Electrical properties of stacks of many long Josephson junctions

We investigated numerically IV-characteristics and power of emission from stacks with various quantities of long Josephson junctions (up to 6 junctions) which interacted inductively with each other. Parameters of junctions were chosen close to those for MoRe-Si(W)-MoRe heterostructures. We set Gaussian spread of about 0.01% of critical currents along junctions. Electrical properties of a stack consisted of three junctions was investigated in details. Zero-fi eld steps at voltages corresponding to frequencies of various modes of electromagnetic waves in the stack were found in the IV-characteristic. We showed that positions of zero-fi eld steps in IV-curves were in good agreement with predictions of the theory. The highest maximum of power of emission corresponded to the so-called in-phase mode at which all voltages over junctions in the stack oscillate in-phase. Considering stacks with many junctions, we showed that power of emission at the voltage of the resonance which corresponds to the in-phase mode is proportional to the square of quantity of long junctions in the stack that is the characteristic of coherent emission.


Introduction
Recently, the huge progress was made in the creation of sources of coherent emission made of many intrinsic Josephson junctions on the base of high-temperature superconductors [1][2][3]. These sources work in the subterahertz and terahertz region of frequencies [1]. This gives the possibility to develop ultrafast nonlinear terahertz spectrometers, terahertz imaging systems, the high-resolution Michelson interferometer [1][2][3]. It is proved experimentally and theoretically that the mechanism which leads to strong synchronization of radiation of junctions in the stack is connected with the interaction of Josephson generation with Вісник ХНУ, серія «Фізика», вип. 29, 2018 Electrical properties of stacks of many long Josephson junctions cavity resonances and the formation of self-resonant steps in the IV-characteristic in the absence of externally applied magnetic fi eld (the so-called zero-fi eld steps) [4,5]. In the present paper we apply basic ideas of the developed theory to stacks of junctions made on the base of low-temperature superconductors which can be more stable sources.
At fi rst we discuss the formation of zero-fi eld steps in the solitary Josephson junction. Zero-fi eld steps are close related to Fiske steps which appear at voltages which correspond to frequencies of geometrical resonances in the system when the Josephson junction is placed in some magnetic fi eld [6,7]. However, when the magnetic fi eld is absent, steps in the IV-curve can not appear in the homogeneous junction [8]. If the distribution of critical currents is inhomogeneous and symmetrical (just this case we discuss in the paper), zero-fi eld steps appear at voltages V p that correspond to even geometrical resonances: where 0  is the quantum of magnetic fl ux, c is the velocity of light in the junction, D is the size of the junction and p=1,2… is integer. It is proven also that some disorder of the distribution of critical currents leads to the formation of zero-fi eld steps [9][10][11].
The theory of electric properties of stacks of Josephson junctions was developed and proved experimentally in Refs. [12][13][14][15]. One of the main results of this theory is that in the stack of m junctions with inductive interaction between superconducting layers the Fiske step is split to m branches. For example, the even Fiske step at the voltage V p calculated from Eq. (1) is split to m branches which positions are determined as follows: is integer and S is the normalized constant of inductive coupling of layers [13]. However, it has been shown that in the stack of two junctions with disorder of critical currents zero-fi eld steps are split also [9][10][11], and positions of steps in the IV-curve can be calculated from Eq. (2). The coeffi cient L L f   which characterizes the relation of the mutual inductance L f between junctions to the inductance L of the junction in a stack [9][10][11] coincides with the constant of inductive coupling S in Eq. (2). Due to the interaction between layers, some modes of excitations are formed in the stack and each of the branches of the zero-fi eld step corresponds to its own mode. Amongst these modes there is the mode of fully coherent oscillations of all junctions (it is the so-called in-phase or coherent mode with m p   which corresponds to the branch with highest voltage). Just at the branch of the coherent mode oscillations of voltages over junctions can be fully synchronized.
In the present paper we investigate numerically IVcharacteristics and power of coherent emission from stacks of long Josephson junctions. Parameters of junctions and the whole stack are chosen close to those for MoRe-Si(W)-MoRe heterostructures which have appropriate values of the characteristic voltage and the density of critical current [16][17][18]. We discuss the IV-curve of the three-junction stack in details and prove the existence of the coherent mode. Then we investigate stacks with diff erent numbers of long junctions and investigate the dependence of emitter power on the quantity of junctions at voltages of the in-phase mode.

The model
For calculations of IV-characteristics and power of emission from the stack of many long junctions we follow the method developed in Ref. [9]. Each of the long junctions with indices i=1…m is divided to n segments with indices j=1…n. Each of the segments has the capacitance C i,j (we suppose C i,j =C for all segments) and the resistance R i,j . Loops between centers of segments in each of the long junctions has the inductance L. Current conservation conditions and conditions of quantization of magnetic fl ux are as follows: I is the bias current (we suppose that the same bias current fl ows through all segments, so the total bias current through the stack is equal to is the ac current in the loop between two segments with indices j-1 and j which are placed within the i-th long junction, L f is the mutual inductance between adjacent long junctions. To describe emission from the system, we attached contours with the resistance R e , the inductance L e and the capacitance C e to the upper and lower superconducting electrodes at both ends of the stack. Then for junctions with indices j=1 and j=n such equations are valid (boundary conditions): where q is the charge that passes the inductance L e . Normalized to the quantity of long junctions voltage over the stack is equal to where the sign ... means the averaging on time which is much large than the period of Josephson generation. Ac power of emission P j from ends of the system was calculated as follows: The system of Eqs. (3)-(7) was solved by the method of Runge-Kutta. IV-characteristics and power of emission fro the left end of the stack P 1 were calculated with the use of Eqs. (8), (9).

Results and Discussion
Now we discuss main conditions for the formation of zero-fi eld steps. One of them is the mentioned in the Section 1 inhomogeneous distribution of critical currents along the junction. In junctions made on the base of lowtemperature superconductors it is impossible to create the inhomogeneous distribution due to self-heating like it is in high-temperature superconductors [11]. The necessary distribution can be created by some disorder of critical currents. Another condition for the appearance of zero-fi eld steps is that the frequency of the geometrical resonance should be inside the interval of frequencies of Josephson generation. To fulfi ll this condition, the stack should have a length of order of hundred micrometers. This means that junctions in such a stack have large capacitances. However, zero-fi eld steps in IV-curves of junctions with large capacitances are hard to reveal. Both mentioned conditions are satisfi ed in heterostructures which consist of superconducting electrodes made of molybdenum-rhenium alloy with the critical temperature of about 9 K and a hybrid tunnel barrier made of silicon with nanoclusters of (the MoRe-Si(W)-MoRe heterostructure) [16][17][18]. These junctions have some capacitance due to non-fully grown nanoclusters. Our fi rst treatment of IV-curves of such structures showed that zero-fi eld steps appear in such structures [19].
The calculated IV-characteristic of the three-junction stack is shown in Fig. 1a. In this plot we used reduced units, namely, i = I b /I ca with I ca is the averaged critical current, ν = V/Vc, p j = P j /(I ca V c ). This IV-curve was obtained by means of sweeping of the bias current, i. e. the bias current in the hysteretic region was decreased to zero at the fi rst sweep, then increased again above the hysteretic region, then was decreased to i=0.34 and then was increased again. According to Eq. (1), the zero-fi eld step in the IV-characteristic of the solitary junction should be at ν p=1 =0.39. For the stack with three junctions this zerofi eld step should split into three steps: ν p=1, p`=1 =0.34, ν p=1, p`=2 =0.39 and ν p=1, p`=3 =0.47. These modes are called as the anti-phase mode, the decoupled mode and the in-phase mode, correspondingly [20]. Calculated positions of steps are shown in Fig. 1a by arrows. It is seen that arrows almost coincide with steps. The plot of emitted power on the voltage is shown in Fig. 1b. As it was expected, the highest Electrical properties of stacks of many long Josephson junctions maximum of p1(v) is at the voltage ν p=1, p`=3 =0.47, which corresponds to the frequency of the in-phase mode. Thus, we investigated electrical properties of the stack with three junctions with the small Gaussian distribution of critical currents and without applied magnetic fi eld and found the good agreement of our numerical results for positions of zero-fi eld steps and emitted radiation with predictions of the theory [12][13][14][15]. Note that earlier this theory was applied for Fiske steps of stacked junctions in magnetic fi eld [13]. In calculations we decrease the bias current in the hysteretic region and defi ne power of emission from maxima of the plot P 1 =f(V) at voltages which correspond to in-phase modes. This method is usually used in experiments when the dc current-biased scheme is applied. The defi ciency of this procedure is clearly seen from Fig.  1a. During the decrease of the bias current there is a jump of voltage from the resistive branch of the IV-curve to the step in the region of its bottom. The maximum of emission power is in the vicinity of the jump. The larger is the McCumber parameter the smaller is the jump, so it is hard to detect the solitary in-phase mode. Because the position of the right edge of the jump is defi ned by the step on the IV-curve, the obtained value of the power of emission at this point is the arbitrary point at the maximum. Therefore, we applied averaging of these values over 5-6 calculations of IV-curves with diff erent steps.
The plot P 1 = f(m) is shown in Fig. 2 (circles). The solid line in this plot is the dependence P 1 (m) = Am 2 with the coeffi cient A = 1.25•10 -9 W that is close to the value of emitted power for the solitary junction (P 1 (1) = 1.5•10 -9 W). It is seen that the approximation describes our data with good accuracy. According to the theory of synchronization [21], in the case of fully coherent emission the ac emitted power is proportional to the square of the quantity of junctions: P 1 (m) = P 1 (1)•m 2 , where P 1 (1) is power of emission of one solitary long junction, and in the case of fully decoupled junctions the power of emission is equal to P 1 (m)=P 1 (1)•m. Thus, we make a conclusion that at the in-phase mode voltage oscillations across long Josephson junctions in stacks are synchronized. With the use of this mode one can create sources of coherent emission.

Conclusions
In this paper we investigated IV-characteristics and emission of stacks with diff erent amount of Josephson junctions which interact inductively with each other. We set small Gaussian spread of critical currents (about 10 -4 •I c or 0.01%) along junctions in stacks and found zero-fi eld steps in IV-characteristics. We showed that voltages at which zero-fi eld steps appeared can be calculated with the use of Eq. (2) of the theory [12][13][14][15] which was earlier applied to stacks placed in external magnetic fi eld. This result is in agreement with our previous investigations [9][10][11] in which we found that the zero-fi eld steps in a stack of two inductively interacting long Josephson junctions was split to two steps and positions of these steps in the IV-curve was defi ned by frequencies of normal modes. We also investigated the dependence of emitted power at the voltage which corresponds to the frequency of the in-phase mode and proved that voltages across long junctions in the stack oscillate in-phase.