to the EM field in the I layer.


Introduction
The dc Josephson effect is characterized by the zero-voltage current through a thin insulating barrier sandwiched by two superconductors. 1) This effect is a macroscopic quantum phenomenon involving phase coherence between two superconductors. When a finite voltage(V )-drop appears in the junction, the difference in the phase of superconducting order parameter, i.e. Josephson-phase (θ), oscillates with time according to ∂θ/∂t = (2e/ )V , and the alternating current with frequency (2e/ )V flows in the junction. This ac Josephson effect is derived by the gauge invariance including θ. The electromagnetic response dominated by θ shows a resonant behavior in the junction. When a dc magnetic field and the dc voltage are applied to the junction, the electromagnetic (EM) field is generated by spatially modulated ac Josephson current. In this case, the current-voltage (I-V ) curve exhibits resonant peaks due to the resonance between the ac Josephson current and the EM field generated by the spatially modulated ac Josephson current itself. This is called Fiske resonance. [2][3][4][5][6] In recent years, a ferromagnetic Josephson junction composed of ferromagnetic metal (F) and superconductors (S's), i.e., S/F/S junction, has received much attention. [7][8][9][10] One of the interesting effects is the formation of π state arising from the Zeeman splitting in F. In addition, the interaction between Cooper pairs and spin waves in F is also of importance in the transport properties in the S/F/S junction. [11][12][13][14][15][16][17][18][19] In a small junction, where the junction width is smaller than the Josephson penetration depth, the spin-wave excitation induced by the ac Josephson effect is observed. 18) In the recent experiment in a S/F/S junction including a nonmagnetic insulator (I) in one of interfaces between S and F, it has been reported that the Fiske resonance has multiple structures that must be associated with the spin-wave excitation. 20) Volkov et al. have theoretically studied collective excitations in such a junction and reported an additional structure in the Fiske resonance induced by spin-waves. 19) In their theory, a nonmagnetic insulator is crucial to obtain the Fiske resonance coupled with spin-waves. On the other hand, another type of ferromagnetic Josephson junction composed of ferromagnetic insulator (FI) and two S's, i.e., S/FI/S junction, is also expected to show the similar multiple structures in the Fiske resonance. It has been reported that the dissipation effect in the S/FI/S junction is smaller than that in the S/F/S junction. 21,22) Such a small dissipation in the S/FI/S junction is due to the small probability of quasi-particle excitation in the FI. 21,22) The damping of spin-waves induced by the similar mechanism is also very small in the FI compared to the case in F. 23,24) Therefore, the coupling between Josephson-phase and spin-waves can be observed more clearly in the S/FI/S junction.
In this paper, we theoretically study a composite excitation of the Josephson-phase and spin-waves in the S/FI/S and S/I/FI/S junctions. First, we calculate the dynamics of Josephson-phase coupled with spin waves by using Maxwell and Landau-Lifshitz-Gilbert (LLG) equations. Second, we derive the dc Josephson current induced by the Fiske resonance.
In the S/FI/S junction, two resonant peaks appear in a current-voltage curve for each mode of the EM field. These two resonant peaks may be associated with the direct coupling between spin-waves and the EM field inside the junction. We also discuss the Fiske resonance in the S/I/FI/S junction. The non-magnetic high resistive layer is sometimes important, since the magnetic dead layer exist in the ferromagnetic insulator near the S/FI interface. Our results clearly show the difference between S/FI/S and S/I/FI/S junctions in the dispersion relations of the Fiske resonance. In such a S/I/FI/S junction, we show that three resonant peaks appear in the I-V curve for each mode of the EM field.
The rest of this paper is organized as follows. In Sec. II, by combining the Maxwell and LLG equations in a S/FI/S junction, we formulate the dc Josephson current induced by the Fiske resonance. In Sec. III, the Fiske resonance is discussed in S/FI/S and S/I/FI/S junctions. Summary is given in Sec. IV.

Formulation of Fiske resonance in S/FI/S junction
The system considered is a Josephson junction with a FI sandwiched by two s-wave superconductors (S's) as shown in Fig. 1. The magnetization in the FI is parallel to the z-direction. 18) A uniform dc magnetic field is applied in the x-direction. In the measurement of the Fiske resonance, the dc magnetic field is smaller than several tens of gauss. Therefore, we can neglect the in-plane magnetization induced by the applied dc magnetic field. Here, we consider that the ac electric and magnetic fields are in the z-and x-direction respectively, both of which are uniform in the x-direction. We consider the situation, in which the z-dependence of the electric and magnetic fields in the FI is negligible due to the very thin thickness of the FI (d FI ). In the S regions, it is assumed that the magnetic field depends on y-and zcomponent. The current density has a nonzero y-component in the superconducting regions (Meissner current) and a nonzero z-component in the ferromagnetic region (quasi-particle and Josephson currents). Based on the above assumptions, the Maxwell equation in each region is given by Here, e i (i = x, y, z) is a unit vector, E z (y, t) is the electric field in the FI, H x (y, z, t) and H x (y, t) are the magnetic fields in the S and the FI, respectively. The electrical flux density, D z (y, t), in FI is given by D z (y, t) = ǫ FI E z (y, t), where ǫ FI is the dielectric constant in FI. J z J (y, t) and J c are the Josephson current and the Josephson critical current densities, respectively. J z Q (y, t) and R FI are the quasi-particle current density and the resistivity of FI, respectively. The x-component of the magnetization, M x (y, t), in the FI is given by Eq. (4).
The motion of magnetization is described by the Landau-Lifshitz-Gilbert (LLG) equation, 23) where M is the magnetization of FI, γ is the gyromagnetic ratio, and α is the Gilbert damping.
The effective field, to which M responds, is given by H eff .
By using Maxwell and LLG equations, we can obtain the voltage coupled with spin-waves (the detail of derivation for Eq. (8) is given in Appendix A.) as follows: and H ex are the magnetization in the FI and the applied dc magnetic field, respectively. where are the effective velocity of light in the FI, the Josephson penetration depth, and the damping factor caused by quasi-particle resistivity, R FI , in the FI, respectively.
We look for the solution of Eq. (8) in the form where V 0 and v(y, t) are the dc bias voltage and ac voltage induced by the ac Josephson current, respectively. In this case, the phase difference, θ(y, t), between two S's is given by where Substituting Eq. (9) and Eq. (11) into Eq. (8), we obtain the equation for θ 1 (y, t) as follows: We expand θ 1 (y, t) in terms of the normal modes of the electromagnetic field generated by the ac Josephson current, where g n is a complex number and k n = nπ/L. This equation of θ 1 (y, t) satisfies [∂θ 1 /∂y] y=0 = [∂θ 1 /∂y] y=L = 0, which corresponds to the open-ended boundary condition for v(y, t). We consider θ 1 (y, t) to be a small perturbation and solve Eq.
where ω n = (c FI π/L)n, and q means k H or k n . In the linearized LLG equation, the magnetic susceptibility in the FI is given by (see Appendix C) Here, Ω S is spin wave frequency whose dispersion relation is given by where . H K and η are the anisotropic field and the stiffness of spin waves in the FI, respectively.

Results and discussion
In this section, we examine the numerical solution for Eq. (21). Figure 2 shows the dc Eqs. (21) and (22). When the denominator of Ψ n in Eq. (22) is minimum with respect to ω J , Ψ n takes a maximum, so that the dc Josephson current shows the resonant behavior as shown in Fig. 2. The dc voltage at which the resonance occurs is determined by neglecting the damping term in Eq. (22) as α = Γ FI = 0. Setting the denominator of Ψ n to be zero, the voltage is given by where ω n is the frequency of the EM field in the FI, and Ω S and γM z Ω S /µ 0 are the frequency of spin-waves and the real part of the magnetic susceptibility with α = 0 in the FI. We have two dc voltages, V + 0 and V − 0 , at which the Fiske resonance occurs for each n. From the analytic formula in Eq. (26), it is found that two dispersions result from the coupling between the EM field and spin waves in the FI. Figure 3 shows a V 0 -n curves obtained by Eq. (26). The vertical axis is the dc voltage normalized by ω L /2e and the horizontal axis is the mode number of EM field. In Fig. 3, V + 0 and V − 0 are shown by open circles and open squares, respectively. For n < 3, V + 0 is nearly constant as a function of n, whereas V − 0 is linear with n. In V + 0 for n < 3, the voltage is nearly equal to Ω S /2e, which relates to the spin wave energy in Fig. 3.
For n ≥ 3, V + 0 increases as a function of n, whereas V − 0 becomes flat with increasing n. For n ≥ 3, V − 0 is nearly equal to Ω S /2e in Fig. 3. Therefore, it is found that the flat behavior Here, we examine the Γ FI -and α-dependence of J dc . Figure 4 (a) shows J dc as a function of V 0 by changing Γ FI /ω L . From this figure, it is found that the Fiske resonance without very sharp peaks around V 0 /(ω L /2e) = 3 exhibits strong damping by increasing Γ FI /ω L .
On the other hand, very sharp peaks around V 0 /(ω L /2e) = 3 almost does not depends on Γ FI /ω L unlike another resonant peaks. Next, we focus on sharp peaks around V 0 /(ω L /2e) = 3. Figure 4 (b) shows J dc as a function of V 0 by changing α. These peaks around V 0 /(ω L /2e) = 3 in Fig. 4 (b) strongly depend on α because these resonant peaks mainly comes from spin-waves.
From Fig. 4, we can easily obtain the Fiske resonance coupled with spin-waves in the S/FI/S junction due to the small Γ FI and α.
The effect of spin-waves having a finite wave number q is neglected in the Fiske resonance because of the following reason: In Eq. (19), the first term Ω B is caused by the anisotropic and demagnetizing fields and finite wave number q is given by nπ/L. In a conventional FI, Ω B is about tens of µeV. 23) On the other hand, ηq 2 is of the order peV due to the small stiffness of spin-waves 28) when the width (L) of the junction is a few mm.
Next, we consider the Fiske resonance in the S/I/FI/S junction. The details of the calcu- The condition of the resonance is given by the minimum in the denominator. To find out the voltage at which the Fiske resonance occurs, we neglect the damping term in the denominator of Eqs. (D·15) and (D·17). As a result, the voltages at the Fiske resonance are given by indicating that there are three dc voltages resulting in the Fiske resonance for each n in the S/I/FI/S junction. Figure 5 shows the V 0 -n characteristic obtained by Eqs. (27) and (28).
The vertical axis is the dc voltage normalized by ω L /2e and the horizontal axis is the mode number of EM field. In In this paper , we assumed that the magnetization of FI is a single domain structure.

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Usually, magnetization structure in the FI possesses a complicated domain structure. The ferromagnet with multidomain structure has multiple magnetic resonance modes different from the single domain system. 29) Therefore, we can expect several additional Fiske resonance peaks in the I-V curve arising from the domain structure in the FI. Those domain structure and their spin dynamics, which depend on our choice of material, have rich variety and wide spectrum.
However, it is difficult to include the multidomain structure of FI in the present theory. The Fiske resonance in the ferromagnetic Josephson junction with the magnetic domains is beyond the scope of the present paper, and will be studied in another one.

Summary
To obtain the voltage equation coupled with spin waves in the FI, we transform the function F (y, t). When we execute the Fourier transformation on χ(y − y ′ , t − t ′ ) and I M (y ′ , t ′ ) referring to y ′ , F (y, t) is given by Making use of the Fourier transformation of y in Eq. (A·6),

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and substituting Eq. (A·10) into Eq. (A·9), F (y, t) is given by Where C ′ = ǫ FI W/d FI . Substituting Eq. (A·11) into Eq. (A·7) and using the relation V (y, t) = d FI E z (y, t), we have the partial differential equation Appendix B: Calculation of factor g n To obtain the complex coefficient g n , substituting Eq. (13) and Eq. (12), we obtain the equation for g n , where ω n = (c FI π/L)n and µ(q, −ω J ) = 1 + χ(q, −ω J )d FI /[(d FI + 2λ J )µ 0 ] and q denotes k H or k n . For both sides of Eq. (B·1), multiplying cos(k n y) and performing integration with respect to y from 0 to L, we get g n as follows: We outline the derivation of the magnetic susceptibility in the FI by using the LLG equation. We consider the situation, in which the direction of magnetization is perpendicular to the junction. In the experimental measurement of the Fiske resonance, the dc magnetic field is smaller than several tens of gauss. Therefore, we can neglect the gradient of the magnetization due to the applied dc magnetic field. Since the precessional angle of spin in the FI is usually very small even at the magnetic resonance, we linearize the LLG equation as follows:

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where H K and M z /µ 0 are an anisotropic field and a demagnetization field in the FI, respectively. We assume the solutions of Eq. (C·1) and (C·2) of the form where χ i (q, ω J ) is a magnetic susceptibility of FI and h x is the amplitude of x-component of ac magnetic field and q is a wave number of spin wave. Substituting Eq. (C·3) into Eq. (C·1) and (C·2), the solutions of the linearized LLG equation are where H K is an anisotropic field in the FI and η is a stiffness of spin waves.
Next we calculate the dc Josephson current coupled with spin waves as a function of the dc voltage and of the external magnetic field. Since we consider V I(FI) (y, t) as a perturbation, we can expand the sine function in terms of V I(FI) (y, t). Within the first order term with respect to V I(FI) (y, t), the dc Josephson current is approximately given by Substituting Eqs. (D·10) and (D·11) into Eq. (D·13), we can obtain the analytic formula of the dc Josephson current in the S/I/FI/S junction as follows: where ω I n and ω n are given by (c I π/L)n and (c FI π/L)n, respectively. F n (φ) is same function as Eq. (25). J dc JI comes from the resonance between the electromagnetic field generated inside the I and the ac Josephson current. J dc JFI originates in the resonance between the electromagnetic field generated inside the FI and the ac Josephson current.