Effect of annealing on a pseudogap state in untwinned YBa2Cu3O7−δ single crystals

The effect of annealing both in the oxygen atmosphere and at room temperatures on physical properties such as the pseudogap (Δ*(T)) and excess conductivity (σ′(T)) of untwined YBa2Cu3O7−δ (YBCO) single crystal with a small deviation from oxygen stoichiometry is studied. It was revealed that as the charge carrier density, nf, increases, Тс also slightly increases, whereas the temperature of the pseudogap opening, T*, decreases noticeably, which is consistent with the phase diagram (PD) of cuprates. The excess conductivity in the vicinity of Tc is represented by the Aslamazov-Larkin and Hikami-Larkin fluctuation theories, illustrating the three-dimensional to two-dimensional (i.e. 3D-2D) crossover with an increase in temperature. The crossover temperature T0 determines the coherence length along the c axis is ξc(0) = 0.86 Å, that is 2.6 times larger than for optimally doped YBCO single crystals with defects. Taking into account the short coherence length in high-temperature superconductors, in the model of free charge carriers the phase relaxation time of fluctuating Cooper pairs is determined, τφ (100 K) = (4.55 ± 0.4) · 10−13 s, which is slightly (1.2 times) larger than in well-structured YBCO films, and as in films, does not depend on nf. It is shown that Δ*(T) at different annealing stages practically does not change its shape. As in the well-structured YBCO films, Δ*(T) demonstrates maximum at Tpair~124 K which depends weakly on nf. However, the maximum value of Δ*(Tpair) increases with increasing nf, as it follows from the PD of cuprates. Comparing the experimental data with the Peters-Bauer theory we estimated the density of local pairs  ≈ 0.3 near Tc that is a common value for high-temperature superconductors.


Results and Discussion
Resistivity. The dependence on temperature of the resistivity (i.e. ρ(T) = ρ ab (T)) of untwined YBa 2 Cu 3 O 7−δ crystals are shown in Fig. 1. As ρ(T) of the samples differ insignificantly, to simplify the figure on the upper panel (Fig. 1a), only the resistive curve of sample A1 is shown. The corresponding dependences of ρ(T) for all three samples are shown as a three-dimensional (3D) graph in Fig. 1b. For temperatures above T* = (185 ± 0.5) K (A1), (182 ± 0.5) K (A2), (179 ± 0.5) K (A3) and up to 300 K, all the ρ(T) dependences are linear and are described by a gradient a = dρ/dT = 0.484 (A1), 0.488 (A2) and 0.478 (A3) μΩ⋅cm/K, which slightly changes with annealing. The gradient was calculated by approximating the experimentally derived curves and confirmed the linear behaviour of ρ(T) with a mean-root-square error of 0.009 ± 0.002 in the specified temperature range for all samples. As mentioned above, the PG temperature T* ≫ T c was defined as a temperature at which the resistive curve deviates downward from the linearity (Fig. 1). The more precise approach to determine T* with accuracy ± 1 K is to explore the criterion [ρ(T) − ρ 0 ]/aT = 1 39 (insert in Fig. 1a), where a designates the slope of the extrapolated normal-state resistivity, ρ N (T), and ρ 0 is its intercept with the y axis. Both methods give the same T*.
In the process of annealing, with increasing the oxygen content, T c slightly increases, and ρ(T) slightly decreases (Fig. 1b). This is not surprising, since the samples are actually on top of the PD. At the same time, T* decreases much more perceptibly, in full agreement with the PD of cuprates 3-5,13,18,21 (and Tables 1, 2 and 3). The main difference between the untwined YBCO and single crystal containing defects, presumably in the form of  www.nature.com/scientificreports www.nature.com/scientificreports/ twin boundaries (TB) 21,40 , is much higher T* value. Usually, in optimally doped YBCO single crystals with T c ~ 91.1 K, but containing defects in the form of TB, T* ~ 140 K 41 . It is assumed that the defects prevent the establishment of phase coherence of LPs (paired fermions) and, thus, effectively reduce T* 21,42 . At the same time, in well-structured YBCO films 11,20 , a sample with T c ~ 88 K has T* ~ 200 K, which is much closer to T* = 185 K, observed for the untwined YBCO single crystal A1 with T c = 91.6 K. Therefore, it can be assumed that, by their properties, YBCO single crystals, which do not contain TB, are closer to well-structured films. This conclusion is supported by the analysis of the results of the FLC and PG study.
Fluctuation conductivity. In the resistive measurements, PG is evident as a deviation of the resistivity ρ ab (T) = ρ(T), determined in the ab plane, from a linear dependence at high temperatures to smaller values (refer to Fig. 1). The result is excess conductivity expressed by σ′(  39 . Panel b. The plot of ρ(T) of the same single crystal for different annealing stages: A1 (blue squares), A2 (green dots) and A3 (red triangles). The solid lines determine ρ N (T), extrapolated to the low-temperature region. www.nature.com/scientificreports www.nature.com/scientificreports/

N N
where ρ N (T) = aT + b is the sample resistivity in the normal state that is extrapolated to the low temperature region 4,5,20,43 . As mentioned above, according to the model 14 , the linear dependence of ρ(T) above T* is the normal state of HTSCs that characterizes by the stability of the FS 2,3,14,19 . According to recent concepts [4][5][6][7][8][9][10][11]21 , a small value of the coherence length in conjunction with a quasi-layered structure of the HTSCs leads to the formation of a noticeable area of SC fluctuations on the ρ(T) in the proximity of T c , where σ′(Т) follows conventional fluctuation theories 11,20,[44][45][46] . At the same time, changes in oxygen content, the presence of impurities, and/or structural defects have a considerable impact on σ′(Т) and, accordingly, on the implementation of various FLC modes above T c 20, [47][48][49] . Fluctuation conductivity for (or of) all the samples was determined by analyzing the excess conductivity, which was calculated in the standard method in accordance with Eq. (1). The FLC analysis was performed within the model of local pairs 1,11,20,21 , in which the presence of paired fermions (LPs) in HTSCs is assumed in the temperature range T c < T < T* 1, [6][7][8][9]20 . Firstly, the mean-field temperature T c mf > T c needs to be determined limiting the region of critical fluctuations near T c , where the mean-field theory does not work 26 . In addition, T c mf determines the reduced temperature c mf that appears in all equations. From this it is clear that the correct determination of T c mf plays a key role in the calculations of both FLC and PG. At the vicinity of T c , the coherence length in the c axis (ξ c (T)) is greater than d. d ≈ 11.7 Å 50 is the c axis lattice parameter of the YBCO unit cell 25,46 . In this case, the FCPs associate throughout the superconductor and form a three-dimensional (3D) state of HTSC 20,25,46 . Therefore, at the proximity of T c , the FLC can be described by the 3D equation of the Aslamazov-Larkin (AL) theory 11,24 with the critical exponent λ = −1/2, which determines the FLC in any 3D system: Here σ′(Т) ~ ε −1/2 . It can be shown that σ′ −2 (Т) ~ ε~ T − T c mf and vanishes at T = T c mf (refer to Fig. 2), which enables the determination of both T c mf and ε with high accuracy 11,20,51 . Also in Fig. 2, the arrows show T c , the Ginzburg temperature T G , down to which the fluctuation theories are valid 46,47 , and the T of the 3D-2D crossover T 0 limiting the area of 3D fluctuations. Notably, above T 0 = 92.34 K (refer to Fig. 2), the data deviate to the right from the linear dependence, which indicates the presence of 2D Maki-Thompson (MT) contribution to the FLC 25,46 . Having determined ε, we construct the dependence lnσ′(lnε) (Fig. 3). Figure 3a shows the corresponding dependence for the base sample A1. Expectedly at the vicinity of T c , in the interval Т G -T 0 (ln(ε 0 ) = −5.21), the FLC is well modelled by the AL fluctuation contribution (3) for the 3D system. In double logarithmic coordinates this is the dashed line (1) with slope λ = −1/2. As mentioned above, it implies that the classical three-dimensional FLC materializes in HTSC for Т → Т с and ξ с (Т) > d 11,20,43 . Above the crossover temperature Т 0 ξ с (Т) < d 11,21,25,46 , and this is no longer a 3D regime. However, as before, ξ с (Т) > d 01 , where d 01 ≈ 3.5 Å is the separation of the conducting planes of CuO 2 in YBCO 50 . Thus, up to temperature Т 01 (ln(ε 01 ) = −2.8, Fig. 3) ξ с (Т) connects the inner planes of CuO 2 by means of the Josephson interaction 11,20,46 . This is the 2D FLC regime, which is perfectly approximated by the Hikami-Larkin theory 2D-MT equation for HTSCs 25 : Above Т 01 , the experimental points deviate downward from the theory (Fig. 3) implying that the classical fluctuation theories are not valid. Thus, Т 01 limits the region of SC fluctuations from above: ΔT fl = T 01 − T G . Conversely, T G limits the region of SC fluctuations from below. As a result, below T G , designated as ln(ε G ) in (Fig. 3a,b), the experimental points also deviate downward from the theory (Fig. 3), suggesting the transition to the range of critical fluctuations or fluctuations of the SC order parameter Δ just near T c , where Δ < kT 11,26 .
The thin curves (3) in the figure are constructed according to the Lawrence-Doniach equation (LD) 44 :      (2)) and LD (solid thin curve (3)). The T 01 (lnε 01 ) determines the range of the SC fluctuations, T 0 (lnε 0 ) is the temperature of the 3D-2D crossover and T G (lnε G ) is the Ginzburg temperature. Panel b: The same dependencies for all three samples: A1 -blue squares, A2 -green dots and A3red triangles.
www.nature.com/scientificreports www.nature.com/scientificreports/ The LD model works in case of defects in samples 21,42,51 . In our case, curves (3) lie far from the experimental points, which confirm the absence of defects (primarily TB) in our samples.
Notably, in this case the maximum distance between the MT curve (2) and the extrapolated AL straight line (1), Δlnσ′~0.1, which is typical for nonmagnetic YBCO 11,20 . In magnetic superconductors, such as SmFeAsO 0.85 , the MT curve (2) always passes much higher than the extrapolated AL straight line (1) 11 , and in this case Δlnσ′~0.8 31 . Such behavior indicates the presence of a magnetic interaction in HTSCs, which is clearly absent in our non-magnetic untwined single crystal.
In Fig. 3b the dependences of lnσ′(lnε) are shown for all three samples A1-A3. It is seen that with increasing T c all characteristic temperatures also vary slightly. The increase in the absolute value of lnε 0 results in decrease in ξ с (0) from 0.86 Å (А1) to 0.81 Å (А3) ( Table 1), in full agreement with the theory of superconductivity, where ξ~1/T c 26 .
In the above equations is the pair-breaking parameter, and the phase relaxation time τ ϕ is determined by the equation where А = 2.998·10 −12 s K. Here the factor β = 1.203(l/ξ ab ) with l being the mean free path and ξ ab the coherence length along the ab plane, takes into account the approximation of the clean limit (l > ξ) that consistently occurs in HTSCs due to the smallness of ξ(T) 25,44-46 .

Comparative analysis of phase relaxation time.
Having determined the parameters of the FLC analysis, it seems interesting to examine the physical meaning of the short coherence length ξ ab (0) in the framework of the simple two-dimensional model of free charge carriers [52][53][54] . This approach allows us to define a set of additional important parameters of the samples, including τ ϕ , which is actually the lifetime of the FCPs in the range of SC fluctuation. In HTSCs all parameters, including τ ϕ and Hall coefficient R H , are functions of temperature. Consequently, the corresponding calculations, including those in YBCO, are performed at T = 100 K, as is customary in the literature 20,[52][53][54] .
From the FLC analysis, using Eq. (7), we find the coupling parameter α, and then the pair-breaking parameter δ of Eq. (8), which is always ~ 2 20,54 , if all other parameters are correctly determined. Next, we calculate the parameter τ ϕ βТ (refer to Table 3), assuming in Eq. (9) ε = ε 0 20 . Since it is assumed that T = 100 K, in order to find τ ϕ (100 K), it is necessary to determine the coefficient β = 1.203(l/ξ ab ). For this it is necessary to know the mean free path l, which is determined by the density of charge carriers n f , and the coherence length in the ab plane (i.e. ξ ab ). The charge carrier density n f can be calculated from the values of the Hall coefficient R H , namely n f = r [l/(e R H )] 20,54,55 . Here e is the electron charge, and the coefficient r = <τ 2 >/<τ> 2 , where τ is the average time between collisions of charge carriers, actually determines the scattering mechanism in the normal state 55 . Since the scattering mechanism of normal charge carriers in HTSCs, especially in the PG state, is still rather uncertain 56-59 , we assume r = 1.
From the literature for optimally doped YBCO single crystals 3,60 and YBCO films with a close value of T c 20 , we find: R H (100 K) ≈ 2.4·10 −9 m 3 /C for sample A1, and, accordingly, n f = 2.6·10 21 см −3 ( Table 2). Continuing the analysis of sample A1, for the carrier density in the planes we obtain n s = n f d = 3.05·10 14 см −3 . Using the corrected value ρ(100 K) = ρ(100 K)·C 3D = 62.9 μΩcm 52,53 , we have μ H = r/(ρne) = 38.2 сm 2 /Vs for the mobility of the Hall carriers. Interestingly, found values of μ H ( Table 2) are in good agreement with those obtained in ref. 60 for YBaCuO untwinned single crystals. It was also shown 60 that the results of Hall-effect measurements are nor affected by the conduction of the Cu-O chains and the in-plane anisotropy of the CuO 2 planes. Now, using the formula l = (ħμ/e)(2πn s ) 1/2 , we find the mean free path of the charge carriers in A1: l = υ F τ ≈ 110 Ǻ, where υ F is the Fermi velocity. To continue the analysis, the mean value ξ ab (0) = 11.0 Ǻ was chosen for A1 from the literature [51][52][53] .
Nevertheless, the mean free path l and the Hall mobility μ H are about 2 times, and τ ϕ (100 K) is ~ 1.2 times more than in OD YBCO films 20 , which is most likely a specific property of untwined single crystals 60 . At the same time τ ϕ (100 K)/τ (100 K) ~ 4 ± 0.2, in excellent agreement with the theory 45 , which takes into account the clean-limit approximation (l > ξ), as mentioned above. It should be emphasized that, as in well-structured YBCO films with different n f 20 , in the untwined single crystals τ ϕ (100 K) is practically independent on n f . This result, apparently, can be considered as a common property of cuprates, at least of compounds based on YBCO.
Pseudogap analysis. In resistive measurements of cuprates the pseudogap is a deviation at T ≤ T* of the longitudinal resistivity ρ(T) from linearity in the normal phase 11,14,64 . This results to the realization of excess conductivity σ′(T) (refer to Eq. (1)). It is established that if there were no processes in HTSC leading to the opening of the PG at T*, then ρ(T) would preserve its linearity up to T ~ T c 4,11,14,[27][28][29][30] . It is obvious that σ′(T) is a consequence of the PG opening and should enclose details about the magnitude and temperature dependence of the PG 11,23 . Conventional fluctuation theories, modified by Hikami-Larkin 25 for HTSCs perfectly describe the experimental σ′(T) but only to about ~110 K 10,11,31 . For complete information about the pseudogap, an equation is needed that would describe the experimental curve in all temperature range from T* to T c and would contain PG explicitly. Such an equation was proposed earlier 12 : Additionally to T*, ε and ξ c (0), already defined above, Eq. 11 includes the numerical coefficient A 4 , which is equivalent to the C-factor in the FLC theory 20,51-53 , the theoretical parameter ε c0 * 65 and Δ* = Δ*(T G ). Here it is presumed that Δ*(T G ) = Δ(0) 66,67 with Δ being the order parameter of the sample in the SC state, as mentioned above. Importantly, all these parameters can be easily determined within the LP model 11,12,20,31 . We consider this for the case of A1 (refer to Figs 4,5). In the region lnε c01 < lnε < lnε c02 or, respectively, ε c01 < ε < ε c02 (113 K < T < 155 K), σ′ −1 ~ exp(ε) 65 . This feature turned out to be one of the main properties of most HTSCs 11,31,54 . As a result, in the specified temperature range, ln (σ′ −1 ) depends linearly with respect to ε with a slope α* = 5.8 (insert to Fig. 4a), which determines the parameter ε c0 * = 1/α* = 0.172 65 . This allows the determination of reliable values of ε c0 * for all samples, which, as established previously 12,31 , significantly impacts the dependence of the theoretical curves shown in Figs 4 and 5, at high T.
To determine A 4 , we approximate the experiment by the dependence σ′(ε) calculated by Eq. (11), in the vicinity of 3D AL-fluctuations near T c (refer to Fig. 4). lnσ′(lnε) is in essence a linear dependence of ε (i.e. the reduced temperature) and has a slope λ = −1/2. To find Δ*(T G ) used in Eq. (11), we construct the curve lnσ′(1/T) using all the parameters found 68 (refer to Fig. 5). Here, the gradient of the theoretical curve (11) is highly influenced by the value of Δ*(T G ) 12,20,31 . The best approximation is achieved when the Bardeen-Cooper-Schrieffer (BCS) ratio D* = 2Δ*(T G )/k B T c is 5.0 ± 0.1, which corresponds to the strong-coupling limit characteristic for YBCO. Accordingly, we obtain: Δ*(T G )/k B ≈ 229 K in good agreement with the experimental value Δ*(T G )/k B ≈ 228 K (see Fig. 6). Similar results were obtained for samples A2 and A3 (refer to Table 4).
Solving equation (11) for the PG, Δ*(T), we obtain 11,12 over the entire temperature range from T* to T c Here σ′(T) is the experimentally determined excess conductivity and the remaining parameters are already defined within the LP model (Tables 1 and 4). The fact that σ′(T) is given by Eq. (11) (refer to Fig. 4) suggests that Eq. (12) gives reliable values of both the magnitude and the temperature dependence of Δ*(T). The dependence Δ*(T) for sample A1, constructed by the formula (12), using the following parameters extracted from the experiment: T c mf = 91.84 K, T* = 185 K, ξ c (0) = 0.86 Å, ε c0 * = 0.17, A 4 = 33, Δ*(T G )/k B = 229 K, shown in Fig. 6. Also shown are the dependencies Δ*(T) for samples A2 and A3, calculated in a similar way with the corresponding set of parameters given in Tables 1 and 4.
All the curves in Fig. 6 have the shape typical for YBCO films, with a maximum at Т = T pair ≈ 124 K, which is close to T pair ≈ 130 K usually observed in well-structured YBCO films 12,54 , and a minimum at Т ≈ Т 01 31,41 . It can be www.nature.com/scientificreports www.nature.com/scientificreports/ seen that, in full accordance with the phase diagram of cuprates, Δ* max (Т pair )/k B expectedly increases from 258 K (A1) to 270 K (A3) along with an increase in n f . The BCS ratio D* = 2Δ*(T G )/k B T c also increases from 5.0 to 5.3. At the same time, T pair practically does not change (Table 4), which is reasonable, given the high T c of the samples.   www.nature.com/scientificreports www.nature.com/scientificreports/ As mentioned above, according to the theory of systems with low n f 8,9,[27][28][29][30] , above T pair , LPs must exist in the form of strongly bound bosons, which obey BEC. Below T pair the LPs must be converted to FCPs, which are subject to BCS theory. Thus, T pair separates both regimes [8][9][10][11][12][28][29][30] . The minimum Δ*(T) at T ≈ Т 01 is also observed on all HTSCs, including pnictides 11 and single crystals of FeSe 69 . Accordingly, when approaching to T c , the maximum Δ*(T) always occurs just below T 0 , and the minimum at T = T G 31,41 (inset in Fig. 6). Below T G , there is an abrupt jump Δ*(T) at Т → T c mf , however, this is already a transition to the region of critical fluctuations, where the LP model does not work. Thus, the approach in the framework of the LP model makes it possible to determine the exact values of T G and, as a consequence, to obtain reliable values of Δ*(T G ) ≈ Δ*(T c ) ≈ Δ(0) 12,54,65-67 (Table 4).
It is noteworthy that the shape of the Δ*(T) curves near T c , shown in the inset to Fig. 6, is very similar to the temperature dependence of the density of local pairs in HTSCs, <n ↑ n ↓ >, calculated within the three-dimensional attractive Hubbard model for different temperatures, interactions U > 0, and filling factors (the Peters-Bauer model (PB) 6 ). Besides, in the calculations the hopping t and the bandwidth W = 12t were used as energy scales. Taking into account the fact that Eq. (12) contains information on the density of local pairs, we tried to compare our data with the results of the PB theory ref. 6 . Having normalized the temperature and PG, respectively, by T*(T) and Δ* max , and having adjusted the parameters, we obtained good agreement with the PB theory (refer to Fig. 7). In the process of fitting, Δ*(T G )/Δ* max was coincided with the minimum value <n ↑ n ↓ > at the lowest T, and Δ*(T 0 )/Δ* max with the maximum value. Both temperatures in Fig. 7 are indicated by arrows. On the theoretical curve 2, these temperatures are indicated by inclined arrows. Importantly, the same fitting factors were used for all samples.
After fitting, good agreement was found between the experimental Δ*(T)/Δ* max and the theory of PB with the lowest interaction parameter U/W = 0.2, which corresponds to the density of local pairs <n ↑ n ↓ > ≈ 0.3 near T c (Fig. 7). As in the PB theory, the density of LPs in our samples increases (from <n ↑ n ↓ > (T G ) ≈ 0.292 (A1) to <n ↑ n ↓ > (T G ) ≈ 0.305 (A3)) with an increase in the interaction energy, which corresponds to an increase in the BCS ratio D* in our case. As the temperature increases both <n ↑ n ↓ > and our data, as expected, decrease (refer to Fig. 7), which seems reasonable. Indeed, the number of FCPs should decrease along with T [3][4][5][6]11 . Importantly, at U/W = 0.2 the experimental data is consistent with theory in a wide temperature range, actually in the whole range of SC fluctuations. Whereas, if we compare the data with the theory for larger values of U/W (curves 2 and 3), the data will deviate from the theory already at T/W ~ 0.2. Notably, <n ↑ n ↓ > ≈ 0.3 was also obtained for FeSe single crystals near T c 69 . Apparently, such a density of LPs near T c is typical of all HTSCs.
Conclusions. Using the LP model, we have studied the effect of annealing on the temperature dependences of FLC and PG in untwined OD YBa 2 Cu 3 O 7−δ (YBCO) single crystal with a slight increase in the oxygen index (7-δ). The increase in oxygen content and, respectively, n f was carried out by annealing the single crystal both in an oxygen atmosphere (sample A1) and by exposure at room temperature (samples A2 and A3). It is found that with increasing n f in the sample, T c increases, and resistivity decreases. As expected, the increase in T c is rather small, since n f actually corresponds to the maximum of the phase diagram (PD). At the same time, T* decreases more significantly (from 185 K to 179 K), which fully corresponds to PD of the cuprates. The first difference from optimally doped YBCO single crystals containing defects in the form of TB, where T* ~ 140 K 41 , is quite large T* = 185 K (A1). It is assumed that defects interfere with the establishment of phase coherence of LPs (paired fermions) and, thus, effectively reduce T* 21,42 . Importantly, in well-structured YBCO films, the sample with T c ~ 88 K has T* ~ 200 K, which is much closer to T* = 185 K, observed for A1. This result suggests the conclusion that the investigated properties of untwined YBCO single crystals are noticeably closer to the well-structured films, which was confirmed by the results of analysis of both FLC and PG. The present study demonstrated that in the range of SC fluctuations near T c FLC is consistent with the fluctuation theories of Aslamazov-Larkin (3D term) and Hikami-Larkin (2D-MT term), and demonstrate a 3D-2D crossover when the temperature is increased. T 0 determines ξ c (0) = 0.86 Å (A1), which is 2.6 times higher than in optimally doped defective YBCO single crystals. This is most likely due to the fact that the range of FLC is very small: ΔT fl = T 01 − T G = 97.4 K− 91.9 K = 5.5 K (А1), which, however, is 3.7 times more than in single crystals containing defects, where ΔT fl ~ 1.5 K 41 . According to the theory 8,9 , in the range of SC fluctuations, cuprates retain the finite value of superfluid density n s (T), and the FCPs behave mainly as SC Cooper pairs, but without long-range order (known as "short-range phase correlations") that is confirmed by a number of experiments 66,70,71 . This result once again underlines a noticeable difference in the behavior of YBCO single crystals with and without defects.
T 01 also determines d 01 (distance between the conducting CuO 2 planes). In this case, regardless of the density of charge carriers, d 01 ~ 3.5 Å, in agreement with the determinations of structural studies 50 . This result, together with the presence of the fluctuation contribution of 2D-MT in FLC, confirms the good structure of the samples. It should be also noted that with increasing Т с , ξ с (0), as it was found, decreases from 0.86 Å (A1) to 0.81 Å (A3) ( Table 1), that is ξ~1/T c . The result is fully consistent with the theory of superconductivity, where ξ~ ћv F /πk B T c 26 , because, as well as in the well-structured YBCO films 20 , v F is almost independent on n f (Table 3).
Having determined the parameters of the FLC analysis, the physical meaning of the short coherence length ξ ab (0) in HTSCs was examined in the framework of the simple two-dimensional model of free charge carriers [52][53][54] . This approach allowed us to define a set of additional important parameters of the samples, including τ ϕ , which is actually the FCPs lifetime in the range SC fluctuations. Most of the calculated parameters are in good agreement with similar data obtained for OD YBCO 20,[52][53][54]58,59 . It is shown that found τ ϕ (100 K) = (4.49 ± 0.06) 10 −13 s (Table 3) is only slightly (~1.2 times) more than in well-structured YBCO films, but, as in films, in fact does not depend on n f . Accordingly, τ ϕ (100 K)/τ(100 K) ~ 4 is in excellent agreement with the Bieri-Maki-Thompson theory, which takes into account the approximation of the clean limit (l > ξ) 41 , which always takes place in HTSCs due to the small value of ξ(T). A certain role in this may be played by the presence of structural and kinematic anisotropy in the system [72][73][74][75] .
The PG analysis has shown that the Δ*(Т) curves (Fig. 6) have the shape characteristic of YBCO films 12,54 , with a clear maximum at Т = T pair ≈ 124 K and a minimum at Т ≈ Т 01 31,41 . According to the theory of systems with low n f 8,9,[27][28][29][30] , T pair separates both BEC and BCS regimes of LPs formation [8][9][10][11][12][27][28][29] . In full accordance with the PD of cuprates, Δ* max (Т pair )/k B expectedly increases from 258 K (A1) to 270 K (A3) along with an increase in n f and T c (Fig. 6). The BCS ratio D* = 2Δ * (T G )/k B T c also increases from 5.0 to 5.3, suggesting the expected increase in bonding energy of the LPs 11,27,30 . At the same time, T pair practically does not change (Table 3), which is understandable due to the high T c of the samples. When approaching T c , the PG curves show behavior being typical for all HTSCs with a maximum Δ*(T) just below T 0 and a minimum value at T = T G (inset in Fig. 6). Thus, the approach within the LP model makes it possible to determine the exact values of T G and, as a consequence, to obtain reliable values of Δ*(T G )≈ Δ(0) 12,54,[65][66][67] (Table 4).
Finally, the shape of the Δ*(T) curves near T c (Fig. 6), was found to be very similar to the temperature dependence of the density of LPs in HTSCs <n ↑ n ↓ > calculated within the three-dimensional attractive Hubbard model for different values of temperature, interaction, and filling factors (the Peters-Bauer model (PB) 6 ). For the first time, an estimation of the density of local pairs <n ↑ n ↓ > in the optimally doped YBCO was carried out by comparing the experimental data of Δ*(T) with the PB theory (Fig. 7). It was determined that <n ↑ n ↓ > ≈ 0.3 near T c , which, likely, is a typical value for HTSCs. www.nature.com/scientificreports www.nature.com/scientificreports/ Experimental methods. The YBa 2 Cu 3 O 7−δ single crystals were grown by the solution-melt technology in a gold crucible, according to the procedure described in 76,77 . As is well known, with an increase in the oxygen content a tetra-ortho structural transition occurs in YBa 2 Cu 3 O 7−δ 78 , which leads to a twinning of the single crystal and the creation of twin boundaries (TB), minimizing its elastic energy 76 . To obtain an untwined sample, the crystal was untwined into a special cell at a temperature 420 °C and a pressure 30-40 GPa, according to the procedure proposed previously 79 . In order to obtain the uniform controlled oxygen content, the crystal after untwined was repeatedly annealed for seven days in an oxygen atmosphere at 420 °C 80 .
Rectangular crystals of about 1.7 × 1.2 × 0.2 mm were selected from the same batch to perform the resistivity measurements. The smallest parameter of the crystal corresponds to the c-axis. The experimental geometry was selected so that the transport current vector was parallel to the ab-plane. The four-point probe technique with stabilized measuring current of up to 10 mA was used to measure the ab-plane resistivity, ρ ab (T) [40, and references therein]. Silver epoxy contacts were glued to the extremities of the crystal in order to produce a uniform current distribution in the central region where voltage probes in the form of parallel stripes were placed. The procedure for making contacts was completed by adding silver wires with a diameter of 0.05 mm and a three-hour annealing at a temperature of 200 °C in an oxygen atmosphere. Contact resistances below 1 Ω were obtained. The temperature was measured using a Pt sensor having an accuracy of about 1 mK. The measurements were carried out in the temperature drift mode on two opposite directions of the transport current to eliminate the influence of the parasitic signal. The critical temperature, T c , was determined by extrapolation of the linear part of the SC transition to its intersection with the axis T 4,5,20,21 .
In order to change the oxygen content and, and obtain the appropriate values of n f and T c , the sample was annealed for two days in an oxygen flow at temperature 620 °C. After annealing, the crystal was cooled to room temperature within 2-3 minutes, mounted in a measuring cell, and cooled to the temperature of liquid nitrogen for 10-15 minutes (sample A1). All measurements were carried out by heating the sample. To study the effect of annealing at room temperature, the sample after the first measurements of ρ(T) was kept for 20 hours at room temperature (sample A2) and then repeated measurements were performed. The following measurements were carried out after additional exposure of the sample at room temperature for three days (sample A3). After this procedure, not only increased T c and decreased ρ(Т), but unlike the data of the previous work 7 , the PG temperature T* also decreased noticeably, whereas the value of PG increased, which is in full agreement with the PD for YBCO (refer to [3][4][5]13,21 and references therein).