The influence of magnetic vortices motion on the inverse ac Josephson effect in asymmetric arrays

Achieving coherent microwave (MW) operation of Josephson junctions (JJ)-based devices operating in the presence of an applied magnetic field B is essential in many applications such as MW generators/detectors,^1,2 Josephson flux-flow-oscillators (FFO) currently used as sub-terahertz integrated-receivers in radio-astronomical research or atmospheric science projects^3,4 or flux-driven Josephson (travelling-wave) parametric amplifiers^5–11 used in MW generation/detection^6–8 or readout of a flux-qubit.⁹ In all these applications, such JJ-based devices are dc current biased and usually operate in an environment where they are simultaneously exposed to both a (remnant) magnetic field B field and microwave (MW) radiation. The exposure to MW can be due to an external source or due to internally generated radiation via the ac Josephson effect. Depending on applications, the presence of either B or MW can be essential for, or detrimental to, their operation. Such JJ-based devices are inherently asymmetrical to some degree (the asymmetry can be structural or current bias) and, therefore, the Lorentz force induces a preferential Josephson vortex flow or intermittent vortex motion. On the other hand, an applied MW induces Shapiro steps on the dc current–voltage characteristics (IVCs). From this perspective, understanding how the Lorentz force-driven asymmetric flux motion impacts on either (1) the height of MW induced Shapiro resonances or (2) on the array's coherent operation is essential. In the first case, it will help minimizing the unwanted interference of Shapiro resonances on the performance of high frequency operation of superconducting devices and their noise spectral density. In the second case, it would be beneficial to our understanding of reaching coherent operation for superconducting MW generators/receivers and FFOs. Here, we address both these cases as explained in the following. The applied magnetic field B penetrates such devices as an ensemble of magnetic flux quanta Φ₀, known as Josephson vortices whose dynamics are very sensitive to applied direct I or/and alternating currents I_ac (and voltages) originating from MW exposure. These devices, being inherently asymmetrical to some degree, operate in a non-zero voltage state where I produces a Lorentz force that induces a preferential Josephson vortex flow. Thus, the lattice of vortices is moving with a speed (proportional to the measured dc voltage V) whose magnitude depends on the value of I: positive or negative. This phenomenon is called a B-field induced magnetic Josephson vortex ratchet effect.^12–20 Josephson ratchets based on very large asymmetric JJ-arrays have shown remarkable features such as an ability to amplify the self-induced electromagnetic radiation.¹⁹ Another fundamental phenomenon in JJ-based superconducting devices is the so-called inverse ac Josephson effect, i.e., the appearance of resonant dc Shapiro steps²¹ on the dc current–voltage characteristics (IVC) in the presence of an applied MW radiation. The Shapiro steps appear at multiple voltages of the ac Josephson effect relation V_n = nhf/2e, with n = 1, 2, 3, …. Surprisingly, the influence of a preferential flux-flow on the strength of the inverse ac Josephson effect, i.e., on the magnitude the Shapiro steps in asymmetrical arrays or on their coherent operation have never been investigated. This is important to fully understand the physics behind the response, coherent operation, and ultimate sensitivity of JJ-based devices. Here, we report on the influence Josephson vortices flow has on the inverse ac Josephson effect in purposely build asymmetric arrays made of 10 YBa₂Cu₃O_7−δ Josephson junctions (JJ) operating in the temperature range (4.2–45) K and irradiated with MW radiation in the range (45–75) GHz. We found that for each particular value of B a preferential direction of vortex flow dramatically impacts on the appearance of Shapiro steps: if, say, for a positive value of I, multiple Shapiro steps are well pronounced, then when we reverse the direction of the current, i.e., for −I, the Shapiro steps are strongly suppressed. Remarkably, when we change the value of the B field slightly the opposite occurs: the Shapiro steps are now strongly suppressed for positive I, while they are well pronounced for negative I (in agreement with general features of dynamical equations which are invariant under B $→$−B, I $→$ −I, x $→$−x transforms). Based on extensive numerical simulations, we were able to explain this behavior by various degree of coherence reached in the 10 JJ-arrays. Our results are particularly relevant where reaching coherent operation in JJ-array-based devices is essential such as flux-flow oscillators for sub-terahertz integrated-receivers^19,22–29 or on-chip superconducting MW generators which usually operate for bias currents at the Shapiro step region (suitable for quantum computers³⁰ and in other applications^29,31,32) Indeed, the Shapiro step is a manifestation of the self-induced locking of the Josephson and resonator dynamics leading to the measurable power emission.²¹ 

The JJ-arrays were fabricated by depositing high-quality epitaxial, 100 nm thick c-axis oriented YBa₂Cu₃O_7−δ (YBCO) films on 10 × 10 mm², 24° symmetric [001] tilt SrTiO₃ bicrystals by pulsed laser deposition. A 200 nm thick Au layer was deposited in situ on top of the YBCO single-layer film to facilitate the fabrication of high-quality electrical contacts for electric transport measurements. Medium-underdoped YBCO films with a critical temperature T_c = 49 K were subsequently patterned by optical lithography and etched by an Ar ion beam to form asymmetric parallel arrays of 10 JJs (see Fig. 1). Within each such parallel array, all 10 JJs are 3 μm wide. The junctions are separated by superconducting loops of identical width of 3 μm but variable length. The loops' length increases linearly from 8 to 16 μm in steps of 1 μm. Since the individual loop inductances, L_n, are proportional to the loop perimeter (1 μm corresponds to approximately 1 pH) $βLn=2πLn Ic$/Φ₀ also increases monotonically by 58% within the 10 JJ-arrays, with n = 1, 2, …, 9, where I_c is the JJs critical current. One can, therefore, define an average value β = $βLn$ for the array, namely, β can be estimated from both the modulation of I_c with B or direct calculations. The bias current I is applied symmetrically via the central top and bottom electrodes and V is measured across the array. B is applied perpendicular to the planar array's structure via a control current I_ctrl through an inductively coupled coil. Consequently, an external magnetic flux, Φ_ex, is coupled into the array. The JJ-array was placed in a Fabry–Pérot resonator which was excited at a TEM00k microwaves resonance in the frequency range, f (45–75) GHz at an input power level up to 30 mW.³³ The coupling, which was controlled by the rotation of the array relative to the electric field in the waveguide, is minimal when E is parallel to the grid and maximal in the perpendicular direction. Due to the inverse ac Josephson effect, the applied MW induce multiple current Shapiro steps at voltages V_n given by the ac Josephson effect relation V_n = nhf/2e. Due to the asymmetry of the 10 JJ-array for each particular value of B, the intensity of the Josephson vortex flow along the array (indicated by the horizontal arrow in Fig. 1) for positive I is strongly enhanced relative to the case when I is negative. This highly preferential Josephson vortex flow in one direction induces a strong asymmetry in the appearance of the Shapiro steps. We fabricated two such devices and both showed a qualitatively similar behavior. These devices showed a strong and robust B-field tunable ratchet effect in the absence of MW.²⁰ 

Families of IVCs and dI/dV's were measured by a four point-contact method at various temperatures T between 10 and 49 K, for different B field values in the range (−53.5, 53.6) μΤ and various MW powers (MW voltage bias V_MW in the range of 1–50 mV). The B field was changed in small steps of 67 nΤ. Shapiro steps are better defined on dI/dV's relative to the IV's data: they appear as peaks as opposed to current steps. For that reason in the following, the dI/dV data will be analyzed. Consequently, we define the height of the Shapiro steps as the height of the Shapiro peak on the dI/dV curves. An example of a complete set of data for a particular MW bias voltage V_MW = 4.5 mV is shown in Fig. 2(a): a 3D plot of dI/dV(V, B) extracted from a family of 301 IVCs measured at 40 K. Two individual dI/dV's for two particular values of the B field are shown in Fig. 2(b). I_c and the height of the first Shapiro step vs B field are shown in Fig. 2(c). The preferential flux-flow-induced asymmetry in the inverse ac Josephson effect is evident for all values of B field measured. Thus, in Fig. 2(b), when B = 9.6 μT, the first five (n = 1–5) Shapiro steps are clearly visible for positive voltages (or current bias I) while they are significantly suppressed for negative voltages (or current bias I). The asymmetry reverses as we change B. Thus, when B = 17.5 μT, the situation is opposite: the first five (n = 1–5) Shapiro steps are now well defined for negative voltages (or current bias I) while they are significantly suppressed for positive voltages (or current bias I). This behavior qualitatively applies for all B field and current values measured: there is always a significant degree of asymmetry in the height of some or all five (n = 1–5) Shapiro steps for positive I relative to negative I [see Fig. 2(a)]. For example, the asymmetry of the first Shapiro step between the case of positive I relative to the case of negative I is shown in Fig. 2(c). The lack of correlation between the value of I_c and the height of first Shapiro step is also evident from Fig. 2(c). We observed a similar qualitative behavior to Fig. 2 for many other values of the MW bias voltages V_MW. For a fixed value of B, the asymmetry stays robust and does not reverse as we change the MW power [see Figs. 3(a) and 3(b)]: that is, multiple Shapiro steps are well pronounced for negative I only, while no Shapiro steps could be resolved for positive I. We observed a similar qualitative behavior to Fig. 3 for many other values of the B field. With increasing MW power, the amplitude of the Josephson critical current decreases, while the Shapiro steps start developing in increasing order: first, the first Shapiro step, then the second one, and so on [see Fig. 3(c)]. Interestingly, this behavior is qualitatively similar to that observed for single Josephson junctions.²¹ 

To better understand the physics behind the asymmetry in the Shapiro steps induced by the preferential directions of vortices, we perform numerical simulations based on a model consisting of an array of resistively shunted junctions connected via superconducting inductances. The asymmetric Josephson transmission line can be described by a set of coupled ordinary differential equations²⁰ 

In order to match the experimentally implemented design, n runs from 1 to 8 in the middle equation of set (1), while n_max = 9 in the last equation of the set. The dimensionless applied current $j=I+IMW cos ωt/Ic$ has a dc component I and a MW component of amplitude I_MW. Here, $Ic$ is the array's average I_c, φ_n (with n = 0, …, 9) is the gauge invariant phase difference across the nth junction, dimensionless time t is measured in the units of the characteristic relaxation time τ = Φ₀/2πcRI_c with the flux quantum Φ₀, the speed of light c, and the junction resistance R (assumed the same for all junctions). The frequency of the applied microwave radiation ω is measured in units of ω_c = 2πRI_c/Φ₀, the Josephson characteristic frequency. The detailed consideration of two-dimensional boundary conditions of the problem suggests the same j in all junctions of the line. We also introduce parameter α = τcR/(4πaS_1/2) with the inter-junction distance a and the smallest area S_1/2 of the array loop between junctions with n = 0 and n = 1. The array loop area linearly increases with junction number and area ratio A_n+1/2 = S_n+1/2/S_1/2 = 1 + n/8, with n + 1/2 refers to the array loop between nth and (n + 1)th junctions. The dimensionless magnetic flux $h=HS1/2/Φ0$⁠, is measured in units of magnetic flux quantum per smallest array loop area of the array. In the simulations, thermal fluctuations have been considered too: we have introduced temporal unbiased δ-correlations Gaussian white noise $ξn$ with intensity D, i.e., $ξn=0, ξn(0)ξn(t)=δ(t)$⁠, which can occur, for instance, due to thermal noise or temporal current fluctuations of the external circuit. We average the time derivative of the gauge invariant phase difference over time and junctions $φ̇¯(j)=⟨⟨dφn/dt⟩n⟩t$⁠. The dynamical equations (1) used here are similar to the ones previously utilized to describe symmetrical 2D arrays of Josephson junctions³⁴ investigated experimentally in Ref. 35. However, there is a significant difference in that we consider not only the applied current and the external magnetic field but also take into account the field gradient trapped in neighboring holes proportional to the electrical current in the junction connecting these holes. The field gradient and self-induction was ignored in Ref. 34, which is a key for our analysis of asymmetric Josephson arrays.

A typical set of data is shown in Figs. 4(a) and 4(b). Neglecting thermal noise in the simulations (D = 0), Shapiro steps are well defined on the IVs and the asymmetric response in the inverse ac Josephson effect is obvious: the first four Shapiro steps are clearly more pronounced for positive I, relative to the case of negative I [see main graph in Fig. 4(a)]. When thermal noise is considered (D ≠ 0), Shapiro steps are suppressed and are better visualized on the dI/dV curves. Now the similarity with the experiments is striking: compare insets of Fig. 3(b) with the left inset in Fig. 4(a). The right inset in Fig. 4(a) suggests that the Shapiro steps can be completely suppressed for negative I, while they are well pronounced for positive I. The asymmetry in the Shapiro steps is very significant for all investigated steps n = 1, 2, 3, 4 and for the vast majority of values in the entire 2D plane investigated (i_ac, h). An example is shown in Fig. 4(b) for the first Shapiro step: n = 1. To understand the asymmetrical response in the inverse ac Josephson effect, we investigated both the dynamic phase coherence/synchronization among the 10 junctions in the array as well as the static magnetic field distribution h_i(I) along all of the nine superconducting loops for current biases that correspond to the middle of first Shapiro steps in Fig. 4(a): I = 1.05 and I = −0.67. It is important to stress that in all simulations performed asymmetric JJ-arrays have always h_i(I) distributions that are highly asymmetric with I in high contrast to the case of symmetric JJ-arrays. Numerical simulations show [see Fig. S1(a) in the supplementary material] that for a positive bias current of I = 1.05 there is a high degree of coherence among the array with 7 out of 10 junctions oscillating in-phase leading to a well pronounced first Shapiro step in Fig. 4(a). In contrast [see Fig. S1(b) in the supplementary material] for a negative bias current of I = −0.67, there is a low degree of coherence among the array with no junctions performing in-phase oscillations, leading to a relatively suppressed first Shapiro step in Fig. 4(a). Interestingly, there is a strong correlation between the degree of dynamic coherence in the oscillations within the 10 JJ-array and the corresponding static flux configuration in the nine holes at the I values corresponding to the Shapiro steps. Thus, for values of I around 1.05 when the first Shapiro step is well pronounced [see Fig. 4(a)], the B field configuration is well structured with significant differences in the values of neighboring loops for most loops in the array [see Fig. S1(c) in the supplementary material). Alternatively, we may say that in this case the JJ-array is significantly polarized, with strong circulating supercurrents around most loops to account for the rather large differences in the B values. This is in high contrast to the case when I take values around −0.67 and the first Shapiro step is strongly suppressed: in this case, the B field configuration is unpolarized, with small differences in the values of B in most loops and, consequently, small corresponding circulating supercurrents [see Fig. S1(c) in the supplementary material]. It follows that the static flux configuration has a strong influence on the intensity of flux flow previously observed in this particular JJ-array²⁰ which, in turns, affects the degree of coherence within the JJ-array leading to an asymmetric response in the height of Shapiro steps for positive I relative to negative I.

We showed experimentally that during the operation of an asymmetrical 10 JJ-array placed in an applied B field, the preferential Josephson vortex-flow induces a strong asymmetry in its response to applied MW power. Thus, the asymmetry in the inverse ac Josephson effect is evident for all values of B measured: multiple robust Shapiro current steps are formed on the IVCs for a particularly positive current bias I, but strongly suppressed when we reverse the current direction, i.e., for a bias −I. Remarkably, by changing B slightly, the situation is reversed: Shapiro steps are now suppressed for a positive I, while well pronounced for a reverse current −I (see Fig. 2). For a fixed B, the asymmetry stays robust and does not reverse as we change the MW power (see Fig. 3): that is, multiple Shapiro steps are well pronounced for negative I only, while no Shapiro steps could be resolved for positive I. The extensive numerical simulations performed are in qualitative agreement with the experiments and strongly suggest that the observed asymmetrical response to MW is due to a different degree of coherence reached in the JJ-array. Our results suggest that vortex motion in asymmetrical devices consisting of either multiple JJs or a single long JJ asymmetrically biased operating under the presence of both a MW and a remnant/applied B field has a fundamental influence in reaching a coherent operation which is essential to consider when designing/fabricating them for various applications such as FFOs, flux-driven Josephson (travelling-wave) parametric amplifiers, or on-chip superconducting MW generators/detectors.

See the supplementary material for the details related to JJ-array fabrication and numerical simulations.