A fine structure of the breakpoint region in the current-voltage
characteristics of the coupled intrinsic Josephson junctions in the layered
superconductors is found. We establish a correspondence between the features in
the current-voltage characteristics and the character of the charge
oscillations in superconducting layers in the stack and explain the origin of
the breakpoint region structure.Comment: 5 pages, 5 figures. Accepted for Phys.Rev.

Mahfouzi, F.

Shukrinov, Yu. M.

Suzuki, M.

Physical Review B

English

arXiv

A fine structure of the breakpoint region in the current-voltage characteristics of the coupled intrinsic Josephson junctions in the layered superconductors is found. We establish a correspondence between the features in the current-voltage characteristics and the character of the charge oscillations in superconducting layers in the stack and explain the origin of the breakpoint region structure

Shukrinov, Yu M.

Kyoto University Research Information Repository

Title Structure of the breakpoint region on current-voltagecharacteristics of intrinsic Josephson junctionsAuthor(s)Shukrinov, Yu M.; Mahfouzi, F.; Suzuki, M.CitationPHYSICAL REVIEW B (2008), 78(13)Issue Date2008-10URL http://hdl.handle.net/2433/84616Right© 2008 The American Physical SocietyType Journal ArticleTextversionpublisherKyoto UniversityStructure of the breakpoint region on current-voltage characteristics of intrinsic JosephsonjunctionsYu. M. Shukrinov,1,2 F. Mahfouzi,1 and M. Suzuki2,31BLTP, JINR, Dubna, Moscow Region 141980, Russia2Photonics and Electronics Science and Engineering Center, Kyoto University, Kyoto 615-8510, Japan3Department of Electronic Science and Engineering, Kyoto University, Kyoto 615-8510, JapanReceived 12 September 2008; published 16 October 2008A fine structure of the breakpoint region in the current-voltage characteristics of the coupled intrinsicJosephson junctions in the layered superconductors is found. We establish a correspondence between thefeatures in the current-voltage characteristics and the character of the charge oscillations in superconductinglayers in the stack and explain the origin of the breakpoint region structure.DOI: 10.1103/PhysRevB.78.134521 PACS numbers: 74.50.r, 74.25.Fy, 74.81.Fa, 74.20.DeA system of Josephson junctions attracts a great interestfrom both scientific and practical sides. To describe it, a sys-tem of coupled nonlinear equations is used, which might besolved numerically only. Recently observed1 powerfulenough coherent radiation from the stack of the intrinsicJosephson junctions IJJ in layered superconductorBi2Sr2CaCu2O8 BSCCO opens possibilities for differentapplications.In Refs. 2–4 we studied the multiple branch structure ofthe current-voltage characteristics CVC of IJJ and showedthat the branches have a breakpoint BP and a breakpointregion BPR before transition to another branch. The BPcurrent is determined by the creation of the longitudinalplasma waves LPW with a definite wave number k, whichdepends on the coupling parameter , the dissipation param-eter , the number of junctions in the stack, and the bound-ary conditions. We generalized the McCumber-Stewards de-pendence of the return current for the case of IJJ in theHTSC, and the oscillation of the breakpoint current as afunction of parameters  and  was found. Based on the ideaof the parametric resonance in the stack of IJJ, we explainedthis oscillation as a result of the creation of longitudinalplasma waves with different wave numbers. A good qualita-tive agreement of the results of modeling of the  depen-dence of the BP current with the results of simulation wasobtained. We demonstrated that the  dependence of the BPcurrent is an instrument to determine the mode of LPW cre-ated at the breakpoint in the stacks with a different numberof junctions. So, the breakpoint in the CVC was explained,but the nature and features of the breakpoint region were notclear at that time.In this paper we study theoretically the phase dynamics ofa stack of IJJ in the high-Tc superconductors. The CVC of IJJare numerically calculated in the framework of capacitivelycoupled Josephson-junction model with diffusion current.5,6From the results of the numerical simulations we predict afine structure of the breakpoint region in the CVC. A corre-spondence between the features in the CVC and the characterof the charge oscillations on different superconducting layersis established and the origin of the breakpoint region struc-ture is explained. In the BPR the plasma mode is a stablesolution of the system and this fact might be used in someapplications, particularly, in high-frequency devices such asterahertz oscillators and mixers. We consider that the break-point phenomenon might be used to understand the mecha-nism of the powerful coherent radiation from the stack of theIJJ in BSCCO.To find the CVC of the stack of IJJ, we solve a system ofdynamical equations for the gauge-invariant phase differ-ences l=l+1−l−2e ll+1dzAzz , between super-conducting layers S layers for the stacks with a differentnumber of IJJ in the framework of the capacitively coupledJosephson-junction model with diffusion current CCJJ+DC model,5,6 where l is the phase of the order parameterin the S-layer l and Az is the vector potential in the barrier.We use a dimensionless time = tp, where p is the plasmafrequency p=2eIc /C, Ic is the critical current, and C isthe capacitance. The system of equations has a form2l /2=lAllI−sin l−l / with matrix A givenin Ref. 3 for periodic and nonperiodic boundary conditionsBC. In our simulations we measure the voltage in units ofV0=p / 2e and the current in units of the Ic. The CVC andtime dependence of the charge oscillation in the S layers aresimulated at =1 and =0.2 and periodic BC. The detailsconcerning the model and numerical procedure are presentedin Refs. 3, 4, and 6 In the inset to Fig. 1a we show theresult of simulation of the CVC for a stack with nine IJJ. Itdemonstrates an equidistant multiple branch structure in thehysteresis region and the breakpoint location. The transitionfrom the state where the outermost branch with all junctionsis in a rotating state R state7 to a state with some junctionsin an oscillating state O state happens in a resonance regionwhere the LPW with a definite wave number is created. Inthe stack with 11 IJJ with mentioned above values of =1and =0.2, the LPW with k=10	 /11d is created, where d isthe lattice period along the c axis.3,4 In Fig. 1a we can seethat the CVC exhibits a fine structure in the BPR. The BPRhas different parts with a different dependence of the voltageon the bias current. For a given stack we can see clear fiveparts in the CVC separated by some points T, Y, C1, and C2where the character of the CVC is changed. The letters B andJ show the breakpoint and jumping point, respectively, to theother branch. Points S on the CVC will be discussed later.To find the origin of this structure and explain its features,we study time dependence of the charge in the superconduct-ing layers. Using Maxwell’s equation div0E=, where PHYSICAL REVIEW B 78, 134521 20081098-0121/2008/7813/1345215 ©2008 The American Physical Society134521-1and 0 are relative dielectric and electric constants, respec-tively, we express the charge density Ql we call it justcharge in the S-layer l by the voltages Vl and Vl+1 in theneighbor insulating layers Ql=Q0Vl+1−Vl, where Q0=0V0 /rD2 and rD is Debay screening length. Solution of thesystem of dynamical equations for the gauge-invariant phasedifferences between S layers gives us the voltages Vl in alljunctions in the stack, and it allows us to investigate the timedependence of the charge on each S layer.The “time dependence” actually consists of time and biascurrent variation. We solve the system of dynamical equa-tions for phase differences at fixed value of bias current I insome time interval 0,Tm of dimensionless time = tp withtime step , where t is real time. This interval is used fortime averaging procedure. Then we change the bias currentby I and repeat the same procedure for the current I+I innew time interval Tm ,2Tm. The values of the phase andderivative of phase in the end of the first time interval areused as initial conditions for second time interval and so on.In our simulations we put Tm=250, =0.05, andI=0.0001, and total recorded time was calculated as+TmI0− I /I, where I0 is an initial value of the bias cur-rent for time-dependence recording.In the Fig. 2 the profile of time dependence of the chargein the S layers in the stack with nine IJJ is combined with theCVC of the outermost branch at =1, =0.2, and periodicBC. We show oscillations in the a first, b third, and cfourth layers only because in the other layers its characterclose to the behavior in these layers. To avoid large data filesin the procedure of time-dependence recording and topresent the CVC and time dependence with the same preci-sion, we have limited the averaging time interval by Tm=250. It leads to a more smeared CVC in this figure than theone shown in Fig. 1a Tm=25000; but the main features ofthe CVC are preserved. Moreover, such an averaging alsoallows us to find one more characteristic point S within theBPR.We have investigated the influence of the noise in biascurrent on the CVC, which is generated arbitrary in eachIJJs. The noise is produced by random number generator andits amplitude is normalized to the critical current value Ic.Figure 3 presents the CVC of the outermost branch in thevicinity of the breakpoint for three values of noise amplitude:10−6, 10−8, and 10−13 of a stack with 11 IJJ. We see that thefirst four parts of the BPR from the point B till the point C1are not affected by noise in current practically. But its strongeffect has a place in the C1−J region. The noise in biascurrent smears the CVC essentially here because the fourfirst parts of the BPR demonstrate a behavior close to theregular one, but the part C1−J shows the chaotic behavior.The voltage value is very sensitive to the small changes inthe value of current here.As we can see in the Fig. 2, the features of the CVC are incorrelation with the features of time dependence of thecharge on the S layer. This fact can be explained by the ideaof the parametric resonance at point B. As was shown in Ref.8, the system of equations for CCJJ has a solution corre-sponding to the LPW propagating along the c axis. A fre-quency of the LPW at I=0 and =0 is LPWk=p1+21−cos kd, where k is wave vector of the LPW.At point B the Josephson oscillations excite the LPW bytheir periodical actions. The frequency of Josephson oscilla-tions is determined by the voltage value in the junction; so atJ=2LPW the parametric resonance is realized and the LPWis created.The sharp increase in the oscillation amplitude in the firstlayer of the stack with nine IJJ in the beginning of the B-Srange is demonstrated in Fig. 1b, where the LPW withwave vector k=8	 /9d is created. This mode is close to the 	mode k=	 /d. In the 	 mode charges on the nearest latticepoints oscillate in the opposite directions and the insetshowed by numbers 1 and 2 that the charges oscillate inlayers 1 and 2 in opposite directions too. But the amplitudeof oscillations in layer 2 is smaller than in layer 1, whichdemonstrate the difference of this mode from the 	 mode.Fast Fourier transformation FFT analysis for the timeI/IcV/Vo0.56 0.572728293031TYC1JBC2Ν =11α = 1, β=0.2SI/IcV/V00.5 102040BN=9tQ1/Q023500 24000 24500-0.002-0.00100.0010.002ωpBtQ1/Q024250 24260 24270-0.000500.000512pω(b)(a)FIG. 1. Color online a The fine structure of the BPR in the outermost branch in the CVC of the stack with 11 IJJ. The insert showsa total branch structure in the CVC of the stack with nine IJJ and the BP location. b Charge oscillation in the first layer in the beginningof B-S part of the BPR. The inset shows the oscillations in layers 1 and 2.SHUKRINOV, MAHFOUZI, AND SUZUKI PHYSICAL REVIEW B 78, 134521 2008134521-2dependence of the charge in third S layer is presented in Fig.4. The time domain samples for FFT are collected across theinterval 0,Tm at each step of the bias current. Figure 4ashows that before the BP at I / Ic=0.573 the Josephson fre-quency J=0.45422	p=2.8538p is observed only. Inaccordance with the parametric resonance, we can see thepeak at =0.22462	p=1.4112p corresponding to theLPW frequency LPW in the B-S region Fig. 4b. We ob-serve also the peak at =0.67382	p=4.2336 corre-sponding to the sum of the Josephson and LPW frequenciesJ+LPW. A sharp increase in the oscillation amplitude atresonance point is observed on other layers in the B-S regionalso. The oscillation in the S-T part is similar to those for alllayers in the stack excluding layer 4. Because we considerthe periodic BC along c axis here, the layer number is usedjust for distinctness. As we can see in Fig. 2c, the ampli-tude of the charge oscillation in layer 4 in the B-Y region isessentially smaller than that in the other layers. The FFTanalysis of the S-T part shows the same peaks as for B-Spart, but here we can see an additional peak 0.43952	p=2.7615p, which value approximately equal to2LPW. As it is demonstrated in Fig. 5b, the character of thetI / IcQ1/Q0V/V020000 30000 40000 50000 600000.5550.560.5650.570.575-1-0.500.512323.52424.52525.526BSTY C1J(a) N9L1ωp tI / IcQ3/Q0V/V020000 30000 40000 50000 600000.5550.560.5650.570.575-1-0.500.512323.52424.52525.526BS TY C1J(b) N9L3ωptI / IcQ4/Q0V/V020000 30000 40000 50000 600000.5550.560.5650.570.575-1-0.500.512323.52424.52525.526BS TY C1J(c) N9L4ωp(b)(a)(c)FIG. 2. Color online a Profile of time dependence of the charge in the S-layers in the stack with nine IJJ: a In the first layer. b Inthe third layer. c In the fourth layer. The red curve shows the CVC of this stack in the BPR.I/IcV/Vo0.56 0.57293031α = 1β = 0.2Ν=11I/IcV/Vo0.56 0.562 0.56429.229.429.6FIG. 3. Color online Superposition of the three CVC with theamplitudes of noise in the current 10−6, 10−8, and 10−13.STRUCTURE OF THE BREAKPOINT REGION ON… PHYSICAL REVIEW B 78, 134521 2008134521-3charge oscillations in T-Y part of the BPR is different fromthat in the S-T part of the BPR. The charge oscillations arecharacterized here by the beating process. A specific featureof these part is the absence of the charge oscillations in thefourth layer. At point Y the charge oscillations in layer 4grow rapidly and are getting the same order as the values inthe other layers. Due to the coupling between junctions theoscillations in the other layers are increasing too, for ex-ample, similar to the third layer shown in Fig. 2b. The Y−C1 part of the BPR Fig. 5c is characterized by the beat-ing process as well but with an additional frequency in com-pare with the T-Y region. The C1−J part of the BPR pre-sented in Fig. 5d display a behavior close to the chaoticone. So, the different character of the charge oscillation inthe S layers is in correlation with the features of the CVCand reflects its nonlinear behavior in the BPR.The question concerning the origin of the BPR and thefine structure in the CVC might be simplified if we take intoaccount the results obtained for stacks with even number ofIJJ in Ref. 3. In that case the LPW with wavelength =2dk=	 is created. This wavelength of the LPW is commen-surable with the length of the stack in the c direction and theBPR in the CVC does not appear. For such stacks with evennumber of junctions the resonance region is observed only,which corresponds to the continuous increase in the ampli-tude of the charge oscillations and followed by jump to theanother branch.Let us now discuss shortly the experimental situation. Aswe mentioned in Ref. 3, the BPR exists in the numericalsimulations of the other authors as well. The points such asbreakpoint are presented in the experimental CVC of Refs. 9and 10 Recently, a new detailed experiment was made, andthe clear manifestation of the BPR structure was found.11 Itmakes promising the possibility to observe experimentallythe features presented in this paper. The following conditionsare important for that. The measurement should be done withhigh precision in bias current and slow sweeping of theCVC. Because the BPR region width is decreasing rapidlywith the number of junctions,3 it should not be too big toobserve the BPR structure. If the step in the bias current isbig enough, it makes possible to pass the bifurcation points12related to the parametric resonance in the system. The influ-ence of the inductive coupling on the BP and BPR is notinvestigated yet, but if the stack’s size is smaller than theJosephson penetration depth, we may neglect it.In summary, we predict a fine structure in the outermostbranch of the current-voltage characteristics in the layeredsuperconductors. We establish a correspondence between thefeatures of this structure and the character of the charge os-cillations in the different superconducting layers in the stack.We demonstrate that the different parts of the BPR corre-spond to the different dynamics of the charge oscillation inthe superconducting layers. Using an idea about parametricresonance at the breakpoint, we explain the origin of theBPR structure. A plasma mode is a stable solution of thesystem in the BPR, and this fact might be used in someapplications, particularly, in high-frequency devices such asterahertz oscillators and mixers. We consider as well that thebreakpoint phenomenon might be used to understand themechanism of the powerful coherent radiation from the stackof the IJJ in BSCCO.We thank T. Hikihara, I. Kakeya, A. Irie, H. J. Lee, K.Kadowaki, N. F. Pedersen, P. Seidel, T. Koyama, M.Machida, T. Hatano, and N. Plakida for helpful discussions.Yu.M.S. thanks J. Ishikawa and S. Noda for their kind hos-pitality during his stay at Kyoto University. This researchwas supported by Russian Foundation for Basic Researchunder Grant No. 08-02-00520-a.Amplitude(arb.units)0 0.5 1 1.5 200.20.40.4542(a)Before BP,I / Ic= 0.573ω/ 2πω pAmplitude(arb.units)0 0.5 1 1.5 200.050.10.15C1(d)- JI / Ic = 0.556ω/ 2πω pAmplitude(arb.units)0 0.5 1 1.5 200.0020.0040.22461.12790.6738(b)B - SI / Ic= 0.5702ω/ 2πω pAmplitude(arb.units)0 0.5 1 1.5 200.10.20.30.21970.65930.4395(c)S - TI / Ic= 0.5691.0940ω/ 2πω pFIG. 4. Color online Results of FFT analysis of time depen-dence of the charge in the third S layer in the different parts of BPR.tQ3/Q025340 25360 25380-0.4-0.200.20.4ωp(a) S-T, I / Ic =0.569tQ3/Q037500 37600 37700-0.500.5ωp(b) T -Y, I / Ic=0.565tQ3/Q043800 43900-0.500.5ωp(c) Y - C1, I / Ic=0.5625t60000 60100 60200ωpQ3/Q0-0.500.5(d) C1 - J, I / Ic=0.556FIG. 5. Color online Charge oscillations in the third S layer inthe stack with nine IJJ in different parts of the BPR specified by thevalue of I / IcSHUKRINOV, MAHFOUZI, AND SUZUKI PHYSICAL REVIEW B 78, 134521 2008134521-41 L. Ozyuzer, A. E. Koshelev, C. Kurter, N. Gopalsami, Q. Li, M.Tachiki, K. Kadowaki, T. Yamamoto, H. Minami, H. Yamaguchi,T. Tachiki, K. E. Gray, W.-K. Kwok, and U. Welp, Science 318,1291 2007.2 Yu. M. Shukrinov and F. Mahfouzi, Supercond. Sci. Technol. 20,S38 2007.3 Yu. M. Shukrinov, F. Mahfouzi, and N. F. Pedersen, Phys. Rev.B 75, 104508 2007.4 Yu. M. Shukrinov and F. Mahfouzi, Phys. Rev. Lett. 98, 1570012007.5 M. Machida, T. Koyama, A. Tanaka, and M. Tachiki, Physica C330, 85 2000.6 Yu. M. Shukrinov, F. Mahfouzi, and P. Seidel, Physica C 449, 622006.7 H. Matsumoto, S. Sakamoto, F. Wajima, T. Koyama, and M.Machida, Phys. Rev. B 60, 3666 1999.8 T. Koyama and M. Tachiki, Phys. Rev. B 54, 16183 1996.9 H. Ishida, K. Okanoue, and K. Hamasaki, Appl. Phys. Lett. 86,122503 2005; K. Okanoue and K. Hamasaki, ibid. 87, 2525062005.10 K. Tanabe, Y. Hidaka, S. Karimoto, and M. Suzuki, Phys. Rev. B53, 9348 1996.11 A. Irie, Yu. M. Shukrinov, and G. Oya, arXiv:0808.3896 unpub-lished.12 M. Machida, T. Koyama, and M. Tachiki, Phys. Rev. Lett. 83,4618 1999.STRUCTURE OF THE BREAKPOINT REGION ON… PHYSICAL REVIEW B 78, 134521 2008134521-5

Structure of the breakpoint region on current-voltage characteristics of intrinsic Josephson junctions

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Structure of the breakpoint region in CVC of the intrinsic Josephson junctions

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