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Physica C 367 (2002) 365±375

www.elsevier.com/locate/physc

Novel features of Josephson ¯ux ¯ow in Bi-2212: contribution of in-plane dissipation, coherent response to mm-wave radiation, size eect
Yu.I. Latyshev
b

a,b

, A.E. Koshelev c, V.N. Pavlenko T. Yamashita a, Yuji Matsuda

a,b,* , d

M.B. Gaifullin d,

RIEC, Tohoku University, Katahira, Aoba-ku, Sendai 980-8577, Japan Institute of Radio-Engineering and Electronics, Russian Academy of Sciences, 11-7 Mokhovaya Str., Moscow 101999, Russia c Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA d Institute for Solid State Physics, University of Tokyo, Kashiwanoha 5-1-5, Kashiwa, Chiba 277-8581, Japan

a

Abstract We studied Josephson ¯ux ¯ow (JFF) in Bi-2212 stacks fabricated from single crystal whiskers by focused ion beam technique. For long junctions with the in-plane sizes 30 á 2 lm2 , we found considerable contribution of the in-plane dissipation to the JFF resistivity, qJff , at low temperatures. According to recent theory [A. Koshelev, Phys. Rev. B 62 (2000) R3616] that results in quadratic type dependence of qJff B with the following saturation. The I±V characteristics in the JFF regime also can be described consistently by that theory. In the JFF regime we found Shapiro-step response to the external mm-wave radiation. The step position is proportional to the frequency of applied microwaves and corresponds to the Josephson emission from all the 60 intrinsic junctions of the stack being synchronized. That implies the coherence of the JFF over the whole thickness of the stack and demonstrates the possibility of synchronization of intrinsic junctions by the magnetic ®eld. We also found a threshold character of the appearance of the JFF branch on the I± V characteristic with the increase of magnetic ®eld, the threshold ®eld Bt being scaled with the junction size perpendicular to the ®eld L L 30±1:4 lm,as Bt % U0 =Ls, where s is the interlayer spacing. On the I±V characteristics of small stacks in the JFF regime we found Fiske-step features associated with resonance of Josephson radiation with the main resonance cavity mode in transmission line formed by stacks. ñ 2002 Elsevier Science B.V. All rights reserved.
PACS: 74.72.Hs; 74.60.Ge Keywords: Josephson ¯ux ¯ow; Josephson emission; Fiske steps

1. Introduction Studies of interlayer tunnelling in layered highTc materials associated with intrinsic Josephson eects [1,2] lately became one of the most interesting issues developed in high-Tc superconductivity. Existence of the intrinsic Josephson eects and the related Josephson plasma oscillations are now quite well documented [3]. Of a particular

* Corresponding author. Address: Institute of Radio-Engineering and Electronics, Russian Academy of Sciences, 11-7 Mokhovaya Str., Moscow 101999, Russia. Tel.: +7-095-2034976; fax: +7-095-203-8414. E-mail address: vit@iname.com (V.N. Pavlenko).

0921-4534/02/$ - see front matter ñ 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 1 0 3 4 - 6


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interest in this ®eld is dynamics of Josephson vortex lattice (JVL) in long stacked junctions. JVL is formed in magnetic ®eld parallel to the layers [4] and can be driven by dc current across the layers. Experimentally that was indicated as a resistive upturned ¯ux-¯ow branch in the I±V characteristics with maximum voltage being proportional to magnetic ®eld [5±7]. Maximum voltage corresponds to a condition, when the JVL velocity approaches the Swihart velocity [8], the velocity of electromagnetic wave propagation in the structure. As it was pointed out recently [9] for highly anisotropic materials, as Bi-2212, the ¯ux-¯ow resistivity at low temperatures can contain considerable contribution of the in-plane quasiparticle dissipation and thus can be used for extraction of the information about both quasiparticle conductivity components in superconducting state. This method still is not well elaborated. The coherent motion of JVL should induce coherent Josephson emission as it occurs in conventional long Josephson junctions [10]. The emission of that type can provide useful information about coherence of JVL motion. Therefore experiments on detection of Josephson emission in sliding JVL are very important. The interesting point in view of possible applications is also the short length limit for an observation of ¯ux-¯ow regime. The present paper has been addressed to clarify the most of questions listed above. We have undertaken the detailed studies of the I±V characteristics and ¯ux-¯ow resistivity in the sliding JVL regime. The data have been shown to be well consistent with recent theories of sliding JVL. We found Shapiro-step-type response of sliding JVL state to the external microwaves and Fiske-step features both pointing out to the existence of coherent Josephson emission in sliding JVL regime. The emission corresponds to the vertical synchronization of the major part or in some cases of all the number of elementary junctions in the stack. We also have studied ¯ux-¯ow behavior with a decrease of the stack length down to $1 lm. 2. Experimental The stacked structures have been fabricated by double-sided processing of high quality Bi-2212

whiskers [11] by focused ion beam. The stages of fabrications were similar to ones described in Ref. [12]. Fig. 1(a) shows schematically the geometry and orientation of the structure with respect to the crystallographic axes. The typical structure sizes L were La 1:5±30 lm, Lb 1±2 lm, Lc 0:05± 0:15 lm. The sizes and other parameters of the structures used in our studies are listed in Table 1. The oxygen doping level of the stacks estimated from qc T measurements above Tc [13] was nearly optimum, d % 0:25. The critical current density Jc at 4.2 K in the absence of magnetic ®eld was 1±2 kA/cm2 for the best samples. Measurements of the I±V characteristics of Bi2212 stacks have been carried out in commercial

Fig. 1. Schematic view of the long Bi-2212 stack in experimental setup (a) and the I±V characteristics of junction #4 in magnetic ®eld Bkb.


Yu.I. Latyshev et al. / Physica C 367 (2002) 365±375 Table 1 Parameters of Bi-2212 stacked junctions No. 4 2 7 5 H-1 H-3 6 8 La lm 28 7 3 2 1.5 1.5 30 28 Lb lm 2 2 2 2 1.5 1.5 2 2.5 N 90 84 81 80 90 80 57 75 Ic lA 550 300 140 18 0.6 0.3 1200 1500 Notes

367

Contains a hole D 0:2 lm Contains a hole D 0:2 lm

cryostat of Quantum Design PPMS facility. The magnetic ®eld has been oriented parallel to the baxis within accuracy 0.1°. Field has been changed in steps of 0.1 T. In each ®xed value of the ®eld the back and forth I±V characteristics have been measured using fast oscilloscope. For the microwave measurements we used more advanced setup, the cryostat with slit pair superconducting magnet in the shielded room. The accuracy of magnetic ®eld orientation Bkb was 0.01°. External microwaves with a maximum incident power of 35 mW at the frequencies ranging from 45 to 142 GHz were applied from the backward-wave oscillators. The samples were mounted on the substrate that was placed at the center of the rectangular waveguides and was capacitively coupled with the ¯ange of a waveguide. Thus the electrical component has been kept parallel to the c-axis.

Ref. [6]). We de®ne JFF resistance, RJff , as initial linear part of the ¯ux-¯ow branch. That can be ®nd from linear extrapolation of the I±V to V 3 0. This extrapolation gives more reliable values for high ®elds above 0.3±0.5 T, when Ic B becomes small. In the following two sections we consider the results of calculations of ¯ux-¯ow resistivity dependence on magnetic ®eld and the results of numerical calculations of ¯ux-¯ow branch on the I±V characteristics, both based on the recent advanced approach [9,14,15] that takes into account the in-plane dissipation channel. Note that in earlier calculations [16±18] this contribution has been ignored. Comparison with experiment presented below proved the importance of the in-plane dissipation channel in JFF dynamics in Bi-2212. 3.1. Flux-¯ow resistivity

3. Flux-¯ow resistivity and the I±V characteristics Fig. 1(b) shows a set of the I±V characteristics of a long stack #4 at T 20 K with subsequent increase of magnetic ®eld B oriented parallel to the b-axis. The ¯ux-¯ow branch and its evolution with magnetic ®eld are clearly seen in the picture. The ¯ux-¯ow branch appears just above Ic on the I±V characteristic. That is characterized by upturn and the maximum voltage, VJff , above which the switch to the multibranched state occurs. As known [5±7] VJff linearly grows with ®eld. We found the coecient dVJff =dB to be 0.42 mV/T per elementary junction. Above approximately 1.7 T the slope of VJff B gradually increases to 0.86 mV/T (see also

The linear ¯ux-¯ow resistivity of the JVL, qJff , is determined by the static lattice structure and linear quasiparticle dissipation. At high ®elds, B > U0 =pcs2 , Josephson vortices homogeneously ®ll all the layers and the static lattice structure is characterized by oscillating patterns of both c-axis and in-plane supercurrents. At small velocities this pattern slowly drifts along the direction of layers, preserving its static structure. This motion pro~ ~ duces oscillating hEz i and in-plane c-axis hEx i electric ®elds leading to extra dissipation, in addition to usual dissipation due to the dc electric ®eld Ez . Total dissipation per unit volume is given by
2 2 ~2 ~2 rJff Ez rc Ez rc hEz i rab hEx i;

1


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where rJff 1=qJff is the ¯ux-¯ow conductivity, hà à ài means time and space average, rc 1=qc and rab 1=qab are the c-axis and in-plane quasiparticle conductivities. An expansion with respect to Josephson current at high ®elds allows to derive a simple analytical formula for the ¯ux-¯ow resistivity [9,14,15]: r B2 rab U0 p qJff B 2 qc ; Br : B B2 rc 2pc2 s2 r 2 A relative importance of the in-plane and c-axis dissipation channels is determined by the dimensionless ratio rab =rc c2 . In high-Tc superconductors at low temperatures, T < 60 K, this ratio is large rab =rc c2 20±50, i.e., the in-plane channel dominates. Eq. (2) describes very well the experimental ®eld dependence of the ¯ux-¯ow resistivity (Fig. 2). Fit of the experimental dependence qJff T by Eq. (2) at 4.2 K gives qc % 480 X cm and Br 3:3 T. This allows to extract the combination

rab =rc c4 % 0:017. Using for critical current density at low temperatures the value 1.7 kA/cm2 we can estimate c as c 500. That gives an estimate for rab 4:2 K % 40 (mX cm)þ1 which is quite close to the value rab % 60 (mX cm)þ1 found at low temperatures from the microwave experiments [19]. Fig. 2 shows that experimental dependence RJff B is well described by the theory in wide temperature region 4.2±60 K. From the ®t of experimental curves to Eq. (2) we can extract the temperature dependence of Br and rc (see insert to Fig. 2). As shown in insert, a dependence rc T found in this way well agrees with dependence for rc T extracted from the independent experiment on small mesas in zero magnetic ®eld [20]. To extract the temperature dependence of rab we need additional knowledge of the temperature dependence of c. In principle that can be extracted from the temperature dependence of linear part of RJff B dependencies, but unfortunately the low ®eld data of RJff B are not so reliable. To demonstrate the importance of in-plane contribution in ¯ux-¯ow resistivity we show in Fig. 2 two theoretical dependencies at 4.2 K, one is ®tted to the experiment with ®nite rab and another one is calculated with the same ®tting parameters, but with zero in-plane dissipation contribution (rab 0). One can see huge inconsistency with experiment in the latter case. 3.2. Flux-¯ow branch At high ®elds in the resistive state the interlayer phase dierences grow approximately linearly in space and time hn t; x % xE t kH x /n ; 3 where xE is the Josephson frequency and kH is magnetic wave vector. In the following we will use reduced parameters: x 3 xE =xp , kH 3 kH cs (see Table 2). The most important degrees of freedom in this state are the phase shifts /n , which describe the structure of the moving JVL. In particular, for the static triangular lattice /n pn. Lattice structure experiences a nontrivial evolution with increase of velocity. The equations for /n can be derived from the coupled sine-Gordon equations for hn t; x by expansion with respect to the

1=2

Fig. 2. Magnetic ®eld dependence of the JFF resistance RJff for long Bi-2212 stack #4 at 4.2, 20, 40, 60, and 75 K. The dashed lines are ®ts to Eq. (2) for each temperature. The dotted curve is a calculated dependence RJff B at 4.2 K in a limit of zero inplane dissipation rab 0. The inset shows temperature dependence of rc : crosses obtained from the ®t of RJff B to Eq. (2), the line corresponds to data obtained from measurements on small mesas in zero magnetic ®eld [20].


Yu.I. Latyshev et al. / Physica C 367 (2002) 365±375 Table 2 Meanings, de®nitions and practical formulas for the reduced parameters used throughout the paper Notation x
E

369

Meaning Reduced Josephson frequency

De®nition (CGS) 2pcsEz U0 xp 2pH cs2 U0 4prc ec xp 4prab k2 xp ab c2 kab =s
ab

Practical formula (BSCCO) U mV=junction 2 á 10þ3 fp GHz

k

H

Magnetic wave vector

mc

c-Axis dissipation parameter

1:8 á 103 ec qc X cmfp GHz 0:79kab lm2 fp GHz qab lX cm

mab

In-plane dissipation parameter

l

Reduced London penetration depth

In practical formulas f xp =2p means plasma frequency, qc and q

are the components of the quasiparticle resistivity.

Josephson current and averaging out fast degrees of freedom. In the case of steady state for a stack consisting of N junctions, this gives the following set of equations:
N 1X Imgn; m expi/m þ /n iJ ; 2 m1

4

where iJ iJ kH ; xE hsin hn t; xi is the reduced Josephson current, which has to be obtained as solution of these equations. The complex function gn; m describes phase oscillations in the mth layer excited by oscillating Josephson current in the nth layer. For a ®nite system it consists of the bulk term Gn þ m and top and bottom re¯ections (multiple re¯ections can be neglected): gn; m Gn þ m BGn m BG2N 2 þ n þ m; where Gn Z dq expiqn x2 þ imc x 2p k 2 1 imab x þ 21 þ cos q1 imab x=l
2

amplitude of re¯ected electromagnetic wave. For the practical case of the boundary between the static and moving Josephson lattices a detailed calculation of Bk ; x is presented in Ref. [15]. In general, quasiparticle conductivities in de®nitions of mc and mab are the complex conductivities at the Josephson frequency. The frequency dependence is especially important for the in-plane conductivity. Recent terahertz spectroscopy measurements of rab x in BSCCO by Corson et al. [21] showed at low temperatures it has Drude frequency dependence with typical relaxation rate 1=s % 1 THz. Maximum Josephson frequency at the termination point of the ¯ux-¯ow branch exceeds this value at ®elds P 2 T. Therefore the frequency dependence has to be taken into account. We use the Drudelike frequency dependence of mab mab x: mab x mab0 ; 1 ixs 6

þ

where s is the quasiparticle relaxation time. Solution of Eq. (4) allows to obtain I±V characteristic as
1

; 5

jEz rc Ez jJ iJ kH ; xE ; where kH and xE has to be expressed via magnetic and electric ®elds (see Table 2). We solved Eq. (4) numerically and calculated the I±V dependencies for the ®rst ¯ux-¯ow branch. We used rc and rab =c4 obtained from the ®t of qJff B, assumed kab 200 nm, and adjusted c to

x xE and k kH are the frequency and the inplane wave vector of the travelling electromagnetic wave generated by moving lattice. Dissipation parameters, mc and mab , and reduced penetration depth l are de®ned in Table 2. B Bk ; x is the


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Table 3 Parameters of BSSCO used in calculation of the I±V dependencies Penetration depth kab (nm) 200 Anisotropy c 500 Josephson current jJ (kA/cm2 ) 1.7 In-plane conductivity rab (mX cmþ1 ) 39 c-Axis conductivity rc (kX cmþ1 ) 2.1 Relaxation rate 1=s (THz) 2p á 0:55

obtain the I±V dependencies most close to experimental ones. At high ®elds we found the ®t can be signi®cantly improved by taking into account frequency dependence of rab via Eq. (6). Obtained parameters are summarized in Table 3. Typically, far away from the boundaries and the center, the solution has the form of a regular lattice, /n1 þ /n j, where j slowly decreases with xE starting from j p at xE 0. Two lattice solutions, selected by the top and bottom boundaries, collide at the center forming defect region (shock). The ®rst ¯ux-¯ow branch terminates when jxE intersects the instability boundary in the j þ xE plane. This corresponds to the maximum in the jEz dependence and happens when xE is somewhat smaller than the minimum frequency of the electromagnetic wave xmin k at ®xed in-plane wave vector k kH . In reduced units xmin k k =2. For our parameters this corresponds to the voltage Vmin H =H % 0:53 mV/junction/T. Structure of the lattice at the instability point is shown in the insert to Fig. 3. Again, as in previous section, we show for contrast a result of the I±V calculation with zero in-plane dissipation, rab 0 (the dashed curve in Fig. 3). One can see a high disagreement with experiment in that case. Theoretical curve is running far below the experimental and the resonance upturn near Vmax is much sharper than in the experiment.

Fig. 3. Comparison of calculated and measured the I±V characteristic at B 2 T. For calculation parameters see text. The inset shows vortex lattice structure near the V Vmax .

4. Shapiro-step response of Josephson ¯ux ¯ow to external microwaves As it was predicted theoretically [8] the coherent sliding of dense lattice of Josephson lattice in layered high-Tc materials should be accompanied by coherent microwave radiation with a frequency proportional to the lattice velocity. Alternatively, one can expect to ®nd coherent DC response of

Shapiro-step type on the I±V characteristics at the JFF regime under microwave radiation. Despite many studies of JFF regime in layered high-Tc materials [5±7], until recently there were practically no experiments on detection of coherent emission or Shapiro steps corresponding to sliding of JVL. Here we report on the observation of Shapirostep-type response on ¯ux-¯ow branch of the I±V characteristics of Bi-2212 ``long'' stacks under microwaves. The I±V characteristics of samples #6 and #8 used for microwave experiments were typical to the Bi-2212 stacks with the multibranched structure [1] and critical current density Jc 1±2 kA/ cm2 at 4.2 K. At ®elds B > 0:03 T we clearly observed ¯ux-¯ow branch on the I±V characteristic. The properties of the ¯ux-¯ow branch were similar to described in previous sections. We found that microwave irradiation induces Shapiro-step structure on the ¯ux-¯ow branch. That is well resolved as a series of peaks on the derivative picture


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371

Fig. 4. The dierential conductance of the stack #6 in ¯ux-¯ow regime (B 2:4 T, Bkb) as a function of microwave power of frequency 90.4 GHz, T 7:4 K. Conductance scale corresponds to the lowest at 8.5 dBm, the other spectra are oset vertically.

dI =dV V (Fig. 4). With an increase of microwave i power P the voltage position of Shapiro steps Vst , does not depend on P and is only a function of microwave frequency f in accordance with the i modi®ed Josephson relation: Vst iNhf =2e, with i p=q; p, q are integers, N the number of the synchronized elementary junctions in the stack (Fig. 5). For one of two samples studied, #6, we found that N (N 57) exactly corresponds to the whole number of the junctions in the stack (N 60 ô 3). As shown in Fig. 4 the amplitudes of both, Shapiro steps DIc and critical current Ic are oscillatory functions of microwave power, the oscillations of the second step and Ic peaks being inphase, whereas oscillations of the ®rst step and Ic being in anti-phase. That resembles behavior of Shapiro steps and critical current for conventional tunnel Josephson junctions [22]. We found also that Shapiro steps are observed at ®elds B above 1 T corresponding to the conditions of dense lattice. However, the step position at ®xed microwave frequency does not depend on B B 1±7 T or temperature up to 75 K. The presence of Shapiro steps on ¯ux-¯ow branch of the I±V characteristics proves the existence of the coherent Josephson emission induced by sliding JVL. The Josephson-type relation between the voltage position of the step and external frequency points out to the in-plane coherency

Fig. 5. 1 tion, Vst Bi-2212 straight

Frequency dependence of the ®rst Shapiro-step posi, as a function of the frequency of microwave ®eld for stacked junction #6, T 7:4 K, B 2:4 T, Bkb. The line corresponds to the Josephson relation for N 57.

of moving JVL, while the big coecient N in Josephson relation proves the vertical coherence of the sliding JVL. Our results demonstrate the principal possibility of synchronization of elementary junctions by magnetic ®eld. Because of the vertical synchronization of many junctions one can expect to get quite high power of the microwave emission. Our rough estimation of the emitted microwave power W at f 120 GHz as W $ DIm Vst with DIm the maximum step height gives the value $10þ6 W.

5. Josephson ¯ux-¯ow regime in short stacks To investigate the JFF regime in a short length limit we fabricated a set of stacks of various lengths ranged from 30 down to 1.5 lm keeping the width of all the stacks the same, close to 2 lm.


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We found that ¯ux-¯ow regime can be achieved for all stacks including the smallest one but at different scale of magnetic ®eld. The general feature found for the stacks of all length was the existence of threshold magnetic ®eld Bt for the ¯ux-¯ow regime. The ¯ux-¯ow branch appears on the I±V characteristic only at ®elds exceeding Bt . Bt was found to increase with sample inverse length. For instance, for sample #4 with L 28 lm Bt was 0.03 T while for the stack #7 with L 3 lm Bt 0:3 T. The log±log scale dependence of threshold ®eld on the stack inverse length is shown in Fig. 6(a). That is very close to the linear dependence of Bt Lþ1 and practically does not depend on temperature at least at the interval 4.2±40 K. As it follows from Fig. 6(a) the found threshold ®eld is close to the characteristic ®eld B0 , B0 U0 =Ls, corresponding to the condition of formation of the dense JVL (dashed line in Fig. 6(a). The threshold ®eld Bt is found to scale with B0 as Bt 0:63B0 . The ¯ux-¯ow branch for short samples was rather linear with slight upturn. The maximum voltage of ¯ux-¯ow branch was generally less than for long stacks but asymptotically approached that for higher ®elds (Fig. 6(b)). For stacks shorter than 5 lm we often observed sharp kinks on the ¯ux-¯ow branch of the I±V characteristics (Fig. 7). The voltage position of the kink, Vk , is ®eld independent and for #5 is equal to %15 mV. Sometimes we observed also second harmonic of the kink at double voltage 2Vk . As shown in Fig. 7 the current amplitude of the kink DIk1 is oscillatory function of parallel magnetic ®eld. The oscillations of DIk1 B are in anti-phase with the Fraunhofer oscillations of critical current (Fig. 8), while DIk2 B oscillates in phase with Ic B. To discuss the threshold behavior of ¯ux-¯ow we analyzed the value and size dependence of the ®rst critical ®eld Bc1 for ¯uxon appearance in the stack. As it was pointed out in Ref. [23] Bc1 increases with a decrease of sample size L as: Bc1 % U0 =L2 kc =kab . We plotted that dependence in Fig. 6(a) using the parameter: c kc =kab 500. The dependence Bc1 Lþ1 is shown to lie below experimental points for Bt Lþ1 . That means that at ®elds Bc1 < B < Bt the ¯uxons exist in the stack but do not contribute to the ¯ux ¯ow. We can

Fig. 6. (a) Experimental dependence of threshold magnetic ®eld for appearance of the ¯ux-¯ow branch on the I±V characteristics as a function of the inverse length of the stack. The dashed line corresponds to condition B U0 =Ls. Solid line corresponds to ®eld Bc1 : Bc1 U0 =L2 kc =kab , (b) magnetic ®eld dependence of maximum JFF voltage for stacks of dierent length.

consider that at those ®elds corresponding to dilute vortex lattice the pinning force is higher than the driving force up to the critical Josephson current across the layers. With ®eld increase the lattice achieves dense limit and becomes rigid enough for collective motion. The increase of transverse rigidity leads to the eective reduction of pinning force, since the pinning force acting on individual ¯uxon will be eectively redistributed over the whole lattice. Thus we consider Bt as a threshold ®eld when the driving force exceeds the pinning force acting on JVL. That turns out to be very near


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373

Fig. 8. current H kb, T Ic =Ic0

( )Magnetic ®eld dependence of normalized critical Ic =Ic0 and ( ) amplitude of the ®rst Fiske step DI =Ic0 . 4:2 K. Solid line is a ®t to Fraunhofer dependence sin x=x with x pBLs=U0 , L 2 lm.

We can identify the kinks on the I±V characteristics of short stacks in JFF regime as the Fiske steps [24], which are known to appear due to the resonance of Josephson radiation with cavity modes in a transmission line formed by the junction. The characteristic features of the Fiske steps are as follows: 1. The step position follows the condition Vm mN U0 c0 =2L with m the integer, N the number of synchronized elementary junctions in the stack, c0 the Swihart velocity, L the stack length. 2. Voltage position of the step is independent of B. 3. Current amplitude of the steps, DIm , is periodic function of B with periodicity DB U0 =Ls, even steps oscillating in-phase with Ic , while odd step oscillations being in anti-phase with Ic B [25]. The conditions (2±3) are valid for our kink structure (see Figs. 7 and 8). The estimation of voltage position also gives reasonable value Vk1 14:2 mV. For estimation we used the following parameters L 2 lm, N 80, kab 200 nm, s 1:5 nm, c0 =c 1:19 á 10þ3 . We estimated Swihart

Fig. 7. A set of the I±V characteristics of a short Bi-2212 stack #5 with variation of parallel magnetic ®eld Bk within 3.7±7.5 T, T 4:2 K. Note a Fiske step at V % 15 mV.

to the condition of formation of dense lattice and corresponds to the transverse ¯uxon density 0.63 ¯uxon per junction.


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velocity as the lowest mode valid for triangular 1=2 lattice as [8] c0 sc=2kab ec with ec 10. Note that the resonance frequency, c0 =2L, corresponding to the ®rst Fiske step is quite high in our case %100 GHz. We observed Fiske steps on even shorter attacks with L 1:4 lm. The ®rst indication of Fiske steps in Bi-2212 stacked junctions has been found in Refs. [26,27] on rather long junctions with L 20±50 lm. The steps found [27] were unstable and not well reproducible. We consider that the better conditions of Fiske-step observation at short stacks are related with an increase of the resonance quality factor with decreasing L.

Acknowledgements We are thankful to A.M. Nikitina for providing us with Bi-2212 single crystal whiskers and S.-J. Kim for technical assistance. This work was supported by Japan Science and Technology Corp. and by Russian State Program on HTS (grant no. 99016).

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6. Conclusion We carried out the detailed measurements of I±V characteristics in long Bi-2212 stacks in Josephson ¯ux-¯ow regime and analyzed the data in the frame of recent theory taking into account both in-plane and out-of-plane channels for dissipation in JFF regime. We found that the experiment is well described by that theory. From the ®t we found a number of Bi-2212 parameters at T 4:2 K as rc , rab and Br as well as the temperature dependence of rc , which agree well with those found by other methods. From that we can conclude that in-plane dissipation plays an important role in JFF regime at low temperatures. By the experiments with external microwave radiation (45±142 GHz) we found a coherent Shapiro-step resonance on the I±V characteristic in JFF regime. That gives a strong evidence of the Josephson emission induced by coherently moving JVL. For short stacks with length 1.5±2 lm we found the Fiske-step resonance which appears due to the resonance of Josephson radiation with cavity modes in transmission line formed by the stack. As in the case of Shapiro step response the Fiske-step position corresponds to the synchronization of all the 80 junction in the stack. We also found that for stacks of dierent length the ¯ux-¯ow branch appears above some threshold ®eld Bt 0:63U0 =Ls.


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