Charging of Superconducting Layers and Novel Type of Hysteresis in Coupled Josephson Junctions
aa r X i v : . [ c ond - m a t . s up r- c on ] A p r Charging of Superconducting Layers and Novel Type of Hysteresis in CoupledJosephson Junctions
Yu. M. Shukrinov and M. A. Gaafar , BLTP, JINR, Dubna, Moscow Region, 141980, Russia Department of Physics, Faculty of Science, Menoufiya University, Egypt.
A manifestation of a novel type of hysteresis related to the parametric resonance in the systemof coupled Josephson junctions is demonstrated. Opposite to McCumber and Steward hysteresis,we find that the width of this hysteresis is inversely proportional to the McCumber parameter anddepends also on coupling between junctions and the boundary conditions. An investigation of timedependence of the electric charge in superconducting layers allow us to explain the origin of thishysteresis by different charge dynamics for increasing and decreasing bias current processes. Theeffect of wavelength of the longitudinal plasma waves created at the resonance on the charging ofsuperconducting layers is demonstrated. We found a strong effect of the dissipation in the systemon the amplitude of the charge oscillations at the resonance.
The hysteresis features of single Josephson junction(JJ) were studied by McCumber and Steward long timeago. [1, 2] Particularly, it was shown that the width ofthe hysteresis is directly proportional to the McCumberparameter β c related to the dissipation parameter β by β c = 1 /β . In compare with the single Josephson junc-tion, the system of the coupled Josephson junctions hasa multiple branch structure and the definition of the re-turn current is more general now: the system can re-turn to the zeroth voltage state from any branch. Thebranches have a breakpoint (BP) and a breakpoint region(BPR) before transition to another branch. [3] The BPcurrent characterizes the resonance point at which thelongitudinal plasma wave (LPW) with a definite wavenumber k is created by Josephson oscillations in stacks.Hysteresis features of intrinsic JJ in high temperaturesuperconductors have a wide interest also due to the ob-served powerful coherent radiation from such system. [4]In Ref. [5] the authors summarized the experimental re-sults and stressed that the strong emission was observednear the unstable point of the return current in the uni-form voltage branch. The radiation is related to the sameregion in the current voltage characteristics (CVC) wherethe BP and the BPR were observed and it made the phasedynamics investigation of intrinsic JJ corresponding tothese parts of CVC an actual problem today.Since thickness of superconducting layer (S-layer) inthe intrinsic JJ is comparable to the Debye screeninglength, the S-layers are in the nonstationary nonequi-librium state due to the injection of quasiparticle andCooper pairs. [6, 7]. The charge neutrality in S-layersis locally broken and this charging effect modifies theJosephson relation between voltage and the phase differ-ence. The question concerning the value of the electriccharge in S-layer and its maximum realized at the para-metric resonance is not investigated yet.In this paper we study the phase dynamics in coupledJosephson junctions. A novel type of hysteresis related tothe parametric resonance in this system is demonstrated.We show that the width of this hysteresis is inversely pro- portional to the McCumber parameter and depends oncoupling parameter of the system and the boundary con-ditions. The origin of this hysteresis is related to thedifferent charge dynamics for increasing and decreasingbias current processes. We discuss the question concern-ing the maximal electric charge in S-layers realized at theresonance and show that it depends on the relation be-tween the wavelength of LPW and the period of lattice.We demonstrate a strong effect of the dissipation in thesystem on the coefficient of the exponential growth of themaximal electric charge in S-layers.First we discuss the phase dynamics in the systemof coupled JJ. The CVC of JJs are numerically calcu-lated in the framework of capacitively coupled Josephsonjunction model with diffusion current (CCJJ+DC) [8].The system of equations for the gauge-invariant phasedifferences ϕ l ( τ ) = θ l +1 ( τ ) − θ l ( τ ) − e ~ R l +1 l dzA z ( z, τ )between S -layers in this model has the form d dt ϕ l =(1 − α ∇ (2) )( I − sin( ϕ l ) − β dϕ l dt ) where θ l is the phase of theorder parameter in the S-layer l , A z is the vector poten-tial in the barrier, α and β are coupling and dissipationparameters, respectively, and ∇ (2) f l = f l +1 + f l − − f l .In our simulations we use both periodic and nonperiodicboundary conditions (BC). At nonperiodic BC it is sug-gested that the first and the last S-layers are in contactwith normal metals and their effective width s and s N may be extended due to the proximity effect into attachedmetals. Nonperiodic BC are characterized by parameter γ = s/s = s/s N and the equations for the first andlast layers in the system for phase differences are differ-ent from the equations for the middle S-layers[7, 9]. Wesolve the system of dynamical equations for phase differ-ences using the fourth order Rungge-Kutta method. Weuse a dimensionless time τ = tω p , where ω p is the plasmafrequency ω p = p eI c / ~ C , I c is the critical current and C is the capacitance. In our simulations we measure thevoltage in units of V = ~ ω p / (2 e ) and the current in unitsof the I c . The details concerning numerical procedure aregiven in Ref. 9. FIG. 1: Demonstration of the hysteresis behavior in the para-metric resonance region in the outermost branch of the CVCfor the stack with 9 JJ at α = 1, β = 0 . γ = 0 . I B ∗ − I B with dissipation parameter β andcoupling parameter α . The inset (c) shows the same withvariation of the nonperiodic parameter γ . The results of simulation of CVC and its features nearparametric resonance region are presented in Fig. 1. Theinset (a) shows the total branch structure of the CVC forthe stack with 9 JJ at α = 1, β = 0 . γ = 0 .
5. TheCVC shows the following features: (i) a jump at
I/I c =1 . I >I c ; (iii) multiple branching in the hysteresis region. Thecircle and arrow with letter B show the BP location onthe outermost branch.Fig. 1 demonstrates a hysteresis in the outermostbranch of the CVC. It is obtained by decreasing the biascurrent till some point in the BPR (curve 1), then weincrease the current to pass the resonance region (curve2). The arrows show direction of the bias current chang-ing. The hysteresis is characterized by its width I B ∗ − I B ,where I B is the value of breakpoint current in decreasingcurrent process and I B ∗ is a characteristic current valuein the increasing current process.The dependence of the hysteresis width on the dis-sipation parameter β is shown in the inset (b). Thewidth is increasing with parameter β (i.e., it is decreas-ing with the McCumber parameter). As we mentionedabove, the McCumber and Steward hysteresis demon-strates an opposite behavior for single JJ. They obtainedthat the return current I r (which characterizes the valueof hysteresis) decreases with increasing of the McCum-ber parameter.[10] So, we have observed a novel type ofhysteresis related to the parametric resonance in coupledJJ . The insets (b) and (s) show as well an increase of the hysteresis width I B ∗ − I B with the coupling parameter α and parameter of nonperiodicity γ , respectively.To make clear the origin of this hysteresis, we studytime dependence of the charge oscillations in the S-layers.Using Maxwell equation div ( εε E ) =Q, where ε and ε are relative dielectric and electric constants, we expressthe charge density Q l = Q α ( V l +1 − V l ) in the S-layer l by the voltages V l and V l +1 in the neighbor insulatinglayers, where Q = εε V /r D , and r D is Debay screeninglength. The charge dynamics in the S-layers determinesthe features of current voltage characteristics of the cou-pled Josephson junctions. FIG. 2: Difference in the charge-time dependence and CVCin (a) decreasing current process and (b) increasing currentprocess. The thick curves show the CVC.
Solution of the system of dynamical equations forphase differences gives us the voltages as a functionsof time V l (t) in all junctions in the stack, and it al-lows us to investigate the time dependence of the chargein each S-layer. Here we investigate the charge-timedependence for two processes: decreasing (Figs. 2(a),3, 4, 5) and increasing (Fig. 2(b)) the bias currentthrough the stack. The recorded time is calculated as t r = tω p + T m ( I − I ) /δI for decreasing bias current pro-cess. For increasing current process (Fig. 2(b)) we recordthe time dependence at bias current value I during thetime interval ( t r , t r − T m ). We put mostly T m = 1000, δτ = 0 .
05 and δI = 0 . α = 1, β = 0 . FIG. 3: Demonstration of the absence of fine structure incharge-time dependence and CVC for the stack with 10 junc-tions at α = 1, β = 0 . Let us now discuss the question concerning the am-plitude of the electric charge oscillations in S-layer atparametric resonance (the parametric resonance corre-sponds to the BP on the outermost branch of CVC).
Doesits maximal value depend on the wavelength of createdLPW?
The Josephson oscillations excite the LPW with k = π/d ( π -mode, wavelength λ = 2 d ) at the parametricresonance in the stack with even number of JJ at α = 1, β = 0 . N of JJ the wave number depends on N and itis equal to k = π ( N − /dN . In the stacks with evennumber of JJ the resonance is ”pure”, i.e., no additionalfine structure appears in CVC.[12] So, it is interesting tocompare the maximal value of the electric charge realizedin S-layers in this case with the case λ = nd , where n isinteger number.In Fig. 3 we present time dependence of the electriccharge in the S-layer for the stack with 10 JJ combinedwith the outermost branch of CVC (the correspondingaxis are shown by arrows). We found that the bias cur-rent interval where the growing region (the beginning ofthat region is demonstrated in the inset (b)) of the elec-tric charge in S-layers is observed, is shorter now in com-pare with N=9 case, where the LPW with k = 8 π/ d is created at the same values of α and β . Compare thisfigure with Fig. 2, we can see that the amplitude of thecharge is larger for the stack with even number of JJ.The inset (a) illustrates the charge distribution amongthe layers and confirms the π -mode of LPW.As we mentioned above, the wavelength of the LPWdepends on the values of dissipation and coupling FIG. 4: Charge-time dependence and CVC at α = 3, β = 0 . π/ d -mode in the stack with 9JJ. parameters.[3] So, we can compare the stacks with 10and 9 JJ when another LPW are created and test theidea concerning the maximal amplitude of charge oscilla-tions. In Fig. 4 the time dependence of the charge in thefirst S-layer at α = 3, β = 0 . k = 3 π/ d is created. In case N = 9 (Fig. 4(b)) the insetillustrates the charge distribution among the layers corre-sponding to the (2 π/ d )-mode ( λ = 3 d ). Also we can seethat the charge value on the S-layers in k = 2 π/ d caseis larger than in case of k = 3 π/ d , which is the sameresult as we got before. In addition to that, we testedthe cases for λ = 4 d and λ = 5 d ( not presented here)and they supported our idea. So, we may conclude thatat fixed α and β the charge value in S-layers is larger forthe stacks with ”pure” parametric resonance where theLPW with λ = nd is created.To demonstrate the character of the charge amplitudeincreasing in the growing region, we enlarged in Fig. 5the charge-time dependence for the stack with 10 JJ at α = 1 and β = 0.2. In inset (a) we present the timedependence of the amplitude of Q A /Q in the logarith-mic scale. The values of amplitude were taken arbitraryat some time moments in total growing region (examplesare shown by circles). We found two parts in this depen-dence: exponential part and transition part (marked bydouble arrow) before jump to another branch. In tran-sition part the amplitude demonstrates a sharp increasein short time interval in compare with exponential part. t r Q / Q ω/ 2πω p (c) A m p li t ud e ω/ 2πω p t r =286000 (b) t r Ln ( Q A / Q ) -6 -4 -2 Ν = 10α =1β = 0.2
K = 0.001I=0.5771 (a)
FIG. 5: Demonstration of the transition part (shown by dou-ble arrow) in charge-time dependence for the stack with 10JJ. Inset (a) shows the amplitude of charge oscillations in thelogarithmic scale. Insets (b) and (c) demonstrates the resultsof FFT analysis of voltage V ( t ) and charge Q ( t ) /Q timedependence in the exponential growth part. t r Ln ( Q A / Q ) -6 -5 -4 -3 -2 -1
1. β = 0 . 2 , Κ = 0 . 0012. β = 0 . 1 , Κ = 0 . 0033. β = 0 . 05 , Κ = 0 . 01Ν = 10α = 1
Periodic
FIG. 6: Influence of dissipation magnitude on time depen-dence of the charge oscillation amplitude in the logarithmicscale for the stack with 10 junctions at α = 1. Insets (b) and (c) show results of FFT analysis of voltage V ( t ) and charge Q ( t ) /Q time dependence in the expo-nential grows part. They prove the parametric resonancecondition ω J = 2 ω LP W . In transition part this conditionis broken. Writing the expression for electric charge by Q l /Q = exp( Kt ), we find K = 0 . β = 0.2, 0.1, and 0.05) on time dependence ofthe charge oscillation amplitude in the logarithmic scalefor the stack with 10 junctions at α = 1. From thisfigure we note the following features which are observedwith decrease in β (increase the McCumber parameter):i) the growing region is getting shorter; ii) the width oftransition part is decreasing; iii) the coefficient of theexponential growth K is increased. We come to the im-portant conclusion that the parametric resonance featuresdepend strongly on dissipation in the system . The widthof the growing region is inversely proportional to the co- efficient K . The value of K is determined by the wavenumber of LPW created at resonance. For all investi-gated stacks with even number of JJ (in our simulationswe checked the stacks with N in the interval (6,14)) at α = 1, β = 0 . K = 0 . π − mode ofcreated LPW (the charge on the nearest neighbor layershas the same value and opposite sign) in the exponentialpart and its modification in the transition region.As a summary, a manifestation of a novel type of hys-teresis related to the parametric resonance in the systemof coupled Josephson junctions is demonstrated. Thewidth of this hysteresis is inversely proportional to theMcCumber parameter and depends on coupling betweenjunctions and the boundary conditions. The origin ofthis hysteresis is related to the different charge dynamicsfor increasing and decreasing bias current processes. Weconsider that these features are common for the systemsdemonstrating the parametric resonance. These featurescan be used to develop new methods for determination ofcoupling and dissipation parameters of the system. Weshow that the maximal value of electric charge ampli-tude realized in superconducting layers at the resonancedepends on the wavelength of the created LPW. A strongeffect of the dissipation in the system on the width of theparametric resonance is demonstrated.We thank I. Rahmonov, M. Hamdipour, H. El Sam-man, S. Maize, M. Elhofy and Kh.Hegab for helpful dis-cussions and support this work. [1] D. E. McCumber, J.Appl.Phys. 39, 3113 (1968).[2] W. C. Steward, Appl.Phys.Lett. , 277 (1968).[3] Yu. M. Shukrinov and F. Mahfouzi, Phys. Rev. Lett. 98,157001 (2007).[4] L. Ozyuzer et al., Science 318, 1291 (2007).[5] T. Koyama, H. Matsumoto, M. Machida, and K. Kad-owaki, Phys. Rev. B 79, 104522 (2009).[6] D. A. Ryndyk, Phys. Rev. Lett. , 3376 (1998).[7] T. Koyama and M. Tachiki, Phys. Rev. B , 16183(1996).[8] Yu. M. Shukrinov, F. Mahfouzi, and P. Seidel, PhysicaC 449, 62 (2006).[9] H. Matsumoto, S. Sakamoto, F. Wajima, T. Koyama, M.Machida, Phys. Rev. B , 3666 (1999).[10] Yu. M. Shukrinov and F. Mahfouzi, Physica C 434 (2006)6-12.[11] Yu. M. Shukrinov and F. Mahfouzi, Supercond. Sci. Tech-nol. 20, S38 (2007).[12] Yu. M. Shukrinov, F. Mahfouzi, M. Suzuki, Phys. Rev.B78