Pseudogap and Local Pairs in High- T c Superconductors

The temperature (T) dependence of pseudogap (PG) Δ * (T) is calculated for Bi2201 within the local pair (LP) model [1, 2]. The model is based on analysis of the excess conductivity derived from resistivity experiments in hightemperature superconductors (HTSs) and supposes the local pairs, which are formed in HTSs at T well above Tc, to generate a pseudogap. To confirm the conclusion, Δ * (T) is compared with the temperature dependence of the loss of the spectral weight W(EF)(T) measured by ARPES for the same Bi2201 sample [3]. A good agreement between Δ * (T) and W(EF)(T) is found, confirming the local pairs to be one of the most likely cause for the PG formation. Температурну залежність псевдощілини Δ * (T) розраховано для Bi2201 в межах моделю локальних пар [1, 2]. Цей модель засновано на аналізі надлишкової провідности, яка одержується з експериментів щодо питомого опору у високотемпературних надпровідниках (ВТНП), і припускає наявність локальних пар, що мають утворюватися у ВТНП за температур T, набагато більш високих, ніж Tc, формуючи псевдощілину. Для підтвердження цього висновку Δ * (T) порівнюється з температурною залежністю втрати спектральної ваги W(EF)(T), що вимірюється ARPES в тому ж самому зразку Bi2201 [3]. Одержано хороший збіг Δ * (T) і W(EF)(T), який підтверджує, що локальні пари є однією з найбільш ймовірних причин утворення псевдощілини. Температурная зависимость псевдощели Δ * (T) рассчитана для Bi2201 в рамках модели локальных пар [1, 2]. Эта модель основана на анализе избыточной проводимости, которая получается из экспериментов по удельному сопротивлению в высокотемпературных сверхпроводниках (ВТСП), и подразумевает наличие локальных пар, которые должны образовываться в ВТСП при температурах T, намного более высоких, чем Tc, формируя


Introduction
The discovery of superconductivity in copper oxides with an active CuO 2 plane [1] at temperatures of the order of 100 K is undoubtedly one of the most important achievements of the modern solid state physics. However, even more than twenty six years later since the discovery the physics of the electronic processes and interactions in high-temperature superconductors (HTS's) and, in particular, the superconducting (SC) pairing mechanism, resulting in such high T c 's, where T c is the superconducting transition (critical) temperature, still remain controversial [2]. This state of affairs is due to the extreme complexity of the electronic configuration of HTS's, where quasi-two-dimensionality is combined with strong charge and spin correlations [3,4,[6][7][8][9][10][11][12].
The Chapter addresses the problem of the PG which is believed to appear most likely due to the ability of a part of conduction electrons to form paired fermions (so-called local pairs) in a high-T c superconductor at T ≤ T * [6,13,[22][23][24][25][26][27]

Theoretical background
There are two different approaches to the question of the mechanisms for SC pairing of charge carriers in cuprates and therefore the physical nature of the PG [2,27]. In the first approach, pairing of charge carriers in HTS's is of a predominantly electronic character, and the influence of phonons is inessential [3-5, 28, 29]. In the second approach, pairing in HTS's can be explained within the framework of the Bardeen-Cooper Shieffer (BCS) theory, if its conclusions are extended to the case of strong coupling [12,[30][31][32]. However, it gradually became clear that aside from the well-known electron-phonon mechanism of superconductivity, due to the inter-electronic attraction by means of phonon exchange [33,34], other mechanisms associated with the inter-electronic Coulomb interaction can also exist in HTS's [3-8, 10, 12, 35, 36]. That is why it is not surprising that the systems of quasiparticle electronic excitations, where factors other than phonons and excitons resulting in inter-electronic attraction and pairing are considered, are studied in a considerable number of theoretical investigations of HTS's. Some examples are charge-density waves [7,37,38], spin fluctuations [3,11,[39][40][41], "spin bag" formation [42,43], and the specific nature of the band structure -"nesting" [44]. The main distinguishing feature of these investigations compared to the conventional superconductors is the more detailed study of the models based on the existence of strong interelectronic repulsion in the Hubbard model which can result in anisotropic d-pairing [35,45]. Attempts have also been made to construct anisotropic models of high-temperature superconducting (HTS) systems with different mechanisms of interelectronic attraction [12,46,47]. Unfortunately, the consensus between both approaches has not been found. As a result, up to now there is still no completed fundamental theory to describe high-T c superconductivity as a whole and to clarify finally the PG phenomenon [2].

Pseudogap in HTS's
Gradually it became evident that a high critical temperature is by no means the only property that distinguishes HTS's from conventional low-temperature superconductors. Another, property of cuprates, which attracts much attention, is the PG state [2,8,15,26,36]. All experiments convincingly show [8] that as the charge-carrier density decreases relative to its value in OD samples, a completely unusual state, where the properties of the normal and superconducting phases appear together [48], arises in HTS systems in a sizable temperature interval above T c [8,15,26,27,[49][50][51][52][53][54]. From the very discovery of the PG state some authors called this state a "pseudogap phase". However, for example, as Abrikosov notes [55], this state actually cannot be interpreted as some new phase state of matter, since the PG is not separated from the normal state by a phase transition. At the same time it can be said that HTS system undergoes a crossover at T ≤ T * [56]. Below T * >> T c , for reasons which have still not been finally established, the density of quasiparticle states at the Fermi level starts to decrease [57][58][59]. That is why the phenomenon has been named a "pseudogap" The number of papers devoted to the problem of the pseudogap in HTS's is extraordinarily large (see Refs. [6-8, 12, 15, 84] and [40,49,50,53,54,56,59] and references therein) and new papers are constantly appearing [38,41,47,[60][61][62][63]. It seems to be reasonable as it is completely clear that a correct understanding of this phenomenon can also provide an answer to the question of the nature of high-temperature superconductivity as a whole. Among many other papers it is worth to mention the most radical model as for the nature of high-temperature superconductivity and a PG in cuprates. It is the Resonating Valence Bonds (RVB) model proposed by Anderson [64,65], which describes a spin liquid of singlet electronic pairs. In this model, largely relies on the results obtained using one-dimensional models of interacting electrons, the low-temperature behavior of electrons differs sharply from the standard behavior in ordinary three-dimensional (3D) systems. An electron possessing charge and spin is no longer a well-defined excitation. So-called charge and spin separation occurs. It is supposed that spin is transferred by an uncharged fermion, called a spinon and charge -a spinless excitation -by a holon. In the RVB model both types of excitations -spinons and holons -contribute to the resistivity. However, the holon contribution is considered to be determining, while spinons, which are effectively coupled with a magnetic field H, must determine the temperature dependence of the Hall effect. Even though the RVB model led to a series of successes [65,66], it is dificult to think up the physics behind the processes which could lead to the charge and spin separation especially in quasi-two-dimensional systems, which cuprate HTS's are. Nevertheless, the RVB model contains at lest one rational idea, namely, it is supposed that two kinds of quasi-particles with different properties have to exist in the high-temperature superconducting (HTS) system at T below T * . In the RVB model such particles are spinons and holons.
However, even though researchers have made great efforts in this direction, the physics of the PG phenomenon is still not entirely understood (see Ref. [2] and references therein). That is why, we have eventually to propose our own Local Pair (LP) model [13,27] developed to study a pseudogap Δ * (T) in high-temperature superconductors and based on analysis of the excess conductivity derived from resistivity experiments. We share the idea of the RVB model as for existence of two kinds of quasi-particles with different properties in HTS's below T * . But in our LP model these are normal electrons and local pairs, respectively. I will frame our discussion in terms of the local pairs, and try to show that this approach allows us to get a set of reasonable and self-consistent results and clarify many of the above questions.

The main considerations as for local pair existence in HTS's
There are several considerations leading to the understanding of the possibility of paired fermions existence in HTS's at temperatures well above T c which I am going to discuss now. It is well known, that a pseudogap in HTS's is manifested in resistivity measurements as a downturn of the longitudinal resistivity ρ xx (T) at T ≤ T * from its linear behaviour above T * (Fig.1). This results in the excess conductivity σ (T) = σ(T) − σ N (T), which can be written as ( 1 ) Here ρ(T) = ρ xx (T) and ρ N (T)=αT+b determines the resistivity of a sample in the normal state extrapolated toward low temperatures.  (Table I); dashed line represents ρ N (T).
This way of determining ρ N (T), which is widely used for calculation σ (T) in HTS's [2], has found validation in the Nearly Antiferromagnetic Fermi Liquid (NAFL) model [11]. The question of whether the appearance of excess conductivity σ (T) in cuprates can be wholly attributed to fluctuating Cooper pairing or whether there are other physical mechanisms responsible for the decrease of ρ xx (T) at T < T * is one of the central questions in the modern physics of HTS's. To clarify the issue it seems reasonable to probe the PG by studying the fluctuation (induced) conductivity (FLC). The study of FLC provides the relatively easy but rather effective method which directly examines the possibility of paired fermions arising at temperatures preceding their transition into the SC state [67,68].
Both Aslamasov-Larkin (AL) [69] and Maki-Thompson (MT) [70,71] coventional FLC theories have been modified for the HTS's by Hikami and Larkin (HL) [72]. In the absence of a magnetic field the AL contribution to the FLC is given by the expression Correspondingly, the HL theory gives for the MT fluctuation contribution the equation In both equations is the coupling parameter, d 11.7 Å in YBCO, is the distance between conducting layers, is the pair-breaking parameter, and ξ c is the coherence length along the c-axis, i.e. perpendicular to the CuO 2 conducting planes. The factor β = 1.203(l / ξ ab ), where l is the electron mean-free path, and ξ ab is the coherence length in the ab plane, takes account of the approach to the clean limit introduced in the Bierei, Maki, and Thompson (BMT) theory [73]. Correspondingly, is the reduced temperature in HTS's.
Here T m f c > T c is the critical temperature in the mean-field approximation, which separates the FLC region from the region of critical fluctuations or fluctuations of the order parameter Δ directly near T c , neglected in the Ginzburg-Landau (GL) theory [74,75]. Hence it is evident that the correct determination of T m f c is decisive in FLC calculations.
Equation (3) actually reproduces the result of the Lawrence-Doniach (LD) model [76], which examines the behavior of the FLC in layered superconductors, which cuprates and FeAs-based superconductors actually are. In the LD model, and hence in the HL theory, it is proposed that a Josephson interaction is present between the conducting layers. This occurs in the 3D temperature region, i.e., near T c , where ξ c (T) > d. Thus, according to the HL theory the AL fluctuation contribution predominates near T c . Correspondingly, the MT mechanism predominates for k(T − T m f c ) >>h/τ φ , where two-particle tunnelling between conducting layers is impossible, since ξ c (T) < d (2D fluctuation region) [77]. Thus, the HL theory predicts the alteration of the electronic dimensionality of the HTS sample leading to a 2D-3D crossover, as T approaches T c . Simultaneously the physical mechanism of superconducting fluctuations changes too resulting in MT-LD crossover. In accordance with the HL theory, the 2D-3D crossover occurs at where it is assumed that α=1/2, i.e.
Thus, now ξ c (0) can be determined, since ε 0 is a measured reduced crossover temperature. Correspondingly, the MT-LD crossover occurs at a temperature at which δ α [72], which gives In accordance with our results [67], it should be the same temperature T 0 . It makes it possible to determine τ φ which is the phase relaxation time (lifetime) of fluctuating pairs. The evaluation of τ φ in comparison with transport relaxation time τ of charge carriers measured by electrical conductivity, is a principal contribution to understanding the physics of transport properties. Whether τ φ > τ or τ φ ≈ τ is important in view of the controversy over the Fermi-liquid or non-Fermi-liquid nature of the electronic state in HTS's [9,64,77]. Thus, the study of FLC can yield information about the scattering and fluctuating pairing mechanisms in HTS's as T draws near T c .
As it was shown in our study of FLC [2,67,68], for optimally doped YBCO the interval T c < T < T c0 = (110 ± 5) K is precisely that temperature region in which the temperature dependence of the resistivity, and consequently of the excess conductivity, is governed by the superconducting fluctuations leading to the onset of fluctuation conductivity which is described by the conventional fluctuation theories [69][70][71][72], as mentioned above. It means that fluctuating Cooper pairs have to exist in cooprates up to very high temperatures, namely, up to ∼ 130 K in YBCO [27,67,68,78] and up to ∼ (140 − 150) K in Bi compounds [16,52,53].
The conclusion has subsequently been shown to be consistent with results of several other research groups which will be briefly discussed now. 1. Kawabata et al. [78] has made a number of small (D ∼ 3μkm) holes in the slightly underdoped YBCO film by means of photolithography and then applied a magnetic field. Expected magnetic flux quantization was observed up to T pair ∼ 130 K. The important point here is that period of oscillations corresponds to the charge of Q = 2e evidently suggesting the electronic pairing in this temperature range. 2. In tunneling experiments by, e.g., Renner et al. [79], the peculiarities of measured differential conductivity dI/dV observed in the SC part of the PG in Bi2212 compounds [13] were found to persist up to temperatures well above T c , and disappeared only at T = T pair ≈ 140 K. But the wide maximum corresponding to the non-SC part of the PG was observed up to T * ≈ 210 K. 3. It has subsequently been shown to be consistent with results of other groups dealing with the tunneling measurements [50][51][52][53][54]. Thus, in tunneling experiments by Yamada et al. performed on Bi compounds too [52], the temperature dependencies of the SC gap and PG, equal to the positions of the tunnel conductivity peaks, similar to that obtained in Ref [79], were studied for the three Bi2223 samples with different doping level. The noticeable increase of the PG in temperature interval from T c up to T pair ≈ 150 K (SC part of the PG), similar to that obtained in our experiments with YBCO films [27], was found for all three samples [52]. However, in accordance with the LP model [13,27], the peaks, which have the SC nature, as well as corresponding PG values, are smeared out above T pair , suggesting expected transition into non-SC part of the PG. 4. Eventually, Yazdani [16] was able to get the direct image of the local pair SC clusters in Bi compounds up to approximately 140 K using the novel STM technique (Fig. 2). As it is clearly seen in the figure obtained for an optimally doped Bi2212 sample (T c = 93 K), the state of the system at 100 K is almost the same as that below T c . At 120 K the picture evidently changes, but the local pairs still form the SC cluster which determines the collective (superconducting-like) behavior of the system in this SC part of the PG. Even at 140 K the SC nuclei are distinctly seen in the figure. But there are no closed clusters now. As a result, no collective behavior of the system is observed above T pair ≥ 140 K, as it was shown in the tunneling experiments [52]. In accordance with the LP model, it is a non-SC part of a PG above T pair [13]. Taking all these experimental results into account, the existence of paired fermions (local pairs) in the SC part of the PG, i.e, in the temperature interval from T c up to T pair , is believed to be well established now [13]. Additionally, T pair is found to amount to 130, 140 and 150 K for Y123, Bi2212, and Bi2223, respectively [13,80].

Properties of the systems with low and reduced current-carrier density
Before to proceed with a question what would happen with the local pairs in the non-SC part of the PG above T pair [13], let us have a look once again at the resistivity curve ( Fig.1) obtained for our slightly underdoped YBCO film (sample F1, Table I). Three representative temperatures, at which ρ(T) noticeably changes it slop, are distinctly seen on the plot. The first temperature is T * = 203 K at which the local pairs are believed to appear [2,27]. The second one is T on ≈ 89, 4 K corresponding to the onset of the superconducting transition. The last one is T c = 87.4 K at which the local pairs have to condense [6,23,24]. One basic question is whether any particularities affect the slop of the experimental curve around T pair ∼ 130 K.
The answer is completely negative. Indeed, the resistivity smoothly evolves with temperature and shows no peculiarities up to T * . The fact suggests that nothing happens with the pairs at T pair . Thus, one may draw a conclusion that if there are paired fermions in the sample below T pair they also have to exist at T > T pair , i.e., up to the very T * . The point of view where the appearance of a PG in HTS's is due to the paired fermions formation at T c < T < T * gradually gaining predominance [15,27,47,[81][82][83][84]. The possibility of the long-lived pair states formation in HTS's in the PG temperature range was justified theoretically in Refs. [26,85,86]. Nevertheless, the question of whether or not paired fermions can form in HTS's in the whole PG temperature range still remains very controversial. Indeed, it seems unlikely that conventional fluctuating Cooper pairs [33] are formed at temperatures T * > 200 K [13,27] especially considering the fact that the coherence length in HTS's is extremely short (ξ ab (T ≤ T * ) (10 ÷ 15) Å) [2,6,8,11,12,15].
We have, however, to keep in mind that we are dealing with the systems with low and reduced charge-carrier density n f , as mentioned above. It has been shown theoretically [6,[23][24][25][26]85] that such systems acquire some unusual properties compared to conventional superconductors. In conventional superconductors it is assumed that the chemical potential μ = ε F , where ε F = E F is the Fermi energy, and their relation actually depends on nothing.
In the systems with low and reduced n f the chemical potential μ becomes a function of n f , T and the energy of a bound state of two fermions, [23][24][25][26]. Here ξ b is the scattering length in the s channel, and m is the mass of fermions with a quadratic dispersion law ε(k) ∼ k 2 [23][24][25]. In the case of HTS's it is believed that ξ b (T) equals to the coherence length of a superconductor in the ab plane, ξ ab (T), and m = m * which is an effective mass of quasi-particles [11,47]. (m * ∼ 4.7m 0 in nearly optimally doped (OD) YBCO [9,11,67,68]. Thus, the ε b becomes an important physical parameter of a Fermi liquid and determines a quantitative criterion for dense ( 1 and μ = ε F in the first case, and k F ε b 1 and μ = −|ε b |/2 ( = ε F ) in the second one which correspond to the strong coupling [61,84,86]. Consequently, in the strong-coupling limit μ is to be equal to approximately ε b /2. It should be also noted, that in this case the paired fermions have to appear in the form of so-called strongly bound bosons (SBB) which satisfy Bose-Einstein condensation (BEC) theory [6,[23][24][25][26][84][85][86][87][88][89]. In accordance with the theory, the SBB are extremely short but very tightly coupled pairs. As a result, the SBB have to be local (i.e. not interacting with one another) objects since the pair size is much less than the distance between the pairs. Besides, they cannot be destroyed by thermal fluctuations, and consequently may form at very high temperatures.
It is clear that some other parameters, including the mean-field critical temperature T m f c and temperature dependence of the SC order parameter Δ(T) have to change too. Analysis shows that in conventional superconductors with a high fermion density T m f c is identical to the BCS theory value [6,23,24,33]. [6,24,25]. The latter relation means that in this case T m f c characterizes not the condensation temperature T c but rather the temperature at which the fermions start to bind into pairs, i.e., T * . An equation for Δ(T) in a form convenient for comparing with experiment was obtained in Ref [87]. In this case the temperature dependence Δ(T) was calculated on the basis of the crossover from BCS to BEC limit for different values of the parameter μ/Δ(0) = x 0 , where Δ(0) is the value of the SC order parameter at T=0: Equation (2) determines how the character of Δ(T) changes when the parameter x 0 changes from 10 (BCS limit) to -10 (BEC limit) (Fig. 12). As it is shown in Ref. [88] the character of the pseudogap temperature dependence Δ * (T) in HTS's has to change in the same manner as the charge-carrier density decreases.

The model of the local pairs (Local Pair model)
For obvious reasons, the question of which density should be regarded as low or high has not been posed for ordinary metals, and for a long time the question of a supposed BCS-BEC transition with decreasing n f [74] was only of theoretical interest. The situation changed dramatically after the discovery of HTS's [1], where the charge-carrier density n f is much lower than in conventional superconductors [8,9,11,12,15,27], as discussed above. It means that in HTS's the mentioned above strongly bound bosons have to exist. Besides, the coherence length in the ab plane, ξ ab (T) is extremely short in HTS's, especially at high temperatures [2,6,8,11,12,15] [67,68]). It leads to the very strong bonding energy ε b in the pair [23][24][25] which is an additional requirement for the formation of the SBB [6,[24][25][26]. Taking all above considerations into account, one may draw a conclusion that at high temperatures (T ≤ T * ) the local pairs in HTS's, which are believed to generate a pseudogap [2,6,13,26,89], have to be in the form of the SBB [2,6,13,26,27,47,81,[85][86][87][88][89]. This is the first basic assumption of the LP model [2,27]. This condition is realized just in the underdoped cuprates (see Ref. [2] and references therein) and new FeAs-based superconductors [17,20]. But, strictly speaking, the presence or absence of a PG in FeAs-based HTS's still remain controversial [17,21].
This assumption is supported by the fact that in accordance with the theory [6,23,24,26] fermions start to bind into pairs at T * , whereas the local pairs (or SBB) formed in the process may condense only for T c << T * , which at first glance seems to be in complete agreement with experimental observations [2,12,80,90,91]. But, the non-interacting SBB cannot be condensed at all [6,23,24,85,86], and this is a point. Eventually, just the value of ξ ab (T) = ξ ab (0) (T/T c − 1) −1/2 will determine the system behavior below T * [2,6,24,25,[86][87][88][89]. As temperature lowers, ξ ab (T) has to noticeably increase whereas the bonding energy ε b in the pair has to decrease. As a result, the paired fermions have to change their state from the SBB into fluctuating Cooper pairs which behave in a good many ways like those of conventional superconductor [2,6,13,26,85,89]. It is just that we call the local pairs. Thus, with decrease of temperature there must be a transition from BEC to BCS state, which is a consequence of a very short ξ ab at high temperatures and its noticeable temperature dependence. The possibility of a such transition is the main assumption of the LP model [2,13,27]. Precisely how this happens is one of the challenging questions in strongly correlated electron systems. Nevertheless, the transition was predicted theoretically in Refs. [23][24][25]89] and approved in our experimental studies [2,27].
Within the LP model a new approach to the analysis of the FLC and PG in HTS's was developed [2,13,27,67]. First, it was convincingly shown that FLC measured for all without exception HTS's always demonstrates a transition from 2D (ξ c (T) < d) into 3D (ξ c (T) > d) state, as T draws near T c . The result is most likely a consequence of Gaussian fluctuations of the order parameter in 2D metals [6,[23][24][25]84], which HTS compounds with pronounced quasi-two-dimensional anisotropy of conducting properties actually are [2,6,9]. The Gaussian fluctuations were found to prevent any phase coherency organization in 2D compounds. As a result, the critical temperature of an ideal 2D metal is found to be zero (Mermin-Wagner-Hoenberg theorem), and a finite value is obtained only when three-dimensional effects are taken into account [6,23,24,85,87]. That is why, the FLC in the 3D state is always extrapolated by the standard 3D equation of the AL theory, which determines the FLC in any 3D system: This means that the conventional 3D FLC is realized in HTS's as T → T c [67,77]. Above the crossover temperature T 0 (Eq. (7)) the FLC in well-structured YBCO films was found to be of the MT FLC type [2,67], in a good agreement with the HL theory [72]. The LD model was found to describe the experimental FLC only in the case of HTS compounds with pronounced structural defects [92,93]. Therefore, we denote the observed 2D-3D crossover also as MT-AL [2,67], unlike the MT-LD one predicted by the HL theory. It is clear on physical grounds that with increasing temperature the 3D fluctuation regime will persist until ξ c > d [77]. Thus, in this case the crossover should occur at ξ c ∼ = d, i.e., at which is larger by a factor of two than is predicted by the LD and HL theories. ξ c (0) is one of the important parameters of the PG analysis.
Third, now as a PG analysis is concern. It is clear, to get information about the PG we need an equation which describes the whole experimental curve from T c up to T * and contains PG in the explicit form. Besides, the dynamics of pair-creation and pair-breaking above T c must be taken into account [2,32,45,72,84]. However, the conventional fluctuation theories [72] well fit the experiment up to approximately 110 K only, whereas T * 200 K even in slightly underdoped cooprates [2,27,67], as discussed above. Unfortunately, so far there is no completed fundamental theory to describe the high-T c superconductivity as a whole and in particular the pseudogap phenomenon. For lack of the theory, such equation for σ (ε) has been proposed in Ref. [27] with respect to the local pairs: where ε is a reduced temperature given by Eq. (6), and T m f c > T c is the mean-field critical temperature, as discussed above. Besides, the dynamics of pair-creation ((1 − T/T * )) and pair-breaking (exp(−Δ * /T)) above T c have been taken into account in order to correctly describe the experiment [2,13,27]. Here A 4 is a numerical factor which has the meaning of the C-factor in the FLC theory [2,92]. All other parameters, including the coherence length along the c-axis, ξ c (0), and the theoretical parameter ε * c0 [2,27], directly come from the experiment. The way of ε * c0 determination is shown in the insert to Fig. 4 and explained below. Thus, the only adjustable parameter remains the coefficient A 4 . To find A 4 we calculate σ (ε) using Eq. (15) and fit the experiment in a range of 3D AL fluctuations near T c where lnσ (lnε) is the linear function of ε with a slope λ = −1/2 (Eq. (11)). Solving Eq. (15) for the pseudogap Δ * (T) one can readily obtain [27] where σ (T) is the experimentally measured excess conductivity over the whole temperature

YBCO films with different oxygen concentration
Within proposed LP model the FLC and PG in YPrBCO films [94], in slightly doped HoBa 2 Cu 3 O 7−δ single-crystals [95], and evev in FeAs-based superconductor SmFeAsO 0.85 with T c =55 K [20] (see division 3.4 below) were successfully studied for the first time. As a result, convincing set of self-consistent and reproducible results was obtained which has to corroborate the LP model approach. But the basic results have been obtained from the analysis of the resistivity data for the set of four YBCO films with different oxygen concentration [2,27,67,68]. The films were fabricated at Max Plank Institute (MPI) in Stuttgart by pulse laser deposition technique [96]. All samples were the well structured c-oriented epitaxial YBCO films, as it was confirmed by studying the correspondent x-ray and Raman spectra [93]. The sample F1 (T c =87.4 K) close to optimally doped systems, the sample F6 (T c =54.2 K) which represents weakly doped HTS systems, and the samples F3 and F4 with T c near 80 K were investigated to obtain the required information. Fig. 3 displays the temperature dependencies of the longitudinal resistivity ρ xx (T) = ρ(T) of the experimental films with parameters shown in Table I, where d 0 is the sample thickness.  Table I.
To simplify our discussion a little bit, I will consider only the sample F1, as an example. The similar results were obtained for all other YBCO films being studied and compared with those obtained for HoBa 2 Cu 3 O 7−δ [95] and SmFeAsO 0.85 [20]. Besides, I will consider mainly the basic aspects of the PG analysis only and touch on the FLC results as far as is necessary. To look for more details, one can see Refs. [2,13,20,27,67,68,94,95].

Pseudogap in YBCO films with different oxygen concentration
We proceed from the fact that the excess conductivity σ (T) arises as a result of the formation of paired fermions (local pairs) at temperatures T c < T < T * [2,6,13,23,26,47]. It is believed that formation of such pairs gives rise to their real binding energy, ε b , which the quantity Δ * characterizes [27]. As a result, the density of states of the normal excitations in this energy range decreases [57], which is referred to as the appearance of a pseudogap in the excitation spectrum [59,61,84,98].
Since the PG is not measured directly in our experiments, the problem reduces to determining Δ * (T) from the experimental dependence σ (T) and comparing it with those obtained from Eq. (10). To perform the analysis, the excess conductivity of every studied YBCO film, measured in the whole temperature interval from T * down to T c , was treated in the framework of the LP model using Eq. (15) and Eq. (16) [2,13,27]. Aside from the parameters, which directly come from experiment (Tables I), we substitute into Eq. (15) the values of Δ * (T c ).
Here, by analogy to the superconducting state, Δ * (T c ) is the value of the PG in the limit T → T c . As it was shown in Ref. [15], Δ * (T c ) ∼ = Δ(0) and, correspondingly, Δ * satisfies the condition 2Δ * ∼ k B T c , as it was demonstrated in Ref's. [99][100][101]. The conclusion has subsequently been confirmed by the tunneling experiments in Bi compounds [52]. To justify the values of Δ * (T c ) used in our analysis we applied an approach proposed in Ref. [88] in which, however, the fluctuation contributions into σ (T) are neglected. But a definite advantage of their representation of the experimental data in the coordinates lnσ versus (1/T) is the fact that the rectilinear part of the resulting plot has turned out to be very sensitive to the value of Δ * (T c ), which makes it possible to adjust the value chosen for this parameter. Curve numbered 3 in the figure reproduces result of Ref. [102] in which, however, both the dynamics of pair-creation and pair-breaking above T c were neglected. But, in accordance to our knowledge, the authors of the Ref. [102] were the first who have paid attention to the experimental fact that in YBCO compounds the reciprocal of the excess conductivity σ −1 (T) is an exponential function of ε in a certain temperature range above T c0 . Correspondingly, the adjustable coefficient A 3 [102] is chosen so that the computed curve matches experiment in the region of exponential behavior of σ −1 (T) (usually from T c01 ≈ 100 K up to T c02 ≈ 150 K). It is clear that lnσ −1 is to be the linear function of ε, as shown in the insert of Fig. 4. The advantage of this approach is that it enables us to determine the parameter ε * c0 which is reciprocal of the slope α of this linear function: ε * c0 = 1/α [2,102]. The same experimental dependencies of σ (T), as shown in Fig. 4, were obtained for all studied compounds, including HoBa 2 Cu 3 O 7−δ [95] and SmFeAsO 0.85 [20], suggesting the similar local pairs behavior in different HTS's. The fact enabled us to analyze the FLC and PG in HoBa 2 Cu 3 O 7−δ single crystals [95] and in SmFeAsO 0.85 FeAs-based superconductor [20] also in terms of the LP model, as will be discussed in the next divisions. Note, the complete coincidence of the given by Eq. (15) theoretical curve and the data (Fig. 4) is not necessary. We fit experiment by the theory to determine mostly the coefficient A 4 [2,27], and coincidence in the 3D fluctuation region near T c is only important, as discussed above. Nevertheless, the good coincidence of both theoretical and experimental curves obtained for all studied compounds [20,67,94,95] means, in turn, that substituting into Eq. (16) the experimentally measured values of the σ (T) with the corresponding set of parameters, we should obtain a result which reflects the real behavior of Δ * (T) quite closely in the experimental samples. The values of Δ * (T) calculated using Eq. (16) for all YBCO films with the similar sets of parameters, as designated above for the film F1, are shown in Fig. 5 which actually displays our principal result. Indeed, despite the rather different Δ * (T c ) and all other parameters, very similar Δ * (T) behavior is observed for all studied films. The main common feature of every plot is a maximum of Δ * (T) observed at the same T max ≈ 130 K. The important point here is that the coherence length ξ ab (T max ) was found to be the same for every studied film, namely, ξ ab (T max ) ≈ 18Å [2,27]. Let us discuss the obtained results (Fig. 5) now. Above 130 K ξ ab (T) is very small (ξ ab (T * ) ≈ 13 Å), whereas the coupling energy ε b is very strong. It is just the condition for the formation of the SBB [6,[24][25][26]. It was found [27] that over the temperature interval T max = T pair < T < T * every experimental Δ * (T)/Δ * (T max ) curve shown in Fig. 5 can be fitted by the Babaev-Kleinert (BK) theory [87] in the BEC limit (low n f ) in which the SBB have to form [6, 24-26, 85, 87]. (See also Fig. 13 as an example). The finding has to confirm the presence of the local pairs in the films at T ≤ T * which are supposed to exist at these temperatures just in the form of SBB. As SBB do not interact with one another, the local pairs demonstrate no SC (collective) behavior in this temperature interval. It has subsequently been shown to be consistent with the tunneling experiments in Bi compounds [52] in which the SC tunneling features are smeared out above T pair . Thus, above T max (Fig. 5), which is also called T pair in accordance with, e.g., Ref. [80], it is the so-called non-superconducting part of the PG.
But the pairs have already formed and exist in the sample even in this temperature range, this is a point. There a few evidences as for paired fermions existence in temperature interval from T * down to T pair [2]. Firstly, from studying the nuclear magnetic resonance (NMR) in weakly doped Y123 systems [57], the anomalous decrease of the Knight shift K(T) was observed on cooling just at T ≤ T * (Fig. 6). In Landau theory [75] K ∼ ρ n (ε) ≡ ρ F , where ρ n (ε) is the energy dependence of the density of Fermi states in the normal phase, which in classical superconductors actually remains constant in the whole temperature range of the normal phase existence, i.e., approximately down to T c . Thus, the decrease of K(T) directly points out at the decrease of density of states most likely because of the local pair formation. Secondly, in measurements of the Hall effect noticeable enhancement of the Hall coefficient R H (T) ∼ 1/e n f was found on cooling just below T * [2,81,97]. Obviously, the increase of R H directly points out at the decrease of the normal charge-carrier density n f . It is believed that it is because of the fact that any part of electrons transform into the local pairs. Thirdly, studying the behavior of the Nb-YBCO point contacts in high-frequency fields, we observed the pair-breaking effect of the microwave power up to T * ≈ 230 K [103]. It is clear, the observation of the pair-breaking effect naturally implies the existence of the paired fermions in the sample at such high temperatures. And, finally, very recently the polar Kerr effect (PKE) in Bi2212 was reported which also appears just below T * ≈ 130 K [90]. To be observable the PKE evidently acquires the presence of two different kinds of the quasi-particles in the sample. It is believed that such particles are most likely the normal electrons and local pairs. Let us turn back to Fig. 5. With decreasing temperature below T max , ξ ab (T) continues to increase whereas ε b (T) becomes smaller. Ultimately, at T ≤ T max = T pair ≈ 130 K, where ξ ab (T) > 18Å, the local pairs begin to overlap and acquire the possibility to interact. Besides, they do can be destroyed by the thermal fluctuations now, i.e. transform into fluctuating Cooper pairs, as mentioned above. The SC (collective) behavior of the local pairs in this temperature region is distinctly observed in many experiments [52,67,[78][79][80], as discussed in details in div. 2.2. Eventually, the direct imaging of the local pair SC clusters persistence up to approximately 140 K in Bi compounds is recently reported in Ref. [16] (Fig.2). Thus, below T max it is the SC part of the pseudogap. Moreover, we consider ξ ab (T max ) ≈ 18Å to be the critical size of the local pair, at least in the YBCO [2,27]. In fact, the local pairs behave like SBB when ξ ab (T) < 18Å, and transform into fluctuating Cooper pairs when ξ ab (T) > 18Å below T max , thus resulting in the BEC-BCS transition [2,27]. That is why, I will call T max as T pair now, as it becomes of common occurrence in the literature [80].

Fluctuation conductivity and pseudogap in HoBa 2 Cu 3 O 7−δ single crystals under pressure
To proceed with further understanding of physical nature of HTS's, it seemed to be rather interesting to compare above results for YBCO films, where no noticeable magnetism is expected, with any other HTS's in which magnetic subsystem could play significant role. One of a such system is HoBa 2 Cu 3 O 7−δ which has magnetic moment μ e f f ≈ 9,7μ B , due to magnetic moment of pure Ho which is μ e f f ≈ 10.6μ B [95]. Thus, when Y is substituted by Ho a qualitatively another behavior of the system is expected because of Ho magnetic properties. Besides, in HTS compounds of the ReBa 2 Cu 3 O 7=δ (Re = Y, Ho, Dy · · · ) type with reduced n f the specific non-equilibrium state can be realized under change of temperature [104] or pressure [105]. In our experiments effect of hydrostatic pressure up to 5 kbar on the fluctuation conductivity σ (T) and pseudogap Δ * (T) of weakly doped high-T c single crystals HoBa 2 Cu 3 O 7−δ (HoBCO) with T c ≈ 62 K and oxygen index 7-δ ≈ 6,65 was studied [95]. The comparison of results with those obtained for YBa 2 Cu 3 O 7−δ (div. 3.2) and FeAs-based superconductor SmFeAsO 0.85 (div. 3.4) is to shed more light on the role of magnetic subsystem in the HTS's.
Measurements were carried out with current flowing in parallel to the twin boundaries when their influence on the charge carriers scattering is minimized [95]. Obtained results are analyzed within the Local Pair model [27]. The FLC and PG analysis of the sample under pressure is mainly discussed, and all results are summarized in the end. As usual, below the PG temperature T * T c resistivity of HoBCO, ρ(T), deviates down from linearity resulting in the appearance of the excess conductivity σ (T) = σ(T) − σ N (T) (Eq.(1)). Resulting ln σ as a function of ln ε near T c is displayed in Fig. 7.  (Fig. 7, dashed line 1). Above T 0 , up to T 01 ≈ 95 K (ln ε 01 ≈ −0.8), the experimental data are well extrapolated by the MT fluctuation contribution (Eq. (3)) (Fig. 7, curve2) of the HL theory [72] Here I would like to emphasize that curve 2 in Fig. 7 is plotted with d 1 = 2.95Å, where d 1 is the distance between conducting CuO 2 planes in HoBaCuO [106], and with τ φ (100K) β = (0.665 ± 0.002)×10 −13 s which is defined by a formula τ φ β T = Aε 01 (Eq. (14)). Accordingly, d 1 corresponds to T 01 ≈ 95 K (ln ε 01 ≈ −0.8) marked by the right arrow in the figure. At T ≤ T 01 , ξ c (T) ≥ d 1 is believed to couple the CuO 2 planes by Josephson interaction, and 2D FLC has to appear [67,77]. This scenario of the FLC appearance reminds that observed for the SmFeAsO 0.85 sample (see div. 3.4), and found lnσ (lnε) is close to that shown in Fig. 10. If we choose d ≈ 11.7Å = c, which is a dimension of the unit sell along the c-axis, as in the case of YBCO films [67], the theory noticeably deviates from experiment (curve 3). Strictly speaking, this curve reflects the temperature dependence of the FLC being close to that obtained for the YBCO film F1 shown in Fig. 4. Thus, compare results one may conclude that HoBaCuO obviously demonstrates the enhanced 2D fluctuation contribution, compared to YBCO, and behaves in a good many ways like the SmFeAsO 0.85 (Fig. 10), most likely due to influence of its magnetic subsystem. Nevertheless, at T = T 0 ≈ 67.3 K (ln ε 0 ≈ −3.55), at which ξ c (T) = d, the habitual dimensional 2D-3D (AL -MT) crossover is distinctly seen on the plot. Below T 0 ξ c (T) >c, and 3D fluctuation behavior is realized in which the fluctuating pairs interact in the whole volume of the sample [67]. It should be emphasized that, as well as it was found for the SmFeAsO 0.85 [20], at both T 01 and T 0 (arrows in Fig. 7) ξ c (0) = d 1 ε 1/2 01 = dε 1/2 0 = (1.98 ± 0.005)Å. The finding is believed to confirm the validity of our analysis.
The similar ln σ vs ln ε was also obtained for P = 0 (T c = 61.4 K, ρ(100K) = 200μΩcm, T 0 = 63.4 K, T 01 = 79.4 K, τ φ (100K)β = 1.13 · 10 −13 s and d 1 = 3.2Å). Thus, the pressure leads to increase in T 0 and T 01 , i.e. increases the interval of 3D and especially 2D fluctuations. Most likely, it is due to decrease of the phase stratification of the single crystal under pressure [104,105]. The value of the C-factor also points out at this effect: C 3D = 1.1 and = 0.53 at P = 4.8 kbar and P = 0 kbar, respectively. The closer C 3D to 1.0, the more homogeneous is the sample structure [2,67].
The experimental dependence of σ (T) in the whole temperature interval from T * and down to T m f c turned out to be very close to that, shown in Fig. 4. The finding enables us to calculate Δ * (T) using Eqs. (15) and (16). Resulting temperature dependence of the PG is shown in Fig. 8. In the figure curve 1 is plotted for P = 0 using Eq. (16)  When P = 0, Δ * (T) exhibits two unexpected maxima at T m1 ≈ 195 K and at T m2 ≈ 210 K (Fig. 8, curve 1), most likely because of two-phase stratification of the single crystal [104,105]. Pressure-induced enhancement of the rising diffusion processes is assumed to cause a redistribution of labile oxygen from the low-temperature phase poor in charge carriers to the high-temperature one [105]. It results in the disappearance of both peaks and linear Δ * (T) dependence with a positive slope at high T (Fig. 8, curve 2). Simultaneously, the sample resistivity, ρ, noticeably decreases, whereas T c and ξ c (0) somewhat increase. Note that the maxima of Δ * (T) and initial values of ρ, T c and ξ c (0) are restored when the pressure is removed suggesting the assumption. Thus, both σ (T) and Δ * (T) are markedly different from those obtained for the YBCO films [27,67]. They resemble similar curves obtained for FeAs-based superconductor SmFeAsO 0.85 [20] (see div. 3.4). The result can be explained by the influence of paramagnetism in HoBa 2 Cu 3 O 7−δ [95,104,105]. Additionally, the linear drop of the PG, found for both HoBa 2 Cu 3 O 7−δ [95] and SmFeAsO 0.85 [20], is believed to reflect the influence of weak magnetic fluctuations in such compounds [21,107,108], as will be discussed in more details in the next division.

Fluctuation conductivity and pseudogap in SmFeASO 0.85
Despite of a huge amount of papers devoted to FeAs-based superconductors, there is an evident lack of the FLC and PG studies in these compounds [20]. This state of affairs is most likely due to the extreme complexity of the electronic configuration of the FeAs-based compounds, where the strong influence of magnetism, being changed with doping, is observed [17][18][19]. It is well known that upon electron or hole doping with F substitution at the O site [17,113], or with oxygen vacancies [17,18] all properties of parent RFeAsO compounds drastically change and evident antiferromagnetic (AFM) order has to disappear [17][18][19]. However, recent results [107][108][109][110] point toward an important role of low-energy spin fluctuations which emerge on doping away from the parent antiferromagnetic state which is of a spin-density wave (SDW) type [107,108,110]. Thus, below T S ∼ 150K the AFM fluctuations, being likely of spin wave type, are believed to noticeably affect the properties of doped RFeAsO 1−x F x systems (Fig. 9). As shown by many studies [108][109][110] the static magnetism persists well into the SC regime of ferropnictides. In SmFeAsO 1−x strongly disordered but static magnetism and superconductivity both are found to coexist in the wide range of doping, and prominent low-energy spin fluctuations are observed up to the highest achievable doping levels where T c is maximal [21,107,108].
The interplay between superconductivity and magnetism has been a long-standing fascinating problem [111,112], and relation between the SDW and SC order is a central topic in the current research on the FeAs-based high-T c superconductors. However, the clear nature of the complex interplay between magnetism and superconductivity in FeAs-based HTS's is still rather controversial [112]. As a result, rather complicated phase diagrams for different FeAs-based high-T c systems [17,109,110], and especially for SmFeAsO 1−x [113][114][115][116] (Fig. 9) are reported. Naturally, rather peculiar normal state behavior of the system upon T diminution is expected in this case [109,110,113], when x is of the order of 0.15, as it is in our sample [20].

Experimental details
To shed more light on the problem, analysis of the FLC and PG was carried out within the LP model using results of resistivity measurements of SmFeAsO 0.85 polycrystal (T c = 55 K) performed on fully computerized set up [20]. The width of the resistive transition into superconducting (SC) state is Δ T ≤ 2K suggesting good phase and structural uniformity of the sample. In accordance with the LP model approach the resistivity curve above T * ∼ 170K is extrapolated by the straight line to get σ (T) = σ(T) − σ N (T) using Eq.(1).
The crossing of measured σ −2 (T) with T-axis denotes the mean-field critical temperature T m f c ∼ = 57K. This is the usual way of T m f c determination within the LP model [2,67]. Resulting lnσ versus lnε is displayed in Fig. 10 in the temperature interval relatively close to T c (dots) and compared with the HL theory [72] in the clean limit (curves 1-3). As expected, up to T 0 ≈ 58.5 K (lnε 0 =-3.6) the data are well extrapolated by the AL fluctuation contribution (Eq. 11) for any 3D system (straight line 2 in the figure). As mentioned above, this 3D fluctuation behavior was found to be typical for all without exception HTS compounds [2,20,67,80,95]. Accordingly, above T 0 , up to T c0 ≈ 69K (lnε c0 ∼ = −1.55), this is 2D MT term (Eq. (3)) of the HL theory (Fig. 10, curve 1) which dominates well above T c in the 2D fluctuation region [2,67], as discussed in details in div. 2.2. As before, ξ c (0) = d ε 1/2 0 is the coherence length along the c-axis, i.e. perpendicular to the conducting planes, and d is the distance between conducting layers in SmFeAsO 1−x . Thus, expected MT-AL (2D-3D) crossover is clearly seen in Fig. 10 at lnε 0 =-3.6. The fact enables us to determine T 0 (Eq. (7)) and therefore the ε 0 (Eq. (7,9)) with adequate accuracy. Now, proceeding in the usual way, i.e., using Eqs. (12,14), and set d = 8.495Å, which is a dimension of the SmFeAsO 0.85 unit sell along the c-axis [114], ξ c (0) = (1.4 ± 0.005)Å, and τ φ (100K) β = (11 ± 0.03) × 10 −13 s are derived from the experiment. As it is seen from the figure, Eq. (11) with measured value of ξ c (0) and the scaling factor C 3 D = 0.083 describes the data fairly well just above T m f c (Fig. 10, dashed line 2) suggesting 3D fluctuation behavior of SmFeAsO 0.85 near T c .
Till now the FeAs-based superconductor behaves like the YBCO films. However, the discrepancy appears when MT contribution is analyzed (Fig. 10, curve 3). Indeed, substituting the measured values of ξ c (0) and τ φ (100K) β into Eq. (3) we obtain curve 3 which looks like that found for YBCO films but apparently does not match the data. The finding suggests that our choice of d is very likely wrong in this case. To proceed with the analysis we have to suppose that SmFeAsO 1−x becomes quasi-two-dimensional just when ξ c (T), getting rise with temperature diminution, becomes equal to d 1 = 3.05 Å at T = T co ≈ 69K (lnε c0 ∼ = −1.55). It is worth to emphasize that (3.1 ÷ 3.0)Å is the distance between As layers in conducting As − Fe − As planes in SmFeAsO 1−x [18,114]. Below this temperature ξ c (T) is believed to couple the As layers in the planes by Josephson interaction [95]. This approach gives the same value of ξ c (0) = d ε 1/2 c0 = (1.4 ± 0.005)Å as calculated above using the usual crossover temperature T 0 . We think this fact is to confirm the supposition. At the same time the phase relaxation time changes noticeably and amounts τ φ (100K) β = (1.41 ± 0.03) × 10 −13 s in this case. Substitution of measured values of ξ c (0) and this new τ φ (100K) β into Eq. (3) enables us to fit the experimental data by the MT term just up to T c0 which is about 15 K above T c (Fig.  10, curve 1). The result suggests enhanced 2D MT fluctuations in SmFeAsO 1− compared to YBCO films. Recall, the similar enhanced 2D MT fluctuations are found for HoBa 2 Cu 3 O 7−δ [95] (div. 3.3)

Pseudogap analysis
Somehow surprisingly, lnσ versus lnε measured for the SmFeAsO 0.85 in the whole temperature range from T * down to T m f c was found to be very close to that of YBCO films (Fig. 4). Even the exponential dependence of the reciprocal of the excess conductivity σ −1 (T) still occurs between lnε c0 and lnε c02 ((69 -100)K). As a result, the function lnσ −1 versus ε appears to be linear in this temperature range (similar to that shown in insert in Fig. 4), and its slope 1/α still determines the parameter ε * c0 used in the PG analysis [20,27]. Taking obtained results into account, one can draw a conclusion that the LP model approach can be applied to analyze the temperature dependence of the PG. To proceed with the PG analysis the value of the coefficient A 4 must be found. As before, to determine A 4 we fit the experimental σ (ε) by the theoretical curve (Eq. (15)) in the region of 3D fluctuations near T c [20,27]. All other parameters directly come from experiment, as discussed above. The theoretical σ (ε) curve is constructed using Eq. (15) with the reasonable set of experimentally measured parameters: ξ *  [95]. The only exception is the 2D MT region where relatively small deviation down from experiment, negligible in the case of YBCO films [67], is observed. It is due to enhanced MT fluctuation contribution in the 2D fluctuation region, as mentioned above. Compare results with those obtained for magnetic superconductor Dy 0.6 Y 0.4 Rh 3.85 Ru 0.15 B 4 [117] we assume this enhancement to be the consequence of weak magnetic fluctuations in SaFeAsO 1−x [107][108][109][110], as will be discussed below. Now we have almost all parameters to calculate the temperature dependence of PG using Eq.(16) except the value of Δ * (T c ) which actually is responsible for the A 4 . The problem is, the value of Δ * (T c ) and in turn the ratio 2Δ * (T c )/k B T c in Fe-based superconductors remain uncertain. Reported in the literature values for Δ(0) and 2Δ(0)/T c in SmFeAsO 1−x range from 2Δ(0) ≈ 37 meV (2Δ/T c ∼ 8, strong-coupling limit) obtained in measurements of far-infrared permittivity [17], down to Δ = (8 -8.5) meV (2Δ/T c ∼ (3.55 -3.8)) measured by a scanning tunneling spectroscopy [118,119] which is very close to standard value 3.52 of the BCS theory (weak-coupling limit). It is believed at present that SmFeAsO 1−x has two superconducting gaps Δ 1 (0) ≈ 6.7 meV and Δ 2 (0) which ranges from ≈ 15 meV up to ≈ 21 meV [120]. Besides we still think that Δ * (T c ) ≈ Δ(0) [20], as is was justified in Ref. [15]. To feel more flexible, four curves are finally plotted in Fig. 11   Very unexpected Δ * (T) behavior is observed (Fig. 11). The most striking result is a sharp drop of Δ * (T) at T s ≈ 147 K, as clearly illustrates the curve with Δ * (T c )/k B =140 K plotted without symbols. Usually T s is treated as a temperature at which a structural tetragonal-orthorhombic transition occurs in parent SmFeAsO. In the undoped FeAs compounds it is also expected to be a transition to SDW ordering regime [17][18][19]. Below T S the pseudogap Δ * (T) drops linearly down to T AFM ≈ 133 K (Fig. 11), which is attributed to the AFM ordering of the Fe spins in a parent SmFeAsO compounds [17,121]. Below T AFM the slop of the Δ * (T) curves apparently depends on the value of Δ * (T c ) [20]. Strictly speaking it is difficult to say at present is T AFM = T N of the whole system or not because the AFM ordering of Sm spins occurs at ≈ 5K only [17,121].
Found Δ * (T) behavior is believed to be explained in terms of the theory by Machida, Nokura and Matsubara (MNM) [111] developed for AFM superconductors in which the AFM ordering with a wave vector Q may coexist with the superconductivity. This assumption was supported in, eg., the theory by Chi and Nagi [122]. In the MNM theory the effect of the AFM molecular field h Q (T) (|h Q | ε F ) on the Cooper pairing was studied. It was shown, that below T N the BCS coupling parameter Δ(T) is reduced by a factor [1 − const · |h Q (T)|/ε F ] due to the formation of energy gaps of SDW on the Fermi surface along Q. As a result the effective attractive interactiong N(0) or, equivalently, the density of states at the Fermi energy ε F is diminished by the periodic molecular field that is Here m(T) is the normalized sublattice magnetization of the antiferromagnetic state and α is a changeable parameter of the theory. Between T c and T N (T c > T N is assumed) the order parameter is that of the BCS theory. Since below T N the magnetization m(T) becomes nonvanishing,g N(0) is weakened that results in turn in a sudden linear drop of Δ(T) immediately below T N . As m(T) saturates at lower temperatures, Δ(T) gradually recovers its value with increasing the superconducting condensation energy (Fig. 12, solid curves). This additional magnetization m(T) was also shown to explain the anomaly in the upper critical field H c2 just below T N observed in studying of RMo 6 S 8 (R = Gd, Tb, and Dy) [111]. However, predicted by the theory decrease of Δ(T) at T ≤ T N was only recently observed in AFM superconductor ErNi 2 B 2 C with T c ≈ 11 K and T N ≈ 6 K, below which the SDW ordering is believed to occur in the system [123]. The result evidently supports the prediction of the MNM theory. Our results are found to be in a qualitatively agreement with the MNM theory as shown in Fig. 12, where the data for Δ * (T c )/k B = 130 K (red •) and Δ * (T c )/k B = 135 K (•) are compared with the MNM theory (solid lines).
The curves are scaled at T/T c = 0.7 and demonstrate rather good agreement with the theory below T/T c = 0.7. Note, that the upper scale is T/T * . Both shown Δ * (T) dependencies suggest the issue that just Δ * (T c )/k B = 133 K, which is just T AFM , would provide the best fit with the theory. Above T/T c = 0.7 the data evidently deviate from the BCS theory. It seems to be reasonable seeing SmFeAsO 0.15 as well as any other ferropnictides is not a BCS superconductor.
It is important to emphasize that in our case we observe the particularities of Δ * (T) in the PG state, i. e. well above T c but just at T ≈ T s , below which the SDW ordering in parent SmFeAsO should occur. However, it seems to be somehow surprising as no SDW ordering in optimally doped SmFeAsO 0.15 is expected. The found very specific Δ * (T) can be understood taking mentioned above weak AFM fluctuations (low-energy spin fluctuations), which should exist in the system, into account. At the singular temperature T s these fluctuations are believed to enhance AFM in the system likely in the form of SDW. After that, in accordance with the MNM theory scenario, the SDW has to suppress the order parameter of the local pairs as shown by our results. Thus, the results support the existence both weak AFM fluctuations and the paired fermions (local pairs) in the PG region, which order parameter is apparently suppressed by these fluctuations [20]. To confirm the conclusion, relation Δ * (T)/Δ * max as a function of T/T * (T/T c in the case of the theory) is plotted in Fig. 13. Black dots represent the studied SmFeAsO 0.85 with Δ * (Tc)/k B = 160 K, which corresponds to the strongly coupled limit (2Δ * (T c )/T c ∼ = 5.82). Blue dots display the data for YBCO film F1 [27]. The solid and dashed curves display the results of the Babaev-Kleinert (BK) theory [87] developed for the SC systems with different charge carrier density n f . For the different curves the different theoretical parameter x 0 = μ/Δ(0) is used, where μ is the chemical potential. Curve 1, with x 0 = +10, gives the BCS limit. Curve 4 with x 0 = -10 represents the strongly coupled BEC limit, which corresponds to the systems with low n f in which the SBB must exist [23,25,85,87]. As well as in the YBCO film, the Δ * (T)/Δ * max in SmFeAsO 0.85 evidently corresponds to the BEC limit (Fig. 13) suggesting the local pairs presence in the FeAs-based superconductor. Below T/T * ≈ 0.4 both experimental curves demonstrate the very similar temperature behavior suggesting the BEC-BCS transition from local pairs to the fluctuating Cooper pairs to be also present in FeAS-based superconductors [20]. But, naturally, no drop of Δ * (T) is observed for the YBCO film (Fig. 13, blue dots) at high temperatures, as no antiferromagnetism is expected in this case. This fact can be considered as an additional evidence of the AFM nature of the Δ * (T) linear reduction below T s in SmFeAsO 0.85 found in our experiment.
It has to be emphasized that recently reported phase diagrams [113][114][115][116] (Fig. 9) apparently take into account a complexity of magnetic subsystem in SmFeAsO 1−x F x and are in much more better agreement with our experimental results. But it has to be also noted that we study the SmFeAsO 1−x system whereas the phase diagrams are mainly reported for the SmFeAsO 1−x F x compounds. Is there any substantial difference between the both compounds has yet to be determined. Evidently more experimental results are required to clarify the question.

Angle-resolved photoemission measurements of the energy pseudogap of high-T c (Bi, Pb) 2 (Sr, La) 2 CuO 6+δ superconductors: A model evidence of local pairs
Taking all above consideration into account, it can be concluded that the FLC and PG description in terms of local pairs gives a set of reasonable and self-consistent results. However, to justify the conclusion it would be appropriate to test the LP model approah using independent results of other research groups who have measured straightforwardly the PG or any other related effects. But for a long time there was a lack of indispensable data.
Fortunately, analysis of the pseudogap in (Bi, Pb) 2 (Sr, La) 2 CuO 6+δ (Bi2201) single-crystals with various T c 's by means of ARPES spectra study was recently reported [80]. The study of Bi2201 allows avoid the complications resulting from the bilayer splitting and strong antinodal bosonic mode coupling inherent to Bi2212 and Bi2223 [90,91]. Symmetrized energy distribution curves (EDCs) were found to demonstrate the opening of the pseudogap on cooling below T * . It was shown that T * , obtained from the resistivity measurements, agrees well with one determined from the ARPES data using a single spectral peak criterion [80]. Finally, from the ARPES experiments information about the temperature dependence of the loss of the spectral weight close to the Fermi level, W(E F ), was derived [80]. W(E F ) versus T measured for optimally doped OP35K Bi2201 (T c =35 K, T * = 160 K) turned out to be rather unexpected, as shown in Fig. 14a taken from Ref. [80]. Above T * the W(E F ) is nonlinear function of T. But below T * , over the temperature range from T * to T pair = (110 ± 5) K (Fig. 14a), the W(E F )(T) decreases linearly which is considered as a characteristic behavior of the "proper" PG state [80]. However, no assumption as for physical nature of this linearity as well as for existence of the paired fermions in the PG region is proposed. Below T pair the W(E F ) vs T noticeably deviates down from the linearity (Fig. 14a). The deviation suggests the onset of another state of the system, which likely arises from the pairing of electrons, since the W(E F )(T) associated with this state smoothly evolves through T c (Fig. 14a).
To compare results and justify our Local Pair model, the ρ ab vs T of the OP35K Bi2201, reported in Ref. [80], was studied within the LP model [13]. The Δ * (T) was calculated by Eq. (16) with the following reasonable set of parameters: T c = 35 K, T m f c = 36,9 K, T * =160 K, ξ c (0)= 2.0Å, ε * c0 = 0.89, and A 4 = 59. σ (T) is the experimentally measured excess conductivity derived from the resistivity data using Eq. (1). Resulting Δ * (T)/Δ * max is plotted in Fig. 14b (green dots). As expected, the shape of the Δ * (T) curve is similar to that found for YBCO films (Fig. 5). Besides, the maximum of Δ * (T)/Δ * max at T max ≈ 100K actually coincides with the change of the W(E F )(T) slop at T pair , measured by ARPES, which seems to be reasonable. In fact, in accordance with our logic, T max is just the temperature which divides the PG region on SC and non-SC parts depending on the local pair state, as described above. Recall, that above T max the local pairs are expected to be in the form of SBB. Most likely just the specific properties of SBB cause the linear W(E F )(T) over this temperature region (Fig. 14a). The two following facts are believed to confirm the conclusion. Firstly, when SBB disappear above T * , the linearity disappears too. Secondly, below T max , or below T pair in terms of Ref. [80], the SBB have to transform into fluctuating Cooper pairs giving rise to the SC (collective) properties of the system. This argumentation coincides with the conclusion of Ref. [80] as for SC part of the pseudogap below T pair . As SBB are now also absent, the linearity of W(E F )(T) again disappears. Thus, we consider the Δ * (T), calculated within our Local Pair model (Fig. 14b), to be in a good agreement with the temperature dependence of the loss of the spectral weight W(E F ) (Fig. 14a) obtained from the ARPES experiments performed on the same sample. In this way, the results of ARPES experiments reported in Ref. [80] are believed to confirm our conclusion as for existence of the local pairs in HTS's, at least in Bi2201 compounds. Fig 14b is the normalized spectral gap (SG(T)) (red dots) equals to the energy of the spectral peaks of EDCs measured by ARPES [80]. Important in this case is that SG(T) smoothly evolves through both T pair and T c . The fact is believed to confirm assumed in the LP model the local pair existence above T pair . Despite the evident similarity there are, however, at least two differences between the curves shown in Fig. 14b. First, there is no direct correlation between the SG(T) and the W(E F )(T) (Fig. 14 a, b). Why the maximum of SG(T) is shifted toward low temperatures compared to T pair , has yet to be understood. The second difference is the absolute value of the SG compared to the pseudogap. The spectral gap has SG max ≈ 40 meV and SG(T c ) ≈ 38 meV [80]. It gives 2SG(T c )/k B T c ≈ 26 which is apparently too high. The PG values are Δ * max ≈ 16.5 meV and Δ * (T c ) ≈ 6.96 meV, respectively. It gives 2Δ * (T c )/k B T c ≈ 6.4 which is a common value for the Bi compounds [124] with respect to relatively low T c in this case. Thus, there is an unexpected lack of coincidence between the SG and PG.

Conclusion
The Chapter presents a detailed consideration of the LP model developed to study the PG in HTS's. In accordance with the model the local pairs have to be the most likely candidate for the PG formation. At high temperatures (T pair < T ≤ T * ) we believe the local pairs to be in the form of SBB which satisfy the BEC theory (non-SC part of a PG). Below T pair the local pairs have to change their state from the SBB into fluctuating Cooper pairs which satisfy the BCS theory (SC part of a PG). Thus, with decrease of temperature there must be a transition from BEC to BCS state [2,27]. The possibility of such a transition is considered to be one of the basic physical principals of the high-T c superconductivity. The transition was predicted theoretically in Ref. [23,24,74] and experimentally observed in our experiments [2,20,27,95].
A key test for our consideration is the comparison of the Δ * (T), calculated within the LP model, with the temperature dependence of the loss of the spectral weight close to the Fermi level W(E F )(T), measured by ARPES for the same sample [80]. The resulting Δ * (T) is found to be in a good agreement with the W(E F )(T) obtained for OP35K Bi2201 (Fig. 14). It allows us to explain reasonably the W(E F )(T) dependence, both above and below T pair , in terms of local pairs.
The obtained results are also in agreement with the conclusions of Ref's. [16,90,91] as for SC and non-SC parts of the PG in Bi systems. Besides, formation of the local pairs is also believed to explain the rise of the polar Kerr effect and response of the time-resolved reflectivity, both observed for Bi systems just below T * [90]. While, the Nernst effect [16], which is likely due to the SC properties of the local pairs, is observed only below T pair , or below T max in terms of our model. All the facts have to support the local pair existence in HTS's at T ≤ T * , Thus, we may conclude, that on the basis of the developed LP model the self-consistent picture of the PG formation in HTS's is obtained. At the same time the issue concerning the pairing mechanism in HTS's still remains controversial [22].