**2. 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, 30–32]. 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, 39–41], "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-*Tc* superconductivity as a whole and to clarify finally the PG phenomenon [2].

## **2.1. 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 *Tc* [8, 15, 26, 27, 49–54]. 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*<sup>∗</sup> [56]. Below *T*<sup>∗</sup> >> *Tc*, for reasons which have still not been finally established, the density of quasiparticle states at the Fermi level starts to decrease [57–59]. That is why the phenomenon has been named a "pseudogap"

2 Will-be-set-by-IN-TECH

doping [2, 8, 9, 11, 12]. But even in an optimally doped YBCO it is an order of magnitude less than in conventional superconductors [2, 8, 9, 11, 12, 15]. There is growing evidence that just the reduced density of charge carriers may be a key feature to account for all other properties

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

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, 30–32]. 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, 39–41], "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-*Tc* superconductivity as a whole and to clarify finally the PG phenomenon [2].

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 *Tc* [8, 15, 26, 27, 49–54]. 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*<sup>∗</sup> [56]. Below *T*<sup>∗</sup> >> *Tc*, for reasons which have

of HTS's [2, 6, 7, 13, 22–26].

**2. Theoretical background**

**2.1. Pseudogap in HTS's**

high-*Tc* superconductor at *T* ≤ *T*<sup>∗</sup> [6, 13, 22–27]

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, 60–63]. 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.

### **2.2. 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 *Tc* 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 *ρxx*(*T*) at *T* ≤ *T*<sup>∗</sup> from its linear behaviour above *T*<sup>∗</sup> (Fig.1). This results in the excess conductivity *σ*� (*T*) = *σ*(*T*) − *σN*(*T*), which can be written as

$$
\sigma'(T) = [\rho\_N(T) - \rho(T)] / [\rho(T)\rho\_N(T)].\tag{1}
$$

Here *ρ*(*T*) = *ρxx*(*T*) and *ρN*(T)=*α*T+b determines the resistivity of a sample in the normal state extrapolated toward low temperatures.

**Figure 1.** Resistivity *ρ* versus temperature T (•) for YBCO film F1 (Table I); dashed line represents *ρN*(*T*).

This way of determining *ρN*(*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 *ρxx*(*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

$$
\sigma\_{AL}' = \frac{\varepsilon^2}{16\,\hbar \, d} (1 + 2\,\mathfrak{a})^{-1/2} \varepsilon^{-1},
\tag{2}
$$

approach to the clean limit introduced in the Bierei, Maki, and Thompson (BMT) theory [73].

the mean-field approximation, which separates the FLC region from the region of critical fluctuations or fluctuations of the order parameter Δ directly near *Tc*, neglected in the Ginzburg-Landau (GL) theory [74, 75]. Hence it is evident that the correct determination of

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 *Tc*, where *ξc*(*T*) > *d*. Thus, according to the HL theory the AL fluctuation contribution predominates near *Tc*. Correspondingly, the MT mechanism

layers is impossible, since *ξc*(*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 *Tc*. Simultaneously the physical mechanism of superconducting fluctuations changes too resulting in MT-LD crossover. In accordance with the HL theory, the 2D-3D

*T*<sup>0</sup> = *Tc*{1 + 2[*ξc*(0)/*d*]

*ξc*(0)=(*d*/2)

Thus, now *ξc*(0) can be determined, since *ε*<sup>0</sup> is a measured reduced crossover temperature. Correspondingly, the MT-LD crossover occurs at a temperature at which *δ* � *α* [72], which

In accordance with our results [67], it should be the same temperature *T*0. It makes it possible to determine *τφ* which is the phase relaxation time (lifetime) of fluctuating pairs. The evaluation of *τφ* 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 *τφ* > *τ* or *τφ* ≈ *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

As it was shown in our study of FLC [2, 67, 68], for optimally doped YBCO the interval *Tc* < *T* < *Tc*<sup>0</sup> = (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 [69–72], 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].

*<sup>c</sup> <sup>f</sup>*) <sup>≈</sup> (*<sup>T</sup>* <sup>−</sup> *<sup>T</sup>m f*

*<sup>c</sup>* ) / *<sup>T</sup>m f*

*<sup>c</sup>* ) >> *h*¯ /*τφ*, where two-particle tunnelling between conducting

*ε*<sup>0</sup> = (*π h*¯)/[1.203(*l*/*ξab*)(8 *kB T τφ*)] (9)

*<sup>c</sup>* (6)

<sup>2</sup>} (7)

<sup>√</sup>*ε*0. (8)

*<sup>c</sup>* > *Tc* is the critical temperature in

Pseudogap and Local Pairs in High-Tc Superconductors 141

*ε* = *ln*(*T*/*T<sup>m</sup>*

is the reduced temperature in HTS's. Here *Tm f*

*<sup>c</sup>* is decisive in FLC calculations.

predominates for *<sup>k</sup>*(*<sup>T</sup>* <sup>−</sup> *<sup>T</sup>m f*

where it is assumed that *α*=1/2, i.e.

in HTS's as *T* draws near *Tc*.

crossover occurs at

gives

Correspondingly,

*Tm f*

Correspondingly, the HL theory gives for the MT fluctuation contribution the equation

$$
\sigma'\_{MT} = \frac{e^2}{8\,d\,\hbar} \frac{1}{1 - \alpha/\delta} \ln\left( (\delta/\alpha) \frac{1 + \mathfrak{a} + \sqrt{1 + 2\,\mathfrak{a}}}{1 + \delta + \sqrt{1 + 2\,\delta}} \right) \,\varepsilon^{-1}.\tag{3}
$$

In both equations

$$\mathfrak{a} = \mathfrak{2}\left[\mathfrak{f}\_{\mathfrak{c}}^{2}(T) \;/\; d^{2}\right] = \mathfrak{2}\left[\mathfrak{f}\_{\mathfrak{c}}(0) \;/\; d\right]^{2} \mathfrak{e}^{-1} \tag{4}$$

is the coupling parameter, d�11.7 Å in YBCO, is the distance between conducting layers,

$$\delta = 1,203 \frac{l}{\tilde{\xi}\_{ab}} \frac{16}{\pi \hbar} \left[ \frac{\tilde{\xi}\_c(0)}{d} \right]^2 k\_B T \,\tau\_\phi \tag{5}$$

is the pair-breaking parameter, and *ξc* is the coherence length along the c-axis, i.e. perpendicular to the *CuO*<sup>2</sup> conducting planes. The factor *β* = 1.203(*l* / *ξab*), where l is the electron mean-free path, and *ξab* 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,

$$\varepsilon = \ln \left( T / T\_{\mathcal{c}}^{m} f \right) \approx \left( T - T\_{\mathcal{c}}^{mf} \right) / \left. T\_{\mathcal{c}}^{mf} \right| \tag{6}$$

is the reduced temperature in HTS's. Here *Tm f <sup>c</sup>* > *Tc* 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 *Tc*, neglected in the Ginzburg-Landau (GL) theory [74, 75]. Hence it is evident that the correct determination of *Tm f <sup>c</sup>* 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 *Tc*, where *ξc*(*T*) > *d*. Thus, according to the HL theory the AL fluctuation contribution predominates near *Tc*. Correspondingly, the MT mechanism predominates for *<sup>k</sup>*(*<sup>T</sup>* <sup>−</sup> *<sup>T</sup>m f <sup>c</sup>* ) >> *h*¯ /*τφ*, where two-particle tunnelling between conducting layers is impossible, since *ξc*(*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 *Tc*. 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

$$T\_0 = T\_\mathfrak{c} \{ 1 + 2[\mathfrak{f}\_\mathfrak{c}(0)/d]^2 \} \tag{7}$$

where it is assumed that *α*=1/2, i.e.

4 Will-be-set-by-IN-TECH

Т\*

0 100 150 200 250 300 350

Т, K

(*T*) in HTS's [2], has

<sup>−</sup>1. (3)

<sup>−</sup><sup>1</sup> (4)

*kB T τφ* (5)

(*T*) in cuprates can be wholly

<sup>−</sup>1/2*ε*<sup>−</sup>1, (2)

 *ε*

**Figure 1.** Resistivity *ρ* versus temperature T (•) for YBCO film F1 (Table I); dashed line represents *ρN*(*T*).

found validation in the Nearly Antiferromagnetic Fermi Liquid (NAFL) model [11]. The

attributed to fluctuating Cooper pairing or whether there are other physical mechanisms responsible for the decrease of *ρxx*(*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

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

16 ¯*h d* (<sup>1</sup> <sup>+</sup> <sup>2</sup> *<sup>α</sup>*)

*<sup>c</sup>* (*T*) / *<sup>d</sup>*2] = <sup>2</sup> [*ξc*(0) / *<sup>d</sup>*]

(*δ*/*α*) <sup>1</sup> <sup>+</sup> *<sup>α</sup>* <sup>+</sup> <sup>√</sup><sup>1</sup> <sup>+</sup> <sup>2</sup> *<sup>α</sup>* <sup>1</sup> <sup>+</sup> *<sup>δ</sup>* <sup>+</sup> <sup>√</sup><sup>1</sup> <sup>+</sup> <sup>2</sup> *<sup>δ</sup>*

2

2 *ε*

Correspondingly, the HL theory gives for the MT fluctuation contribution the equation

is the coupling parameter, d�11.7 Å in YBCO, is the distance between conducting layers,

16 *π h*¯ *ξc*(0) *d*

is the pair-breaking parameter, and *ξc* is the coherence length along the c-axis, i.e. perpendicular to the *CuO*<sup>2</sup> conducting planes. The factor *β* = 1.203(*l* / *ξab*), where l is the electron mean-free path, and *ξab* is the coherence length in the *ab* plane, takes account of the

*ξab*

This way of determining *ρN*(*T*), which is widely used for calculation *σ*�

question of whether the appearance of excess conductivity *σ*�

temperatures preceding their transition into the SC state [67, 68].

*σ*�

*α* = 2 [*ξ*<sup>2</sup>

*σ*�

In both equations

*MT* <sup>=</sup> *<sup>e</sup>*<sup>2</sup> 8 *d h*¯

magnetic field the AL contribution to the FLC is given by the expression

1 <sup>1</sup> <sup>−</sup> *<sup>α</sup>*/*<sup>δ</sup> ln*

*<sup>δ</sup>* <sup>=</sup> 1, 203 *<sup>l</sup>*

*AL* <sup>=</sup> *<sup>e</sup>*<sup>2</sup>

500

400

300

200

xx,

мкОм·см

100

$$
\mathfrak{F}\_{\varepsilon}(0) = (d/2)\sqrt{\varepsilon}\_0. \tag{8}
$$

Thus, now *ξc*(0) can be determined, since *ε*<sup>0</sup> is a measured reduced crossover temperature. Correspondingly, the MT-LD crossover occurs at a temperature at which *δ* � *α* [72], which gives

$$
\varepsilon\_0 = (\pi \,\hbar) / [1.203 (l / \zeta\_{ab}) (8 \, k\_B \, T \, \tau\_{\phi})] \tag{9}
$$

In accordance with our results [67], it should be the same temperature *T*0. It makes it possible to determine *τφ* which is the phase relaxation time (lifetime) of fluctuating pairs. The evaluation of *τφ* 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 *τφ* > *τ* or *τφ* ≈ *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 *Tc*.

As it was shown in our study of FLC [2, 67, 68], for optimally doped YBCO the interval *Tc* < *T* < *Tc*<sup>0</sup> = (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 [69–72], 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].

#### 6 Will-be-set-by-IN-TECH 142 Superconductors – Materials, Properties and Applications Pseudogap and Local Pairs in High-*Tc* Superconductors <sup>7</sup>

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* ∼ 3*μkm*) 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 *Tpair* ∼ 130 *K*. The important point here is that period of oscillations corresponds to the charge of *Q* = 2*e* 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 *Tc*, and disappeared only at *T* = *Tpair* ≈ 140 *K*. But the wide maximum corresponding to the non-SC part of the PG was observed up to *T*<sup>∗</sup> ≈ 210 *K*. 3. It has subsequently been shown to be consistent with results of other groups dealing with the tunneling measurements [50–54]. 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 *Tc* up to *Tpair* ≈ 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 *Tpair*, suggesting expected transition into non-SC part of the PG.

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 *Tpair* ≥ 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 *Tpair* [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 *Tc* up to *Tpair*, is believed to be well established now [13]. Additionally, *Tpair* is found to amount to � 130, 140

Pseudogap and Local Pairs in High-Tc Superconductors 143

Before to proceed with a question what would happen with the local pairs in the non-SC part of the PG above *Tpair* [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 *Ton* ≈ 89, 4 *K* corresponding to the onset of the superconducting transition. The last one is *Tc* = 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 *Tpair* ∼ 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 *Tpair*. Thus, one may draw a conclusion that if there are paired fermions in the sample below *Tpair* they also have to exist at *T* > *Tpair*, 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 *Tc* < *T* < *T*∗ gradually gaining predominance [15, 27, 47, 81–84]. 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 (*ξab*(*T* ≤

We have, however, to keep in mind that we are dealing with the systems with low and reduced charge-carrier density *n <sup>f</sup>* , as mentioned above. It has been shown theoretically [6, 23–26, 85] that such systems acquire some unusual properties compared to conventional superconductors. In conventional superconductors it is assumed that the chemical potential *μ* = *εF*, where *ε<sup>F</sup>* = *EF* is the Fermi energy, and their relation actually depends on nothing. In the systems with low and reduced *n <sup>f</sup>* the chemical potential *μ* becomes a function of *n <sup>f</sup>* ,

scattering length in the *s* channel, and *m* is the mass of fermions with a quadratic dispersion law *<sup>ε</sup>*(*k*) <sup>∼</sup> *<sup>k</sup>*<sup>2</sup> [23–25]. In the case of HTS's it is believed that *<sup>ξ</sup>b*(*T*) equals to the coherence length of a superconductor in the *ab* plane, *ξab*(*T*), and *m* = *m*<sup>∗</sup> which is an effective mass of quasi-particles [11, 47]. (*m*<sup>∗</sup> ∼ 4.7*m*<sup>0</sup> in nearly optimally doped (OD) YBCO [9, 11, 67, 68]. Thus, the *ε<sup>b</sup>* becomes an important physical parameter of a Fermi liquid and determines a quantitative criterion for dense (*ε<sup>F</sup>* � |*εb*|) or dilute (*ε<sup>F</sup>* � |*εb*|) Fermi liquid. Accordingly, *kFε<sup>b</sup>* � 1 and *μ* = *ε<sup>F</sup>* in the first case, and *kFε<sup>b</sup>* �1 and *μ* = −|*εb*|/2 (�= *εF*) 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 *εb*/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

*<sup>b</sup>* )−<sup>1</sup> [23–26]. Here *<sup>ξ</sup><sup>b</sup>* is the

*<sup>T</sup>* and the energy of a bound state of two fermions, *<sup>ε</sup><sup>b</sup>* <sup>=</sup> <sup>−</sup>(*<sup>m</sup> <sup>ξ</sup>*<sup>2</sup>

and 150 *K* for Y123, Bi2212, and Bi2223, respectively [13, 80].

*T*∗) � (10 ÷ 15) Å) [2, 6, 8, 11, 12, 15].

*2.2.1. Properties of the systems with low and reduced current-carrier density*

**Figure 2.** STM image of Bi2212 with *Tc* =93 K at different temperatures [[16]].

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 (*Tc* = 93 K), the state of the system at 100 K is almost the same as that below *Tc*. 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 *Tpair* ≥ 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 *Tpair* [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 *Tc* up to *Tpair*, is believed to be well established now [13]. Additionally, *Tpair* is found to amount to � 130, 140 and 150 *K* for Y123, Bi2212, and Bi2223, respectively [13, 80].

### *2.2.1. Properties of the systems with low and reduced current-carrier density*

6 Will-be-set-by-IN-TECH

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* ∼ 3*μkm*) 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 *Tpair* ∼ 130 *K*. The important point here is that period of oscillations corresponds to the charge of *Q* = 2*e* 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 *Tc*, and disappeared only at *T* = *Tpair* ≈ 140 *K*. But the wide maximum corresponding to the non-SC part of the PG was observed up to *T*<sup>∗</sup> ≈ 210 *K*. 3. It has subsequently been shown to be consistent with results of other groups dealing with the tunneling measurements [50–54]. 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 *Tc* up to *Tpair* ≈ 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 *Tpair*, suggesting expected transition into non-SC part of the PG.

**Figure 2.** STM image of Bi2212 with *Tc* =93 K at different temperatures [[16]].

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 (*Tc* = 93 K), the state of the system at 100 K is almost the same as that below *Tc*. 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

Before to proceed with a question what would happen with the local pairs in the non-SC part of the PG above *Tpair* [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 *Ton* ≈ 89, 4 *K* corresponding to the onset of the superconducting transition. The last one is *Tc* = 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 *Tpair* ∼ 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 *Tpair*. Thus, one may draw a conclusion that if there are paired fermions in the sample below *Tpair* they also have to exist at *T* > *Tpair*, 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 *Tc* < *T* < *T*∗ gradually gaining predominance [15, 27, 47, 81–84]. 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 (*ξab*(*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 <sup>f</sup>* , as mentioned above. It has been shown theoretically [6, 23–26, 85] that such systems acquire some unusual properties compared to conventional superconductors. In conventional superconductors it is assumed that the chemical potential *μ* = *εF*, where *ε<sup>F</sup>* = *EF* is the Fermi energy, and their relation actually depends on nothing. In the systems with low and reduced *n <sup>f</sup>* the chemical potential *μ* becomes a function of *n <sup>f</sup>* , *<sup>T</sup>* and the energy of a bound state of two fermions, *<sup>ε</sup><sup>b</sup>* <sup>=</sup> <sup>−</sup>(*<sup>m</sup> <sup>ξ</sup>*<sup>2</sup> *<sup>b</sup>* )−<sup>1</sup> [23–26]. Here *<sup>ξ</sup><sup>b</sup>* is the scattering length in the *s* channel, and *m* is the mass of fermions with a quadratic dispersion law *<sup>ε</sup>*(*k*) <sup>∼</sup> *<sup>k</sup>*<sup>2</sup> [23–25]. In the case of HTS's it is believed that *<sup>ξ</sup>b*(*T*) equals to the coherence length of a superconductor in the *ab* plane, *ξab*(*T*), and *m* = *m*<sup>∗</sup> which is an effective mass of quasi-particles [11, 47]. (*m*<sup>∗</sup> ∼ 4.7*m*<sup>0</sup> in nearly optimally doped (OD) YBCO [9, 11, 67, 68]. Thus, the *ε<sup>b</sup>* becomes an important physical parameter of a Fermi liquid and determines a quantitative criterion for dense (*ε<sup>F</sup>* � |*εb*|) or dilute (*ε<sup>F</sup>* � |*εb*|) Fermi liquid. Accordingly, *kFε<sup>b</sup>* � 1 and *μ* = *ε<sup>F</sup>* in the first case, and *kFε<sup>b</sup>* �1 and *μ* = −|*εb*|/2 (�= *εF*) 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 *εb*/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, 23–26, 84–89]. 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.

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 *ξab*(*T*) = *<sup>ξ</sup>ab*(0) (*T*/*Tc* <sup>−</sup> <sup>1</sup>)−1/2 will determine the system behavior below *<sup>T</sup>*<sup>∗</sup> [2, 6, 24, 25, 86–89]. As temperature lowers, *ξab*(*T*) has to noticeably increase whereas the bonding energy *ε<sup>b</sup>* 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 *ξab* 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. [23–25, 89] and approved in our experimental

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 (*ξc*(*T*) < *d*) into 3D (*ξc*(*T*) > *d*) state, as T draws near *Tc*. The result is most likely a consequence of Gaussian fluctuations of the order parameter in 2D metals [6, 23–25, 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

studies [2, 27].

determines the FLC in any 3D system:

this case the crossover should occur at *ξc* ∼= *d*, i.e., at

of FLC. To find *τφ* we proceed as follows: we denote

the important parameters of the PG analysis.

*σ*�

*AL*3*<sup>D</sup>* <sup>=</sup> *<sup>e</sup>*<sup>2</sup>

32 ¯*h ξc*(0)

This means that the conventional 3D FLC is realized in HTS's as *T* → *Tc* [67, 77]. Above the crossover temperature *T*<sup>0</sup> (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<sup>c</sup>* > *d* [77]. Thus, in

*ξc*(0) = *d*

which is larger by a factor of two than is predicted by the LD and HL theories. *ξc*(0) is one of

Second, observation of the 2D-3D (MT-AL) crossover allows us to determine *ε*<sup>0</sup> quite accurately and, using Eq.(12), to obtain reliable values of *ξc*(0) [2, 67, 68]. However, *τφ* [(see Eq.(9)] still remains unknown, since neither *l* nor *ξab*(0) is measured experimentally in a study

*ε*

<sup>−</sup>1/2, (11)

Pseudogap and Local Pairs in High-Tc Superconductors 145

<sup>√</sup>*ε*0, (12)

*β* = [1.203 (*l*/*ξab*)]; (13)

It is clear that some other parameters, including the mean-field critical temperature *Tm 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 *Tm f <sup>c</sup>* ≈ *Tc*, i.e. it is identical to the BCS theory value [6, 23, 24, 33]. Moreover, *Tm f <sup>c</sup>* << *ε<sup>F</sup>* in this case. For the systems with low density *Tm f <sup>c</sup>* ∼ |*εb*|, whence *<sup>T</sup>m f <sup>c</sup>* >> *ε<sup>F</sup>* [6, 24, 25]. The latter relation means that in this case *Tm f <sup>c</sup>* characterizes not the condensation temperature *Tc* 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:

$$\Delta(T) = \Delta(0) - \left( (8\sqrt{\pi})\sqrt{-\text{x}\_0(\Delta(0)/T)^{3/2}} \right) \exp\left[ - (\mu^2 + \Delta^2(0))^{1/2}/T \right] \tag{10}$$

Equation (2) determines how the character of Δ(*T*) changes when the parameter *x*<sup>0</sup> 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.

### *2.2.2. 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 <sup>f</sup>* [74] was only of theoretical interest. The situation changed dramatically after the discovery of HTS's [1], where the charge-carrier density *n <sup>f</sup>* 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, *ξab*(*T*) is extremely short in HTS's, especially at high temperatures [2, 6, 8, 11, 12, 15] (*ξab*(*T* ≤ *T*∗) � 13Å in YBCO [67, 68]). It leads to the very strong bonding energy *ε<sup>b</sup>* in the pair [23–25] which is an additional requirement for the formation of the SBB [6, 24–26]. 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, 85–89]. *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 *Tc* << *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 *ξab*(*T*) = *<sup>ξ</sup>ab*(0) (*T*/*Tc* <sup>−</sup> <sup>1</sup>)−1/2 will determine the system behavior below *<sup>T</sup>*<sup>∗</sup> [2, 6, 24, 25, 86–89]. As temperature lowers, *ξab*(*T*) has to noticeably increase whereas the bonding energy *ε<sup>b</sup>* 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 *ξab* 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. [23–25, 89] and approved in our experimental studies [2, 27].

8 Will-be-set-by-IN-TECH

Bose-Einstein condensation (BEC) theory [6, 23–26, 84–89]. 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

It is clear that some other parameters, including the mean-field critical temperature *Tm f*

and temperature dependence of the SC order parameter Δ(*T*) have to change too. Analysis

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

−*x*0(Δ(0)/*T*)3/2

Equation (2) determines how the character of Δ(*T*) changes when the parameter *x*<sup>0</sup> 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

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 <sup>f</sup>* [74] was only of theoretical interest. The situation changed dramatically after the discovery of HTS's [1], where the charge-carrier density *n <sup>f</sup>* 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, *ξab*(*T*) is extremely short in HTS's, especially at high temperatures [2, 6, 8, 11, 12, 15] (*ξab*(*T* ≤ *T*∗) � 13Å in YBCO [67, 68]). It leads to the very strong bonding energy *ε<sup>b</sup>* in the pair [23–25] which is an additional requirement for the formation of the SBB [6, 24–26]. 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, 85–89]. *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

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 *Tc* << *T*∗, which at first glance seems to be in complete agreement

shows that in conventional superconductors with a high fermion density *Tm f*

*<sup>c</sup>* ∼ |*εb*|, whence *<sup>T</sup>m f*

is identical to the BCS theory value [6, 23, 24, 33]. Moreover, *Tm f*

*c*

(10)

*<sup>c</sup>* ≈ *Tc*, i.e. it

*<sup>c</sup>* << *ε<sup>F</sup>* in this case. For

*<sup>c</sup>* >> *ε<sup>F</sup>* [6, 24, 25]. The latter relation

<sup>−</sup>(*μ*<sup>2</sup> <sup>+</sup> <sup>Δ</sup>2(0))1/2/*<sup>T</sup>*

*<sup>c</sup>* characterizes not the condensation temperature *Tc* but rather the

 *exp* 

consequently may form at very high temperatures.

the systems with low density *Tm f*

Δ(*T*) = Δ(0) −

charge-carrier density decreases.

 (8 <sup>√</sup>*π*) 

*2.2.2. The model of the local pairs (Local Pair model)*

PG in FeAs-based HTS's still remain controversial [17, 21].

means that in this case *Tm f*

order parameter at T=0:

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 (*ξc*(*T*) < *d*) into 3D (*ξc*(*T*) > *d*) state, as T draws near *Tc*. The result is most likely a consequence of Gaussian fluctuations of the order parameter in 2D metals [6, 23–25, 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:

$$
\sigma'\_{AL3D} = \frac{e^2}{32\,\hbar \,\tilde{\xi}\_c(0)} e^{-1/2} \,\prime \,\tag{11}
$$

This means that the conventional 3D FLC is realized in HTS's as *T* → *Tc* [67, 77]. Above the crossover temperature *T*<sup>0</sup> (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<sup>c</sup>* > *d* [77]. Thus, in this case the crossover should occur at *ξc* ∼= *d*, i.e., at

$$
\mathfrak{F}\_{\mathcal{L}}(0) = d\sqrt{\varepsilon}\_{0\prime} \tag{12}
$$

which is larger by a factor of two than is predicted by the LD and HL theories. *ξc*(0) is one of the important parameters of the PG analysis.

Second, observation of the 2D-3D (MT-AL) crossover allows us to determine *ε*<sup>0</sup> quite accurately and, using Eq.(12), to obtain reliable values of *ξc*(0) [2, 67, 68]. However, *τφ* [(see Eq.(9)] still remains unknown, since neither *l* nor *ξab*(0) is measured experimentally in a study of FLC. To find *τφ* we proceed as follows: we denote

$$\mathcal{B} = [1.203 \, (l/\mathbb{Z}\_{ab})];\tag{13}$$

we assume as before that *τφ*(*T*) ∝ 1/*T* [11, 67], and for our subsequent estimate of *τφ*(100 *K*) we assume that *τφ T* =const. Finally, equation (9) can be rewritten as

$$
\pi\_{\Phi} \mathcal{S} \mathcal{T} = (\pi \hbar) / (8 \, k\_B \, T \varepsilon\_0) = A \, \varepsilon\_0 \tag{14}
$$

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 (*Tc*=87.4 K) close to optimally doped systems, the sample F6 (*Tc*=54.2 K) which represents weakly doped HTS systems, and the samples F3 and F4 with *Tc* near 80 K were investigated to obtain the required information. Fig. 3 displays the temperature dependencies of the longitudinal resistivity *ρxx*(*T*) = *ρ*(*T*) of the experimental films with

Pseudogap and Local Pairs in High-Tc Superconductors 147

**Figure 3.** Temperature dependencies of *ρxx* for the samples F1(1), F3 (2), F4 (3), and F6 (4). Inset: *ρxx*(T)

The inset shows *ρ*(*T*) for the sample F4 (*Tc*=80.3 K) in zero magnetic field **H**=0 (curve 1), showing how *Tc* was determined, and at **H** = 0.6*T* (curve 2), confirming the phase uniformity of the samples. Comparing the results with similar dependencies obtained for single crystals [97], the oxygen index of our samples can be estimated as follows: Δ *y* = (7 − *y*) ∼= 6.85 (F1), Δ *y* ∼= 6.8 (F3), Δ *y* ∼= 6.78 (F4), and Δ *y* ∼= 6.5 (F6). The resistivity parameters of the films are

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−*<sup>δ</sup>* [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

for sample F4 (*Tc*=80.3 K) in zero magnetic field (1) and for **H** = 0.6 T (2).

look for more details, one can see Refs.[2, 13, 20, 27, 67, 68, 94, 95].

listed in Table I.

parameters shown in Table I, where *d*<sup>0</sup> is the sample thickness.

where *<sup>A</sup>* = (*<sup>π</sup> <sup>h</sup>*¯)/(<sup>8</sup> *kB*) = 2.988 <sup>×</sup> <sup>10</sup>−12. Now the parameter *τφ*(<sup>100</sup> *<sup>k</sup>*) *<sup>β</sup>* is also clearly determined by the measured value of *ε*<sup>0</sup> and eventually enables us to determine *τφ*(100 *k*) [2, 67, 68].

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 *Tc* up to *T*∗ and contains PG in the explicit form. Besides, the dynamics of pair-creation and pair-breaking above *Tc* 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*<sup>∗</sup> � 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-*Tc* 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:

$$\sigma'(\varepsilon) = \frac{e^2 A\_4 \left(1 - \frac{T}{T^\*}\right) \left(\exp\left(-\frac{\Delta^\*}{T}\right)\right)}{(16\,\hbar\,\tilde{\xi}\_c(0)\,\sqrt{2\,\varepsilon\_{c0}^\* \,\sinh(2\,\varepsilon / \,\varepsilon\_{c0}^\*)}}\,\text{}\tag{15}$$

where *<sup>ε</sup>* is a reduced temperature given by Eq. (6), and *<sup>T</sup>m f <sup>c</sup>* > *Tc* 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 *Tc* have been taken into account in order to correctly describe the experiment [2, 13, 27]. Here *A*<sup>4</sup> 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, *ξc*(0), and the theoretical parameter *ε*∗ *<sup>c</sup>*<sup>0</sup> [2, 27], directly come from the experiment. The way of *ε*∗ *<sup>c</sup>*<sup>0</sup> 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*<sup>4</sup> we calculate *σ*� (*ε*) using Eq. (15) and fit the experiment in a range of 3D AL fluctuations near *Tc* 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]

$$\Delta^\*(T) = T \ln \frac{e^2 A\_4 \left(1 - \frac{T}{T^\*}\right)}{\sigma'(T) \operatorname{16} \hbar \xi\_c^\*(0) \sqrt{2 \varepsilon\_{c0}^\* \sinh(2 \,\varepsilon \,/\ \varepsilon\_{c0}^\*)}},\tag{16}$$

where *σ*� (T) is the experimentally measured excess conductivity over the whole temperature interval from *T*<sup>∗</sup> down to *Tm f <sup>c</sup>* .

### **3. Experimental results with respect to the Local Pair model**

### **3.1. 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−*<sup>δ</sup>* single-crystals [95], and evev in FeAs-based superconductor *SmFeAsO*0.85 with *Tc* =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 (*Tc*=87.4 K) close to optimally doped systems, the sample F6 (*Tc*=54.2 K) which represents weakly doped HTS systems, and the samples F3 and F4 with *Tc* near 80 K were investigated to obtain the required information. Fig. 3 displays the temperature dependencies of the longitudinal resistivity *ρxx*(*T*) = *ρ*(*T*) of the experimental films with parameters shown in Table I, where *d*<sup>0</sup> is the sample thickness.

10 Will-be-set-by-IN-TECH

we assume as before that *τφ*(*T*) ∝ 1/*T* [11, 67], and for our subsequent estimate of *τφ*(100 *K*)

where *<sup>A</sup>* = (*<sup>π</sup> <sup>h</sup>*¯)/(<sup>8</sup> *kB*) = 2.988 <sup>×</sup> <sup>10</sup>−12. Now the parameter *τφ*(<sup>100</sup> *<sup>k</sup>*) *<sup>β</sup>* is also clearly determined by the measured value of *ε*<sup>0</sup> and eventually enables us to determine *τφ*(100 *k*)

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 *Tc* up to *T*∗ and contains PG in the explicit form. Besides, the dynamics of pair-creation and pair-breaking above *Tc* 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*<sup>∗</sup> � 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-*Tc* superconductivity as a whole and in particular the pseudogap phenomenon. For lack of the theory, such equation for *σ*�

> 2 *ε*∗

temperature, as discussed above. Besides, the dynamics of pair-creation ((1 − *T*/*T*∗)) and pair-breaking (*exp*(−Δ∗/*T*)) above *Tc* have been taken into account in order to correctly describe the experiment [2, 13, 27]. Here *A*<sup>4</sup> 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

linear function of *ε* with a slope *λ* = −1/2 (Eq. (11)). Solving Eq. (15) for the pseudogap Δ∗(*T*)

Within proposed LP model the FLC and PG in YPrBCO films [94], in slightly doped *HoBa*2*Cu*3*O*7−*<sup>δ</sup>* single-crystals [95], and evev in FeAs-based superconductor *SmFeAsO*0.85 with *Tc* =55 K [20] (see division 3.4 below) were successfully studied for the first time. As

*<sup>c</sup>*<sup>0</sup> sinh(2 *ε* / *ε*<sup>∗</sup>

*<sup>c</sup>*<sup>0</sup> determination is shown in the insert to Fig. 4 and explained below. Thus, the

 2 *ε*∗

(T) is the experimentally measured excess conductivity over the whole temperature

*<sup>T</sup>*<sup>∗</sup> )

*<sup>c</sup>*<sup>0</sup> sinh(2 *ε* / *ε*<sup>∗</sup>

*c*0)

*τφ β T* = (*π h*¯)/(8 *kB T ε*0) = *A ε*0, (14)

(*ε*) has

(*ε*) using Eq.

, (16)

(*lnε*) is the

, (15)

*<sup>c</sup>* > *Tc* is the mean-field critical

*<sup>c</sup>*<sup>0</sup> [2, 27], directly come from the experiment.

*c*0)

we assume that *τφ T* =const. Finally, equation (9) can be rewritten as

been proposed in Ref. [27] with respect to the local pairs:

*e*<sup>2</sup> *A*<sup>4</sup> <sup>1</sup> <sup>−</sup> *<sup>T</sup> T*∗ *exp* <sup>−</sup> <sup>Δ</sup><sup>∗</sup> *T* 

(16 ¯*h ξc*(0)

only adjustable parameter remains the coefficient *A*4. To find *A*<sup>4</sup> we calculate *σ*�

<sup>Δ</sup>∗(*T*) = *T ln <sup>e</sup>*<sup>2</sup> *<sup>A</sup>*<sup>4</sup> (<sup>1</sup> <sup>−</sup> *<sup>T</sup>*

(*T*) 16 ¯*h ξc*(0)

*σ*�

**3. Experimental results with respect to the Local Pair model**

*<sup>c</sup>* .

**3.1. YBCO films with different oxygen concentration**

(15) and fit the experiment in a range of 3D AL fluctuations near *Tc* where *lnσ*�

*σ*� (*ε*) =

the *c*-axis, *ξc*(0), and the theoretical parameter *ε*∗

where *<sup>ε</sup>* is a reduced temperature given by Eq. (6), and *<sup>T</sup>m f*

[2, 67, 68].

The way of *ε*∗

where *σ*�

one can readily obtain [27]

interval from *T*<sup>∗</sup> down to *Tm f*

**Figure 3.** Temperature dependencies of *ρxx* for the samples F1(1), F3 (2), F4 (3), and F6 (4). Inset: *ρxx*(T) for sample F4 (*Tc*=80.3 K) in zero magnetic field (1) and for **H** = 0.6 T (2).

The inset shows *ρ*(*T*) for the sample F4 (*Tc*=80.3 K) in zero magnetic field **H**=0 (curve 1), showing how *Tc* was determined, and at **H** = 0.6*T* (curve 2), confirming the phase uniformity of the samples. Comparing the results with similar dependencies obtained for single crystals [97], the oxygen index of our samples can be estimated as follows: Δ *y* = (7 − *y*) ∼= 6.85 (F1), Δ *y* ∼= 6.8 (F3), Δ *y* ∼= 6.78 (F4), and Δ *y* ∼= 6.5 (F6). The resistivity parameters of the films are listed in 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−*<sup>δ</sup>* [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].


**Table 1.** The parameters of the YBCO films with different oxygen concentration (sample F1−F6).

### **3.2. 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 *Tc* < *T* < *T*∗ [2, 6, 13, 23, 26, 47]. It is believed that formation of such pairs gives rise to their real binding energy, *εb*, which the quantity Δ<sup>∗</sup> 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 *Tc*, 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∗(*Tc* ). Here, by analogy to the superconducting state, Δ∗(*Tc* ) is the value of the PG in the limit *T* → *Tc*. As it was shown in Ref. [15], Δ∗(*Tc*) ∼= Δ(0) and, correspondingly, Δ<sup>∗</sup> satisfies the condition 2Δ<sup>∗</sup> ∼ *kB Tc*, as it was demonstrated in Ref's. [99–101]. The conclusion has subsequently been confirmed by the tunneling experiments in Bi compounds [52]. To justify the values of Δ∗(*Tc*) 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∗(*Tc* ), which makes it possible to adjust the value chosen for this parameter. As expected, matching is achieved for values of Δ∗(*Tc*) which are determined by the relation 2Δ∗(*Tc*)/*kB Tc* ∼= 5 [99–101]. For the sample F4 Δ∗(*Tc*) ≈ 190*K*, i.e., 2Δ∗(*Tc*)/(*kB Tc*) � 4.75 [2, 27].

**Figure 4.** *σ*�

extrapolation of the rectilinear section [2, 27].

The same experimental dependencies of *σ*�

slope *α* of this linear function: *ε*∗

measured values of the *σ*�

(*T*) in the coordinates *lnσ*� versus *lnε* (solid curve **I**) for sample F1 for T from *Tc* to *T*∗ in

*<sup>c</sup>*<sup>0</sup> which is reciprocal of the

(*T*), as shown in Fig. 4, were obtained for all

Pseudogap and Local Pairs in High-Tc Superconductors 149

(*T*) with the corresponding set of parameters, we should obtain

comparison with theory: curve 1-Maki-Thompsom contribution; 2-Aslamasov-Larkin contribution; 3 theory [102]; 4 - Eq. (15) (short dashed segment). Inset: ln *σ*�−<sup>1</sup> versus *ε* (solid line); dashed line -

region of exponential behavior of *<sup>σ</sup>*�−1(*T*) (usually from *Tc*<sup>01</sup> <sup>≈</sup> <sup>100</sup> *<sup>K</sup>* up to *Tc*<sup>02</sup> <sup>≈</sup> <sup>150</sup> *<sup>K</sup>*). It is clear that *lnσ*�−<sup>1</sup> is to be the linear function of *ε*, as shown in the insert of Fig. 4. The advantage

studied compounds, including *HoBa*2*Cu*3*O*7−*<sup>δ</sup>* [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−*<sup>δ</sup>* 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*<sup>4</sup> [2, 27], and coincidence in the 3D fluctuation region near *Tc* 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

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∗(*Tc*) 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 *Tmax* ≈ 130 K. The important point here is

of this approach is that it enables us to determine the parameter *ε*∗

*<sup>c</sup>*<sup>0</sup> = 1/*α* [2, 102].

The curve constructed for F1 using Eq. (15) with the parameters *ξ*∗ *<sup>c</sup>*<sup>0</sup> =0.233, *<sup>ξ</sup>c*(0) = 1.65 Å, *<sup>T</sup>m f c* =88.46 K, *T*∗=203 K, Δ∗(*Tc*)=218 K, and *A*<sup>4</sup> = 20 is labeled with the number 4 in Fig. 4. As one can see from the figure, the equation (15) describes well the experimental curve (thick solid line marked by **I**) over the whole temperature interval from *T*∗ down to *Tc*. Similar results were obtained for the all other films studied.

Curve numbered 3 in the figure reproduces result of Ref. [102] in which, however, both the dynamics of pair-creation and pair-breaking above *Tc* 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 *Tc*<sup>0</sup> . Correspondingly, the adjustable coefficient *A*<sup>3</sup> [102] is chosen so that the computed curve matches experiment in the

12 Will-be-set-by-IN-TECH

F1 1050 87.4 88.46 148 476 203 1.65 F3 850 81.4 84.55 237 760 213 1.75 F4 850 80.3 83.4 386 1125 218 1.78 F6 650 54.2 55.88 364 1460 245 2.64

of paired fermions (local pairs) at temperatures *Tc* < *T* < *T*∗ [2, 6, 13, 23, 26, 47]. It is believed that formation of such pairs gives rise to their real binding energy, *εb*, which the quantity Δ<sup>∗</sup> 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

Since the PG is not measured directly in our experiments, the problem reduces to determining

Eq. (10). To perform the analysis, the excess conductivity of every studied YBCO film, measured in the whole temperature interval from *T*∗ down to *Tc*, 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∗(*Tc* ). Here, by analogy to the superconducting state, Δ∗(*Tc* ) is the value of the PG in the limit *T* → *Tc*. As it was shown in Ref. [15], Δ∗(*Tc*) ∼= Δ(0) and, correspondingly, Δ<sup>∗</sup> satisfies the condition 2Δ<sup>∗</sup> ∼ *kB Tc*, as it was demonstrated in Ref's. [99–101]. The conclusion has subsequently been confirmed by the tunneling experiments in Bi compounds [52]. To justify the values of Δ∗(*Tc*) used in our analysis we applied an approach proposed in Ref. [88]

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∗(*Tc* ), which makes it possible to adjust the value chosen for this parameter. As expected, matching is achieved for values of Δ∗(*Tc*) which are determined by the relation 2Δ∗(*Tc*)/*kB Tc* ∼= 5 [99–101]. For the sample F4 Δ∗(*Tc*) ≈ 190*K*, i.e., 2Δ∗(*Tc*)/(*kB Tc*) � 4.75

=88.46 K, *T*∗=203 K, Δ∗(*Tc*)=218 K, and *A*<sup>4</sup> = 20 is labeled with the number 4 in Fig. 4. As one can see from the figure, the equation (15) describes well the experimental curve (thick solid line marked by **I**) over the whole temperature interval from *T*∗ down to *Tc*. Similar results

Curve numbered 3 in the figure reproduces result of Ref. [102] in which, however, both the dynamics of pair-creation and pair-breaking above *Tc* 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 *Tc*<sup>0</sup> . Correspondingly, the adjustable coefficient *A*<sup>3</sup> [102] is chosen so that the computed curve matches experiment in the

**Table 1.** The parameters of the YBCO films with different oxygen concentration (sample F1−F6).

**3.2. Pseudogap in YBCO films with different oxygen concentration**

() (*K*) (*K*) (*μ*Ω*cm*) (*μ*Ω*cm*) (*K*) ()

*<sup>c</sup> ρ*(100*K*) *ρ*(300*K*) T\* *ξc*(0)

(*T*) arises as a result of the formation

(*T*) are neglected. But a definite

*<sup>c</sup>*<sup>0</sup> =0.233, *<sup>ξ</sup>c*(0) = 1.65 Å, *<sup>T</sup>m f*

*c*

(*T*) and comparing it with those obtained from

Sample *<sup>d</sup>*<sup>0</sup> *Tc <sup>T</sup>m f*

We proceed from the fact that the excess conductivity *σ*�

in which, however, the fluctuation contributions into *σ*�

The curve constructed for F1 using Eq. (15) with the parameters *ξ*∗

Δ∗(*T*) from the experimental dependence *σ*�

were obtained for the all other films studied.

spectrum [59, 61, 84, 98].

[2, 27].

**Figure 4.** *σ*� (*T*) in the coordinates *lnσ*� versus *lnε* (solid curve **I**) for sample F1 for T from *Tc* to *T*∗ in comparison with theory: curve 1-Maki-Thompsom contribution; 2-Aslamasov-Larkin contribution; 3 theory [102]; 4 - Eq. (15) (short dashed segment). Inset: ln *σ*�−<sup>1</sup> versus *ε* (solid line); dashed line extrapolation of the rectilinear section [2, 27].

region of exponential behavior of *<sup>σ</sup>*�−1(*T*) (usually from *Tc*<sup>01</sup> <sup>≈</sup> <sup>100</sup> *<sup>K</sup>* up to *Tc*<sup>02</sup> <sup>≈</sup> <sup>150</sup> *<sup>K</sup>*). It is clear that *lnσ*�−<sup>1</sup> 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*∗ *<sup>c</sup>*<sup>0</sup> which is reciprocal of the slope *α* of this linear function: *ε*∗ *<sup>c</sup>*<sup>0</sup> = 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−*<sup>δ</sup>* [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−*<sup>δ</sup>* 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*<sup>4</sup> [2, 27], and coincidence in the 3D fluctuation region near *Tc* 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∗(*Tc*) 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 *Tmax* ≈ 130 K. The important point here is

#### 14 Will-be-set-by-IN-TECH 150 Superconductors – Materials, Properties and Applications Pseudogap and Local Pairs in High-*Tc* Superconductors <sup>15</sup>

that the coherence length *ξab*(*Tmax*) was found to be the same for every studied film, namely, *ξab*(*Tmax*) ≈ 18Å [2, 27].

**Figure 6.** Temperature dependence of the Knight shift K(T) in classical superconductors (solid line) and

Pseudogap and Local Pairs in High-Tc Superconductors 151

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 *RH*(*T*) ∼ 1/*e n <sup>f</sup>* was found on cooling just below *T*<sup>∗</sup> [2, 81, 97]. Obviously, the increase of *RH* directly points out at the decrease of the normal charge-carrier density *n <sup>f</sup>* . 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*<sup>∗</sup> ≈ 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*<sup>∗</sup> ≈ 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.

as it becomes of common occurrence in the literature [80].

Let us turn back to Fig. 5. With decreasing temperature below *Tmax*, *ξab*(*T*) continues to increase whereas *εb*(*T*) becomes smaller. Ultimately, at *T* ≤ *Tmax* = *Tpair* ≈ 130 *K*, where *ξab*(*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, 78–80], 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 *Tmax* it is the SC part of the pseudogap. Moreover, we consider *ξab*(*Tmax*) ≈ 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 *ξab*(*T*) < 18Å, and transform into fluctuating Cooper pairs when *ξab*(*T*) > 18Å below *Tmax*, thus resulting in the BEC-BCS transition [2, 27]. That is why, I will call *Tmax* as *Tpair* now,

in HTS's (dashed line) [[57]].

**Figure 5.** Dependencies of Δ∗/*kB* on T calculated by Eq. (16) for samples F1 (upper curve, circles), F3 (squares), F4 (dots) and F6 (low curve, triangles). *Tmax* = *Tpair* ≈ 130 *K*.

Let us discuss the obtained results (Fig. 5) now. Above 130 K *ξab*(T) is very small (*ξab*(*T*∗) ≈ 13 Å), whereas the coupling energy *ε<sup>b</sup>* is very strong. It is just the condition for the formation of the SBB [6, 24–26]. It was found [27] that over the temperature interval *Tmax* = *Tpair* < *T* < *T*∗ every experimental Δ∗(*T*)/Δ∗(*Tmax*) curve shown in Fig. 5 can be fitted by the Babaev-Kleinert (BK) theory [87] in the BEC limit (low *n <sup>f</sup>*) 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*<sup>∗</sup> 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 *Tpair*. Thus, above *Tmax* (Fig. 5), which is also called *Tpair* 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*<sup>∗</sup> down to *Tpair* [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*<sup>∗</sup> (Fig. 6). In Landau theory [75] *K* ∼ *ρn*(*ε*) ≡ *ρF*, where *ρn*(*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 *Tc*. Thus, the decrease of K(T) directly

14 Will-be-set-by-IN-TECH

that the coherence length *ξab*(*Tmax*) was found to be the same for every studied film, namely,

**Figure 5.** Dependencies of Δ∗/*kB* on T calculated by Eq. (16) for samples F1 (upper curve, circles), F3

Let us discuss the obtained results (Fig. 5) now. Above 130 K *ξab*(T) is very small (*ξab*(*T*∗) ≈ 13 Å), whereas the coupling energy *ε<sup>b</sup>* is very strong. It is just the condition for the formation of the SBB [6, 24–26]. It was found [27] that over the temperature interval *Tmax* = *Tpair* < *T* < *T*∗ every experimental Δ∗(*T*)/Δ∗(*Tmax*) curve shown in Fig. 5 can be fitted by the Babaev-Kleinert (BK) theory [87] in the BEC limit (low *n <sup>f</sup>*) 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*<sup>∗</sup> 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 *Tpair*. Thus, above *Tmax* (Fig. 5), which is also called *Tpair* 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*<sup>∗</sup> down to *Tpair* [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*<sup>∗</sup> (Fig. 6). In Landau theory [75] *K* ∼ *ρn*(*ε*) ≡ *ρF*, where *ρn*(*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 *Tc*. Thus, the decrease of K(T) directly

(squares), F4 (dots) and F6 (low curve, triangles). *Tmax* = *Tpair* ≈ 130 *K*.

*ξab*(*Tmax*) ≈ 18Å [2, 27].

**Figure 6.** Temperature dependence of the Knight shift K(T) in classical superconductors (solid line) and in HTS's (dashed line) [[57]].

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 *RH*(*T*) ∼ 1/*e n <sup>f</sup>* was found on cooling just below *T*<sup>∗</sup> [2, 81, 97]. Obviously, the increase of *RH* directly points out at the decrease of the normal charge-carrier density *n <sup>f</sup>* . 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*<sup>∗</sup> ≈ 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*<sup>∗</sup> ≈ 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 *Tmax*, *ξab*(*T*) continues to increase whereas *εb*(*T*) becomes smaller. Ultimately, at *T* ≤ *Tmax* = *Tpair* ≈ 130 *K*, where *ξab*(*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, 78–80], 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 *Tmax* it is the SC part of the pseudogap. Moreover, we consider *ξab*(*Tmax*) ≈ 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 *ξab*(*T*) < 18Å, and transform into fluctuating Cooper pairs when *ξab*(*T*) > 18Å below *Tmax*, thus resulting in the BEC-BCS transition [2, 27]. That is why, I will call *Tmax* as *Tpair* now, as it becomes of common occurrence in the literature [80].

### **3.3. Fluctuation conductivity and pseudogap in** *HoBa*2*Cu*3*O*7−*<sup>δ</sup>* **single crystals under pressure**

3D fluctuation term of the AL theory (Eq.(11)) (Fig. 7, dashed line 1). Above *T*0, up to *T*<sup>01</sup> ≈ 95 K (ln *ε*<sup>01</sup> ≈ −0.8), the experimental data are well extrapolated by the MT fluctuation

Here I would like to emphasize that curve 2 in Fig. 7 is plotted with *d*<sup>1</sup> = 2.95Å, where *d*<sup>1</sup> is the distance between conducting CuO2 planes in HoBaCuO [106], and with *τφ* (100K) *β* = (0.665 <sup>±</sup> 0.002)×10−13s which is defined by a formula *τφ <sup>β</sup>* T=A*ε*<sup>01</sup> (Eq. (14)). Accordingly, *d*<sup>1</sup> corresponds to *T*<sup>01</sup> ≈ 95 K (ln *ε*<sup>01</sup> ≈ −0.8) marked by the right arrow in the figure. At *T* ≤ *T*01, *ξc*(*T*) ≥ *d*<sup>1</sup> is believed to couple the CuO2 planes by Josephson interaction, and 2D FLC has to appear [67, 77]. This scenario of the FLC appearance reminds that observed for the

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

**Figure 8.** Δ∗/*kB* as a function of T. Curve 1 - P = 0; 2 - P = 4.8 kbar. Arrows designate maxima

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*<sup>0</sup> ≈ 67.3 K (ln *ε*<sup>0</sup> ≈ −3.55), at which *ξc*(*T*) = *d*, the habitual dimensional 2D-3D (AL - MT) crossover is distinctly seen on the plot. Below *T*<sup>0</sup> *ξc*(*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

The similar ln *σ*� vs ln *ε* was also obtained for *P* = 0 (*Tc* = 61.4 K, *ρ*(100K) = 200*μ*Ωcm, *<sup>T</sup>*<sup>0</sup> <sup>=</sup> 63.4 K, *<sup>T</sup>*<sup>01</sup> <sup>=</sup> 79.4 K, *τφ*(100K)*<sup>β</sup>* <sup>=</sup> 1.13 · <sup>10</sup>−13s and *<sup>d</sup>*<sup>1</sup> <sup>=</sup> 3.2Å). Thus, the pressure leads to increase in *T*<sup>0</sup> 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

1/2 <sup>01</sup> = *dε*

1/2

<sup>0</sup> = (1.98 ± 0.005)Å. The

(*lnε*) is close to that shown in Fig. 10. If we

Pseudogap and Local Pairs in High-Tc Superconductors 153

contribution (Eq. (3)) (Fig. 7, curve2) of the HL theory [72]

*SmFeAsO*0.85 sample (see div. 3.4), and found *lnσ*�

YBCO film F1 shown in Fig. 4.

temperatures at P = 0: *Tm*<sup>1</sup> and *Tm*2.

[20], at both *T*<sup>01</sup> and *T*<sup>0</sup> (arrows in Fig. 7) *ξc*(0) = *d*1*ε*

finding is believed to confirm the validity of our analysis.

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−*<sup>δ</sup>* which has magnetic moment *<sup>μ</sup>eff* ≈ 9,7*μB*, due to magnetic moment of pure Ho which is *μeff* ≈ 10.6*μ<sup>B</sup>* [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=*<sup>δ</sup>* (Re = Y, Ho, *Dy* ··· ) type with reduced *n <sup>f</sup>* 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 HoBa2Cu3O7−*<sup>δ</sup>* (HoBCO) with T*<sup>c</sup>* ≈ 62 K and oxygen index 7-*δ* ≈ 6,65 was studied [95]. The comparison of results with those obtained for YBa2Cu3O7−*<sup>δ</sup>* (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*<sup>∗</sup> � T*<sup>c</sup>* resistivity of HoBCO, *ρ*(T), deviates down from linearity resulting in the appearance of the excess conductivity *σ*� (*T*) = *σ*(*T*) − *σN*(*T*) (Eq.(1)). Resulting ln *σ*� as a function of ln *ε* near *Tc* is displayed in Fig. 7.

**Figure 7.** ln *σ*� vs ln *�* for *P* = 4.8 kbar (dots). Curve 1 - AL term; 2 - MT term with *d*<sup>1</sup> = 2.95Å; 3 - MT term with *d* = 11.68Å. Arrows designate *T*<sup>0</sup> and *T*01.

As expected in the LP model, above the mean field critical temperature *Tm f <sup>c</sup>* ≈ 65.4 K and up to *T*<sup>0</sup> ≈ 67.3 K (ln *ε*<sup>0</sup> ≈ −3.55), experimental *lnσ*� (*lnε*) can be well extrapolated by the 3D fluctuation term of the AL theory (Eq.(11)) (Fig. 7, dashed line 1). Above *T*0, up to *T*<sup>01</sup> ≈ 95 K (ln *ε*<sup>01</sup> ≈ −0.8), the experimental data are well extrapolated by the MT fluctuation contribution (Eq. (3)) (Fig. 7, curve2) of the HL theory [72]

16 Will-be-set-by-IN-TECH

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−*<sup>δ</sup>* which has magnetic moment *<sup>μ</sup>eff* ≈ 9,7*μB*, due to magnetic moment of pure Ho which is *μeff* ≈ 10.6*μ<sup>B</sup>* [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=*<sup>δ</sup>* (Re = Y, Ho, *Dy* ··· ) type with reduced *n <sup>f</sup>* 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

HoBa2Cu3O7−*<sup>δ</sup>* (HoBCO) with T*<sup>c</sup>* ≈ 62 K and oxygen index 7-*δ* ≈ 6,65 was studied [95]. The comparison of results with those obtained for YBa2Cu3O7−*<sup>δ</sup>* (div. 3.2) and FeAs-based superconductor *SmFeAsO*0.85 (div. 3.4) is to shed more light on the role of magnetic subsystem

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*<sup>∗</sup> � T*<sup>c</sup>* resistivity of HoBCO, *ρ*(T), deviates down from linearity resulting

**Figure 7.** ln *σ*� vs ln *�* for *P* = 4.8 kbar (dots). Curve 1 - AL term; 2 - MT term with *d*<sup>1</sup> = 2.95Å; 3 - MT

As expected in the LP model, above the mean field critical temperature *Tm f*

(T) and pseudogap Δ∗(T) of weakly doped high-T*c* single crystals

(*T*) = *σ*(*T*) − *σN*(*T*) (Eq.(1)). Resulting ln *σ*� as

*<sup>c</sup>* ≈ 65.4 K and

(*lnε*) can be well extrapolated by the

**3.3. Fluctuation conductivity and pseudogap in** *HoBa*2*Cu*3*O*7−*<sup>δ</sup>* **single crystals**

**under pressure**

fluctuation conductivity *σ*�

in the appearance of the excess conductivity *σ*�

a function of ln *ε* near *Tc* is displayed in Fig. 7.

term with *d* = 11.68Å. Arrows designate *T*<sup>0</sup> and *T*01.

up to *T*<sup>0</sup> ≈ 67.3 K (ln *ε*<sup>0</sup> ≈ −3.55), experimental *lnσ*�

in the HTS's.

Here I would like to emphasize that curve 2 in Fig. 7 is plotted with *d*<sup>1</sup> = 2.95Å, where *d*<sup>1</sup> is the distance between conducting CuO2 planes in HoBaCuO [106], and with *τφ* (100K) *β* = (0.665 <sup>±</sup> 0.002)×10−13s which is defined by a formula *τφ <sup>β</sup>* T=A*ε*<sup>01</sup> (Eq. (14)). Accordingly, *d*<sup>1</sup> corresponds to *T*<sup>01</sup> ≈ 95 K (ln *ε*<sup>01</sup> ≈ −0.8) marked by the right arrow in the figure. At *T* ≤ *T*01, *ξc*(*T*) ≥ *d*<sup>1</sup> is believed to couple the CuO2 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.

**Figure 8.** Δ∗/*kB* as a function of T. Curve 1 - P = 0; 2 - P = 4.8 kbar. Arrows designate maxima temperatures at P = 0: *Tm*<sup>1</sup> and *Tm*2.

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*<sup>0</sup> ≈ 67.3 K (ln *ε*<sup>0</sup> ≈ −3.55), at which *ξc*(*T*) = *d*, the habitual dimensional 2D-3D (AL - MT) crossover is distinctly seen on the plot. Below *T*<sup>0</sup> *ξc*(*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*<sup>01</sup> and *T*<sup>0</sup> (arrows in Fig. 7) *ξc*(0) = *d*1*ε* 1/2 <sup>01</sup> = *dε* 1/2 <sup>0</sup> = (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 (*Tc* = 61.4 K, *ρ*(100K) = 200*μ*Ωcm, *<sup>T</sup>*<sup>0</sup> <sup>=</sup> 63.4 K, *<sup>T</sup>*<sup>01</sup> <sup>=</sup> 79.4 K, *τφ*(100K)*<sup>β</sup>* <sup>=</sup> 1.13 · <sup>10</sup>−13s and *<sup>d</sup>*<sup>1</sup> <sup>=</sup> 3.2Å). Thus, the pressure leads to increase in *T*<sup>0</sup> 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*3*<sup>D</sup>* = 1.1 and = 0.53 at *P* = 4.8 kbar and *P* = 0 kbar, respectively. The closer *C*3*<sup>D</sup>* to 1.0, the more homogeneous is the sample structure [2, 67].

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-*Tc* 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-*Tc* systems [17, 109, 110], and especially for *SmFeAsO*1−*<sup>x</sup>* [113–116] (Fig. 9)

Pseudogap and Local Pairs in High-Tc Superconductors 155

Naturally, rather peculiar normal state behavior of the system upon T diminution is expected

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 (*Tc* = 55 K) performed on fully computerized set up [20]. The width of the resistive transition into superconducting (SC) state is Δ *T* ≤ 2*K* suggesting good phase and structural uniformity of the sample. In accordance with the LP model approach the resistivity curve above *T*<sup>∗</sup> ∼ 170*K*

The crossing of measured *σ*�−2(*T*) with T-axis denotes the mean-field critical temperature

ln*σ*� versus ln*ε* is displayed in Fig. 10 in the temperature interval relatively close to *Tc* (dots) and compared with the HL theory [72] in the clean limit (curves 1-3). As expected, up to *T*<sup>0</sup> ≈

(*T*) = *σ*(*T*) − *σN*(*T*) using Eq.(1).

*<sup>c</sup>* determination within the LP model [2, 67]. Resulting

in this case [109, 110, 113], when x is of the order of 0.15, as it is in our sample [20].

are reported.

**Figure 9.** Phase diagram of *SmFeAsO*1−*xFx* [[113]].

is extrapolated by the straight line to get *σ*�

*<sup>c</sup>* <sup>∼</sup><sup>=</sup> <sup>57</sup>*K*. This is the usual way of *<sup>T</sup>m f*

*3.4.1. Experimental details*

*Tm f*

The experimental dependence of *σ*� (*T*) in the whole temperature interval from *T*∗ and down to *Tm f <sup>c</sup>* 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) with the following parameters derived from experiment: *ε*∗ *<sup>c</sup>*<sup>0</sup> <sup>=</sup> 0.88, *<sup>ξ</sup>c*(0) <sup>=</sup> 1.57Å, *<sup>T</sup>m f <sup>c</sup>* = 62.3 K, *T*<sup>∗</sup> = 242 K, *A*<sup>4</sup> = 4.95 and Δ∗(*T*)/*k* = 155 K. Curve 2 is plotted for *P* = 4.8 kbar with *ε*∗ *<sup>c</sup>*<sup>0</sup> = 0.67, *ξc*(0) = 1.98Å, *Tm f <sup>c</sup>* = 65.4 K, *T*<sup>∗</sup> = 243 K, and *A*<sup>4</sup> = 18, Δ∗(*T*)/*k* = 160 K, respectively. In this case *Tc* = 64 K and *ρ*(100)K = 172*μ*Ωcm.

When *P* = 0, Δ∗(*T*) exhibits two unexpected maxima at *Tm*<sup>1</sup> ≈ 195 K and at *Tm*<sup>2</sup> ≈ 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 *Tc* and *ξc*(0) somewhat increase. Note that the maxima of Δ∗(*T*) and initial values of *ρ*, T*<sup>c</sup>* and *ξc*(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 SmFeAsO0.85 [20] (see div. 3.4). The result can be explained by the influence of paramagnetism in HoBa2Cu3O7−*<sup>δ</sup>* [95, 104, 105]. Additionally, the linear drop of the PG, found for both HoBa2Cu3O7−*<sup>δ</sup>* [95] and SmFeAsO0.85 [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.
