Boson–fermion fluctuation dominated model of the superconducting transition in hole-underdoped cuprates

Physica C 411 (2004) 107–113 www.elsevier.com/locate/physc

Boson–fermion fluctuation dominated model of the superconducting transition in hole-underdoped cuprates A.C. Bo´di a

a,*

, R. Laiho b, E. La¨hderanta

b,c

Institute of Experimental Physics, University of Debrecen, P.O. Box 81, H-4010 Debrecen, Hungary b Wihuri Physical Laboratory, Turku University, FIN-20014 Turku, Finland c Physics, University of Vaasa, FIN-65101 Vaasa, Finland Received 27 January 2003; received in revised form 16 March 2004; accepted 21 June 2004 Available online 6 August 2004

Abstract Reliable correlation of simultaneously measured resistively, susceptibility and specific heat critical parameters of hole-underdoped YBCO permitted to establish a phenomenological model of normal-superconductor transition. After the model, fluctuations of quasifree electrons drives at T*
TC the appearance of paired charge carriers (pseudogapPG) characterized by partial coherence of the order parameterÕs phase. The evolution of PG state from T*  140 K to the critical temperature TC  92 K is connected to the average increase of the statistically distributed partial coherence lifetime (from s  ps to 1) and coherence length (from k  nm to 1) of the fluctuating precursor islands. In this dynamic process the specific heatÕs maximum signals (at temperature TCC  93.6 K) the appearance of the first mesoscopic coherent islands. TC marks the percolation of these islands. We believe that the presented concept of dynamical evolution (in coherence time and space from 0 to 1) of the paired charge structures is a promising scenario while describing the properties of hole doped high-TC superconductors between the temperatures T* and TC.
2004 Elsevier B.V. All rights reserved. PACS: 74.25.Dw; 74.30.Ek; 74.60.Ec Keywords: Superconductivity phase diagrams; Thermodynamic properties; Mixed state

1. Introduction In metallic superconductors––according to the theory BCS––the pairing of charge carriers and *

Corresponding author. Tel.: +36 52 415 222; fax: +36 52 315 087. E-mail address: [email protected] (A.C. Bo´di).

the setting of coherence occur at the same critical temperature TC. Xu et al. reported evidence for vortices in La2  xSrxCuO4 at temperatures significantly above the critical temperature [1]. Meingast et al. observed precursor phase-incoherent Cooper pairing in YBa2Cu3Ox crystals [2]. Diamagnetic activity above TC as a precursor to superconductivity in La2xSrxCuO4 thin films was also

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2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.06.014

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demonstrated [3]. These and other experiments suggest that in hole doped HTSC the process of pair formation and their coherence take place at different temperatures: pairing at the pseudogap (PG) critical temperature T* and phase coherence at TC  T* [4]. It has been recognized early that the small superfluid density in HTSC (ns / TC) leads to fluctuations of the order parameterÕs phase [5]. It is important to know how these fluctuations drive the basic phenomenon of superconducting transition in underdoped cuprates. The experimental evidence of a pseudogap together with numerous indications suggest that possibly a two component system composed out of itinerant electrons (fermions) and electron pairs (bosons) is involved [6]. Different investigation methods supply different characteristic parameters––between them––different critical temperatures for the same type of sample. For correct correlation of all critical temperature data of YBCO ceramics we measured––using one common apparatus––first the dc resistance after that simultaneously the specific heat (using differential scanning calorimetry [7]), and the complex ac susceptibility [8] parameters of a hole-underdoped sample. We addressed the problems without having to resort to a specific mechanism for electron pairing.

2. Experimental The important physical and chemical parameters of the investigated under-doped bar-shaped sample SS (Y1.01Ba1.95Cu2.97O6.86), density  4.82 g/cm3, dimensions 6 · 6 · 11 mm3, have been published earlier [9]. Measurements on ceramic samples (in zero magnetic field) are found to be as appropriate as those on single crystals [10]. SS was inserted in the (thermally isolated) secondary (pick-up) coil (Fig. 1). The reference material RS––necessary for the differential calorimetry––was cut from the same pellet as SS, with the same dimensions but its superconductivity was suppressed by de-oxygenation. Both SS (with its thermocouple, electrodes and pick-up coil), and RS (with its thermocouple) were hermetically surrounded by epoxy resin. SS and RS were in quasi-identical thermal conditions. The bath temperature (T) was measured between SS

TC TS

TC T

TC TR

sample SS

reference RS epoxy

χ ' ; χ"

LN

pick-up coil

Fig. 1. Schematic diagram of the experimental setup. TC–– thermocouples, SS–– superconducting sample, RS––normal reference specimen, LN––liquid nitrogen. The primary coil and the dc current electrodes are not marked.

and RS. The epoxy isolated SS–RS system was inserted into an axial primary coil (D  10 cm). For cooling (heating) the bottom end of the thermal insulator epoxy was submerged in (or raised from) liquid nitrogen in a controlled way to obtain a constant rate of cooling or heating (jDT/Dtj  0.01 K/s). For dc resistance measurements the four probe method, for ac susceptibility measurements audiofrequency (f = 5 kHz) lock-in amplifier and the two-coil system was used. At susceptibility measurements the primary current produced an ac magnetic field of the order H  0.1 Oe on the surface of the sample. In this field, the in-phase (/v 0 ) and out-of-phase (/v00 ) components of the secondary voltage were recorded in function of the epoxyÕs center (bath) temperature T. For simultaneous specific heat determinations the temperatures of the sample TS and of the referSS

RS

M1' TS B1 bath

TR B2

M1

B2'

B1'

T

Fig. 2. Linear model of the setupÕs thermal system in which the sample SS (TS), the reference material RS (TR), the bath (T) are considered as one body without temperature distribution in it. M1, M 0 1,B1, and B 0 1 are thermal conductance. The arrows mark the direction of heat flow.

A.C. Bo´di et al. / Physica C 411 (2004) 107–113

ence TR and their difference DT = TR  TS were monitored in function of the bath temperature T (Fig. 2). Our thermocouple measurements show a typical temperature resolution, and accuracy of 0.05 and 2 K, respectively.

3. Results

YBCO 6x6x11 mm heating 6 f=5kHz

0

3

-2 -4

4 -6 2

χ' (arb. units)

8

ρ(10−4 Ω cm)

χ" (arb. units)

SS and RS were zero-field cooled. Under cooling––from the dc resistance variation of the sample––the TC  92 K critical temperature (zero resistance) value was obtained (Fig. 3(a)). The figure also shows the temperature dependent variation of the susceptibility components v 0 (T) and v00 (T) of SS upon heating. The susceptibility variations begin very slowly and smoothly at T  88 K where the increase of v 0 , and v00 becomes observable. The sigmoid shape of the curve v 0 and the lognormal shape of v00 (with a peak at 92 K) are usual. Over temperature T  94.7 K the deviation from zero of both susceptibility components became unobservable. Fig. 3(b) shows the simultaneously

-8

0 88

-10 90

(a)

92

94

96

T(K)

0.5 0.0 -0.5

∆T ( K )

TS ( K) ; TR ( K )

TR TS ∆T

90 -1.0 -1.5 88

(b)

90

92

recorded variation of temperatures TS and TR together with their difference DT. The deviation of DT from its linear variation begins at T  88 K. During the transition SS––having a larger specific heat––warms up more slowly than RS. Consequently DT increases and presents a k type peak. After the k-peak, the temperature TS increases faster than TR and DT presents a fast downward jump (  0.5 K/s). The v00 and DT curves partly overlap but their shapes are different and their peaks appear at different temperatures (92 and 93.6 K approximately the highest TC attainable for optimum doping). At low frequencies the temperature of the maximum v00 (or of the midpoint of the curve v 0 , or of the maximum of ov 0 /oT) is sometimes taken as the critical one (TC). Similarly, the DT curve maximumÕs temperature will be used as the calorimetric critical temperature (93.6 K = TCC). It is worth using Ce instead of the temperature difference: DT ¼ T R  T S ¼ ½mðC S  C R Þ þ nT / ðC S  C R Þ ¼ DC ¼ C e

ð1Þ

Here m and n are constants, CS, CR are the apparent specific heats of the sample SS, of the reference RS, and Ce is the apparent electronic specific heat of the sample. At the deduction of the formula (1) we started from the equations (Fig. 2): C S oT S =ot ¼ ðM 1 þ B1 ÞT S þ M 01 T R þ B2 T

ð2Þ

C R oT R =ot ¼ ðM 01 þ B01 ÞT R þ M 1 T S þ B02 T ;

ð3Þ

0

1.0 95

109

94

96

T(K)

Fig. 3. (a) Resistivity (q), real (v 0 ), and imaginary (v00 ) component of ac susceptibility as a function of bath temperature T, during heating of the sample 6 · 6 · 11 mm3. (b) The corresponding variation of the temperatures TS, TR and their difference DT = TR  TS / Ce.

0

where M1, B1, M 1, and B 1 are thermal conductance, and the temperatures were expressed in the forms: TS = T0 exp (bt), TR = T0 exp (b 0 t), and T = T0t. We assumed that in the used temperature interval the thermal conductance bath-SS and bath-RS are equals B2 = B 0 2), and under transition-free conditions we have: b = b 0 . In this case the deviation from the symmetric thermal system remains negligible. So, at a constant rate of cooling or heating (oT/ot = constant), the transition can be extracted from the apparent heat capacity (CS) by subtracting the appropriate baseline of the transition-free heat capacity (CR) [7]. Susceptibility and DT curves recorded at cooling are very similar to results obtained at heating.

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110

4. Phenomenological model of the transition Irrespective of the underlying microscopic mechanism of superconductivity we try to present a possible phenomenological scenario of the transition. Our description starts from the nearly anti-ferromagnetic model of under-doped cuprates at room temperature. At cooling the PG phenomena appears at microscopic scale at the onset temperature T*
TC [11–13]. There is no consensus at the present time in the origin of the PG [14]. After different studies the PG can be interpreted in terms of oxygen related pairing associated with minor ( 6 pm) local structural changes [15], not only in YBCO but in each of the high-TC materials [16]. While in our SS both the distance and the dimension of the Cooper-pairs are  nm sized, at T* first the appearance of nanometre-sized diamagnetic clusters is favorised [10,17–20]. As a con-

TC

TC*

92 K

93.6 K

T << TC

sequence of quasifree electron (fermion) and tightly bound electron pair (boson) dominated fluctuations, the paired charges possess time and spatially dependent (partial) phase coherence [21–26]. This is an open question but we suppose that (at cooling) the continuous evolution in time (from s  ps coherence lifetime to 1) and space (from k  nm coherence length to 1) of the fluctuating (statistically distributed) paired charge structures (precursor superconducting clusters) could form the intermediary steps of the NS transitions (Fig. 4), which induces a PG in the single particle spectra [27–38]. With decreasing temperature the statistical distribution of incoherent (normal) and partial coherent clusters evolve in favor of coherence in time and size. Only when the total volume of precursor diamagnetic structures reaches the sensibility limit

Tonset χ 94.7...110 K

T*

T >> T*

140 K

(a)

volume ( % )

100

partial-coherent paired fermions (pseudogap) coherent incoherent hard core bosons nonbonding state of fermions (superconductor)

(normal)

λ

nm

mm

cm

µm

nm

0

τ

ps

s

min

s

ps

0

8

(b) 0

cm

(c)

ps

cm

8

time

mm

min

8

nm

8

size

mm

8

8

λ τ

Fig. 4. Stages of the normal-superconductor transition between T* and TC in hole-underdoped high-TC cuprates using as parameters: s––the coherence lifetime of the local paired charge carrier structures, and k––their coherence length, that is their size.

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of the used setup (here at T  94.7 K, but with other setup we measured 110 K susceptibility onset temperature [39]), the non-phase-sensitive macroscopic experimental methods begin to observe the presence of the partial coherent state. So, macroscopically T* can be only asymptotically determined. As a result of cooperative phenomena (coupling through domain walls by the proximity effect, or stimulation [40]) from the partial phase-coherent clusters bigger (mesoscopic: k  lm) and more stable (s  1 s) domains appear. The further development both in volume and stability, switches (at T C ) some domains from the partial to the stable (phase-locked) local (mesoscopic) coherent state. We suppose that this T C  T CC temperature is connected to a lattice distorsion. The crossover temperature T C separates two temperature region with different excitation dynamics [41,42] showing distinct differences from and similarities to the superconducting state. The Gaussian component of the spin-echo decay rate decreases below T C and undergoes a similar variation with TC as a function of doping level [43]. This result is corroborated by nuclear spin-lattice relaxation rate and Knight shift data [44,45]. Lowering further the thermal energy, the domains (passing over the granularity created Josephson junctions) condense into even bigger islands with longer coherence time. Due to interplay between mesoscopic precursors and normal regions the observed (even adiabatically developed) oscillations demonstrate that these states are in dynamic equilibrium not only at microscopic but also at mesoscopic level [39,46]. The fluctuations concerning the number, the dimension, and the degree of partial coherence of the fragments may be responsible for the broad hump of quasi-particle excitations [47]. The onset of macroscopic phase coherence (k = 1, s = 1 at R  0, or at oR/oT maximum, or at v00 maximum, or at ov 0 /oT maximum or at about 50% superconducting volume percentage [48–50]) is considered to be determined by random percolation between the already coherent mesoscopic islands. Continuing the temperature decrease below TC, the Meissner volume extends further to the detriment of the still remained pre-

111

cursor (partial coherent) and normal state regions (R remains zero, v 0 ,v00 and DT decrease).

5. Discussion and conclusions In our model the appearance of paired charge carriers with partial phase coherence is linked to the fluctuation of fermionic excitations. This is strongly supported by the reflectivity measurements of Kaindl et al. [51], and by [5], suggesting that fluctuation play a prominent role for coupling in the superconducting condensate. Mihailovic et al. in their model of phase coherence by percolating considered that in superconducting cuprates the onset of macroscopic phase coherence is determined by random percolating between mesoscopic Jahn–Teller clusters [50]. All these considerations are included in our model. But they give not a picture how the preformed pairs arises from fluctuation, and how they arrive by increasing partial phase coherence (in time and space) to the establishment of percolation. Chakraverty and Ramachrishnan suppose that the Cooper-pair droplets start forming at T  T* [21]. But they did not mention the dimension of droplets and did not consider the existence of partial coherence. By us the evolution of PG in time end space (between T* and TC) is connected only to the continuous increase of the coherence lifetime and coherence volume of the––by fluctuation statistically distributed––local paired charge carrier structures (partial coherent islands). This picture explains naturally also the appearance of a second characteristic PG temperature [52]. We suppose that the calorimetric critical temperature (DT maximum at TCC = 93.6 K) coincide with it. This temperature marks the appearance of the first (isolated) mesoscopic superconducting clusters. The topologically fixed grain structure may play a role in the initial formation of islands. But the dynamic evolution of coherent islands exceeds–– through Josephson junctions––the granularity created borders. It is to add that a normal-superconductor mosaic structure (without partial coherence) cannot explain neither the PG phenomena nor the observed difference between the diamagnetic level of

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precursor islands and the Meissner state region [38]. Since the PG begins with microscopic phenomena, its onset temperature can be determined only asymptotically with (our) macroscopic methods. The simultaneous measurements of different critical parameters of the pseudogap-superconductor transition permitted to detect that the calorimetric and susceptibility data give two different critical temperatures. The calorimetric critical temperature (TCC) marks the appearance of the first superconducting islands and by susceptibility measurements determined critical temperature (TC) marks the percolation of these islands.

Acknowledgment This work was partly supported by the Hungarian National Foundation for Scientific Research through the contract no. 037212.

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