Stationary charge imbalance effect in a system of coupled Josephson junctions

Nonequilibrium effects in different types of superconducting electronic devices are of considerable interest [1], since they could affect both the efficiency and the performance of these devices. At finite temperature, there are always nonpaired quasiparticles in the superconductor, the so-called equilibrium quasiparticles. The injection of excess quasiparticles in this case may lead to deviations from equilibrium of charge, energy and/or spin degrees of freedom. Rapid development of micro- and nano-technologies facilitated the fabrication of devices and circuits with dimensions of the order of relaxation lengths of these nonequilibrium modes. The nonequilibrium quasiparticles might limit the performance of a variety of nanoscale superconducting devices with dimensions comparable to the corresponding relaxation scales, such as superconducting resonators [2–4] and superconducting qubits [5–7], because they cause a dissipation or excess noise that can limit qubit coherence and reduce the sensitivity of amplifier and magnetometer devices.

One of the most affected materials by different kinds of nonequilibrium effects is the layered superconducting material, in which the thickness of the superconducting layer is comparable to the Debye screening length. So the electrical charge does not screen in the superconducting layers and they could form a system of coupled Josephson junctions (JJs) [8,9]. The phase dynamics of such a system has attracted great interest because of rich and interesting physics, on the one hand, and perspectives of application on the other hand [10–14]. In particular, the nonequilibrium effects created by stationary current injection in high-T_c materials have been studied very intensively [15–21]. This could bring to the so-called charge imbalance, which is an asymmetrical occupation of the excitation spectrum. However, the charge imbalance in the systematic perturbation theory is considered only indirectly as far as it is induced by fluctuations of the scalar potential [15,16,19]. In ref. [22], the charge imbalance is considered as an independent degree of freedom and, therefore, the results are different from those of earlier treatments. The change of the critical current in the Josephson junction, caused by the charge imbalance, has been stressed in refs. [23,24].

The data in refs. [25–27] provided experimental evidence of the charge imbalance effect for low-temperature superconductors. In these experiments, a difference between the quasiparticle potential and the Cooper pair chemical potential in the nonequilibrium region is detected. In ref. [28], the current biased experiments are carried out for the mesa structure of BSCCO. In these experiments two important effects are observed for the stationary case and they are explained as a result of the charge imbalance in the superconductor layers. The first effect is the shifting of the Shapiro step from its canonical value $\hbar \omega_R/(2e)$. While, the second effect occurs for the two-mesa structure on the same base crystal, the results show the influence of the current through one mesa on the voltage drop across the other mesa.

In ref. [29], we showed that in the underdamped system of intrinsic Josephson junctions of high-temperature superconductors under external electromagnetic radiation the charge imbalance was responsible for a slope in the Shapiro step in the IV-characteristic. We found that the value of the slope increased with the nonequilibrium parameter, and the coupling between the junctions led to the distribution of the slope values along the stack. Furthermore, the nonperiodic boundary conditions shift the Shapiro step from its canonical position.

In this paper, we consider the current biased coupled overdamped Josephson junctions in the presence of the stationary charge imbalance. We show that the stationary charge imbalance leads to a change in the Josephson frequency in JJs of the stack. The difference in frequency along the stack brings to the appearance of spikes on the IV-characteristics below and above the Shapiro step. We also demonstrate a slope and shift of the step caused by the stationary charge imbalance.

The key point of the theory is the nonequilibrium character of the Josephson effect in layered superconductors [8,9,15]. Superconducting layers with thickness $d_s^l$ less than the characteristic length of the nonequilibrium relaxation l_E and the Debye screening length r_D are in a nonstationary nonequilibrium state due to the injection of quasiparticles, Cooper pairs, and the nonzero invariant potential formed inside them,

$$\begin{equation} \Phi_l(t)= \phi_l + \frac{\hbar}{2e}\dot{\theta}_{l}, \end{equation} \tag{ 1 }$$

where $\phi_l$ is the electrostatic potential, θ is the phase of superconducting condensate, $\Phi=0$ in the equilibrium state (here and below $e=\mid e \mid$). It is important to note that in the nonequilibrium mode, the usual Josephson ratio $\text{d}\varphi_l/\text{d}t = (\hbar/2e)V$, which connects the Josephson phase difference $\varphi_{l}(t)$ between the layers l –1, l and the voltage $V_l=\phi_{l-1}-\phi_l$ [8,9,15] is violated and instead a generalized Josephson relation is used,

$$\begin{equation} \frac{\text{d} \varphi_l}{\text{d}t}= \frac{2e}{\hbar}V_l + \frac{2e}{\hbar}(\Phi_{l-1}-\Phi_l). \end{equation} \tag{ 2 }$$

Thus, $\Phi_l$ become the new important dynamical variables of the theory. Note that the shift of the chemical potential of a superconducting condensate from its equilibrium value is $\delta \mu_s =e \Phi$ and is determined from the expression of the charge density inside the superconducting layer [30,31]

$$\begin{equation} \rho=-2e^2 N(0)(\Phi_l - \Psi_l)= -\frac{1}{4\pi r^2_D}(\Phi_l - \Psi_l), \end{equation} \tag{ 3 }$$

where N(0) is the density of states in J^-1cm^-3 and $\Psi_l$ is the charge imbalance potential determined from the imbalance between the quasi-electron and quasi-hole branches of the superconductor excitation spectrum.

A stack of N + 1 superconducting layers (S-layers) forming a system of N coupled JJs is presented in fig. 1. The Josephson relation can be rewritten in terms of the coupling between the S-layers and the stationary charge imbalance potential as [32,33]

$$\begin{eqnarray} \frac{\text{d}\varphi_l(t)}{\text{d}t}=\,& \frac{2e}{\hbar} \Bigl( (1+2 \alpha ) V_{l}(t)-\alpha (V_{l-1}(t) + V_{l+1}(t)) \nonumber\\ & {}+ \Psi_{l}(t) - \Psi_{l-1}(t) \Bigr), \end{eqnarray} \tag{ 4 }$$

with a voltage V_l between the layers l –1 and $l, V_l(t) \equiv V_{l,l-1}(t)$, $\varphi_{l}(t)$ is the phase difference across the layers l –1 and $l, \Psi_{l}(t)$ is the charge imbalance potential on the layer l and $\alpha = \epsilon\epsilon_{o} /2e^{2} N(0) d_s d_i$ is the coupling parameter, is the dielectric constant, $\epsilon_{o}$ is the vacuum permittivity, d_s is the thickness of superconducting layers, d_i is the thickness of insulating layers.

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The total current density $J_{l-1,l} \equiv J_{l}$ through each S-layer is given by the sum of superconducting, quasiparticle and diffusion currents:

$$\begin{equation} J_{l}=J_{c}\sin\varphi_{l}+\frac{\hbar}{2eR}\dot{\varphi}_{l}+\frac{\Psi_{l-1}-\Psi_{l}}{R}, \end{equation} \tag{ 5 }$$

where (· ) shows the derivative with respect to time, J_c is the critical current density, and R is the junction resistance measured in $\Omega \text{cm}^2$. This equation together with the generalized Josephson relation and the kinetic equations for $\Psi_l$,

$$\begin{eqnarray} &&\Psi_l= \frac{\tau_{qp}}{ 2 e^{2} N(0) d_s} (J_{l}^{qp}-J_{l-1}^{qp}), \end{eqnarray} \tag{ 6 }$$

describe the dynamics of coupled JJs. Note that $\tau_{qp}/(2 e^{2} N(0) d_s)$ has the unit of resistance in $\Omega \text{cm}^{2}$. In formula (6) $\tau_{qp}$ is the quasiparticle relaxation time. We use the normalized equation for the current to calculate the phase dynamics

$$\begin{equation} \dot{\varphi}_{l}=I -\sin\varphi_{l} + A \sin{\omega} \tau + I_{noise} +\psi_{l}-\psi_{l-1}, \end{equation} \tag{ 7 }$$

where $I=J/J_{c}$ is the dimensionless current density and $\omega_c=\frac{2eJ_cR}{\hbar}$ is the characteristic frequency. This equation is solved numerically using the fourth order Runge-Kutta method. In parallel, at each time step we use the normalized kinetic equations at nonperiodic boundary conditions to calculate the nonequilibrium potential by means of the Gauss method,

$$\begin{eqnarray} & (1+\eta_0)\psi_0 - \eta_0 \psi_1 = \eta_0 (I + A\sin{\omega} \tau - \dot{\varphi}_1),\nonumber\\ & {}-\eta \psi_{i-1} + (1+2\eta)\psi_{i} - \eta \psi_{i+1} = \eta (\dot{\varphi}_i - \dot{\varphi}_{i+1}),\\ & {}\eta_N \psi_{N-1} + (1-\eta_N)\psi_N = \eta_N(\dot{\varphi}_N - I - A\sin{\omega \tau}),\nonumber \hspace{0.3cm} \end{eqnarray} \tag{ 8 }$$

where $\eta=\frac{\tau_{qp}}{2 e^2 N(0)R d_s}$ is the nonequilibrium parameter. The nonequilibrium potential is normalized to J_c R. Also, we can write η as $\tau_{qp} \nu$, where ν is the tunneling frequency $\nu=1/(2 e^2 N(0)R d_s)$.

Then, we calculate the voltage across the JJ using the normalized Josephson relation at nonperiodic boundary condition,

$$\begin{eqnarray} \dot{\varphi}_{1} &= v_{1} - \alpha (v_{2}-(1+\gamma)v_{1}) +\psi_{1}-\psi_{0},\nonumber\\ \dot{\varphi}_{l} &= (1+2\alpha)v_{l} - \alpha (v_{l-1}+v_{l+1}) +\psi_{l}-\psi_{l-1},\nonumber\\ \dot{\varphi}_{N}&= v_{N} - \alpha (v_{N-1}-(1+\gamma)v_{N}) + \psi_{N}-\psi_{N-1}, \end{eqnarray} \tag{ 9 }$$

where the voltage v_l is normalized to J_c R and $\gamma=d_s^{0,N}/d_s$ is the ratio between the thickness of the first (last) and middle S-layers. We assume that due to the proximity effect, the thickness of the first and the last S-layer is larger than the middle one. Therefore, the nonequilibrium parameters depend on the parameter of boundary conditions γ, $\eta_{0,N}=\gamma\eta$, where $l=1,2,\ldots,N-1$ and η is the nonequilibrium parameter for the middle layers.

The main purpose of our paper is to demonstrate a specific influence of the stationary charge imbalance on the phase dynamics of the system of coupled JJs. The IV-characteristic together with the average nonequilibrium potential for the current bias system of coupled JJs is shown in fig. 2. The IV-characteristic has no hysteresis since there is no capacitance in the system. The inset of fig. 2(a) enlarges the current interval I < I_c. The presence of the nonequilibrium potential gives a nonzero voltage at each current step. So, the finite slope appears in the IV-characteristics at I < I_c. In fig. 2(b), the raise of the average nonequilibrium potential of the 0th layer $\Psi_0$ is shown. Here, we show only $\Psi_0$ because the charge imbalance potential has almost negligible value in the S-layers of the middle JJs and $\Psi_N$ is antisymmetrical to $\Psi_0$ with a negative sign. It is important to notice that the averaged superconducting current in the overdamped junctions exists at values much larger than the critical one. So the charge imbalance potential is nonzero when all JJs are in the resistive-state (R-state, I  > I_c), as is shown in fig. 2(b).

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Important fact here is that a presence of the charge imbalance in the R-state gives a shift of the phase derivative, as can be seen from eqs. (9). In our normalization the derivative of the phase is equal to the Josephson frequency $\omega_j$. We show the Josephson frequency as a function of the current in fig. 3(a). The inset in fig. 3(a) shows that for a fixed value of current, the Josephson frequency of the first and last JJs is smaller than the middle junctions. Therefore, the nonequilibrium potential decreases the frequency of oscillations in the JJs at the edges of the stack. In addition, the critical currents of the first and last JJs are larger than the middle one. The increase in the critical current in JJs was discussed in refs. [23,24].

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Figure 3(b) demonstrates the distribution of $\omega_j$ along the stack. The maxima of the Ψ-potentials occur in the first and the last S-layers where the connections to the normal leads take place. According to this, the minima of the Josephson frequencies occur for the JJs at the edges of the stack. The distribution of $\omega_j$ along the stack depends on the value of the nonequilibrium parameter.

We emphasize that the existence of the charge imbalance in the resistive state of overdamped JJ affects the Josephson oscillations. So it can affect the performance of superconductor electronic devices, it can destroy the synchronization properties of the stack of coupled overdamped JJs.

Let us discuss now the influence of the stationary charge imbalance on the Shapiro step which appears in the IV-characteristics under external electromagnetic radiation. The simulated IV-characteristic for this case is represented in fig. 4. In the lower inset of fig. 4 the Shapiro step region is enlarged to demonstrate a slope of the step. The voltage value of the slope δ is marked by two dashed lines. The value of δ depends on the nonequilibrium parameter and has a maximum in the first and last JJs, decreasing in the middle of the stack with decreasing Ψ-potential [29].

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The upper inset of fig. 4 demonstrates another interesting effect, namely, small spikes of voltage below and above the Shapiro step. Those spikes appear due to the fact that JJs in the stack are not switched simultaneously to the locked regime. Particularly, the junctions in the middle of the stack are switched to the locked regime at a smaller current value. This brings to the situation in which the frequency of the middle junction is equal to $\omega_R$ but the frequency of the junctions at the edges of the stack is still smaller than $\omega_R$. The details of this process can be seen in fig. 5. It demonstrates an enlarged part of the IV-characteristic with the Shapiro step (solid lines) together with the Josephson frequency (dashed lines) as a function of current. Figure 5(a) demonstrates that the third JJ is locked at $I=1.071$ with increasing bias current. Furthermore, junctions 2 and 4 are also in the locked regime at this value of current, since $\omega_j$ (see dashed line in fig. 5(b)) is equal to the radiation frequency $\omega_R$ in those junctions at this current. On the other hand, the voltage value is smaller than $V=N\omega_R$ until the value of current $I=1.079$. The cause of this is the coupling of the middle JJs to the first and last ones, which are outside of the Shapiro step region (see fig. 5(c)). Both the first and last junctions are switched to the locked regime at the current $I=1.079$, as it shown in fig. 5(c). Therefore, the middle part of the stack "jumps" to the Shapiro step at a smaller current than JJs at the edges and it gives small corrections of the total voltage. A reverse situation occurs above the Shapiro step. Middle JJs "jump" off the Shapiro step at smaller current when the junctions at the edges of the stack are still on the Shapiro step.

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We assume that the observed spikes might be used for the experimental measurement of the coupling between JJS and the value of the nonequilibrium parameter, because the distance and the size of those spikes depend on η and α.

In the previous theoretical [34] and experimental [28] works the influence of the stationary charge imbalance on the Shapiro step properties were investigated in the voltage bias and current bias coupled JJs, respectively. It was shown that the Shapiro step shifted down when the first or last JJ was in the resistive state in comparison with the position of the Shapiro step where one of the middle JJs was in the resistive state. The results were obtained for the overdamped JJs.

In our study we concentrate on the current bias coupled JJs. We model the same situation when one of the JJs has a smaller critical current $I_c=0.5$, in this case one JJ switches to the R-state at smaller current and the internal branch appears in the IV-characteristic. We investigate the Shapiro step features on the internal branch taking into account the stationary charge imbalance effect. In fig. 6 the enlarged part of IV-characteristic of the five coupled JJs with the obtained internal branch is shown. The inset enlarges the Shapiro step region in the IV-characteristic where one of the junctions is in the resistive state and others are in the superconducting-state. The upper curves correspond to the case when one of the middle JJs (second or third) has a smaller critical current. The lower one corresponds to the case when the first JJ is in the R-state. Notice that the Shapiro step is shifted down when the first JJ is in the R-state. This effect is in perfect agreement with the results obtained in refs. [28,34].

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Furthermore, the Shapiro steps demonstrate a finite slope (dashed line is horizontal in the inset) due to the difference of the nonequilibrium potentials at the opposite edges of the step [29]. We note that it is impossible to obtain the slope of the Shapiro step in the case of voltage bias JJs [34]. Therefore, we reproduced the experimental results obtained in the case of the current bias system of coupled JJs and were able to notice additional manifestation of the charge imbalance.

For the experimental observation of coupled Josephson junctions we consider an artificial layered system based on Al or Nb [35,36] and natural crystals of intrinsic Josephson junctions. Although the HTSC materials such as BSCCO mesa are strongly underdamped, we note that at temperature close to the critical one the McCumber parameter decreases and the BSCCO mesa is getting close to the overdamped system. The main nonequilibrium parameter, which determines the features presented in this article, is η. It is rather difficult to make its accurate estimations for intrinsic JJs in HTSC due to the absence of the experimental data. However, one can make its rough estimation using the formula for the tunnel frequency at zero temperature in the BCS model [35],

$$\begin{equation*} \nu=\frac{J_{c}}{2\pi e \Delta N(0) d_{s}}. \end{equation*}$$

Based on the results in refs. [35,37] with $\Delta\approx20 \ \text{meV}$, and $N(0)\approx10^{22}\ \text{eV}^{-1}\text{ cm}^{-3}$, $d_{s}\approx80\ \text{\AA}$ and $J_{c}\approx10000 \ \text{A cm}^{-2}$ the tunneling frequency ν will be $\approx 6.2 \times 10^{7}\ \text{s}^{-1}$. In general, the relaxation time can be $1-1000\ \text{ps}$, in this case $\eta\approx0.02$ for $\tau_{qp}= 400 \ \text{ps}$, while for the artificial structure to estimate η, we can use [35]

$$\begin{equation*} \nu=\frac{J_{c}}{4e^{2} N(0) R d_{s}}. \end{equation*}$$

In this case for structures [35,36] based on Al and Nb, with $N(0) \approx 0.81 \times 10^{41}\ \text{J}^{-1}\text{ cm}^{-3}$, $R\approx 4 \times 10^{-7}\ \Omega \text{cm}^{2}$ and $d_{s}\approx80 \ \text{\AA}$, the tunneling frequency ν will be $\approx 3 \times 10^{8}\ \text{s}^{-1}$. In this structure the relaxation time is around $10^{-8}\ \text{s}$ and $\eta \approx 3$. So the results obtained in this paper at $\eta=0.01$ should be relevant and reproducible in the experiment.

We have investigated the effect of the stationary charge imbalance in the system of coupled JJs. It has been shown that it leads to a decrease in the Josephson frequencies in the junctions of the stack. The difference in frequencies appears due to the nonzero averaged superconducting current in the R-state of the overdamped JJs, which gives nonzero nonequilibrium potential. In addition, the distribution of the Josephson frequencies along the stack arises if the thickness of the S-layers is smaller than the relaxation length of quasiparticles. This distribution strongly depends on the value of the nonequilibrium parameter. In our case, the charge imbalance potential Ψ decays very fast along the stack, since $\eta=0.01$. In particular, for $\Psi_{0,5} \gg \Psi_{2,3,4}$ a noticeable difference in $\omega_j$ occurs for the first and last JJs. This difference leads to the nonuniform switch to the Shapiro step regime in the presence of external radiation and the appearance of voltage spikes on the IV-characteristics of the stack. The distance between those spikes on the IV-characteristics and their size strongly depends on the frequency difference and the coupling between JJs. We tested if the spikes appear in the IV-characteristics of the underdamped Josephson junctions and found that the spikes in voltage appear in the hysteresis region of IV-characteristics in the underdamped case as well. Therefore, we assume that our results can be used as a basis for the experimental measurements of the nonequilibrium potential and coupling parameter.

Also, we have shown that the stationary charge imbalance brings to the slope in the Shapiro step due to the difference of the Ψ-potentials at the edges of the step. The voltage value of the Shapiro step slope depends on the value of the nonequilibrium parameter. This slope was investigated in detail for the system of coupled JJs in the presence of nonstationary charge imbalance in ref. [29]. We assume that the slope of the Shapiro steps in the experimental results of ref. [38] may be also caused by nonequilibrium effects such as charge imbalance, but the authors explained it as a result of phase diffusion. Furthermore, a small slanting for the Shapiro step can be seen in the results of ref. [28] (see fig. 3 there).

Finally, we have found that the Shapiro step on the inner branch of the IV-characteristic is shifted from the canonical value in the current bias case when one of the JJs is in the resistive state. The shift value is smaller in the situations when the first JJ in the stack has a smaller critical current $I_c=0.5$ than the second or the third JJ which have $I_c=0.5$. The obtained results are in good agreement with the theoretical [34] results of the voltage bias system of coupled JJs and experimental results [28]. Additionally, it has been shown that the Shapiro step on the inner branch has a finite slope due to the influence of the charge imbalance. This slope cannot be obtained in the voltage bias coupled JJs.

The authors thank I. Rahmonov for fruitful discussions. This work is supported by RFBR grant 18-32-00950.