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Shunt-capacitor-assisted synchronization of oscillations in intrinsic Josephson junctions stack

We show that shunt capacitor stabilizes synchronized oscillations in intrinsic Josephson junction stacks biased by DC current. This synchronization mechanism has an effect similar to the previously discussed radiative coupling between junctions, however, it is not defined by the geometry of the stack. It is particularly important in crystals with smaller number of junctions, where radiation coupling is week, and is comparable with the effect of strong super-radiation in crystal with many junctions. The shunt also helps to enter the phase-locked regime in the beginning of oscillations, after switching on the bias current. Shunt may be used to tune radiation power, which drops as shunt capacitance increases.

Recently THz radiation was obtained from mesa-type layered crystals with intrinsic Josephson junctions (IJJ) 1 . The number of junctions in that case was not very large, about 600, and power of radiation was enhanced by exciting resonance modes inherent to the crystal, which acts as a cavity.
Part of energy stored in an excited mode leaked outside the crystal as radiation. The limitation of such a design is that the radiation frequency is fixed by the crystal resonances and cannot be continuously tuned. A general design for high power tunable source of THz radiation based on IJJ in layered superconductors was discussed in Ref. approach formulated in Ref. 4. The Lagrangian for the system with shunt shown in Fig. 1 is Here ϕ n (r, t) is the gauge-invariant phase difference between the layers n and n + 1, the coordinates inside layer are r = x, y, phase difference gradients are ∇ϕ n = (∇ x ϕ n , ∇ y ϕ n ), the London penetration lengths are λ c and λ ab for currents between layers and inside layers, respectively, ǫ c and ǫ ab are high-frequency dielectric constant for electric fields perpendicular to layers (along the z-axis) and along layers, where γ = λ c /λ ab is the anisotropy ratio, and Φ 0 = π c/e.
The first two terms account for the electro-chemical and the Josephson energies of the IJJ.
The factor (1 − α∇ 2 n ) −1 , with second discrete derivative ∇ 2 n A n = A n+1 + A n−1 − 2A n and α = e −1 s −2 (4π) −1 ∂µ/∂ρ, originates from the relation between gauge invariant time derivative of the phase difference and difference in the chemical potentials, see Ref. 5: where V n , ρ n = (E zn − E z,n−1 )/(4πs), and µ n are the potential, the charge density, and the chemical potential in the layer n, while E n is the electric field in the junction n. Eq. (2) results in the relation The next terms in the square brackets account for the kinetic energy of the intralayer currents.
The intralayer current is where we introduced the phase φ n of the superconducting order parameter and the vector potential A n (r) in the layer n. The fourth term in the Lagrangian is the energy of magnetic field inside the crystal.
The term withQ n is omitted because its contribution is negligible at the low frequencies discussed here.
The electric energy inside the shunt capacitor C s is accounted for by the last term in Lagrangian, written in terms of the average phase difference The dissipative function is Here σ c and σ ab are the quasiparticle conductivities perpendicular and along the layers, respectively.
The Lagrangian and the dissipative function result in the equations of motion for the phases φ n and the vector potential A d dt and similar equation for A. We write them in the form Here we use reduced coordinates u = x/λ J and v = y/λ J , reduced time τ = tω p and frequency Further, T ab = 1 + ν ab ∂/∂τ , and C J = ǫ c L x L y /(4πs) is the junction capacitance. We introduce reduced dissipative parameters ν ab = 4πσ ab /(γ 2 ǫ c ω p ) and ν c = 4πσ c /(ǫ c ω p ). The shunt capacitor effectively enhances the capacitance of the Josephson junctions, but only for synchronized oscillations; this enhancement is proportional to the number of junctions.
The differential equations (10) should be completed by the boundary conditions at x = ±L x /2 and y = ±L y /2. Using Eqs. (10) and continuity of B x and B y for the time-independent part of the phase difference we obtain at these boundaries The outside magnetic field is created by the bias DC current and by the induced alternating current.
For L y ≫ L x we estimate B y (x = ±L x /2, y = ±L y /2) ≈ ±2πI/(cL y ), while B x B y . Here I = jL x L y is the total interlayer bias current and j is the bias current density. Hence, we estimate time-independent phase difference, Here we used the relation j ≈ σ c E z in the resistive state when voltage is present. The phase difference estimated here is very small for crystals with dimensions smaller than cm. Neglecting it we use approximation with y-independent phase difference.
For alternating part of the phase difference, we find boundary conditions by matching electromagnetic fields inside and outside the crystal. 2 Outside fields, in half spaces |x| > L x /2, obey the Maxwell equations, which fix the ratio between electric and magnetic field. Inside the crystal, B n (r, t) given by Eq. (10) and the electric by Eq. (3). When L y ≫ L x , c/ω J , the predominant radiation is along the x-axis. For weak radiation in the y-direction, we can use the boundary conditions ∇ v ϕ n = 0. Thus we can omit dependence of ϕ n on the y-coordinate also for alternating part of the phase differences. Then we obtain the boundary conditions at x = ±L x /2: where we use the Fourier transforms with respect to time, k ω = ω/c, and J 0 (x) and N 0 (x) are the Bessel functions.
The following calculations are similar to those in Ref. 2 but accounting for shunt contribution.
We consider high frequencies ω ≫ 1 and N ≫ ℓ and neglect finite-size effects along the c-axis.
The equation for yand n-uniform phase is HereL x = L x /λ J . In the limit ω ≫ 1 we look for solution The equation for η is which should be solved with the boundary conditions at u = ±L x /2 The solution is where we approximate sin(k ωLx /2) ≈ k ωLx /2. The first term in η is the amplitude of synchronized (y-and n-independent) Josephson oscillations. It drops as β increases. The second term describes nonuniform electromagnetic wave inside the crystal. It is generated at the boundaries due to radiation field.
To analyze stability of synchronized Josephson oscillations we consider a small perturbation θ n (u, τ ) to the solution uniform in n, ϕ, Equations for θ n (u, τ ) are obtained by linearization of Eqs. (10) with respect to θ n (u, τ ). The term cos[η(τ )]θ n (u, τ ) in the linearized equation couples harmonics with small frequency Ω to highfrequency terms at Ω ± ω. At ω ≫ 1 we can neglect coupling to the higher frequency harmonics Ω ± mω with m > 1 and represent the phase perturbation as where q = πk/N, k = 1, 2, ...N. The complex eigenfrequencies Ω(q) are assumed to be small, |Ω| ≪ ω. We will find them and also conditions when Im[Ω] < 0 for all q (stability condition). Substituting θ n into linearized equations (8)-(10), excluding oscillating magnetic fields and separating the fast and slow parts, we obtain for q = 0 the coupled equations ( (21) Here G q = G q,0 . The boundary conditions for slow and fast components at u = ±L x /2 and q ≫ π/N follow from Eq. (13).
The first and the second terms in Eq. (28) represent the stabilization effects of shunt and radiation, and of the long-range interlayer capacitance, respectively. The last term, W 2 , describes the effect of modesθ q± induced inside the crystal due to radiation (parametric excitation of Fiske modes). This term leads to instability in the limit of zero dissipation and in the absence of other stabilizing terms.
It is much smaller than unity. Capacitive shunt and radiation introduce the gap in the spectrum of weak distortions and are most effective in stabilization. Their contributions can both reach order one for ǫ c L x < L z = Ns (in the super-radiation regime) and NC s ≈ C J . As C J = 60 cm for S = 1200 µm 2 , it is easy to reach this condition. In order to achieve the maximal stabilization without sacrificing the radiation power, one needs to choose β ≈ 1.
We conclude that shunt capacitor stabilizes synchronized oscillations in IJJ stack. The effect is particularly useful in crystals with the small junction number or at the initial stages of radiation.
Shunt may also be used to tune the radiation power.