**3.4. Fluctuation conductivity and pseudogap in** *SmFeASO*0.85

Despite of a huge amount of papers devoted to FeAs-based superconductors, there is an evident lack of the FLC and PG studies in these compounds [20]. This state of affairs is most likely due to the extreme complexity of the electronic configuration of the FeAs-based compounds, where the strong influence of magnetism, being changed with doping, is observed [17–19]. It is well known that upon electron or hole doping with F substitution at the O site [17, 113], or with oxygen vacancies [17, 18] all properties of parent RFeAsO compounds drastically change and evident antiferromagnetic (AFM) order has to disappear [17–19]. However, recent results [107–110] point toward an important role of low-energy spin fluctuations which emerge on doping away from the parent antiferromagnetic state which is of a spin-density wave (SDW) type [107, 108, 110]. Thus, below *TS* ∼ 150*K* the AFM fluctuations, being likely of spin wave type, are believed to noticeably affect the properties of doped *RFeAsO*1−*xFx* systems (Fig. 9). As shown by many studies [108–110] the static magnetism persists well into the SC regime of ferropnictides. In *SmFeAsO*1−*<sup>x</sup>* strongly disordered but static magnetism and superconductivity both are found to coexist in the wide range of doping, and prominent low-energy spin fluctuations are observed up to the highest achievable doping levels where *Tc* is maximal [21, 107, 108].

The interplay between superconductivity and magnetism has been a long-standing fascinating problem [111, 112], and relation between the SDW and SC order is a central topic in the current research on the FeAs-based high-*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) are reported.

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

Naturally, rather peculiar normal state behavior of the system upon T diminution is expected in this case [109, 110, 113], when x is of the order of 0.15, as it is in our sample [20].

### *3.4.1. Experimental details*

18 Will-be-set-by-IN-TECH

[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

*<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

*<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

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

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

Despite of a huge amount of papers devoted to FeAs-based superconductors, there is an evident lack of the FLC and PG studies in these compounds [20]. This state of affairs is most likely due to the extreme complexity of the electronic configuration of the FeAs-based compounds, where the strong influence of magnetism, being changed with doping, is observed [17–19]. It is well known that upon electron or hole doping with F substitution at the O site [17, 113], or with oxygen vacancies [17, 18] all properties of parent RFeAsO compounds drastically change and evident antiferromagnetic (AFM) order has to disappear [17–19]. However, recent results [107–110] point toward an important role of low-energy spin fluctuations which emerge on doping away from the parent antiferromagnetic state which is of a spin-density wave (SDW) type [107, 108, 110]. Thus, below *TS* ∼ 150*K* the AFM fluctuations, being likely of spin wave type, are believed to noticeably affect the properties of doped *RFeAsO*1−*xFx* systems (Fig. 9). As shown by many studies [108–110] the static magnetism persists well into the SC regime of ferropnictides. In *SmFeAsO*1−*<sup>x</sup>* strongly disordered but static magnetism and superconductivity both are found to coexist in the wide range of doping, and prominent low-energy spin fluctuations are observed up to the highest achievable doping

*<sup>c</sup>*<sup>0</sup> <sup>=</sup> 0.88, *<sup>ξ</sup>c*(0) <sup>=</sup> 1.57Å, *<sup>T</sup>m f*

and Δ∗(*T*)/*k* = 155 K. Curve 2 is plotted for *P* = 4.8 kbar with *ε*∗

is removed suggesting the assumption. Thus, both *σ*�

**3.4. Fluctuation conductivity and pseudogap in** *SmFeASO*0.85

in more details in the next division.

levels where *Tc* is maximal [21, 107, 108].

(*T*) in the whole temperature interval from *T*∗ and down

*<sup>c</sup>* = 62.3 K, *T*<sup>∗</sup> = 242 K, *A*<sup>4</sup> = 4.95

(*T*) and Δ∗(*T*) are markedly different

*<sup>c</sup>*<sup>0</sup> = 0.67, *ξc*(0) = 1.98Å,

sample structure [2, 67].

to *Tm f*

*Tm f*

The experimental dependence of *σ*�

derived from experiment: *ε*∗

and *ρ*(100)K = 172*μ*Ωcm.

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* is extrapolated by the straight line to get *σ*� (*T*) = *σ*(*T*) − *σN*(*T*) using Eq.(1).

The crossing of measured *σ*�−2(*T*) with T-axis denotes the mean-field critical temperature *Tm f <sup>c</sup>* <sup>∼</sup><sup>=</sup> <sup>57</sup>*K*. This is the usual way of *<sup>T</sup>m f <sup>c</sup>* determination within the LP model [2, 67]. Resulting 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> ≈

same value of *ξc*(0) = *d ε*

*3.4.2. Pseudogap analysis*

= 0.616, *<sup>ξ</sup>* (0) = 1.4Å, *<sup>T</sup>m f*

*T*<sup>∗</sup> down to *Tm f*

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

slope 1/*α* still determines the parameter *ε*∗

*SaFeAsO*1−*<sup>x</sup>* [107–110], as will be discussed below.

whereas the other above parameters remain unchangeable.

[95] (div. 3.3)

1/2

*<sup>c</sup>*<sup>0</sup> = (1.4 ± 0.005)Å as calculated above using the usual crossover

Pseudogap and Local Pairs in High-Tc Superconductors 157

*<sup>c</sup>* was found to be very close to that of YBCO films

*<sup>c</sup>*<sup>0</sup> used in the PG analysis [20, 27]. Taking obtained

(*ε*)

*c*0

(*ε*) curve is

temperature *T*0. We think this fact is to confirm the supposition. At the same time the phase relaxation time changes noticeably and amounts *τφ*(100*K*) *<sup>β</sup>* <sup>=</sup> (1.41 <sup>±</sup> 0.03) <sup>×</sup> <sup>10</sup>−13s in this case. Substitution of measured values of *ξc*(0) and this new *τφ*(100*K*) *β* into Eq. (3) enables us to fit the experimental data by the MT term just up to *Tc*<sup>0</sup> which is about 15 K above *Tc* (Fig. 10, curve 1). The result suggests enhanced 2D MT fluctuations in *SmFeAsO*1<sup>−</sup> compared to YBCO films. Recall, the similar enhanced 2D MT fluctuations are found for HoBa2Cu3O7−*<sup>δ</sup>*

Somehow surprisingly, *lnσ*� versus *lnε* measured for the *SmFeAsO*0.85 in the whole

(Fig. 4). Even the exponential dependence of the reciprocal of the excess conductivity *σ*�−1(*T*) still occurs between *lnεc*<sup>0</sup> and *lnεc*<sup>02</sup> ((69 - 100)K). As a result, the function *lnσ*�−<sup>1</sup> versus *ε* appears to be linear in this temperature range (similar to that shown in insert in Fig. 4), and its

results into account, one can draw a conclusion that the LP model approach can be applied to analyze the temperature dependence of the PG. To proceed with the PG analysis the value of the coefficient *A*<sup>4</sup> must be found. As before, to determine *A*<sup>4</sup> we fit the experimental *σ*�

by the theoretical curve (Eq. (15)) in the region of 3D fluctuations near *Tc* [20, 27]. All other

constructed using Eq. (15) with the reasonable set of experimentally measured parameters: *ξ*∗

160 K and found to describe the experimental data well in the whole temperature interval from

down from experiment, negligible in the case of YBCO films [67], is observed. It is due to enhanced MT fluctuation contribution in the 2D fluctuation region, as mentioned above. Compare results with those obtained for magnetic superconductor *Dy*0.6*Y*0.4*Rh*3.85*Ru*0.15*B*<sup>4</sup> [117] we assume this enhancement to be the consequence of weak magnetic fluctuations in

Now we have almost all parameters to calculate the temperature dependence of PG using Eq.(16) except the value of Δ∗(*Tc* ) which actually is responsible for the *A*4. The problem is, the value of Δ∗(*Tc*) and in turn the ratio 2Δ∗(*Tc*)/*kB Tc* in Fe-based superconductors remain uncertain. Reported in the literature values for <sup>Δ</sup>(0) and 2Δ(0)/*Tc* in *SmFeAsO*1−*<sup>x</sup>* range from 2Δ(0) ≈ 37 meV (2Δ/*Tc* ∼ 8, strong-coupling limit) obtained in measurements of far-infrared permittivity [17], down to Δ = (8 - 8.5) meV (2Δ/*Tc* ∼ (3.55 - 3.8)) measured by a scanning tunneling spectroscopy [118, 119] which is very close to standard value 3.52 of the BCS theory (weak-coupling limit). It is believed at present that *SmFeAsO*1−*<sup>x</sup>* has two superconducting gaps Δ1(0) ≈ 6.7 meV and Δ2(0) which ranges from ≈ 15 *meV* up to ≈ 21 *meV* [120]. Besides we still think that Δ∗(*Tc* ) ≈ Δ(0) [20], as is was justified in Ref. [15]. To feel more flexible, four curves are finally plotted in Fig. 11 with Δ∗(*Tc*)/*kB* = 160 K (2Δ∗(*Tc*)/*Tc* ∼= 5.82), = 140 K (2Δ∗(*Tc*)/*Tc* ∼= 5.0), 120 K (2Δ∗(*Tc*)/*Tc* ∼= 4.36) and = 100 K (2Δ∗(*Tc*)/*Tc* ∼= 3.63) from top to bottom, respectively. Naturally, different values of coefficients *A*<sup>4</sup> correspond to each curve,

*<sup>c</sup>* = 56.99 K, *T*<sup>∗</sup> = 175*K*, *A*<sup>4</sup> = 1.98 and optionally chosen Δ∗(*Tc*)/*kB* =

*<sup>c</sup>* [95]. The only exception is the 2D MT region where relatively small deviation

parameters directly come from experiment, as discussed above. The theoretical *σ*�

**Figure 10.** *lnσ*� as a function of *lnε* near *Tc* (dots) in comparison with the HL theory: 1-MT contribution (d = 3.05Å); 2 - 3D AL contribution; 3-MT contribution (d = 8.495Å).

58.5 K (*lnε*<sup>0</sup> =-3.6) the data are well extrapolated by the AL fluctuation contribution (Eq. 11) for any 3D system (straight line 2 in the figure). As mentioned above, this 3D fluctuation behavior was found to be typical for all without exception HTS compounds [2, 20, 67, 80, 95]. Accordingly, above *T*0, up to *Tc*<sup>0</sup> ≈ 69*K* (ln*εc*<sup>0</sup> ∼= −1.55), this is 2D MT term (Eq. (3)) of the HL theory (Fig. 10, curve 1) which dominates well above *Tc* in the 2D fluctuation region [2, 67], as discussed in details in div. 2.2. As before, *ξc*(0) = *d ε* 1/2 <sup>0</sup> is the coherence length along the *c*-axis, i.e. perpendicular to the conducting planes, and d is the distance between conducting layers in *SmFeAsO*1−*x*. Thus, expected MT-AL (2D-3D) crossover is clearly seen in Fig. 10 at *lnε*<sup>0</sup> =-3.6. The fact enables us to determine *T*<sup>0</sup> (Eq. (7)) and therefore the *ε*<sup>0</sup> (Eq. (7, 9)) with adequate accuracy. Now, proceeding in the usual way, i.e., using Eqs. (12, 14), and set d = 8.495Å, which is a dimension of the *SmFeAsO*0.85 unit sell along the *c*-axis [114], *<sup>ξ</sup>c*(0)=(1.4 <sup>±</sup> 0.005)Å, and *τφ*(100*K*) *<sup>β</sup>* = (<sup>11</sup> <sup>±</sup> 0.03) <sup>×</sup> <sup>10</sup>−<sup>13</sup> s are derived from the experiment. As it is seen from the figure, Eq. (11) with measured value of *ξc*(0) and the scaling factor *<sup>C</sup>*3*<sup>D</sup>* = 0.083 describes the data fairly well just above *<sup>T</sup>m f <sup>c</sup>* (Fig. 10, dashed line 2) suggesting 3D fluctuation behavior of *SmFeAsO*0.85 near *Tc*.

Till now the FeAs-based superconductor behaves like the YBCO films. However, the discrepancy appears when MT contribution is analyzed (Fig. 10, curve 3). Indeed, substituting the measured values of *ξc*(0) and *τφ*(100*K*) *β* into Eq. (3) we obtain curve 3 which looks like that found for YBCO films but apparently does not match the data. The finding suggests that our choice of d is very likely wrong in this case. To proceed with the analysis we have to suppose that *SmFeAsO*1−*<sup>x</sup>* becomes quasi-two-dimensional just when *<sup>ξ</sup>c*(*T*), getting rise with temperature diminution, becomes equal to *d*<sup>1</sup> = 3.05 Å at *T* = *Tco* ≈ 69*K* (*lnεc*<sup>0</sup> ∼= −1.55). It is worth to emphasize that (3.1 ÷ 3.0)Å is the distance between *As* layers in conducting *As* − *Fe* − *As* planes in *SmFeAsO*1−*<sup>x</sup>* [18, 114]. Below this temperature *<sup>ξ</sup>c*(*T*) is believed to couple the *As* layers in the planes by Josephson interaction [95]. This approach gives the same value of *ξc*(0) = *d ε* 1/2 *<sup>c</sup>*<sup>0</sup> = (1.4 ± 0.005)Å as calculated above using the usual crossover temperature *T*0. We think this fact is to confirm the supposition. At the same time the phase relaxation time changes noticeably and amounts *τφ*(100*K*) *<sup>β</sup>* <sup>=</sup> (1.41 <sup>±</sup> 0.03) <sup>×</sup> <sup>10</sup>−13s in this case. Substitution of measured values of *ξc*(0) and this new *τφ*(100*K*) *β* into Eq. (3) enables us to fit the experimental data by the MT term just up to *Tc*<sup>0</sup> which is about 15 K above *Tc* (Fig. 10, curve 1). The result suggests enhanced 2D MT fluctuations in *SmFeAsO*1<sup>−</sup> compared to YBCO films. Recall, the similar enhanced 2D MT fluctuations are found for HoBa2Cu3O7−*<sup>δ</sup>* [95] (div. 3.3)

### *3.4.2. Pseudogap analysis*

20 Will-be-set-by-IN-TECH

**Figure 10.** *lnσ*� as a function of *lnε* near *Tc* (dots) in comparison with the HL theory: 1-MT contribution

58.5 K (*lnε*<sup>0</sup> =-3.6) the data are well extrapolated by the AL fluctuation contribution (Eq. 11) for any 3D system (straight line 2 in the figure). As mentioned above, this 3D fluctuation behavior was found to be typical for all without exception HTS compounds [2, 20, 67, 80, 95]. Accordingly, above *T*0, up to *Tc*<sup>0</sup> ≈ 69*K* (ln*εc*<sup>0</sup> ∼= −1.55), this is 2D MT term (Eq. (3)) of the HL theory (Fig. 10, curve 1) which dominates well above *Tc* in the 2D fluctuation region

along the *c*-axis, i.e. perpendicular to the conducting planes, and d is the distance between conducting layers in *SmFeAsO*1−*x*. Thus, expected MT-AL (2D-3D) crossover is clearly seen in Fig. 10 at *lnε*<sup>0</sup> =-3.6. The fact enables us to determine *T*<sup>0</sup> (Eq. (7)) and therefore the *ε*<sup>0</sup> (Eq. (7, 9)) with adequate accuracy. Now, proceeding in the usual way, i.e., using Eqs. (12, 14), and set d = 8.495Å, which is a dimension of the *SmFeAsO*0.85 unit sell along the *c*-axis [114], *<sup>ξ</sup>c*(0)=(1.4 <sup>±</sup> 0.005)Å, and *τφ*(100*K*) *<sup>β</sup>* = (<sup>11</sup> <sup>±</sup> 0.03) <sup>×</sup> <sup>10</sup>−<sup>13</sup> s are derived from the experiment. As it is seen from the figure, Eq. (11) with measured value of *ξc*(0) and the

Till now the FeAs-based superconductor behaves like the YBCO films. However, the discrepancy appears when MT contribution is analyzed (Fig. 10, curve 3). Indeed, substituting the measured values of *ξc*(0) and *τφ*(100*K*) *β* into Eq. (3) we obtain curve 3 which looks like that found for YBCO films but apparently does not match the data. The finding suggests that our choice of d is very likely wrong in this case. To proceed with the analysis we have to suppose that *SmFeAsO*1−*<sup>x</sup>* becomes quasi-two-dimensional just when *<sup>ξ</sup>c*(*T*), getting rise with temperature diminution, becomes equal to *d*<sup>1</sup> = 3.05 Å at *T* = *Tco* ≈ 69*K* (*lnεc*<sup>0</sup> ∼= −1.55). It is worth to emphasize that (3.1 ÷ 3.0)Å is the distance between *As* layers in conducting *As* − *Fe* − *As* planes in *SmFeAsO*1−*<sup>x</sup>* [18, 114]. Below this temperature *<sup>ξ</sup>c*(*T*) is believed to couple the *As* layers in the planes by Josephson interaction [95]. This approach gives the

1/2

<sup>0</sup> is the coherence length

*<sup>c</sup>* (Fig. 10, dashed line 2)

(d = 3.05Å); 2 - 3D AL contribution; 3-MT contribution (d = 8.495Å).

[2, 67], as discussed in details in div. 2.2. As before, *ξc*(0) = *d ε*

scaling factor *<sup>C</sup>*3*<sup>D</sup>* = 0.083 describes the data fairly well just above *<sup>T</sup>m f*

suggesting 3D fluctuation behavior of *SmFeAsO*0.85 near *Tc*.

Somehow surprisingly, *lnσ*� versus *lnε* measured for the *SmFeAsO*0.85 in the whole temperature range from *T*<sup>∗</sup> down to *Tm f <sup>c</sup>* was found to be very close to that of YBCO films (Fig. 4). Even the exponential dependence of the reciprocal of the excess conductivity *σ*�−1(*T*) still occurs between *lnεc*<sup>0</sup> and *lnεc*<sup>02</sup> ((69 - 100)K). As a result, the function *lnσ*�−<sup>1</sup> versus *ε* appears to be linear in this temperature range (similar to that shown in insert in Fig. 4), and its slope 1/*α* still determines the parameter *ε*∗ *<sup>c</sup>*<sup>0</sup> used in the PG analysis [20, 27]. Taking obtained results into account, one can draw a conclusion that the LP model approach can be applied to analyze the temperature dependence of the PG. To proceed with the PG analysis the value of the coefficient *A*<sup>4</sup> must be found. As before, to determine *A*<sup>4</sup> we fit the experimental *σ*� (*ε*) by the theoretical curve (Eq. (15)) in the region of 3D fluctuations near *Tc* [20, 27]. All other parameters directly come from experiment, as discussed above. The theoretical *σ*� (*ε*) curve is constructed using Eq. (15) with the reasonable set of experimentally measured parameters: *ξ*∗ *c*0 = 0.616, *<sup>ξ</sup>* (0) = 1.4Å, *<sup>T</sup>m f <sup>c</sup>* = 56.99 K, *T*<sup>∗</sup> = 175*K*, *A*<sup>4</sup> = 1.98 and optionally chosen Δ∗(*Tc*)/*kB* = 160 K and found to describe the experimental data well in the whole temperature interval from *T*<sup>∗</sup> down to *Tm f <sup>c</sup>* [95]. The only exception is the 2D MT region where relatively small deviation down from experiment, negligible in the case of YBCO films [67], is observed. It is due to enhanced MT fluctuation contribution in the 2D fluctuation region, as mentioned above. Compare results with those obtained for magnetic superconductor *Dy*0.6*Y*0.4*Rh*3.85*Ru*0.15*B*<sup>4</sup> [117] we assume this enhancement to be the consequence of weak magnetic fluctuations in *SaFeAsO*1−*<sup>x</sup>* [107–110], as will be discussed below.

Now we have almost all parameters to calculate the temperature dependence of PG using Eq.(16) except the value of Δ∗(*Tc* ) which actually is responsible for the *A*4. The problem is, the value of Δ∗(*Tc*) and in turn the ratio 2Δ∗(*Tc*)/*kB Tc* in Fe-based superconductors remain uncertain. Reported in the literature values for <sup>Δ</sup>(0) and 2Δ(0)/*Tc* in *SmFeAsO*1−*<sup>x</sup>* range from 2Δ(0) ≈ 37 meV (2Δ/*Tc* ∼ 8, strong-coupling limit) obtained in measurements of far-infrared permittivity [17], down to Δ = (8 - 8.5) meV (2Δ/*Tc* ∼ (3.55 - 3.8)) measured by a scanning tunneling spectroscopy [118, 119] which is very close to standard value 3.52 of the BCS theory (weak-coupling limit). It is believed at present that *SmFeAsO*1−*<sup>x</sup>* has two superconducting gaps Δ1(0) ≈ 6.7 meV and Δ2(0) which ranges from ≈ 15 *meV* up to ≈ 21 *meV* [120]. Besides we still think that Δ∗(*Tc* ) ≈ Δ(0) [20], as is was justified in Ref. [15]. To feel more flexible, four curves are finally plotted in Fig. 11 with Δ∗(*Tc*)/*kB* = 160 K (2Δ∗(*Tc*)/*Tc* ∼= 5.82), = 140 K (2Δ∗(*Tc*)/*Tc* ∼= 5.0), 120 K (2Δ∗(*Tc*)/*Tc* ∼= 4.36) and = 100 K (2Δ∗(*Tc*)/*Tc* ∼= 3.63) from top to bottom, respectively. Naturally, different values of coefficients *A*<sup>4</sup> correspond to each curve, whereas the other above parameters remain unchangeable.

**Figure 11.** Δ∗/*kB* versus T dependencies for *SmFeAsO*0.85 with four different values of Δ∗(*Tc*)/*kB* (see the text).

Very unexpected Δ∗(*T*) behavior is observed (Fig. 11). The most striking result is a sharp drop of Δ∗(*T*) at *Ts* ≈ 147 K, as clearly illustrates the curve with Δ∗(*Tc*)/*kB* =140 K plotted without symbols. Usually *Ts* is treated as a temperature at which a structural tetragonal-orthorhombic transition occurs in parent SmFeAsO. In the undoped FeAs compounds it is also expected to be a transition to SDW ordering regime [17–19]. Below *TS* the pseudogap Δ∗(*T*) drops linearly down to *TAFM* ≈ 133 K (Fig. 11), which is attributed to the AFM ordering of the Fe spins in a parent SmFeAsO compounds [17, 121]. Below *TAFM* the slop of the Δ∗(*T*) curves apparently depends on the value of Δ∗(*Tc*) [20]. Strictly speaking it is difficult to say at present is *TAFM* = *TN* of the whole system or not because the AFM ordering of Sm spins occurs at ≈ 5*K* only [17, 121].

Found Δ∗(*T*) behavior is believed to be explained in terms of the theory by Machida, Nokura and Matsubara (MNM) [111] developed for AFM superconductors in which the AFM ordering with a wave vector *Q* may coexist with the superconductivity. This assumption was supported in, eg., the theory by Chi and Nagi [122]. In the MNM theory the effect of the AFM molecular field *hQ*(*T*) (|*hQ*| � *ε<sup>F</sup>* ) on the Cooper pairing was studied. It was shown, that below *TN* the BCS coupling parameter Δ(*T*) is reduced by a factor [1 − *const* · |*hQ*(*T*)|/*εF*] due to the formation of energy gaps of SDW on the Fermi surface along *Q*. As a result the effective attractive interaction *g N*˘ (0) or, equivalently, the density of states at the Fermi energy *ε<sup>F</sup>* is diminished by the periodic molecular field that is

$$\circlearrowleft N(0) = \emptyset \, N(0) \, [1 - \alpha \, m(T)].\tag{17}$$

**Figure 12.** Δ∗(*T*)/Δ*max* in *SmFeAsO*0.85: (red ◦) - Δ∗(*Tc*)/*kB*=130 K; (◦) - 135 K. Solid curves Δ(*T*)/Δ*max* correspond to MNM theory with the different *α* ∼ 1/[*g N*˘ (0)]: (1) - *α* = 0.1, (2) - *α* = 0.2, (3) - *α* = 0.3, (4) - *α*

Pseudogap and Local Pairs in High-Tc Superconductors 159

immediately below *TN*. As *m*(*T*) saturates at lower temperatures, Δ(*T*) gradually recovers its value with increasing the superconducting condensation energy (Fig. 12, solid curves). This additional magnetization *m*(*T*) was also shown to explain the anomaly in the upper critical field *Hc*<sup>2</sup> just below *TN* observed in studying of *RMo*6*S*<sup>8</sup> (R = Gd, Tb, and Dy) [111]. However, predicted by the theory decrease of Δ(*T*) at *T* ≤ *TN* was only recently observed in AFM superconductor *ErNi*2*B*2*C* with *Tc* ≈ 11 K and *TN* ≈ 6 K, below which the SDW ordering is believed to occur in the system [123]. The result evidently supports the prediction of the MNM theory. Our results are found to be in a qualitatively agreement with the MNM theory as shown in Fig. 12, where the data for Δ∗(*Tc* )/*kB* = 130 K (red ◦) and Δ∗(*Tc*)/*kB* = 135

The curves are scaled at *T*/*Tc* = 0.7 and demonstrate rather good agreement with the theory below *T*/*Tc* = 0.7. Note, that the upper scale is *T*/*T*∗. Both shown Δ∗(*T*) dependencies suggest the issue that just Δ∗(*Tc*)/*kB* = 133 K, which is just *TAFM*, would provide the best fit with the theory. Above *T*/*Tc* = 0.7 the data evidently deviate from the BCS theory. It seems to be reasonable seeing *SmFeAsO*0.15 as well as any other ferropnictides is not a BCS

It is important to emphasize that in our case we observe the particularities of Δ∗(*T*) in the PG state, i. e. well above *Tc* but just at *T* ≈ *Ts*, below which the SDW ordering in parent SmFeAsO should occur. However, it seems to be somehow surprising as no SDW ordering in optimally doped *SmFeAsO*0.15 is expected. The found very specific Δ∗(*T*) can be understood taking mentioned above weak AFM fluctuations (low-energy spin fluctuations), which should exist in the system, into account. At the singular temperature *Ts* these fluctuations are believed to enhance AFM in the system likely in the form of SDW. After that, in accordance with the MNM theory scenario, the SDW has to suppress the order parameter of the local pairs as shown by our results. Thus, the results support the existence both weak AFM fluctuations and the paired fermions (local pairs) in the PG region, which order parameter is apparently

= 0.6 , (5) - *α* = 1.0; *TN*/*Tc* = 0.7 [[111]].

superconductor.

suppressed by these fluctuations [20].

K (◦) are compared with the MNM theory (solid lines).

Here *m*(*T*) is the normalized sublattice magnetization of the antiferromagnetic state and *α* is a changeable parameter of the theory. Between *Tc* and *TN* (*Tc* > *TN* is assumed) the order parameter is that of the BCS theory. Since below *TN* the magnetization *m*(*T*) becomes nonvanishing, *g N*˘ (0) is weakened that results in turn in a sudden linear drop of Δ(*T*)

22 Will-be-set-by-IN-TECH

**Figure 11.** Δ∗/*kB* versus T dependencies for *SmFeAsO*0.85 with four different values of Δ∗(*Tc*)/*kB* (see

Very unexpected Δ∗(*T*) behavior is observed (Fig. 11). The most striking result is a sharp drop of Δ∗(*T*) at *Ts* ≈ 147 K, as clearly illustrates the curve with Δ∗(*Tc*)/*kB* =140 K plotted without symbols. Usually *Ts* is treated as a temperature at which a structural tetragonal-orthorhombic transition occurs in parent SmFeAsO. In the undoped FeAs compounds it is also expected to be a transition to SDW ordering regime [17–19]. Below *TS* the pseudogap Δ∗(*T*) drops linearly down to *TAFM* ≈ 133 K (Fig. 11), which is attributed to the AFM ordering of the Fe spins in a parent SmFeAsO compounds [17, 121]. Below *TAFM* the slop of the Δ∗(*T*) curves apparently depends on the value of Δ∗(*Tc*) [20]. Strictly speaking it is difficult to say at present is *TAFM* = *TN* of the whole system or not because the AFM ordering of Sm spins occurs at

Found Δ∗(*T*) behavior is believed to be explained in terms of the theory by Machida, Nokura and Matsubara (MNM) [111] developed for AFM superconductors in which the AFM ordering with a wave vector *Q* may coexist with the superconductivity. This assumption was supported in, eg., the theory by Chi and Nagi [122]. In the MNM theory the effect of the AFM molecular field *hQ*(*T*) (|*hQ*| � *ε<sup>F</sup>* ) on the Cooper pairing was studied. It was shown, that below *TN* the BCS coupling parameter Δ(*T*) is reduced by a factor [1 − *const* · |*hQ*(*T*)|/*εF*] due to the formation of energy gaps of SDW on the Fermi surface along *Q*. As a result the effective attractive interaction *g N*˘ (0) or, equivalently, the density of states at the Fermi energy *ε<sup>F</sup>* is

Here *m*(*T*) is the normalized sublattice magnetization of the antiferromagnetic state and *α* is a changeable parameter of the theory. Between *Tc* and *TN* (*Tc* > *TN* is assumed) the order parameter is that of the BCS theory. Since below *TN* the magnetization *m*(*T*) becomes nonvanishing, *g N*˘ (0) is weakened that results in turn in a sudden linear drop of Δ(*T*)

*g N*˘ (0) = *g N*(0) [1 − *α m*(*T*)]. (17)

the text).

≈ 5*K* only [17, 121].

diminished by the periodic molecular field that is

**Figure 12.** Δ∗(*T*)/Δ*max* in *SmFeAsO*0.85: (red ◦) - Δ∗(*Tc*)/*kB*=130 K; (◦) - 135 K. Solid curves Δ(*T*)/Δ*max* correspond to MNM theory with the different *α* ∼ 1/[*g N*˘ (0)]: (1) - *α* = 0.1, (2) - *α* = 0.2, (3) - *α* = 0.3, (4) - *α* = 0.6 , (5) - *α* = 1.0; *TN*/*Tc* = 0.7 [[111]].

immediately below *TN*. As *m*(*T*) saturates at lower temperatures, Δ(*T*) gradually recovers its value with increasing the superconducting condensation energy (Fig. 12, solid curves). This additional magnetization *m*(*T*) was also shown to explain the anomaly in the upper critical field *Hc*<sup>2</sup> just below *TN* observed in studying of *RMo*6*S*<sup>8</sup> (R = Gd, Tb, and Dy) [111]. However, predicted by the theory decrease of Δ(*T*) at *T* ≤ *TN* was only recently observed in AFM superconductor *ErNi*2*B*2*C* with *Tc* ≈ 11 K and *TN* ≈ 6 K, below which the SDW ordering is believed to occur in the system [123]. The result evidently supports the prediction of the MNM theory. Our results are found to be in a qualitatively agreement with the MNM theory as shown in Fig. 12, where the data for Δ∗(*Tc* )/*kB* = 130 K (red ◦) and Δ∗(*Tc*)/*kB* = 135 K (◦) are compared with the MNM theory (solid lines).

The curves are scaled at *T*/*Tc* = 0.7 and demonstrate rather good agreement with the theory below *T*/*Tc* = 0.7. Note, that the upper scale is *T*/*T*∗. Both shown Δ∗(*T*) dependencies suggest the issue that just Δ∗(*Tc*)/*kB* = 133 K, which is just *TAFM*, would provide the best fit with the theory. Above *T*/*Tc* = 0.7 the data evidently deviate from the BCS theory. It seems to be reasonable seeing *SmFeAsO*0.15 as well as any other ferropnictides is not a BCS superconductor.

It is important to emphasize that in our case we observe the particularities of Δ∗(*T*) in the PG state, i. e. well above *Tc* but just at *T* ≈ *Ts*, below which the SDW ordering in parent SmFeAsO should occur. However, it seems to be somehow surprising as no SDW ordering in optimally doped *SmFeAsO*0.15 is expected. The found very specific Δ∗(*T*) can be understood taking mentioned above weak AFM fluctuations (low-energy spin fluctuations), which should exist in the system, into account. At the singular temperature *Ts* these fluctuations are believed to enhance AFM in the system likely in the form of SDW. After that, in accordance with the MNM theory scenario, the SDW has to suppress the order parameter of the local pairs as shown by our results. Thus, the results support the existence both weak AFM fluctuations and the paired fermions (local pairs) in the PG region, which order parameter is apparently suppressed by these fluctuations [20].

**3.5. Angle-resolved photoemission measurements of the energy pseudogap of high-***Tc* (*Bi*, *Pb*)2(*Sr*, *La*)2*CuO*6+*<sup>δ</sup>* **superconductors: A model evidence of local**

Taking all above consideration into account, it can be concluded that the FLC and PG description in terms of local pairs gives a set of reasonable and self-consistent results. However, to justify the conclusion it would be appropriate to test the LP model approah using independent results of other research groups who have measured straightforwardly the PG or any other related effects. But for a long time there was a lack of indispensable data.

Fortunately, analysis of the pseudogap in (*Bi*, *Pb*)2(*Sr*, *La*)2*CuO*6+*<sup>δ</sup>* (Bi2201) single-crystals with various *Tc*'s by means of ARPES spectra study was recently reported [80]. The study of Bi2201 allows avoid the complications resulting from the bilayer splitting and strong antinodal bosonic mode coupling inherent to Bi2212 and Bi2223 [90, 91]. Symmetrized energy distribution curves (EDCs) were found to demonstrate the opening of the pseudogap on cooling below *T*∗. It was shown that *T*∗, obtained from the resistivity measurements, agrees well with one determined from the ARPES data using a single spectral peak criterion [80]. Finally, from the ARPES experiments information about the temperature dependence of the loss of the spectral weight close to the Fermi level, *W*(*EF*), was derived [80]. *W*(*EF*) versus *T* measured for optimally doped OP35K Bi2201 (*Tc*=35 K, *T*∗ = 160 K) turned out to be rather unexpected, as shown in Fig. 14a taken from Ref. [80]. Above *T*∗ the *W*(*EF*) is nonlinear function of *T*. But below *T*∗, over the temperature range from *T*<sup>∗</sup> to *Tpair* = (110 ± 5) K (Fig. 14a), the *W*(*EF*)(*T*) decreases linearly which is considered as a characteristic behavior of the "proper" PG state [80]. However, no assumption as for physical nature of this linearity as well as for existence of the paired fermions in the PG region is proposed. Below *Tpair* the *W*(*EF*) vs *T* noticeably deviates down from the linearity (Fig. 14a). The deviation suggests the onset of another state of the system, which likely arises from the pairing of electrons, since the

*W*(*EF*)(*T*) associated with this state smoothly evolves through *Tc* (Fig. 14a).

the following reasonable set of parameters: *Tc* = 35 K, *<sup>T</sup>m f*

(Fig. 5). Besides, the maximum of Δ∗(*T*)/Δ∗

from the resistivity data using Eq. (1). Resulting Δ∗(*T*)/Δ∗

To compare results and justify our Local Pair model, the *ρab* vs *T* of the OP35K Bi2201, reported in Ref. [80], was studied within the LP model [13]. The Δ∗(*T*) was calculated by Eq. (16) with

dots). As expected, the shape of the Δ∗(*T*) curve is similar to that found for YBCO films

change of the *W*(*EF*)(*T*) slop at *Tpair*, measured by ARPES, which seems to be reasonable. In fact, in accordance with our logic, *Tmax* is just the temperature which divides the PG region on SC and non-SC parts depending on the local pair state, as described above. Recall, that above *Tmax* the local pairs are expected to be in the form of SBB. Most likely just the specific properties of SBB cause the linear *W*(*EF*)(*T*) over this temperature region (Fig. 14a). The two following facts are believed to confirm the conclusion. Firstly, when SBB disappear above *T*∗, the linearity disappears too. Secondly, below *Tmax*, or below *Tpair* in terms of Ref. [80], the SBB have to transform into fluctuating Cooper pairs giving rise to the SC (collective) properties of the system. This argumentation coincides with the conclusion of Ref. [80] as for SC part of the pseudogap below *Tpair*. As SBB are now also absent, the linearity of *W*(*EF*)(*T*) again disappears. Thus, we consider the Δ∗(*T*), calculated within our Local Pair model (Fig. 14b), to be in a good agreement with the temperature dependence of the loss of the spectral weight

*<sup>c</sup>* = 36,9 K, *T*∗ =160 K, *ξc*(0)= 2.0Å,

Pseudogap and Local Pairs in High-Tc Superconductors 161

*max* at *Tmax* ≈ 100*K* actually coincides with the

*max* is plotted in Fig. 14b (green

(*T*) is the experimentally measured excess conductivity derived

**pairs**

*ε*∗

*<sup>c</sup>*<sup>0</sup> = 0.89, and *A*<sup>4</sup> = 59. *σ*�

**Figure 13.** Δ∗(*T*)/Δ∗ *max* in *SmFeAsO*0.85 with Δ∗(*Tc*)/*kB* = 160 K (•) and in YBCO film with *Tc* = 87.4 K (blue •) [27] as a function of T/*T*<sup>∗</sup> (T/*Tc* in the case of the theory). Curves 1 ··· 4 correspond to BK theory with different *x*<sup>0</sup> = *μ*/Δ(0):1- *x*<sup>0</sup> = 10.0 (BSC limit), 2 - *x*<sup>0</sup> = -2.0, 3- *x*<sup>0</sup> = -5.0, 4 - *x*<sup>0</sup> = -10.0 (BEC limit) [[87]].

To confirm the conclusion, relation Δ∗(*T*)/Δ∗ *max* as a function of *T*/*T*<sup>∗</sup> (*T*/*Tc* in the case of the theory) is plotted in Fig. 13. Black dots represent the studied *SmFeAsO*0.85 with Δ∗(*Tc*)/*kB* = 160 K, which corresponds to the strongly coupled limit (2Δ∗(*Tc*)/*Tc* ∼= 5.82). Blue dots display the data for YBCO film F1 [27]. The solid and dashed curves display the results of the Babaev-Kleinert (BK) theory [87] developed for the SC systems with different charge carrier density *n <sup>f</sup>* . For the different curves the different theoretical parameter *x*<sup>0</sup> = *μ*/Δ(0) is used, where *μ* is the chemical potential. Curve 1, with *x*<sup>0</sup> = +10, gives the BCS limit. Curve 4 with *x*<sup>0</sup> = -10 represents the strongly coupled BEC limit, which corresponds to the systems with low *n <sup>f</sup>* in which the SBB must exist [23, 25, 85, 87]. As well as in the YBCO film, the Δ∗(*T*)/Δ<sup>∗</sup> *max* in *SmFeAsO*0.85 evidently corresponds to the BEC limit (Fig. 13) suggesting the local pairs presence in the FeAs-based superconductor. Below *T*/*T*<sup>∗</sup> ≈ 0.4 both experimental curves demonstrate the very similar temperature behavior suggesting the BEC-BCS transition from local pairs to the fluctuating Cooper pairs to be also present in FeAS-based superconductors [20]. But, naturally, no drop of Δ∗(*T*) is observed for the YBCO film (Fig. 13, blue dots) at high temperatures, as no antiferromagnetism is expected in this case. This fact can be considered as an additional evidence of the AFM nature of the Δ∗(*T*) linear reduction below *Ts* in *SmFeAsO*0.85 found in our experiment.

It has to be emphasized that recently reported phase diagrams [113–116] (Fig. 9) apparently take into account a complexity of magnetic subsystem in *SmFeAsO*1−*xFx* and are in much more better agreement with our experimental results. But it has to be also noted that we study the *SmFeAsO*1−*<sup>x</sup>* system whereas the phase diagrams are mainly reported for the *SmFeAsO*1−*xFx* compounds. Is there any substantial difference between the both compounds has yet to be determined. Evidently more experimental results are required to clarify the question.

## **3.5. Angle-resolved photoemission measurements of the energy pseudogap of high-***Tc* (*Bi*, *Pb*)2(*Sr*, *La*)2*CuO*6+*<sup>δ</sup>* **superconductors: A model evidence of local pairs**

24 Will-be-set-by-IN-TECH

*max* in *SmFeAsO*0.85 with Δ∗(*Tc*)/*kB* = 160 K (•) and in YBCO film with *Tc* = 87.4 K

*max* as a function of *T*/*T*<sup>∗</sup> (*T*/*Tc* in the case of the

*max*

(blue •) [27] as a function of T/*T*<sup>∗</sup> (T/*Tc* in the case of the theory). Curves 1 ··· 4 correspond to BK theory with different *x*<sup>0</sup> = *μ*/Δ(0):1- *x*<sup>0</sup> = 10.0 (BSC limit), 2 - *x*<sup>0</sup> = -2.0, 3- *x*<sup>0</sup> = -5.0, 4 - *x*<sup>0</sup> = -10.0 (BEC

theory) is plotted in Fig. 13. Black dots represent the studied *SmFeAsO*0.85 with Δ∗(*Tc*)/*kB* = 160 K, which corresponds to the strongly coupled limit (2Δ∗(*Tc*)/*Tc* ∼= 5.82). Blue dots display the data for YBCO film F1 [27]. The solid and dashed curves display the results of the Babaev-Kleinert (BK) theory [87] developed for the SC systems with different charge carrier density *n <sup>f</sup>* . For the different curves the different theoretical parameter *x*<sup>0</sup> = *μ*/Δ(0) is used, where *μ* is the chemical potential. Curve 1, with *x*<sup>0</sup> = +10, gives the BCS limit. Curve 4 with *x*<sup>0</sup> = -10 represents the strongly coupled BEC limit, which corresponds to the systems with low *n <sup>f</sup>* in which the SBB must exist [23, 25, 85, 87]. As well as in the YBCO film, the Δ∗(*T*)/Δ<sup>∗</sup>

in *SmFeAsO*0.85 evidently corresponds to the BEC limit (Fig. 13) suggesting the local pairs presence in the FeAs-based superconductor. Below *T*/*T*<sup>∗</sup> ≈ 0.4 both experimental curves demonstrate the very similar temperature behavior suggesting the BEC-BCS transition from local pairs to the fluctuating Cooper pairs to be also present in FeAS-based superconductors [20]. But, naturally, no drop of Δ∗(*T*) is observed for the YBCO film (Fig. 13, blue dots) at high temperatures, as no antiferromagnetism is expected in this case. This fact can be considered as an additional evidence of the AFM nature of the Δ∗(*T*) linear reduction below

It has to be emphasized that recently reported phase diagrams [113–116] (Fig. 9) apparently take into account a complexity of magnetic subsystem in *SmFeAsO*1−*xFx* and are in much more better agreement with our experimental results. But it has to be also noted that we study the *SmFeAsO*1−*<sup>x</sup>* system whereas the phase diagrams are mainly reported for the *SmFeAsO*1−*xFx* compounds. Is there any substantial difference between the both compounds has yet to be determined. Evidently more experimental results are required to clarify the

**Figure 13.** Δ∗(*T*)/Δ∗

To confirm the conclusion, relation Δ∗(*T*)/Δ∗

*Ts* in *SmFeAsO*0.85 found in our experiment.

limit) [[87]].

question.

Taking all above consideration into account, it can be concluded that the FLC and PG description in terms of local pairs gives a set of reasonable and self-consistent results. However, to justify the conclusion it would be appropriate to test the LP model approah using independent results of other research groups who have measured straightforwardly the PG or any other related effects. But for a long time there was a lack of indispensable data.

Fortunately, analysis of the pseudogap in (*Bi*, *Pb*)2(*Sr*, *La*)2*CuO*6+*<sup>δ</sup>* (Bi2201) single-crystals with various *Tc*'s by means of ARPES spectra study was recently reported [80]. The study of Bi2201 allows avoid the complications resulting from the bilayer splitting and strong antinodal bosonic mode coupling inherent to Bi2212 and Bi2223 [90, 91]. Symmetrized energy distribution curves (EDCs) were found to demonstrate the opening of the pseudogap on cooling below *T*∗. It was shown that *T*∗, obtained from the resistivity measurements, agrees well with one determined from the ARPES data using a single spectral peak criterion [80]. Finally, from the ARPES experiments information about the temperature dependence of the loss of the spectral weight close to the Fermi level, *W*(*EF*), was derived [80]. *W*(*EF*) versus *T* measured for optimally doped OP35K Bi2201 (*Tc*=35 K, *T*∗ = 160 K) turned out to be rather unexpected, as shown in Fig. 14a taken from Ref. [80]. Above *T*∗ the *W*(*EF*) is nonlinear function of *T*. But below *T*∗, over the temperature range from *T*<sup>∗</sup> to *Tpair* = (110 ± 5) K (Fig. 14a), the *W*(*EF*)(*T*) decreases linearly which is considered as a characteristic behavior of the "proper" PG state [80]. However, no assumption as for physical nature of this linearity as well as for existence of the paired fermions in the PG region is proposed. Below *Tpair* the *W*(*EF*) vs *T* noticeably deviates down from the linearity (Fig. 14a). The deviation suggests the onset of another state of the system, which likely arises from the pairing of electrons, since the *W*(*EF*)(*T*) associated with this state smoothly evolves through *Tc* (Fig. 14a).

To compare results and justify our Local Pair model, the *ρab* vs *T* of the OP35K Bi2201, reported in Ref. [80], was studied within the LP model [13]. The Δ∗(*T*) was calculated by Eq. (16) with the following reasonable set of parameters: *Tc* = 35 K, *<sup>T</sup>m f <sup>c</sup>* = 36,9 K, *T*∗ =160 K, *ξc*(0)= 2.0Å, *ε*∗ *<sup>c</sup>*<sup>0</sup> = 0.89, and *A*<sup>4</sup> = 59. *σ*� (*T*) is the experimentally measured excess conductivity derived from the resistivity data using Eq. (1). Resulting Δ∗(*T*)/Δ∗ *max* is plotted in Fig. 14b (green dots). As expected, the shape of the Δ∗(*T*) curve is similar to that found for YBCO films (Fig. 5). Besides, the maximum of Δ∗(*T*)/Δ∗ *max* at *Tmax* ≈ 100*K* actually coincides with the change of the *W*(*EF*)(*T*) slop at *Tpair*, measured by ARPES, which seems to be reasonable. In fact, in accordance with our logic, *Tmax* is just the temperature which divides the PG region on SC and non-SC parts depending on the local pair state, as described above. Recall, that above *Tmax* the local pairs are expected to be in the form of SBB. Most likely just the specific properties of SBB cause the linear *W*(*EF*)(*T*) over this temperature region (Fig. 14a). The two following facts are believed to confirm the conclusion. Firstly, when SBB disappear above *T*∗, the linearity disappears too. Secondly, below *Tmax*, or below *Tpair* in terms of Ref. [80], the SBB have to transform into fluctuating Cooper pairs giving rise to the SC (collective) properties of the system. This argumentation coincides with the conclusion of Ref. [80] as for SC part of the pseudogap below *Tpair*. As SBB are now also absent, the linearity of *W*(*EF*)(*T*) again disappears. Thus, we consider the Δ∗(*T*), calculated within our Local Pair model (Fig. 14b), to be in a good agreement with the temperature dependence of the loss of the spectral weight

relatively low *Tc* in this case. Thus, there is an unexpected lack of coincidence between the SG

Pseudogap and Local Pairs in High-Tc Superconductors 163

The Chapter presents a detailed consideration of the LP model developed to study the PG in HTS's. In accordance with the model the local pairs have to be the most likely candidate for the PG formation. At high temperatures (*Tpair* < *T* ≤ *T*∗) we believe the local pairs to be in the form of SBB which satisfy the BEC theory (non-SC part of a PG). Below *Tpair* the local pairs have to change their state from the SBB into fluctuating Cooper pairs which satisfy the BCS theory (SC part of a PG). Thus, with decrease of temperature there must be a transition from BEC to BCS state [2, 27]. The possibility of such a transition is considered to be one of the basic physical principals of the high-*Tc* superconductivity. The transition was predicted theoretically in Ref. [23, 24, 74] and experimentally observed in our experiments [2, 20, 27, 95]. A key test for our consideration is the comparison of the Δ∗(*T*), calculated within the LP model, with the temperature dependence of the loss of the spectral weight close to the Fermi level *W*(*EF*)(*T*), measured by ARPES for the same sample [80]. The resulting Δ∗(*T*) is found to be in a good agreement with the W(*EF*)(*T*) obtained for OP35K Bi2201 (Fig. 14). It allows us to explain reasonably the W(*EF*)(*T*) dependence, both above and below *Tpair*, in terms of

The obtained results are also in agreement with the conclusions of Ref's. [16, 90, 91] as for SC and non-SC parts of the PG in Bi systems. Besides, formation of the local pairs is also believed to explain the rise of the polar Kerr effect and response of the time-resolved reflectivity, both observed for Bi systems just below *T*∗ [90]. While, the Nernst effect [16], which is likely due to the SC properties of the local pairs, is observed only below *Tpair*, or below *Tmax* in terms of our model. All the facts have to support the local pair existence in HTS's at *T* ≤ *T*∗, Thus, we may conclude, that on the basis of the developed LP model the self-consistent picture of the PG formation in HTS's is obtained. At the same time the issue concerning the pairing mechanism

The author is grateful to V. M. Loktev for valuable discussions and to T. Kondo for critical

*B. I. Verkin Institute for Low Temperature Physics and Engineering of National Academy of Science of*

[1] J. G. Bednorz , K. A. Mueller. "Possible High - *Tc* Superconductivity in the Ba-La-Cu-O

System", *Z. Phys. B. - Condensed Matter*, vol. 64, pp. 189-193, 1986.

and PG.

local pairs.

in HTS's still remains controversial [22].

*Ukraine, Lenin ave. 47, Kharkov 61103, Ukraine*

**Acknowledgments**

**Author details** Andrei L. Solovjov

**5. References**

remarks.

**4. Conclusion**

**Figure 14.** a. Spectral weight W(*EF* ) vs T (blue dots) for OP35K Bi2201. The solid line is the guidance for eyes only [[80]]. b. Pseudogap Δ∗(*T*)/Δ∗ *max* (green dots) and spectral gap SG/*SGmax* (red dots) [[80]] as the functions of temperature for the same sample.

W(*EF*) (Fig. 14a) obtained from the ARPES experiments performed on the same sample. In this way, the results of ARPES experiments reported in Ref. [80] are believed to confirm our conclusion as for existence of the local pairs in HTS's, at least in Bi2201 compounds.

Also plotted in Fig 14b is the normalized spectral gap (*SG*(*T*)) (red dots) equals to the energy of the spectral peaks of EDCs measured by ARPES [80]. Important in this case is that *SG*(*T*) smoothly evolves through both *Tpair* and *Tc*. The fact is believed to confirm assumed in the LP model the local pair existence above *Tpair*. Despite the evident similarity there are, however, at least two differences between the curves shown in Fig. 14b. First, there is no direct correlation between the *SG*(*T*) and the *W*(*EF*)(*T*) (Fig. 14 a, b). Why the maximum of *SG*(*T*) is shifted toward low temperatures compared to *Tpair*, has yet to be understood. The second difference is the absolute value of the SG compared to the pseudogap. The spectral gap has *SGmax* ≈ 40 meV and *SG*(*Tc*) ≈ 38 meV [80]. It gives 2*SG*(*Tc*)/*kBTc* ≈ 26 which is apparently too high. The PG values are Δ∗ *max* ≈ 16.5 meV and Δ∗(*Tc*) ≈ 6.96 meV, respectively. It gives 2Δ∗(*Tc*)/*kBTc* ≈ 6.4 which is a common value for the Bi compounds [124] with respect to relatively low *Tc* in this case. Thus, there is an unexpected lack of coincidence between the SG and PG.
