Magnetization relaxation, critical current density and vortex dynamics in a Ba 0.66 K 0.32 BiO 3+δ single crystal
Jian Tao, Qiang Deng, Huan Yang, Zhihe Wang, Xiyu Zhu, Hai-Hu Wen
aa r X i v : . [ c ond - m a t . s up r- c on ] M a y Magnetization relaxation, critical current density and vortex dynamics in aBa . K . BiO δ single crystal Jian Tao, Qiang Deng, Huan Yang, Zhihe Wang, Xiyu Zhu, and Hai-Hu Wen ∗ National Laboratory of Solid State Microstructures and Department of Physics,Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China (Dated: September 10, 2018)We have conducted extensive investigations on the magnetization and its dynamical relaxation ona Ba . K . BiO δ single crystal. It is found that the magnetization relaxation rate is rather weakcompared with that in the cuprate superconductors, indicating a higher collective vortex pinningpotential (or activation energy), although the intrinsic pinning potential U c is weaker. Detailed anal-ysis leads to the following discoveries: (1) A second-peak effect on the magnetization-hysteresis-loopwas observed in a very wide temperature region, ranging from 2K to 24K. Its general behavior lookslike that in YBa Cu O ; (2) Associated with the second peak effect, the magnetization relaxationrate is inversely related to the transient superconducting current density J s revealing a quite generaland similar mechanism for the second peak effect in many high temperature superconductors; (3) Adetailed analysis based on the collective creep model reveals a large glassy exponent µ and a smallintrinsic pinning potential U c ; (4) Investigation on the volume pinning force density shows that thedata can be scaled to the formula F p ∝ b p (1 − b ) q with p = 2 .
79 and q = 3 .
14, here b is the reducedmagnetic field to the irreversible magnetic field. The maximum normalized pinning force densityappears near b ≈ .
47. Finally, a vortex phase diagram is drawn for showing the phase transitionsor crossovers between different vortex phases.
PACS numbers: 74.70.-b, 74.25.Ha, 74.25.Wx, 74.25.Uv
I. INTRODUCTION
Investigation on vortex physics is very important con-cerning the potential high-power applications of a su-perconductor. In the cuprate superconductors, due tothe very high superconducting transition temperature,layered structure, short coherence length, strong ther-mal fluctuation etc., the vortex physics is extremelyrich, which has led to the unprecedented prosperousdevelopment on the vortex physics . Many new con-cepts and phenomena, such as collective vortex creep ,vortex glass , first order vortex transitions , vor-tex melting , second peak effect of magnetization etc. have been proposed or discovered. In the ironbased superconductors, the vortex physics looks quitesimilar to the cuprate although the anisotropyof ξ ab /ξ c is only about 2-5 which is much smallerthan that in the cuprate system . Preliminary ex-perimental studies have revealed that the vortex dy-namics in iron pnictide may be understood with themodel of thermally activated flux motion within thescenario of collective vortex pinning . A second-peak (SP) effect on the magnetization-hysteresis-loop(MHL) has also been observed in Ba − x K x Fe As andBa(Fe − x Co x ) As single crystals . Beside thethree typical high temperature superconductors, namelythe cuprate, MgB and the iron based superconductors,the Ba − x K x BiO (hereafter abbreviated as BKBO)superconductor is quite special in terms of its relativelyhigh transition temperature (The highest T c can reachabout 34 K ) and almost three-dimensional feature .In the BKBO superconductors, the coherence length de-tected from scanning tunneling microscope (STM) and other measurements is about 3-7 nm . The structuralcharacteristics , the coherence length, the Ginzburg-Landau parameter κ = λ L /ξ seem to be very dif-ferent from those in the cuprate superconductors .These peculiarities may bring about new ingredients tous in understanding the vortex physics in high temper-ature superconductors. Therefore we have grown theBa − x K x BiO single crystals and investigated the vortexphysics extensively with measurements of magnetizationand its dynamical relaxation. II. EXPERIMENT
The single crystals with high quality studied in thiswork were prepared by the molten salt electrochemi-cal method presented previously . In the process ofelectrochemical growth, the working electrode was madefrom 0.5 mm-diameter platinum wire (Alfa Aesar, 4N),and the working current was 2 mA. In addition, we placed43 g of KOH (J&K Chemical Ltd.) in a 100 cm Telfoncontainer, and heat it up to 250 ◦ C , staying for severalhours until KOH was completely melted, then added 1.49g of Ba(OH) · O (J&K Chemical Ltd., 2N) and 3.22g of Bi O to the molten KOH solution, the growth be-gins after stirring the solution for almost two hours. Inthis way, the crystals can be successfully obtained withthe size up to several millimeters if the growing time islong enough. The best growth time in the experiment isaround 48 hours. Inset of Fig. 1 shows the photograph ofsamples we synthesized through the electrochemical re-action method. By the way, all the samples we measuredwere polished to a proper thickness in order to guarantee FIG. 1: (Color online) X-ray diffraction pattern of a crystalwith composition Ba . K . BiO δ . The inset shows thephotograph of BKBO crystals grown by the electrochemicalmethod. the homogeneity. The lattice structure of the sample wascharacterized by x-ray diffraction (XRD) at room tem-perature with a Bruck-D8-type diffractometer. The XRDpattern of a sample is shown in Fig. 1, the vast value ofthe intensity of the ( l
00) indices from the XRD patterndemonstrates the a -axis orientation of the single crys-tal. The a -axis lattice constant is 4.2995˚A through cal-culating the indexed peaks. The sample composition wasanalyzed by using the energy dispersive x-ray spectrom-eter (EDX/EDS). We concluded that the composition ofmeasured sample is about Ba . K . BiO δ , where theoxygen content cannot be accurately determined.The electric transport and magnetization measurementwere performed by a physical property measurement sys-tem (PPMS, Quantum Design) and SQUID vibratingsample magnetometer (SQUID-VSM, Quantum Design),respectively. Fig. 2 (a) and (b) show the temperature de-pendence of resistivity for the crystal Ba . K . BiO δ under different magnetic fields ranging from 0 T to 9 T.The onset transition temperature at zero field is about27 K by taking a criterion of 90% ρ n , here ρ n is the nor-mal state resistivity. The diamagnetic moment of thesample is shown in Fig. 2(c) which was measured inthe zero-field-cooled (ZFC) and field-cooled (FC) modeunder a DC magnetic field of 20 Oe. The ZFC curve dis-plays a transition with an onset temperature around 26.5K. The transition temperature of the present sample isin agreement with the phase diagram reported by othergroup . From the results of XRD, resistivity and dia-magnetic measurements, the quality of the sample hasbeen proved to be good enough to do further study ofthe vortex dynamics. In Fig. 3 we show the MHL curveswith a magnetic field sweeping rate dB/dt of 200 Oe/sand 50 Oe/s at different temperatures ranging from 2 Kto 24 K (the magnetic field is vertical to the ab plane ofthe sample). The symmetric MHL curves demonstratethat the measured sample is bulk superconductive and (b) T (K) M ( e m u / g ) H =
20 Oe
FC ZFC
22 24 26 28-101234567 H = 0 T H = 0.1 T H = 0.2 T (c) ( m c m ) T (K) H = 0.4 T H = 0.6 T H = 0.8 T
30 60 90-1012345678
T (K) ( m c m ) H = 0 T H = 1 T H = 3 T H = 5 T H = 7 T H = 9 T (a)
FIG. 2: (Color online)(a) Temperature dependence of resis-tivity at different magnetic fields ranging from 0 T to 9 T. (b)Temperature dependence of resistivity at different magneticfields ranging from 0 T to 0.8 T. (c) Temperature dependenceof the diamagnetic moment measured in processes of ZFC andFC at a magnetic field of 20 Oe. the vortex pinning is bulk in nature. Undoubtedly, thedia-magnetization here is not due to the surface shieldingeffect, and the Bean critical state model can be used.
III. MODELS AND ANALYSIS METHODA. Thermally activated flux motion and collectiveflux creep
To fully understand the vortex motion in the BKBOsingle crystal, we start from the model of thermally acti-vated flux motion : E = v B exp( − U ( J s , T, B e ) k B T ) . (1)Here E is the electric field induced by the vortex mo-tion, v is the attempting moving velocity of the hop-ping vortex lines, U ( J s , T, B e ) is the effective activationenergy, and B e is the external magnetic field, B is thelocal averaged magnetic induction. Based on the vortexglass and the collective pinning models , it is predictedthat U ( J s , T, B e ) is positively related to [ J c ( T, B e ) /J s ] µ ,where µ is the glassy exponent describing the activationenergy. In order to ensure that U ( J s , T, B e ) reaches zerowhen the external applied current J s approaches the crit-ical current J c , then Malozemoff proposed to rewrite theactivation energy in a very general way as U ( J s , T, B e ) = U c ( T, B e ) µ ( T, B e ) [( J c ( T, B e ) J s ( T, B e ) ) µ ( T,B e ) −
1] (2) -4 -3 -2 -1 0 1 2 3 4-20-15-10-505101520
T = 18 K
T = 20 K
T = 22 K
T = 23 K T = 24 K(b) dH/dt = 50 Oe/s dH/dt = 200 Oe/s M ( e m u / c m ) H (T) -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7-150-100-50050100150 T = 6 K
T = 8 K
T = 2 K
T = 4 K
T = 10 K
T = 15 K H (T) dH/dt = 50 Oe/s dH/dt = 200 Oe/s M ( e m u / c m ) (a) FIG. 3: (Color online) Magnetization hysteresis loops of theBKBO single crystal at various temperatures ranging from 2K to 15 K (a), 18 K to 24 K (b). The solid lines stand forthe magnetization measured with field sweeping rate of 200Oe/s while the dash lines represent those measured with 50Oe/s. We need to note that the magnetization at 4 K in thefield ascending process with 50 Oe/s experienced a small fluxjump at a low field. This flux jump keeps giving influence onthe MHL at high field on this curve. For calculating the ∆ M , J s and Q for 4K, we used the data in the left-hand side andsecond quadrant with negative magnetic field at 4K. where U c and J c are the characteristic (or called as the in-trinsic) pinning energy and initial critical current density(unrelaxed), respectively. The glassy exponent µ givesdifferent values for different regimes of flux creep. Fromthe elastic manifold theory , it is predicted that µ = 1 / U ( J s ). For instance, when µ = −
1, it will go back to theKim-Anderson model , and µ = 0 corresponds to theZeldov’s logarithmic model . Furthermore, when the J c is much larger than J s , Eq. (2) will return to the formof collective pinning models. Therefore, the value of µ will play a significant role in understanding the vortexmotion. B. Models for analyzing the magnetizationrelaxation
For the sake of discussion, we will calculate the tran-sient current density J s from the width ∆ M of MHLs,where ∆ M = M − − M + with M − ( M + ) the magne-tization at a certain magnetic field in the increasing(decreasing)-field process. According to the Bean criti-cal state model , the transient superconducting currentdensity J s can be expressed as J s = 20 ∆ Mw (1 − w l ) , (3)where the unit of ∆ M is emu/cm , w , l are the widthand length of the sample measured in cm ( w < l ), re-spectively. In this work, we utilized the dynamical re-laxation method to study the vortex dynamics, insteadof using the conventional relaxation method . Thecorresponding physical quantity is the magnetization-relaxation rate Q which is defined as: Q ≡ d ln J s d ln( dB/dt ) = d ln(∆ M )d ln(d B/ d t ) . (4)The dynamical relaxation measurements are followed inthis way: the sample is cooled down to a certain temper-ature at ambient magnetic field, and then we measure theMHL curves with two different magnetic field sweepingrates.From the general formulas Eqs. (1) and (2) mentionedabove and the definition of Q , we will employ the fol-lowing expression to quantify the characteristic pinningenergy as derived by Wen et al. TQ ( T, B e ) = U c ( T, B e ) k B + µ ( T, B e ) CT, (5)here C = ln(2 ν B/ ( ldB e /dt )) is a parameter which isweakly temperature dependent, l is the lateral dimen-sion of the sample. According to Eq.(5), in the lowtemperature region, the term of µ ( T, B e ) CT is muchsmaller than the term of U c ( T, B e ) /k B , so we could ig-nore µ ( T, B e ) CT , and get T /Q ( T, B e ) ≈ U c ( T, B e ) /k B ,and Q should show a linear dependence on T . However,as we will show below, in BKBO, the term U c is not big,but the glassy exponent µ is sizeable, therefore the secondterm becomes quite important. In the low temperaturelimit, we could use this approximation to extrapolate thecurve T /Q down to zero temperature, the intercept gives U c (0) /k B . The relaxation rate Q is related to the bal-ance of the two terms U c and µCT and shows a complextemperature dependent behavior.
10 K8 K6 K4 K M ( e m u / c m ) H (T) FIG. 4: (Color online) Field dependence of ∆ M with differentfield sweeping rates at various temperatures ranging from 2K to 10 K. J s ( A / c m ) dH/dt = 50 Oe/s T = 2 K
T = 4 K
T = 6 K
T = 8 K
T = 10 K (a)
24 K 22 K 20 K
18 K H (T) J s ( A / c m ) dH/dt = 50 Oe/s (b)
15 K
FIG. 5: (Color online) (a) Magnetic field dependence of thecalculated transient superconducting current density J s basedon the Bean critical state model at temperatures ranging from2 K to 10 K. (b) Magnetic field dependence of the calculatedtransient superconducting current density based on the Beancritical state model at temperatures ranging from 15 K to 24K. IV. RESULTS AND DISCUSSIONA. Transient superconducting current density andsecond peak effect
In Fig. 4, we show the field dependent ∆ M with differ-ent field sweeping rates of 200 Oe/s and 50 Oe/s respec-tively at different temperatures ranging from 2 K to 10K. Fig. 5 shows the field dependence of J s with the mag-netic field sweeping rate of 50 Oe/s by using Eq. (3). Thecalculated J s at 2 K at zero magnetic field can reach upto 10 A/cm , which is an order of magnitude larger thanthe values reported in literatures , while the J s is muchsmaller than the cuprate and iron-based superconduc-tors. As presented in Fig. 5(b), in the high temperatureregion J s decreases greatly due to severe flux motion.From the MHL curves in Fig. 3 and J s - B curve inFig. 5, we can observe second peak (SP) effect (fish-taileffect) in a very wide temperature region ranging from 2K to 22 K. The second peak is relative to the first onenear zero magnetic field in MHL or J s - B curve, and themagnetic field of the second peak position is defined as H sp which is dependent on temperature. We can see thisclearly from Fig. 3 that the peak position of SP moves to-ward lower magnetic field as the temperature rises. Thisfeature was also observed in the cuprate superconduc-tors, such as YBa Cu O (YBCO) , as well as in theiron-based superconductors . The second peak effecthas been intensively studied previously, and several pos-sible mechanisms have been proposed. These include (1)Inhomogeneities in the sample, such as nonuniform oxy-gen distribution in the cuprates, which generates oxygen-deficient regions acting as extra pinning centers in highmagnetic field , and thus enhances the superconductingcurrent density; (2) A crossover from fast relaxation toslow relaxation, the transition occurs between the sin-gle vortex regime and the collective pinning and creepregimes with slower relaxation at sufficiently high mag-netic fields ; (3) A crossover in flux dynamics from elas-tic to plastic vortex creep . It is well-known that theSP effect in YBCO and Bi Sr CaCu O (Bi2212) sam-ples exhibits in different ways. In Bi2212, the SP field islow (usually a few hundreds of oersted) and weakly tem-perature dependent, but SP in YBCO occurs at a highmagnetic field (usually a few or more than ten Tesla) andis strongly temperature dependent. Our present resultsin BKBO show that it is more like the SP in YBCO. Thisprobably suggests that the SP in BKBO and YBCO isdue to the similar reasons. One of possible explanationsis that both systems are more three dimensional and con-taining oxygen vacancies. The oxygen content may notbe very uniform leading to many local random pinningcenters. B. Magnetization relaxation and its correlationwith J s It can be clearly seen from Fig. 4 that the MHLcurves demonstrate a difference in magnitude with dif-ferent field sweeping rates at a certain temperature. Thelarger the field sweeping rate is, the bigger of the MHLwidth will be. As we know the magnitude of the dif-ference between the MHL curves measured at differentfield sweeping rates reflects how strong the vortex creepis. In the cuprate high temperature superconductors, itwas found that the separation between different MHLs isquite large . At high temperatures, however, the magni-tude of the diamagnetic moment greatly decreases withincreasing magnetic field. Based on Eq. (4) for treat-ing the magnetization relaxation, we calculated the mag-netic field and temperature dependence of the dynamicalrelaxation rate Q from the data shown in Fig. 4. Aspresented in Fig. 6(a), it can be seen that Q decreasesfirst and then increases with increasing magnetic field attemperatures of 2 K and 10 K, showing a minimum of Q in the intermediate magnetic field region. This cor-responds very well with the width of the MHLs or thetransient superconducting current density qualitativelyas shown in Fig. 6(b). However, not like that reportedin Ba(Fe . Co . ) As , we do not observe a clearcrossover of Q value near zero field, which was inter-preted as a crossover from the strong intrinsic pinningnear zero field to a collective pinning at a high magneticfield. This may suggest that there is just one kind of pin-ning mechanism for the measured BKBO single crystal.As we know that magnetization relaxation rate Q isaffected by not only the vortex pinning but also the inter-action between vortices. In the normal circumstance, themagnetization relaxation Q will increase with increasingmagnetic field because of the enhanced interaction be-tween vortices at a certain temperature. The irreversiblemagnetization or the transient critical current density J s will decrease with the magnetic field as the flux creepis enhanced. However, in Fig. 6 we can observe non-monotonic magnetic field dependence of both Q and J s .Indeed, one can clearly see that when the MHL is gettingwider versus magnetic field, the relaxation rate is gettingsmaller, showing a slower magnetization relaxation, i.e.,there is a close correlation between field dependent be-haviors of Q and ∆ M (or J s ). This feature is very similarto those in cuprate and iron based superconductors .What we want to notice is that the positions of mini-mum Q are roughly corresponding to the location where∆ M takes the maximum. This also reminds us that thesecond peak effect appearing here is just reflecting a dy-namical process of the vortex entity, but not the detailedcharacteristics of the pinning centers, because one cannotguarantee the pining mechanism is the same in all thesedifferent superconductors. It would be very interestingto measure the magnetization relaxation in a long timescale, to reveal whether the SP effect corresponds to astatic vortex phase transition .
10 K 2 K M ( e m u / c m ) H (T) dH/dt = 200 Oe/s dH/dt = 50Oe/s (b) Q (a)
10 K 2 K
FIG. 6: (Color online)(a) The magnetic field dependence ofmagnetization-relaxation rate Q at temperatures of 2 K and10 K obtained from curves in (b) with Eq.(4). (b) The MHLcurves at temperatures of 2 K and 10 K. Solid lines stand forthe magnetization measured with a magnetic field sweepingrate of 200 Oe/s while the dashed line to 50 Oe/s. Severalsharp peaks of magnetization in the field ascending processbelow 0.5 T at 2K are induced by the flux jump effect. Fig. 7 shows the temperature dependence of transientsuperconducting current density J s and dynamical relax-ation rate Q taken from field dependent values at var-ious temperatures, respectively. We notice that in thelow and intermediate temperature region, the curves oflog J s ( T ) at different magnetic fields almost merge to-gether below 6 T, which disperses clearly in high tem-perature region. This behavior is similar to that ofBa(Fe . Co . ) As , and may be caused by the SPeffect which prevents the rapid decreasing of J s . Associ-ated with the SP effect, the magnetization relaxation rate Q is inversely related to transient superconducting cur-rent density J s as shown in Fig. 7(b). As we mentionedalready that there is a plateau in Q − T curve below 18 K,and below 2 T the Q value decreases with increasing field,all these features demonstrate that the Q -plateau and theSP effect are closely related. When it is in the collectivecreep regime, the magnetization relaxation rate is lowand exhibits a plateau, thus the transient critical currentdensity J s decays slowly with temperature. It seems thatall these features can be explained coherently.As addressed above, a plateau of Q appears in theintermediate temperature region which is followed by a = 0.6 T = 1.0 T = 1.5 T = 2.0 T = 3.0 T = 4.0 T = 5.0 T Q T (K) dH/dt = 50 Oe/s = 0.6 T = 1.0 T = 2.0 T = 3.0 T = 4.0 T = 5.0 T = 6.0 T J s ( A / c m ) FIG. 7: (Color online) (a) Temperature dependence of log J s at different magnetic fields ranging from 0.6 T to 6 T,the data is the same as that presented in Fig. 5. (b) Tem-perature dependence of magnetization relaxation rate Q atvarious magnetic fields from 0.6 T to 5.0 T obtained from thecorresponding curves in Fig. 3 with Eq.(4). severe increase in the high temperature region. Thisbehavior of magnetization relaxation rate was also ob-served in cuprate superconductor YBCO and iron-based superconductors, such as Ba(Fe . Co . ) As and SmFeAsO . F . . This plateau cannot be under-stood within the picture of single vortex creep with arigid hopping length as predicted by the Kim-Andersonmodel, but perhaps due to the effect of collective fluxcreep. The reason is that, in the Kim-Anderson model,itis quite easy to derive that T /Q ( T ) = U c ( T ). Suppose U c ( T ) is a weak temperature dependent function, we have Q ( T ) ∝ T , which contradicts the observation of a Q -plateau in the intermediate temperature region. As de-scribed in Eq. (5), the relaxation rate Q is dependenton both U c ( T ) and µCT . In the intermediate tempera-ture region, the term of µCT becomes much larger than U c /k B and we will get almost constant value of Q , whichprovides a simple explanation of the plateau.A coherent picture to interpret the complex tempera-ture and magnetic field dependence of Q and J s is thusproposed in the following way. When the magnetic fieldincreases from zero, the vortex system runs into the vor- T / Q ( K ) T (K)
FIG. 8: (Color online) Temperature dependence of the ratio
T /Q at different magnetic fields ranging from 0.6 T to 5.0 T. tex glass regime with much smaller relaxation rate. Inthe high field region, the magnetic relaxation rate Q goesup drastically, meanwhile the transient superconductingcurrent J s drops down quickly as shown in Fig. 7(a),which could be interpreted as a crossover from the elas-tic motion to a plastic one of the vortex system. C. Analysis based on the collective pinning/creepmodel
Now let’s have a quantitative consideration on thevortex dynamics based on the collective pinning/creepmodel. According to Eq. (5), the intercept of
T /Q - T curve will give U c /k B and the slope gives rise to µC ifwe assume U c and µ is weakly temperature dependent.In Fig.8 we present the T /Q vs. T at different mag-netic fields ranging from 0.6 T to 5 T. One can see thatthe intercept U c /k B is actually very small, about 111 Kat 1 T, and 198 K at 2 T, which is much smaller thanthe value of over 3000 K in MgB and also smallerthan about 300-500 K in YBCO . However, since J s is quite large and the relaxation rate is very small, thecollective pinning should play the role here. As we dis-cussed above, in the model of collective creep, the glassyexponent µ becomes influential on the vortex dynamics.From Eq.(5), one can see that µC can be determinedfrom the slope of T /Q vs T . From our data, we ob-tained the value µC = 63 in the low temperature regimeat 1 T from Fig. 8. So in the intermediate tempera-ture region, U c /k B ≪ µC and we will get the plateau of Q ≈ /µC as shown in Fig. 7(b). Meanwhile, the pa-rameter C can be determined by the slope of − d ln J s /dT vs Q/T , and we get the C = 24 . µ ≈ .
54. The value of µ is much larger than1 and clearly shows a collective creep effect. Actually, µ can also be estimated from µ = − Qd ln J s /d ln E .From this equation, one can imagine that large µ meansa stronger glassy effect, showing a strong downward cur- F p / F m a x p b = H / H irr T = 18 K
T = 20 K
T = 22 K
T = 24 K
Fitting line b = 0.47
FIG. 9: (Color online) The scaling of normalized pinning forcedensity F p /F max p as a function of the reduced field b = H/H irr .Curves at different temperatures are almost scaled together.A fitting result with Eq. (6) is presented as the red line, andthe maximum locates at b ≈ . vature in the ln E vs ln J s curve. All these indicate thatthe vortex pinning and dynamics in BKBO can be de-scribed by the collective pinning/creep model with weakcharacteristic pinning energy U c but large glassy expo-nent µ . D. Characteristics of pinning force density
In order to get further insight into the origin of thesecond peak effect, we need to analyze the pinning forcedensity F p which is proportional to J s H . In Fig. 9 weshow the F p normalized to its maximum value as a func-tion of reduced field b = H/H irr , and H irr is the irre-versible magnetic field determined using a criterion of J s = 20 A/cm . Although there is uncertainty to de-termine H irr , the curves of normalized pinning force atdifferent temperatures seem to scale well and have thesimilar maximum value, which is different from the poorscaling results observed in the thick films of BKBO .The maximum of the pinning force density locates at b ≈ .
47. We use the following expression to study thepinning mechanism for the sample Ba . K . BiO δ F p F max p = Ab p (1 − b ) q , (6)where A , p and q are the parameters, and the values of p and q can tell us the characteristic properties of thevortex pinning mechanism in the sample. We use thisequation to fit the data, which is presented as the redline in Fig. 9. It can be seen that the fitting curvewith p = 2 .
79 and q = 3 .
14 catches the main feature ofthe experimental data. As we know the cuprate high-temperature superconductors are usually satisfied withthe relation q > p , e.g., q = 4, p = 2 were obtained for H c2 H ( T ) T (K) H sp H min H irr J c ( A / c m ) H (T) H min H SP H irr FIG. 10: (Color online) The phase diagram of the BKBOsample. The open symbols are taken from the resistive mea-surements shown in Fig. 2, while the solid ones are taken fromthe J s - µ H curve in Fig. 5. The inset shows a typical exam-ple for how to determine characteristic fields H irr , H sp and H min . All red lines in this figure are the fitting curves withthe formula H ( T ) = H (0)(1 − T /T c ) n . For the irreversibilityfield H irr ( T ), the open circles correspond to the data deter-mined from resistivity, the filled green squares represent thedata determined from the irreversible magnetization. YBCO single crystals , and q = 2, p = 0 . . In this sample, b for the maximum valueof the pinning force density is near 0.5 which correspondsto the δκ -type pinning , however this kind of pinningrequests p = q = 1. Further studies are still required toresolve this discrepancy. V. VORTEX PHASE AND GENERALDISCUSSION
In Fig. 10, we present the vortex phase diagram ofthe sample Ba . K . BiO δ . The upper critical field H c2 and the irreversible field H irr shown as open sym-bols in Fig. 10 are determined by a criterion of 90% ρ n and 10% ρ n in Fig. 2. The three characteristic fields areshown as solid symbols in Fig. 10, i.e., the second mag-netization peak field H sp , H min determined at the mini-mum point of ∆ M or J s between the first and the secondmagnetization peak, and the irreversibility field H irr de-termined from the field dependent J s curve in Fig. 5.In order to get more information, we use the expression H ( T ) = H (0)(1 − T /T c ) n to fit these curves. We got thefollowing values: n = 1 .
32 for H min and H min (0) = 1 . n = 0 .
723 for H sp and H sp (0) = 3 .
59 T, n = 1 .
07 for H c2 and H c2 (0) = 21 . n from the H sp - T curve is almost half less than that of YBCO . It isclear that in BKBO, the separation between H irr - T and H c2 - T is small, which is similar to YBCO and indicatesa rather weak vortex fluctuation. In addition, one cansee that the separation between H sp - T and H irr - T curvesis quite large. One cannot interpret this large region asdue to the non-uniform distribution of disorders or pin-ning centers. However, it is reasonable to understandthis region as the plastic flux motion since this phase cangradually “melt” through losing the rigidity of the vortexmanifold. It is interesting to note that the H sp ( T ) looksvery similar to that in YBCO, but very different fromthat in Bi2212. Therefore we believe that the secondpeak effect, at least in BKBO and YBCO, may be in-duced by the similar reason. It is quite possible that theelastic energy which depends on the shear module C is an influential factor for the occurrence of the secondpeak effect. VI. CONCLUDING REMARKS
We have investigated the vortex dynamics and phasediagram through measuring the magnetization and itsrelaxation on a Ba . K . BiO δ single crystal withtransition temperature T c = 27 K. Second magnetiza-tion peak has been observed in wide temperature regionfrom 2 K to 24 K. It is found that through out the non-monotonic magnetic field dependence of the magnetiza- tion, the relaxation rate is inversely related to the tran-sient critical current density, indicating that the SP effectis dynamical in nature. It is found that many observedfeatures can be coherently described by the collective pin-ning/creep model. Through the fitting and analysis, wefind that the characteristic pinning energy U c is quitesmall (about 198 K at 2 T), but the glassy exponent µ is quite large, which induces a relatively small mag-netization relaxation rate. A universal scaling law forthe pinning force density F p /F maxp vs H/H irr is found,which suggests that the pinning mechanism is probably δκ -type. The characteristics of the SP effect and magne-tization relaxation as well as the vortex dynamics of thesystem allow us to conclude that it is more like those inthe cuprate superconductor YBCO. VII. ACKNOWLEDGMENTS
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