Vortex thermal fluctuations in heavily underdoped Bi2Sr2CaCu2O8+d (Tc=69.4 K)
are studied using Josephson plasma resonance (JPR). From the data in zero
magnetic field, we obtain the penetration depth along the c-axis,
lambda_{L,c}(0) = 229 micrometers and the anisotropy ratio gamma(0) = 600. The
low plasma frequency allows us to study phase correlations over the whole
vortex solid (Bragg-glass) state. The JPR results yield a wandering length
r_{w} of vortex pancakes. The temperature dependence of r_{w} as well as its
increase with applied dc magnetic field can only be explained by the
renormalization of the tilt modulus by thermal fluctuations, and suggest the
latter is responsible for the dissociation of the vortices at the first order
transition.Comment: 4 pages, 5 figures. Submitted to Phys. Rev. Let

A. E. Koshelev

Cornelis J. van der Beek

E. Zeldov

E. H. Brandt

L. N. Bulaevskii

M. B. Gaifullin

Marat B. Gaifullin

Marcin Konczykowski

Ming Li

N. Morozov

Peter H. Kes

Piotr Gierłowski

R. Cubitt

S. Colson

Sylvain Colson

T. R. Goldin

Y. Matsuda

Yuji Matsuda

English

arXiv

Crossref

Vortex Fluctuations in Underdoped
                    
                      
                        
                          B

Colson, S.

Konczykowski, M.

Gaifullin, M.B.

Matsuda, Y.

Gierlowski, P.

Li, M.

Kes, P.H.

Beek, C.J. van der

Leiden University Scholary Publications

P H Y S I C A L R E V I E W L E T T E R S week ending4 APRIL 2003VOLUME 90, NUMBER 13Vortex Fluctuations in Underdoped Bi2Sr2CaCu2O8 CrystalsSylvain Colson,1 Marcin Konczykowski,1 Marat B. Gaifullin,2 Yuji Matsuda,2 Piotr Gierłowski,3 Ming Li,4Peter H. Kes,4 and Cornelis J. van der Beek11Laboratoire des Solides Irradiés, CNRS-UMR 7642 and CEA/DSM/DRECAM, Ecole Polytechnique, 91128 Palaiseau, France2Institute for Solid State Physics, University of Tokyo, Kashiwanoha, Kashiwa, Chiba 277-8581, Japan3Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland4Kamerlingh Onnes Laboratorium, Leiden University, P.O. Box 9506, 2300 RA Leiden, the Netherlands(Received 15 April 2002; published 3 April 2003)137002-1Vortex thermal fluctuations in heavily underdoped Bi2Sr2CaCu2O8 (Tc  69:4 K) are studiedusing Josephson plasma resonance. From the zero-field data, we obtain the c-axis penetration depthL;c0  230  10 m and the anisotropy ratio 	T. The low plasma frequency allows us to studyphase correlations over the whole vortex solid state and to extract a wandering length rw of vortexpancakes. The temperature dependence of rw as well as its increase with dc magnetic field is explainedby the renormalization of the vortex line tension by the fluctuations, suggesting that this softening isresponsible for the dissociation of the vortices at the first order transition.DOI: 10.1103/PhysRevLett.90.137002 PACS numbers: 74.25.Op, 74.40.+k, 74.25.Qt, 74.72.–hc direction, by a distance rn;n1  un1  un. Here un is sample position was verified with a Hall probe of activeVortex thermal fluctuations are considered essential indetermining the B; T phase diagram of layered hightemperature superconductors and notably the first ordertransition (FOT) in which the ordered vortex crystaltransforms to a liquid state without long range phasecoherence [1,2]. Many scenarios, varying from vortexlattice melting described by a Lindemann criterion [3]to layer decoupling [4–6], all considering various degreesof coupling between the superconducting layers, havesuccessfully been used to describe the position of theFOT in the B; T plane. However, such fits to the FOTline have not been able to convincingly discriminatebetween the different models. Here, we aim to do justthat, through a direct measurement of the amplitude, aswell as the field and temperature dependence of vortexthermal excursions in the vortex solid phase that lead tothe FOT.For this study, we use the layered Bi2Sr2CaCu2O8(BSCCO) compound, in which vortex excursions areaccessible using the Josephson plasma resonance (JPR)technique [7,8]. Briefly, the interlayer Josephson currentJcm is measured through the JPR frequency !pl  Jc1=2m ,at which the equality of charging and kinetic energy leadsto a collective excitation of Cooper pairs across the layers.In turn, !2plB; T  !2pl0; T  hcosn;n1i intimatelydepends on the gauge-invariant phase difference n;n1between adjacent layers n and n 1 [9]. Here, h  istands for thermal and disorder averaging. Thus, JPR isa probe of the interlayer phase coherence. The fluctuationsof vortex lines created by a dc magnetic field appliedperpendicularly to the layers modify the relative phasedifference between adjacent layers and thus depress !pl.In Bi2Sr2CaCu2O8, the ensemble of vortex lines shouldbe described as stacks of two-dimensional pancake vor-tices. Thermal fluctuations shift the individual vortexpancakes with respect to their nearest neighbors in the0031-9007=03=90(13)=137002(4)$20.00the ab-plane displacement of the pancake vortex in layern with respect to the equilibrium position of the stackit belongs to. The wandering length of vortex lines, whichis related directly to the JPR frequency !2pl, is defined asrw  hr2n;n1i1=2 [10,11]. Below, we shall consider onlytemperatures above T  42 K, at which vortex pinning(quenched disorder) is unimportant [2,12].Underdoped BSCCO single crystals (Tc  69:4 0:6 K) were grown by the traveling solvent floating zonemethod in 25 mbar O2 partial pressure at the FOM-ALMOS center, the Netherlands [13]. The samples werepostannealed for one week at 700 C in flowing N2. Theadvantage of using heavily underdoped BSCCO is that!pl0; 0  61 GHz turns out to be very low, which al-lows us to measure the vortex meandering over the entireB; T phase diagram. Samples A and B (cut from thesame crystal) have dimensions 1:35  1  0:04 mm3 and0:7  0:47  0:04 mm3, respectively. The FOT tempera-ture TFOT of these and of a third crystal (C) from the samebatch, was determined by measuring the paramagneticpeak at the FOT with a miniature Hall probe magne-tometer [14].The JPR measurements were carried out using thecavity perturbation technique (on sample A) and thebolometric method (on samples A and B). For the cavityperturbation technique, the sample was glued in thecenter of the top cover of a cylindrical Cu cavity usedin the different TM01i (i  0; . . . ; 4) modes. These providethe correct configuration of the microwave field at thesample location, in which Erf k c axis and Hrf  0 [15].The unloaded quality factor Q0 is measured as a functionof temperature and field to obtain the power absorbed bythe sample (Fig. 1). The bolometric method [16] consistsof measuring the heating of the sample induced by theabsorption of the incident microwave power when the JPRis excited [8,15]. The homogeneity of the dc field at the 2003 The American Physical Society 137002-102004006000.4 0.6 0.8 1γ T / Tc01020304050600 10 20 30 40 50 60 70A - cavityA - bolometricB - bolometricB - cavityf JPR  ( GHz )T ( K )FIG. 2. JPR frequency vs temperature in B  0 for samples Aand B, measured by both the bolometric method and the cavityperturbation technique. We use spline fits (solid lines) in theextraction of the wandering length (see text). Inset: experimen-tal temperature dependence of 	, obtained by the division ofthe experimental L;cT by the L;abT from reversible mag-netization.FIG. 1. Field sweep experiment on sample A at T  66 K inthe TM012 mode of the cavity (f  22:9 GHz). At BJPR  5:3 G(open arrow), for which !  !pl, the power absorbed in thesample () has a maximum and the resonance frequency of thecavity () shows a double-peak structure (closed arrows), anda jump (arrow between dashed lines). Inset: meandering ofpancakes along a vortex line in a layered superconductor.Thermal fluctuations shift pancakes (full circles) away fromtheir equilibrium positions (open circles).P H Y S I C A L R E V I E W L E T T E R S week ending4 APRIL 2003VOLUME 90, NUMBER 13area 10  10 m2. Magneto-optical imaging of the fluxdistribution in the samples revealed that field inhomoge-neity and irreversibility due to surface barriers can beneglected. The reversible magnetization of sample A wasmeasured using a commercial superconducting quantuminterference device magnetometer in order to extract theab-plane London penetration depth L;abT [13].Figure 2 shows the JPR frequency fJPR  !pl=2 inzero field obtained by the above-mentioned methods onsamples A and B. !2pl is proportional to the maximuminterlayer Josephson current along the c axis [9],!2plB; T  !2pl0; Thcosn;n1i 2s$0Jcm B; T;(1)where Jcm B; T  Jcm 0; Thcosn;n1i is the maxi-mum Josephson current, s  1:5 nm is the interlayerspacing,  the high-frequency dielectric constant, and$0 the flux quantum. Using !pl0; T  1=L;cT0pand   11:50 [17], we obtain the London penetrationdepth for currents along the c axis, L;cT (0 is thepermittivity of the vacuum). When divided by theL;abT data from reversible magnetization, this yields,without any model assumptions, the anisotropy parame-ter 	T  L;cT=L;abT, shown in Fig. 2. Typically, atT  0:5Tc, L;c  240 m and L;ab  400 nm, so that	  600, consistent with other data for the same material[8]. Note that 	 decreases as a function of temperature.To analyze our JPR data in nonzero magnetic fields, weshould divide!plB; T by the zero-field result depicted inFig. 2. In the absence of a model that satisfactorily de-scribes !pl0; T over the whole temperature range, weresort to a spline fit to the experimental data. Slightdifferences between samples A and B were found to137002-2have a negligible influence. The vortex wandering lengthrw is extracted as follows. In the single-vortex regime, atvery low fields B< BJ  $0=2J, B< B  $0=42L;ab,Bulaevskii and Koshelev derived [10,11]1 !2plB; T!2pl0; TB2$0r2w lnJrw; (2)where the Josephson length J  	s. This relation ismeaningful only for small excursions rw & 0:6J, i.e.,for hcosn;n1i  !2plB; T=!2pl0; T & 1. More gener-ally, one expects an increase of 1  hcosn;n1i with rwup to a plateau for large rw, as was found in recentsimulations of the evolution of 1  hcosn;n1i versushui=a0  rw=a0 for a pancake gas (a0 $0=Bpis theintervortex spacing) [18]. The numerical data show that1  hcosn;n1i is almost quadratic in rw for 0 & 1 hcosn;n1i & 0:7–0:8, in agreement with Eq. (2), if theweak logarithmic dependence on J=rw is disregarded.Thus, we user2w 2$0B1  hcosn;n1i (3)to obtain an approximation of the wandering length.Since rw  hun1  un2i1=2  2u2  hun  un1i1=2,one has, in the case of completely uncorre-lated layers (e.g., for a pancake gas), rw  h2u2ni1=2 2pu. Disregarding the ‘‘anticorrelated’’ situation withun  un1 < 0, correlations between pancake positionsin different layers yield rw <2pu, i.e., rw=2pis a lowerlimit for the root mean squared (rms) displacement u ofthe vortex line.Figure 3 shows 1 !2plB; T=!2pl0; T  1 hcosn;n1i as a function of temperature in differentdc fields. The temperature dependence of the wanderinglength rw, obtained by applying Eq. (3), is represented in137002-2200400600800100012000.7 0.75 0.8 0.85 0.9 0.95 1 1.05r w (nm)T/TFOT(b)11010030 35 40 45 50 55 60 65 70B FOT ( A )Bsp    ( A )BFOT ( C )BFOT ( B )Bsp    ( B )B ( G )T  ( K )200400600800100012000.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 12.9 G5.4 G7.5 G12.4 G15.3 G22.4 G27.5 G9.9 G32.6 Gr w (nm) T / Tc0.9s (kBTγ2/ε0s)1/2α = 0.65α = 0.8(a)FIG. 4. (a) Experimental rw vs T=Tc for different magneticfields in strongly underdoped BSCCO. For B  27:5, 22.4, 15.3,12.4, and 9.9 G, arrows show the temperature of the FOT. Thethick line shows the evolution of 0:9skBT	2="0s1=2. Thinlines are fits to Eq. (5) with (  0:8 (for B< 15 G) and ( 0:65, omitting the term 4=(x2  14 (B > 15 G). (b) rw vsT=TFOT. Solid lines are guides to the eye. Inset: phase diagramof the samples used in this study, showing the position of theFOT line as revealed by the paramagnetic peak in the localsusceptibility (BFOT) or the second peak in the dc magneti-zation (Bsp).00.20.40.60.810.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 12.9 G5.4 G7.5 G9.9 G12.4 G15.3 G22.4 G27.5 G32.6 G1-<cosΦn,n+1>T/TcFIG. 3. 1  hcosn;n1i vs temperature for different mag-netic fields. We extract rw from these data using Eq. (3).P H Y S I C A L R E V I E W L E T T E R S week ending4 APRIL 2003VOLUME 90, NUMBER 13Fig. 4. For every field, we observe an increase of rw withT. At constant temperature, rw increases with magneticfield, implying that the single-vortex part of the tiltmodulus dominates the elastic energy (see below).Another interesting feature of the rwT curve is the breakin the slope which appears at a field-dependent tempera-ture close to the FOT and above which all the rw curvesmerge into one. Alternatively, one may plot the samevalues of rw vs T=TFOT [Fig. 4(b)]. Here, two regimesappear clearly: for T < 0:96TFOT, rwT=TFOT roughlyoverlaps for all fields, whereas for T > 0:96TFOT, thecurves deviate from each other. This shows that in thevortex solid, rw depends on temperature as rwT=TFOT.We now discuss the temperature and field dependenceof rw in the vortex solid. The rms thermal vortex displace-ment u can be obtained by equipartition of the associ-ated elastic energy with the thermal energy, Uel c44a20u2=s  kBT. The vortex lattice tilt modulusc44k B2=01  2ck2k 2abQ2z"02	2a20lnk2maxK20  Qz=	2"022abQ2za20ln1 a2021:3r2w; (4)calculated by Koshelev and Vinokur [19] and Goldin andHorovitz [20], consists of three terms: the nonlocal col-lective (lattice) term, the vortex line tension term, deter-mined by Josephson coupling between layers, and a thirdterm due to the electromagnetic dipole interaction be-tween pancakes. Of particular interest here is the loga-rithmic correction to the temperature dependence of thesecond term, introduced by the cutoff kmax  =rw,which corresponds to the smallest meaningful deforma-tion [19,20]. To proceed, we evaluate Uel at the typicalvortex line deformation wave vectors parallel and per-pendicular to the layers, kk  =u and Qz  	=2a0 2=s. Writing K0 4p=a0, r2w  (u2 (with (  1)and x  a0=rw, equipartition yields137002-3r2w  (s2kBT	2"0s4(x2  1412ln0:66x222a0L;ab2ln1 x221:31: (5)All parameters in Eq. (5), and notably "0T=	2T $20=402L;c, are known from experiment, which allowsa direct comparison to the rwT data. For the lowest fourfields (B< 10 G), the line tension term is largest all theway to the FOT. Equation (5), which then reduces toEq. (40) of Ref. [20] with Qz  	=2a0 instead of137002-3P H Y S I C A L R E V I E W L E T T E R S week ending4 APRIL 2003VOLUME 90, NUMBER 132=s, very well describes the magnitude, the tempera-ture, as well as the field dependence of rw, using thesingle free parameter (  0:8. For higher fields, the non-local collective contribution to c44 increases, eventuallyexceeding the Josephson coupling (line tension) termclose to the FOT for B > 20 G. However, the inclusionof the nonlocal term leads to too weak a temperaturedependence of rwT. Rather, the experimental wanderinglength behaves as if the line tension term dominates thevortex response at all fields [11]: excellent fits of both thetemperature and field dependence are obtained by omit-ting the nonlocal collective term [i.e., 4=(x2  14 inEq. (5)] and using (  0:65; see Fig. 4(a). Note that whilethe main rwT dependence comes from the prefactor	2T="0 in Eq. (5) [thick line in Fig. 4(a)], the behaviorof rw in the vortex solid can be understood as the result ofthe logarithmic correction arising from the softening ofthe line tension term by thermal fluctuations [19,20]. Thefield dependence comes from the zone-boundary vectorK0 and the small Qz, indicating that vortex lines arecorrelated (linelike) on distances that well exceed thelayer spacing s.An increase of rw with increasing vortex density canarise from the suppression of Josephson coupling by vor-tex fluctuations or from the weaker interpancake dipolecoupling, but is incompatible with a dominant role of theshear energy or of compressional or collective tilt modesand thus with Lindemann-like melting [3]. Moreover,using our experimental data to compare the differentcontributions to the elastic energy, we find that the dipolecoupling and the shear energy are, under all circumstan-ces, negligible. Thus, dislocation-mediated (Kosterlitz-Thouless–like) melting, as well as vortex line evapora-tion [6] are also unlikely. Rather, the large thermal ex-cursions of pancake vortices soften the line tensioncontribution to c44 for the large-wave vector modes thatlead to the FOT. This complies with recent measurementsshowing that vortex lattice order is not a prerequisite forthe FOT [21]. For deformations with smaller wave vec-tors, Josephson coupling still contributes to the line ten-sion even in the vortex liquid, leading to, e.g., theanisotropic vortex response to columnar defects inheavy-ion irradiated samples. These conclusions, arrivedat for extremely anisotropic underdoped BSCCO, willalso hold for less anisotropic layered superconductors.Summarizing, JPR measurements on heavily under-doped BSCCO crystals yield the c-axis penetration depth,the anisotropy parameter 	T, and the wandering lengthrw of vortex lines. The observed temperature and fielddependences of rw suggest that thermal fluctuationssoften the Josephson coupling contribution to the tiltmodulus for short wavelengths [20], a softening that webelieve drives the FOT.We thank the European Science Foundation VORTEXprogram and the Nederlandse Organisatie voor Weten-schappelijk Onderzoek for financial support. P. G. ac-knowledges support of the Polish Government, Grant137002-4No. PBZ-KBN-013/T08/19.[1] R. Cubitt, E. M. Forgan, G. Yang, S. L. Lee, D. McK. Paul,H. A. Mook, M. Yethiraj, P. H. Kes, T.W. Li, A. A.Menovsky, Z. Tarnawski, and K. Mortensen, Nature(London) 365, 407 (1993).[2] E. Zeldov, D. Majer, M. Konczykowski, V. B.Geshkenbein, V. M. Vinokur, and H. Shtrikman, Nature(London) 375, 373 (1995).[3] G. Blatter, V. B. Geshkenbein, A. I. Larkin, and H.Nordborg, Phys. Rev. B 54, 72 (1996).[4] L. Glazman and A. E. Koshelev, Phys. Rev. B 43, 2835(1991).[5] L. L. Daemen, L. N. Bulaevskii, M. P. Maley, and J.Y.Coulter, Phys. Rev. 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Vortex fluctuations in underdoped Bi2Sr2CaCu2O8+δ crystals

Vortex fluctuations in underdoped Bi2Sr2CaCu2O8+d crystals

Vortex fluctuations in underdoped Bi2Sr2CaCu2O8+d crystals

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