Normal-state conductivity of underdoped to overdoped cuprate superconductors: Pseudogap effects on the in-plane and c-axis charge transports

Physica B 440 (2014) 17–32

Contents lists available at ScienceDirect

Physica B journal homepage: www.elsevier.com/locate/physb

Normal-state conductivity of underdoped to overdoped cuprate superconductors: Pseudogap effects on the in-plane and c-axis charge transports S. Dzhumanov n, O.K. Ganiev, Sh.S. Djumanov Institute of Nuclear Physics, Uzbek Academy of Sciences, Ulugbek, Tashkent 100214, Uzbekistan

art ic l e i nf o

a b s t r a c t

Article history: Received 6 November 2013 Received in revised form 24 December 2013 Accepted 13 January 2014 Available online 22 January 2014

We have developed a theory of the unusual in-plane and c-axis charge transports in hole-doped cuprate superconductors and explain the temperature- and doping-dependent in-plane resistivity ρab, c-axis resistivity ρc and resistivity anisotropy ρc =ρab seen experimentally above Tc. We argue that the relevant current carriers in these materials above Tc are hole-like. The in-plane conductivity of underdoped to overdoped cuprates is considered as the conductivity of hole polarons and preformed Cooper pairs at their scattering by lattice vibrations in hole-rich CuO2 layers (with nonzero thickness). The appropriate Boltzmann transport equations were used to calculate the conductivity of polaronic carriers and bosonic Cooper pairs above and below the pseudogap (PG) temperature Tn in the relaxation time approximation. We show that the linearity of ρab(T) above Tn is associated with the polaron–phonon scattering, while different deviations from the T-linear behavior in ρab(T) below Tn are caused by transition to the BCS-like PG regime. The specific model for layered cuprates is used to simulate the c-axis transport and to calculate the c-axis conductivity associated with the thermal dissociation of localized bipolarons in carrier-poor regions between the CuO2 layers into hole polarons which subsequently move by hopping along the c-axis. It is shown that the bipolaronic PG and carrier-confinement together cause the insulating ρc(T) behavior in the cuprates. The calculated results for ρab(T), ρc(T) and ρc ðTÞ=ρab ðTÞ were compared with the experimental data obtained for various hole-doped cuprates. For all the considered cases, a good quantitative agreement was found between theory and experimental data. & 2014 Elsevier B.V. All rights reserved.

Keywords: High-Tc cuprates Polarons Polaronic Cooper pairs Bipolarons Pairing pseudogaps Transport properties

1. Introduction After the discovery of the doped cuprate superconductors [1,2], it has become clearer that the mechanism of superconductivity is related to their unusual normal state properties observed from the underdoped to overdoped regime and not encountered before in conventional superconductors. In particular, the normal state transport properties of underdoped to overdoped cuprates show striking deviations from the standard Fermi-liquid behavior due to the pseudogap (PG) which appears in the excitation spectra of these materials [3]. Despite a large number of experimental [4–12] and theoretical [13–19] studies considering the in-plane and c-axis transports, especially the in-plane and c-axis resistivities (ρab and ρc) in layered cuprates, many aspects of this issue are still undetermined. There is much experimental evidence that in hole-doped cuprates ρab(T) shows various anomalous behaviors below the

n

Corresponding author. Fax: þ 998 71 150 30 80. E-mail address: [email protected] (S. Dzhumanov).

0921-4526/$ - see front matter & 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2014.01.017

crossover temperature Tn, which increases with decreasing the doping level, and above this temperature ρab(T) exhibits T-linear behavior. Below Tn, ρab(T) deviates either downwards (i.e. ρab(T) shows a bending behavior around Tn) or upwards from the hightemperature linear behavior attributed to the PG opening [3,5,9]. In some high-Tc cuprates, the T dependence of ρab shows a positive curvature around Tn and a maximum (i.e. abnormal resistivity peak) between Tc and Tn [12,20,21]. Sometimes, anomalous resistive transitions (i.e. a sharp drop [22] and a clear jump [12,21] in the temperature dependence of the resistivity) were also observed at Tn. Specifically, ρab shows unusual metallic behavior above Tc in underdoped to overdoped cuprates, while the normal-state charge transport along the c-axis is incoherent and ρc in most of cases shows insulating behavior. The charge transport in these systems is essentially anisotropic due to their layered structure and a substantial anisotropy exists between ρab(T) and ρc(T) in the low-doping and low-temperature region. The anisotropy of the normal-state resistivity is characterized by the ratio ρc =ρab which decreases with increasing the doping level p and temperature T [10]. The c-axis resistivity quickly increases with decreasing of p and T, indicating that there is some carrier localization mechanism

18

S. Dzhumanov et al. / Physica B 440 (2014) 17–32

in the c-direction, which is responsible for the carrier confinement between the CuO2 layers of the cuprates. The physics of the cuprates in the intermediate doping regime is very complex and an agreement on how one should treat cuprates in the underdoped to overdoped region, where two distinct pseudogaps (PGs) and high-Tc superconductivity appear, has not yet been achieved. As for the origin of the PG in cuprates, a number of theoretical explanations have been proposed so far, but they are still controversial. The PG is usually attributed either to the precursor Cooper pairing above Tc or to other effects such as spin-charge separation, antiferromagnetic fluctuations, stripe formation, circulating currents and spin-vortex-induced loop currents (see Refs. [3,23–26]). Many experimental results indicate (see Refs. [27–30]) that strong electron–phonon interactions and polaronic effects may be involved in the PG phenomena and high-Tc superconductivity in these systems. The opening of a PG in the normal state of the cuprates should affect their transport properties. Several charge transport mechanisms based on non-Fermi-liquid models [13–15] or unusual Fermi-liquid models (see Ref. [16]) have been proposed to account for the observed T-linear behavior of ρab(T) above Tn and these models explain the available experimental data qualitatively or sometimes quantitatively with varying degrees of success. A theory capable of a unified description of the T-linear resistivity and other anomalies in ρab(T) found in various high-Tc cuprates below Tn is therefore desirable. A number of theoretical studies have also attempted to model the c-axis transport in cuprates by considering different physical scenarios. However, the underlying mechanism of c-axis transport in layered cuprates is not well understood yet. Thus, a detailed theoretical investigation of the origin of various resistivity anomalies, such as T-linear behavior of ρab(T) above Tn, nonlinear or nonmonotonic temperature dependence of ρab(T) and peculiar insulating temperature dependence of ρc ðTÞ in the PG state of underdoped to overdoped cuprates, within the proper theoretical approaches can contribute to the understanding of these intricate materials. In this work, we address the above unresolved issues of the charge transport in the normal state of underdoped to overdoped cuprates and discuss the intrinsic mechanisms of the unusual inplane and c-axis charge transports. We will address the question of how the electron–phonon interactions (which describe the scattering of carriers at acoustic and optical lattice vibrations, the Cooper pairing of polaronic carriers in the CuO2 layers below the PG temperature Tn and the real-space pairing of such carriers leading to the carrier localization in the c-direction) affect the normal-state transport properties of these high-Tc materials. Here we expect that in doped cuprates the charge inhomogeneity, the real- and k-space pairing in the strong and intermediate electron– phonon coupling regimes and the nanoscale phase separation into two different domains (i.e. carrier-rich CuO2 layers and carrierpoor regions between the CuO2 layers) play essential roles. We consider two specific charge transport mechanisms and obtain the expressions for the in-plane and c-axis conductivities (resistivities) as a function of temperature and doping. We argue that the doped carriers in the CuO2 layers are the noninteracting hole polarons above Tn and their scattering at acoustic and optical phonons is responsible for the T-linear resistivity, while some fraction of doped carriers is confined by the bipolaron potential wells between the CuO2 layers and the c-axis conductivity is associated with the thermal dissociation of localized bipolarons into separate polarons which subsequently move by hopping along the c-axis. We demonstrate that the opening of the BCS-like PG and the effective conductivity of bosonic Cooper pairs in cuprates below Tn lead to the nonlinear or nonmonotonic temperature dependence of ρab and different downward and upward deviations from the T-linear behavior in ρab(T) below Tn, which increases with decreasing of the doping level. We also show that the BCS-like

transitions above Tc are manifested as the different resistive transitions at Tn, which are similar to existing experimental data. We conclude that the two phenomena, the bipolaronic PG and the carrier confinement between the CuO2 layers, together cause the peculiar insulating behavior of ρc(T) in the cuprates. The rest of the paper is organized as follows. In Section 2 we discuss the nature of charge carriers (below and above Tc) and the existence of the BCS-like pairing PG above Tc in layered holedoped cuprates. We argue that the relevant charge carriers in doped cuprates are large polarons and polaronic Cooper pairs in the CuO2 layers and also localized large bipolarons lying between the CuO2 layers. The scattering of polaronic carriers and bosonic Cooper pairs at acoustic and optical phonons is considered in Section 3 and the appropriate expressions are presented for the quasiparticle relaxation rates, which are used to obtain the transport coefficients. In Section 4 we calculate the in-plane conductivity of the cuprates in the relaxation time approximation by using the appropriate Boltzmann transport equations (above and below Tn) and considering the BCS-like PG in the quasiparticle energy spectrum below Tn and obtain the temperature- and doping-dependent in-plane resistivity. We compare the calculated results with the experimental ρab(T) data obtained for various high-Tc cuprates. In Section 5 we describe the carrier confinement and formation of localized bipolarons in the low-doping regions between the CuO2 layers of high-Tc cuprates and study the c-axis hopping transport of polarons at the thermal dissociation of localized bipolarons into polarons. We show that the bipolaronic pairing PG is responsible for the activation behavior of the c-axis conductivity in the cuprates, from the underdoped to the overdoped regime. Results obtained from the analysis of c-axis resistivity will be compared with the experimental ρc(T) data obtained for various high-Tc cuprates. In Section 6 we present results of numerical calculations of ρab(T) and ρc(T) for the realistic set of parameters, and compare the results for the resistivity anisotropy ratio ρc =ρab with the existing experimental data. Finally, in Section 7 we summarize our conclusions.

2. Relevant charge carriers and BCS-like pairing pseudogap in layered cuprates The undoped cuprates are charge-transfer (CT)-type insulators [31]. Upon doping the oxygen valence band of these anisotropic three-dimensional (3D) cuprates is occupied by holes. These charge carriers being placed in a polar crystal will interact with the acoustic and optical phonons and the ground states of the doped carriers interacting with lattice vibrations are their selftrapped (polaronic) states lying in the CT gap of the cuprates. The polaron concept was introduced by Landau [32] to describe the electronic properties of polar materials and the quasi-particle, which consists of the electron (or hole) together with the lattice distortion (phonon cloud) and electronic polarization induced by it, called a polaron. Theoretical [27–29,33–35] and experimental [36–40] studies show that a polaronic transition from a quasi-free state to the self-trapped one occurs in doped cuprates and the electron–phonon interaction is responsible for the enhanced polaron masses mp ¼ ð2–3Þme [38,39,41] (where me is the free electron mass). As the doping (or polaron concentration n) grows towards the underdoped region, the Coulomb repulsion between polarons increases and the binding energy Ep of polarons decreases, so that the polaronic effect weakens with increasing doping and disappears in the overdoped region. Indeed, the binding energies of polarons Ep ¼0.12 eV and Ep ¼0.06 eV were observed experimentally in underdoped and optimally doped cuprates, respectively [37]. The radii of polarons in high-Tc cuprates vary from 6 to 10 Å [29,38]. Such polarons having

S. Dzhumanov et al. / Physica B 440 (2014) 17–32

where ɛ is the energy of large polarons measured from the Fermi n energy ɛ F , λ ¼ Dp ðɛ F ÞV~ p is the BCS-like coupling constant, Dp ðɛ F Þ is the DOS at the polaronic Fermi level, V~ p ¼ V ph  V~ c , V~ c ¼ V c =½1 þ Dp ðɛ F ÞV c lnðɛc =ɛ A Þ is the screened Coulomb interaction between two polarons, ɛ A ¼ Ep þℏω0 . For T ¼ T n and ɛ c ¼ ɛF 4 ɛ A , Eq. (1) becomes Z ɛA 1 dɛ ɛ tanh ¼ : ð2Þ ɛ 2kB T n λn 0 At ɛ A c kB T n , we obtain the following general expression for Tn:   1 kB T n C 1:134ðEp þ ℏω0 Þ exp  n : ð3Þ

λ

The transition from the intermediate- to the weak-coupling regime occurs at Ep ¼0 (i.e. in the absence of polaronic effects). Therefore, the usual BCS picture ðT c ¼ T n Þ as the particular case is recovered in the weak coupling regime, and the prefactor in Eq. (3) is replaced by Debye energy ℏωD for ordinary metals or by ℏω0 for heavily overdoped cuprates. In this case, a low T c ð ¼ T n Þ is

n

expected for even λ ¼ 0:3 in metallic superconductors. At Ep a0 the cuprate superconductors are unlike the BCS superconductors and the polaronic effects are responsible for the formation of incoherent Cooper pairs in the temperature range T c oT o T n [24,30] (cf. Ref. [47]). The underdoped, optimally doped and moderately overdoped cuprates are actually non-BCS superconductors, where the separation between the two temperatures Tn (the onset of the BCS-like transition) and Tc (the onset of the λ-like superconducting (SC) transition) occurs due to the polaronic n effects. To determine the doping dependence of Δ and Tn, we can approximate the polaronic DOS in a simple form ( 1=ɛ F for ɛ oɛ F ¼ ℏ2 ð3π 2 nÞ2=3 =2mp Dp ðɛ F Þ ¼ ð4Þ 0 otherwise: Then we obtain the following equations for Δ and Tn: ɛA ; Δn ðpÞ ¼ sinh½ℏ2 ð3π 2 na pÞ2=3 =2mp V~ p  n

and

ð5Þ

"

# ℏ2 ð3π 2 na pÞ2=3 kB T ðpÞ C 1:134ɛ A exp  ; 2mp V~ p n

ð6Þ

from which it follows that both Δ ðpÞ and T n ðpÞ increase with decreasing the doping level p ¼ n=na (where na ¼ 1=V a is the density of the host lattice atoms and Va is the volume per CuO2 unit in the n cuprates). Such doping dependences of the PG, Δ ðpÞ and the n characteristic temperature T ðpÞ were observed experimentally in high-Tc cuprates [11,46]. As shown in Fig. 1, the predicted behavior of Tn as a function of doping p is fairly consistent with the experimental results reported for Bi2 Sr2 CaCu2 O8 (Bi-2212). From the above considerations, it follows that the in-plane conductivity of underdoped to overdoped cuprates will be associated with the band-like metallic transport of hole polarons and polaronic Cooper pairs in the carrier-rich CuO2 layers. The presence of polaronic effects in these high-Tc materials raises some fundamental questions. Among the issues raised, three are particularly important: (i) the transformation of the Fermi surface (FS) from the ordinary large FS to a small polaronic one [48] and the additional presence of some electron-like carriers in hole-doped cuprates [49]; (ii) the existence of the quantum criticality (i.e. quantum critical point (QCP)) near or somewhat above optimal doping (where the breakdown of the one-electron band theory and the usual Fermi-liquid and BCS pairing theories occurs [48] (cf. Varma's work [13] and Tallon and Loram's work [3])) and the n

350 300 250

T(K)

relatively small binding energies, sizes and effective masses are nearly large polarons. According to Refs. [41,42], the formation of nearly small polarons in the cuprates might be also relevant. The hole-doped cuprates are inhomogeneous systems (where the charge carriers are distributed inhomogeneously) and the underdoped cuprates are more inhomogeneous than the overdoped ones [43]. In these systems the electronic inhomogeneities and electron–phonon interactions play an important role and are responsible for the carrier segregation,which may manifest itself via local nanoscale phase separation in the form of alternating metallic and insulating domains with mobile and immobile (i.e. localized) carriers,respectively [44] (see also Ref. [29]). We believe that the inhomogeneous spatial distribution of charge carriers leads to their aggregation in carrier-rich CuO2 layers together with charge depletion in spatially separated carrier-poor regions between the CuO2 layers. The mobile polaronic carriers are confined in thin quasi-two dimensional (2D) CuO2 layers (with nonzero thickness) and they have well-defined momentum k at W p ≳Ep (where Wp and Ep are the bandwidth and binding energy of large polarons,respectively),while the immobile carriers are distributed over the interlayer regions and reside between the CuO2 layers (along the c-axis). In the carrier-rich metallic regions the normal state (precursor) Cooper pairing of large polarons may occur in the intermediate coupling regime [24,30]. As will be discussed below,the formation of incoherent (i.e. nonsuperconducting) polaronic Cooper pairs becomes possible at T n 4 T c in the CuO2 layers of underdoped to overdoped cuprates. In this case the unconventional electron–phonon interactions (i.e. the combined and more effective BCS- and Fröhlich-type attractive interactions) are believed to be responsible for the pairing correlation above Tc in these materials. Here we note that the Migdal approximation is applied in the extreme adiabatic regime ℏωD =EF 51,which is valid only for ordinary metals with the Fermi energy EF c 1 eV. Whereas the high-Tc cuprates with the low Fermi energy ðɛF 5 1 eVÞ [45] and high-frequency optical phonons ðℏω0 ¼ 0:04–0:07 eV [38,46]) are in the nonadiabatic regime (i.e. the ratio ℏω0 =ɛ F is no longer small). By applying the BCS formalism to the interacting Fermi-gas of large polarons and using the Bogoliubov-like model of interpolaron interaction potential [24] (which has both an attractive and a repulsive part), we obtain the following BCS-like equation for n determining the pairing PG Δ and mean-field pairing temperan ture T : qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ɛA n2 ɛ 2 þ Δ ðTÞ 1 dɛ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ; ð1Þ ¼ tanh 2kB T n2 λn 0 ɛ 2 þ Δ ðTÞ

19

T*

200 150

Tp

100

Tc

50 0

0.05

0.1

0.15

0.2

0.25

carrier concentration, p Fig. 1. Temperature-doping ðT–pÞ phase diagram for a Bi2 Sr2 CaCu2 O8 (Bi-2212) film [11]. Tn is the temperature below which the PG opens, as determined by analyzing ρab(T) data. The temperature Tp corresponds to the onset of superconductivity along the c-axis in Bi-2212. The solid line is the fit to the experimental data (open circles) using T n ðpÞ (Eq. (6)) with parameters mp ¼ 2:2me , ɛA ¼ 0:138 eV, V~ p ¼ 0.09 eV and na ¼ 1  1022 cm  3 .

20

S. Dzhumanov et al. / Physica B 440 (2014) 17–32

PG state, which were not encountered in conventional superconductors; (iii) the unconventional superconductivity in the PG state [24,26,30,50], which is radically different from the conventional superconductivity in ordinary metals. Apparently, the current carriers are holes in hole-doped cuprates just as in p-type semiconductors. However, in p-type cuprates, there is a QCP that separates two distinct metallic states (at T ¼0, in the absence of superconductivity) [48]. Experimentally, the existence of an electron-like pocket in the FS of a hole-doped cuprate implies that the large hole-like FS of the overdoped cuprate superconductor [49] undergoes a transformation at a QCP located somewhere between p¼ 0.25 and p ¼0.15, in the normal state at T¼0 (once superconductivity is suppressed). We believe that the possible mechanism for producing a small electron pocket out of a large hole FS is a FS reconstruction caused by the onset of a polaronic transition from a free hole state to a self-trapped one. Actually, a polaron is defined as a carrier wrapped in both the phonon cloud and the electronic polarization cloud [51]. Therefore, the electronic polarization cloud around the hole carrier can be regarded as a small electron-like pocket in a hole-like FS of underdoped to overdoped cuprates. Other mechanisms can be also invoked for a reconstruction of the FS that would result in hole-like and electron-like pockets (see Refs. [49,50]). It seems likely that under certain conditions, holes give negative, and electrons positive, values of the Hall coefficient [52]. If the current carrier is strongly localized in the strong coupling regime, the characteristic relaxation times for hole carriers and electronic polarizations are about τh  10  14 s and τe  10  15 s, respectively. In this case, the electronic polarization cloud (electron-like pocket) induced by the hole carrier should be taken into account in the polaron transport. At τh  τe the current carrier is sufficiently delocalized in the intermediate coupling regime and the electronic polarization becomes unimportant [51]. Such a situation for large hole polarons is probably realized below and above Tc in high-Tc cuprates at zero magnetic field. However, in magnetic fields large enough to suppress superconductivity, the situation is completely different for the hole carriers in the underdoped cuprates at sufficiently low temperatures due to the charge carrier localization by strong magnetic fields (see, e.g., Ref. [53]). For example, in holedoped cuprates the Hall coefficient measured below Tc by destroying the SC state with a strong magnetic field is negative [49], which suggests that the electron-like carriers with high enough mobilities dominate the sign of the Hall coefficient, while the hole polarons are essentially localized in strong magnetic fields and have lower mobilities at low temperatures. A similar explanation of these experiments was recently proposed based on the analysis of the mobility of hole carriers [50], which is much reduced at low temperatures due to the small polaron formation. Thus, polaronic effects control the essential physics of underdoped to overdoped cuprates, which are in the non-BCS (bosonic) limit. In contrast, conventional superconductors and heavily overdoped cuprates are in the BCS (fermionic) limit due to the absence of the polaronic effects. In these weak-coupling systems, Cooper pairing of electrons and BCS condensation of loosely correlated Cooper pairs into a superfluid Fermi-liquid state occur simultaneously at Tc. Aside from others [13–16,47], two theoretical scenarios were proposed for explaining the novel superconductivity in cuprates. One of them is the spin-vortex scenario for superconductivity [26,50]. This theory explains the novel superconductivity in the cuprates based on the spin-vortex formation and the existence of localized polarons. In this theory, the cuprate system might be in the SC state even without Cooper pairs and doped holes become almost localized polarons which in turn become stabilizing centers of spin-vortices. The new criterion for the occurrence of spin-vortex superconductivity was discussed in Ref. [50]. The spin-vortex model was used to describe the change of the Hall coefficient sign from

positive to negative as the temperature is lowered in the presence of a magnetic field; the positive sign indicates that the dominant charge carriers are doped holes, and the negative means that they are electron-like carriers. The other scenario suggests that preformed polaronic Cooper pairs exist below the PG formation temperature Tn and become supercurrent carriers below the SC transition temperature Tc [24]. In the non-BCS cuprate superconductors, the important difference between the normal-state Cooper pairs and SC state Cooper pairs is that the preformed Cooper pairs like composite bosons may undergo the Bose–Einstein condensation (BEC) in the noninteracting particle approximation above Tc without superconductivity (i.e. according to the Landau criterion for superfluidity, the BEC state of an ideal Bose-gas of preformed Cooper pairs is not the superfluid state), while the interacting composite bosons (SC state Cooper pairs) condense into a superfluid Bose-liquid state (that is like the superfluid state of liquid 4He) below Tc [24]. Superconductivity of polaronic Cooper pairs just as superfluidity of 4He atoms is described by the mean-field theory of boson pairing [24,30] and determined by the excitation spectrum of a Bose-liquid of such composite bosons. It follows that the unconventional cuprate superconductivity is not associated with a BCSlike gap (or PG) at the FS and its formation does not necessarily lead the system to a SC state, as argued in Refs. [24,26]. In the following, we will concentrate on the charge transport properties in the normal state of hole-doped cuprates, while the properties in the SC state are not considered in this paper. The Hall coefficient of these materials is found to be positive above Tc [49] and the contribution from hole-like parts of the FS to the conductivity of the CuO2 layers in cuprates becomes dominating. This suggests that the generally positive Hall coefficient seen in these materials above Tc is the result of the change in the balance between hole-like and electron-like carriers, i.e. a fraction of the electron-like carriers becomes small even in the presence of a strong magnetic field. Apparently, at zero magnetic field the current carriers in the normal state of hole-doped cuprates are basically hole polarons and polaronic Cooper pairs in the CuO2 layers. In this case, the fraction of electron-like carriers is small enough and can be safely neglected.

3. Relaxation time for lattice scattering The charge carriers in polar crystals are scattered at their interaction with the acoustic and optical lattice vibrations and these scattering processes are major sources of temperaturedependent resistivity in the cuprates above Tc and can describe better the normal-state transport properties. Other scattering mechanisms such as the scattering of charge carriers by impurities and lattice defects may be important at sufficiently low temperatures below Tc. In this section we consider the scattering of charge carriers by the acoustic and optical lattice vibrations, in order to find the variation of the relaxation times with the energy of the carrier and with the temperature of the crystal. We discuss the behavior of the energy- and temperature-dependent relaxation times determining the rate at which polaronic carriers and ! incoherent Cooper pairs are caused to change their k -vectors and also their mobility in electric fields. Knowledge of the relaxation times for lattice scattering will allow us to calculate the in-plane conductivity (resistivity) of high-Tc cuprate superconductors in the next sections. From the Boltzmann transport equation and the principle of detailed balance, the relaxation time τðkÞ of a carrier for any type of scattering is generally given by [54] Z 1 V 3 0 0 ¼ ð7Þ d k Pðk; k Þð1  cos θÞ; τðkÞ ð2π Þ3

S. Dzhumanov et al. / Physica B 440 (2014) 17–32 0

where V ¼ N Ω is the crystal volume, Pðk; k Þ is the scattering 0 probability for charge carriers from jk〉 to jk 〉 states and θ is the 0 angle between k and k , N is the number of unit cells and Ω is the volume of a unit cell. When we consider the lattice scattering, the transition rate of a 0 carrier from the initial state jk〉 to the final state jk 〉 is given by  2π  0  Pðk; k Þ ¼ 〈k ; q0 H eL k; q〉j2 ℏ 0 δ½ɛðk Þ  ɛðkÞ 8ℏωq ; 0

ð8Þ

where HeL is the electron-lattice (or phonon) interaction Hamiltonian, and jk; q〉 is given by the product of the electron wave function and the wave function of the scattering center, which is a 0 phonon jq〉 with wave vector q, ɛðkÞ and ɛðk Þ are the energies of the initial and final states, ℏωq is the phonon energy.

where Ed is the deformation potential, M is the mass of atoms of unit cell, aqþ ðaq Þ is the creation (annihilation) operator of a phonon, r is the position vector of the carrier. In order to calculate the matrix element for this interaction Hamiltonian we replace jq〉 by jnq 〉 (where nq is the phonon number) and use the Bloch function ð10Þ

Then the wave function of the initial and final states are given by ð11Þ

and 0

jk ; q0 〉 ¼ jnq þ 1〉ψ k0 ðrÞ;

ð12Þ

respectively. 0 Further the scattering probability Pðk; k Þ is calculated by using the cyclic boundary conditions for the Bloch function Z 0 1 3 ∑eiðk  k  qÞrl ¼ N; unk0 ðrÞuk ðrÞd r ¼ 1 ð13Þ

Ω

l

and by taking into account the scattering processes corresponding to the phonon emission and absorption, and also the integral in 0 Eq. (7) with respect to k is replaced by the integral over q. For acoustic phonons the relation ωq ¼ vs q (where vs is the sound velocity) holds, and the allowed values of q are determined 0 on the basis of the relevant conservation laws ðk ¼ k 7 q and 0 ɛðk Þ ¼ ɛðkÞ 7 ℏωq Þ as  mp vs  q ¼ 2k 8 cos β 7 ð14Þ ℏk the condition where β is the angle between k and q. When pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mp vs =ℏk 5 1 is fulfilled (e.g., if ℏk C mp v ¼ 3mp kB T , mp ¼ 2me , vs ¼ 4  105 cm=s and T ¼100 K, the condition mp vs =ℏk o0:084 is fulfilled), the carrier-acoustic phonon scattering may be treated as elastic scattering. Under this condition the allowed values of q range from 0 to 2k. Above Tn the expression for τac ðkÞ is then written as Z 2k  1 V ¼ q2 dqVðqÞj2 τac ðkÞ 2π ℏ 0 Z βmax  sin β dβ ½ðnq þ 1Þδðɛðk qÞ  ɛðkÞÞ βmin

þ nq δðɛðk þ qÞ  ɛðkÞÞð1  cos θÞ;

0

As far as we consider the elastic scattering, the vectors k and k lie on the same energy surface (i.e. on the same sphere), and thus in this case the following relation holds: 1  cos θ ¼ 

q cos β; k

0 r θ r π;

π 2

rβ rπ

ð15Þ

ð17Þ

Inserting the relations (16) and (17) into Eq. (15), using the approximation nq þ 1  nq  kB T=ℏωq 4 4 1, and performing the integration over q and β, we obtain

π ℏ4 ρM v2s

pffiffiffi; 3=2 2E2d mp kB T ɛ

The carrier-acoustic phonon interaction Hamiltonian is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ℏ  iqr iqðaq eiqr  a þ Þ; ð9Þ H eL ¼ Ed ∑  qe 2MN ωq q

jk; q〉 ¼ jnq 〉ψ k ðrÞ

where VðqÞ ¼ Ed ðℏ=2MN ωq Þ1=2 iq and the δ-function is written in the form " # 2 i mp h q ℏ2 ðk 7 qÞ2 ℏ2 k ð16Þ 7 cos β δ  ¼ 2 δ 2mp 2mp ℏ kq 2k

τac ðɛÞ ¼ pffiffiffi

3.1. The relaxation time for carrier-acoustic phonon scattering

jk〉 ¼ ðN ΩÞ  1=2 U k ðrÞeikr

21

ð18Þ

where ρM is the material density. 3.2. The relaxation time for carrier-optical phonon scattering In order to determine the relaxation time of large polarons at their scattering by longitudinal optical phonons in an ionic model, we consider the polarization of a crystal due to an ionic movement or longitudinal optical lattice vibrations given by [54,55] PðrÞ ¼ en uðrÞ=Ω

or

PðrÞ ¼ ðN=VÞen uðrÞ;

ð19Þ

where en is the effective charge of ions, uðrÞ is their relative displacement, Ω ¼ 2a3 , a is the distance between the nearest neighbor ions. The effective charge of ions is given by [55] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi en ¼ M Ωω20 =16π 2 ɛ~ ; ð20Þ where M ¼ M 1 M 2 =ðM 1 þ M 2 Þ is the reduced mass of ions (anions and cations) in the CuO2 layer of the cuprates, ω0 is the optical phonon frequency, ɛ~ ¼ ð1  ηÞ=ɛ 1 is the effective dielectric constant, ɛ 1 and ɛ0 are the high frequency and static dielectric constants, respectively. When the condition kB T o o ℏω0 is satisfied, the relaxation time of large polarons scattered by optical phonons having the specific frequency ω0 ¼ ω01 may be determined from the relation [54] 1

τop

¼

  ℏω01 exp  3 kB T 8π 2 ℏ2 k NMa6 ω01 2 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ k k þk 0 6 7 2 2 2 42k k þ k0  k0 ln qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5; 2 2 k þ k0 k Vmnp ð2π Þ3 en2 e2

ð21Þ

where k0 is the wave vector corresponding to the polaron energy ℏω01: 2

ℏ2 k0 ¼ ℏω01 : 2mp

ð22Þ

For small k, the Taylor expansion of the expression in the square 3 bracket with respect to ðk=k0 Þ 5 1 gives k =k0 . Next we use the 3 relation Ω ¼ V=N ¼ 2a and Eq. (20). Then the relaxation time for polaron-optical phonon scattering is independent of the polaron energy and expressed as

τop ¼

pffiffiffi   4 2π ɛ~ ðℏω01 Þ3=2 ℏω01 : exp pffiffiffiffiffiffiffi 2 2 kB T ω01 e mp

ð23Þ

22

S. Dzhumanov et al. / Physica B 440 (2014) 17–32

given by

3.3. The total relaxation time for carrier-phonon scattering above and below Tn

sp ðT 4 T n Þ ¼ 

The total scattering probability of polaronic carriers at their scattering by acoustic and optical phonons is defined by the sum of two possible scattering probabilities. In particular, the total relaxation time above Tn is given by 1

τp ðɛÞ

¼

1

þ

1

τac ðɛÞ τop

;

ð24Þ

pffiffiffi 3=2 pffiffiffi where τac ðɛÞ ¼ Ap =t pɛffiffiffi, Ap ¼ π ℏ2 ρM v2s =E2d 2mp kB T n ,t ¼ T=T n; τop ¼ n pffiffiffiffiffiffiffi Bp eℏω01 =kB T t , Bp ¼ 4 2π ɛ~ ðℏω01 Þ3=2 =ω201 e2 mp . n As mentioned in Section 2, below T the polaronic carriers in the energy layer of width ɛc around the Fermi surface take part in the BCS-like pairing and form Cooper pairs in the CuO2 layers. If 0 0 we use the property of δ- function δ½Eðk Þ  EðkÞ ¼ ðdɛ=dEÞδ½ɛðk Þ  n ɛðkÞ in the expression for τp ðkÞ below T , the relaxation time of large polarons at their BCS-like pairing is given by EðkÞ τp ðɛÞ; ð25Þ jξðkÞj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 n2 where EðkÞ ¼ ξ ðkÞ þ Δ is the excitation spectrum of quasipar-

τBCS ðɛÞ ¼

ticles in the BCS-like PG state, ξðkÞ ¼ ɛðkÞ  μ, μ is the chemical potential of a polaronic Fermi-gas. The relaxation time of preformed Cooper pairs, which are scattered as the composite bosons at the acoustic phonons below Tn, is determined from the relation

τcac ðɛÞ ¼ pffiffiffi

π ℏ4 ρM v2s

pffiffiffi 3=2 2E2d mB kB T n t ɛ

ð26Þ

where mB ¼ 2mp is the mass of polaronic Cooper pairs. One can assume that the preformed Cooper pairs are scattered effectively by optical phonons having the distinctive frequency ω0 ¼ ω02 and the relaxation time of such bosonic carriers below Tn is given by pffiffiffi   4 2π ɛ~ ðℏω02 Þ3=2 ℏω02 : ð27Þ τcop ðt o 1Þ ¼ exp pffiffiffiffiffiffiffi kB T n t ðω02 Þ2 ð2eÞ2 mB

4. Normal state conductivity of charge carriers in the CuO2 layers of the cuprates Now we consider the layered cuprate superconductor with a simple ellipsoidal energy surface and the normal state conductivity of large polarons in the quasi-two-dimensional (2D) CuO2 layers (with nonzero thickness) above and below their characteristic Cooper pairing temperature Tn. We will take this approach, since it seems more natural. In the following, we will take the effective-mass components mp1 ¼ mp2 ¼ mab for the ab-plane and mp3 ¼ mc for the c-axis in the cuprates. Then the effective mass mp of polarons in the layered cuprates is ðm2ab mc Þ1=3 . In this section we calculate the normal-state conductivities of underdoped to overdoped cuprates as a function of doping and temperature. Using the Boltzmann transport equations in the relaxation time approximation, we obtain appropriate expressions for the conductivity of large polarons above Tn and for the conductivities of the excited Fermi components of polaronic Cooper pairs and bosonic Cooper pairs below Tn in the CuO2 layers. Further we compare our theoretical results with the experimental data. 4.1. The conductivity of large polarons above Tn

e2 4π 3

Z

τp ðɛÞv2x

∂f p 3 d k; ∂ɛ

where f p ðɛÞ ¼ ðeðɛ  μÞ=kB T þ 1Þ  1 is the Fermi distribution function, 2 ɛ ¼ ℏ2 k =2mp and vx ¼ ð1=ℏÞ∂ɛ=∂kx are the energy and velocity of polarons, respectively, mp ¼ ðm2ab mc Þ1=3 . In order to find the DOS in the case of an ellipsoidal energy surface, we make the following transformations similar to Ref. 1=2 0 1=2 0 1=2 0 [56]: kx ¼ mp1 kx , ky ¼ mp2 ky , kz ¼ mp3 kz . Then the energy ɛ is 2

transformed from ℏ2 kα =2mα to ℏ2 kα0 2 =2 and the mean value of 2 ℏ2 kα =2mα over the energy layer Δɛ in k-space is equal to the mean 0 value of ℏ2 kα0 2 =2 over the spherical layer in k -space, and therefore does not depend on αð ¼ 1; 2; 3Þ. It follows that the average kinetic energy of a carrier over the layer Δɛ along three directions kx, ky and kz is the same and equal to one-third of the total energy ɛ. Therefore, replacing v2x by ℏ2 kα0 2 =mα and using the relation 3 3 0 d k ¼ ðm2ab mc Þ1=3 d k , we may write Eq. (28) in the form Z e2 ℏ2 k0 2 ∂f sp ðT 4 T n Þ ¼  2 ðm2ab mc Þ1=2 τp ðɛÞ α p d3 k0 ð29Þ mα ∂ɛ 4π Replacing k0 α2 by 2ɛ=3ℏ2 and using further the carrier density n given by Z Z ð2m2ab mc Þ1=2 2 3 n¼ f f p ðɛÞɛ1=2 dɛ; ðkÞ d k ¼ ð30Þ p π 2 ℏ3 ð2π Þ3 the expression (29) is written as R1 2ne2 0 τp ðɛÞɛ 3=2 ð  ∂f p =∂ɛÞ dɛ n R1 : sp ðT 4 T Þ ¼ 1=2 dɛ 3mab 0 f p ðɛÞɛ

ð31Þ

When the Fermi energy of large polarons ɛ F is much larger than their thermal energy kBT, we deal with a degenerate polaronic gas. For a degenerate polaronic Fermi-gas, we have approximately f p ðɛ o ɛ F Þ ¼ 1 and f p ðɛ 4 ɛF Þ ¼ 0. In this case the function  ∂f p =∂ɛ is nonzero only near ɛ ¼ ɛ F ¼ μ and close to the δ-function. Therefore, we may replace ∂f p =∂ɛ by δðɛ  ɛ F Þ and the integral in (31) may be evaluated as  Z 1 Z 1 ∂f ɛ 3=2 pffiffiffi τp ðɛÞɛ3=2  p dɛ ¼ Bp eα=t ∂ɛ 1 þ c p ðtÞ ɛ 0 0  δðɛ  ɛ F Þ dɛ ¼ Bp eα=t

ɛ 3=2 pffiffiffi; 1 þ cp ðtÞ ɛ

ð32Þ

where αp ¼ ℏω01 =kB T n , cp ðtÞ ¼ ðBp =Ap Þteαp =t . Using the above property of the Fermi distribution function, the integral in the denominator in Eq. (31) is evaluated as Z 1 Z ɛF 2 3=2 f p ðɛÞɛ 3=2 dɛ ¼ ɛ 1=2 dɛ ¼ ɛF : ð33Þ 3 0 0 Inserting the relations (32) and (33) into (31), we obtain

sp ðt 41Þ ¼

ne2 Bp eαp =t pffiffiffiffiffi : mab ð1 þ cp ðtÞ ɛF Þ

ð34Þ

4.2. The conductivity of the excited polaronic components of Cooper pairs and the bosonic Cooper pairs below Tn Below Tn, large polarons in the energy layer of width ɛc around the Fermi surface take part in the Cooper pairing and form polaronic Cooper pairs. The total number of the excited Fermi components of Cooper pairs and composite bosonic Cooper pairs is given by n

n

n ¼ nnp þ 2nB ¼ 2∑½u2k f p ðkÞ þ v2k ð1  f p ðkÞÞ; When the electric field is applied in the x-direction, the conductivity of large polarons in the anisotropic 3D or quasi-2D cuprate superconductors above Tn in the relaxation time approximation is

ð28Þ

ð35Þ

k

n

where nnp ¼ 2∑k u2k f p ðkÞ is the number of the excited polaronic n components of Cooper pairs, nB ¼ ∑k v2k ð1  f p ðkÞÞ is the number of

S. Dzhumanov et al. / Physica B 440 (2014) 17–32

bosonic Cooper pairs, f p ðkÞ ¼ ðeE=kB T þ 1Þ  1 , uk ¼ 12ð1 þ ξ=EÞ, vk ¼ 1 2ð1  ξ=EÞ. The contribution of the excited polaronic components of Cooper pairs to the conductivity in quasi-2D cuprate superconductors below Tn in the relaxation time approximation is given by  n Z e2 ξ ξ ∂f p 3 d k: snp ðT o T n Þ ¼  3 τBCS ðξÞv2α 1 þ ð36Þ E E ∂E 8π n

When we consider a thin CuO2 layer of the doped cuprate superconductor with an ellipsoidal energy surface, the expression for snp ðT o T n Þ can be written as !  n ∂f p R ɛc ξ 3=2 ξ 1þ  dξ  ɛ c τ BCS ðξ þ μÞðξ þ μÞ E E ∂E ne2 R1 : snp ðt o 1Þ ¼ 1=2 dɛ 3mab 0 f p ðɛÞɛ ð37Þ The pairing PG Δ and characteristic temperature Tn are determined from the BCS-like gap equation (1). In the calculation of the contribution of bosonic Cooper pairs to the normal-state conductivity of the cuprates, the mass of the Cooper pair in layered cuprates can be defined as mB ¼ ðM 2ab M c Þ1=3 , where M ab ¼ 2mab and M c ¼ 2mc are the in-plane and out-of-plane (c-axis) masses of the polaronic Cooper pairs, respectively. Below Tn the density of Cooper pairs is determined from the equation   Z ðm2ab mc Þ1=2 ɛc ξ eE=kB T ffiffiffi ðξ þ μÞ1=2 E=k T 1  ð38Þ dξ: nB ¼ p 3 2 E e B þ1 2 2π ℏ  ɛc n

Numerical calculations of the concentration nB and the BEC 2=3 temperature of bosonic Cooper pairs T 0 ¼ 3:31ℏ2 nB =kB mB show n n that just below T the value of T0 is very close to T (i.e. T 0 ≳T n Þ, but somewhat below Tn, T 0 4 4 T n . Therefore, we can consider polaronic Cooper pairs below Tn as an ideal Bose-gas with chemical potential μB ¼ 0. Below T0 the total number of bosonic Cooper pairs with zero and non-zero momenta K or energies ɛ is given by nB ¼ nB ðɛ 4 0Þ þ nB ðɛ ¼ 0Þ;

ð39Þ

where ðM 2 M c Þ1=2 ffiffiffi nB ðɛ 4 0Þ ¼ pab 2π 2 ℏ3

Z

1 0

 3=2 ɛ1=2 dɛ T : ¼ n B T0 eɛ=kB T  1

ð40Þ

Obviously, bosonic Cooper pairs with zero center-of-mass momentum (K ¼0) or velocity do not contribute to the current and only the Cooper pairs with K a 0 and density nB ðT=T 0 Þ3=2 contribute to the normal-state conductivity of the layered cuprate superconductors with the ellipsoidal constant-energy surfaces. Again, one 0 can make the transformation K ¼ M 1=2 α K . Then the expression for n the conductivity sB ðT o T Þ of bosonic Cooper pairs in the anisotropic cuprate superconductor at their scattering by acoustic and optical phonons can be written as

sB ðT oT n Þ ¼

e2 ðM 2 M c Þ1=2 2π 3 ab  Z ℏ2 K α0 2 ∂f 3  τ B ðɛÞ  B d K0; Mα ∂ɛ

ð41Þ

where f B ðɛÞ ¼ ðeɛ=kB T  1Þ  1 is the Bose distribution function, τB ðɛÞ is the relaxation time of Cooper pairs scattered by acoustic and optical phonons and determined as τB ðɛÞ p ¼ffiffiffiτcac ðɛÞτcop ðɛÞ=ðτcac ðɛÞ þ pffiffiffi 3=2 τcop ðɛÞÞ, τcac ðɛÞ ¼ p Acffiffiffi=t ɛ, Ac ¼ π ℏ4 ρM v2s = E2d 2mB kB T n , τcop ðɛÞ ¼ pffiffiffiffiffiffiffi 3=2 ℏω02 =kB T n 2 2 Bc e , Bc ¼ 2π ɛ~ ðℏω02 Þ =ω02 e mB . Using Eq. (40) and the relation K α0 2 ¼ 2ɛ=3ℏ2 and after replacing M α in Eq. (41) by Mab, the above expression for sB ðT o T n Þ has the following form:

sB ðT oT n Þ ¼

8nB ðT=T 0 Þ3=2 e2 3M ab

R1 0

τB ðɛÞɛ3=2 ð  ∂f B =∂ɛÞ dɛ R1 0

f B ðɛÞɛ 1=2 dɛ

:

ð42Þ

23

After evaluating the integral in the denominator in this equation, we can write Eq. (42) in the form  Z 3=2 3=2 m e2 1 m e2 B eα=t ∂f sB ðt o 1Þ ¼ 0:19 B 3 τB ðɛÞɛ3=2  B dɛ ¼ 0:19 B 3 c n ∂ɛ M ab ℏ 0 M ab ℏ kB T t Z 1 n ɛ 3=2 eɛ=kB T t ð43Þ  pffiffiffi dɛ; n ɛ=k T t B ðe  1Þ2 ð1 þ βc ðtÞ ɛ Þ 0 where β c ðtÞ ¼ Bc teαc =t =Ac , αc ¼ ℏω02 =kB T n . The resulting conductivity of the excited polaronic components of Cooper pairs and the bosonic Cooper pairs below Tn in the CuO2 layers is calculated as

sab ðt o 1Þ ¼ snp ðt o 1Þ þ sB ðt o1Þ:

ð44Þ

By using the resistivity data from various experiments, we were able to obtain both qualitative and quantitative agreement with the experimental data presented in Sections 4.3 and 4.4. 4.3. Anomalous resistive transitions above Tc and their experimental verification It is well known that the resistivity ρðTÞ of the layered cuprates represents essentially the ab-plane value due to relatively high conductivity in the CuO2 layers as compared to that along the caxis. Any microscopic theory that tries to explain unconventional superconductivity in high-Tc cuprates must be able to consistently and quantitatively explain the observed temperature and doping dependences of the normal-state transport properties in these materials. The experimental observations indicate [3,5,7,9–12] that the temperature dependences of the measured bulk resistivity ρ (including in-plane resistivity ρab) above and below the PG formation temperature Tn (which systematically shifts to lower temperatures with increasing the doping level p, and finally merges with Tc in the overdoped cuprates [57,58] (see also Refs. [3,30])) are strikingly different. In the PG regime, the situation is very complicated and somewhat unclear due to the distinctly different deviations from the T-linear resistive behavior, which produce the pronounced non-linear or even non-monotonic temperature dependence of ρ (or ρab). In some cases, the resistivity varies very rapidly near Tn. It should be noted that none of the existing theoretical models that explain the high-temperature linear behavior of ρðTÞ and ρab(T) can satisfactorily describe distinctly different deviations from the linear dependence of the resistivity below the PG opening temperature. Here and in the next section, we show that our theory of normal-state charge transport in the CuO2 layers of high-Tc cuprates can describe adequately and consistently the different temperature dependences of the resistivity above and below Tn and the anomalous resistive transitions at Tn, from the underdoped to the overdoped cases. The above-mentioned combined and more effective BCSand Fröhlich-type attractive interactions (Section 2) are responsible for the pairing correlation above Tc in the cuprates and the unconventional phonon-mediated Cooper pairing at some temperature T n 4 T c should occur in the intermediate-coupling regime. Eq. (34) allows us to calculate the in-plane resistivity ρab ðT 4 T n Þ ¼ ρ0 þ 1=sab ðT 4 T n Þ, where ρ0 is the residual resistivity, due presumably to impurity or disorder in samples of high-Tc cuprates. We further use the solution of Eq. (1) to calculate ρab ðT o T n Þ ¼ ρ0 þ1=sab ðT o T n Þ by numerically integrating Eqs. (37) and (43). The Fermi energy for undoped cuprates is about EF ¼7 eV [59,60] and Ed is estimated as Ed ¼ ð2=3ÞEF . For high-Tc cuprates, the experimental values of ρM , vs, ɛ 1 , ɛ0 and ℏω0 lie in the ranges ρM C ð4–7Þ g=cm3 [4], vs C ð4–7Þ  105 cm=s [4], ɛ1 C 3–7 [33,61], ɛ0 C 22–50 [33,38] and ℏω0 C 0:03–0:08 eV [38,46,61,62]. To illustrate the competing effects of two contributions from snp ðt o1Þ and sB ðt o 1Þ on the resultant conductivity

24

S. Dzhumanov et al. / Physica B 440 (2014) 17–32

(resistivity), we show in Fig. 2 results of our calculations for two different samples of underdoped cuprates with T n ¼ 170 K n n ðλ ¼ 0:585Þ and T n ¼ 110 K ðλ ¼ 0:541Þ obtained using the relevant 5 parameters vs ¼ 5:5  10 cm/s, ρM ¼ 6:0 g=cm3 , ɛ~ ¼ 5:6, mab ¼ 1:82  10  27 g, mp ¼ 2:003  10  27 g, n ¼ 0:5  1021 cm  3 , ℏω01 ¼ 0:05 eV, ℏω02 ¼ 0:056 eV, ρ0 ¼ 0:22 mΩ cm and vs ¼ 5:4  105 cm=s, ρM ¼ 5:8 g=cm3 , ɛ~ ¼ 4:6, mab ¼ 1:88  10  27 g, mp ¼ 2:14  10  27 g, n ¼ 0:45  1021 cm  3 , ℏω01 ¼ 0:04 eV, ℏω02 ¼ 0:041 eV, ρ0 ¼ 0:04 mΩ cm, respectively. As can be seen in Fig. 2, ρab(T) shows T-linear behavior above Tn as observed in various underdoped cuprates. This strange metallic T-linear behavior of the resistivity arises from the scattering of large polarons by acoustic and optical phonons. Below Tn the resistivity ρab(T) shows nonlinear T dependence and starts to deviate either downward or upward from the T-linear behavior, depending on specific materials’ parameters. Our calculations show that the anomalous resistivity behavior in the PG regime, which is in fact characteristic of underdoped to overdoped cuprates and not very sensitive to changes in the carrier concentration, depends sensitively on the two distinctive frequencies of optical phonons ω01 and ω02. Quite generally, in different hole-doped cuprates, the downward deviation of ρab(T) from linearity occurs below Tn, which indicates the appearance of some excess conductivity due to the transition to the PG state and the effective conductivity of bosonic Cooper pairs. The crossover between the high- and lowtemperature regimes occurs near Tn where the value of ρab(T) changes rapidly due to the sensitivity of the resulting conductivity sab ðt o 1Þ to the rapid temperature variation of snp ðt o1Þ and sB ðt o 1Þ below Tn. Fig. 2 shows clearly that ρab(T) in high-Tc cuprates exhibits a sharp drop or jump at Tn. These results for the resistive transitions closely resemble those found in some high-Tc cuprates [12,21,22]. Indeed, the calculated resistivity curves shown in Fig. 2 exhibit a crossover at Tn, 1

ρab (mΩ cm)

0.8

1

0.6

T* 2

0.4

T*

0.2

0

0

1

2

3

T/T* Fig. 2. Temperature dependences of ρab calculated for two different underdoped samples of the cuprates with different PG opening temperatures Tn below which ρab(T) deviates either downward (curve 1) or upward (curve 2) from linearity.

similar to that observed experimentally at Tn in various high-Tc cuprates. The detailed explanation of the different anomalous behaviors of the normal-state resistivity observed above Tn, below Tn and at Tn in underdoped to overdoped cuprates is given in Section 4.4 in terms of the above charge transport theory as applied to these materials.

4.4. Comparison with other experimental results For the comparison with other existing experimental resistivity data we also present our results for T-dependent resistivity in underdoped, optimally doped and moderately overdoped cuprates with realistic sets of fitting parameters, which in many cases have been previously determined experimentally and are not entirely free parameters. Experimentally, in these materials one encounters a crossover from linear-in-T behavior of the resistivity to nonlinear (including nonmonotonic)-in-T behavior below Tn even though the anomaly near Tn is weak. It is possible that the inhomogeneity and other imperfections in the samples of the doped high-Tc cuprates have an effect on this crossover which may be obscured due to such extrinsic factors and may become almost masked or less pronounced BCS-type resistive transition. In fact, a variety of different crossovers in resistivity have been observed in underdoped, optimally doped and even overdoped materials near Tn, where ρab(T) displays a finite negative or positive curvature. It is often incorrectly assumed that optimally doped cuprates possess a T-linear resistivity over a wide temperature region which extends down to Tc. However, close examination of the experimental resistivity data in various optimally doped cuprates shows that the resistivity will be linear-in-T from 300 K down to Tn and then different deviations from linearity occur below Tn in these materials. Cuprate superconductors are very complicated and characterized by many intrinsic parameters. Clearly, the minimal model, which uses fewer parameters of the cuprates, does not describe the real physical picture especially in inhomogeneous high-Tc cuprates and fails to reproduce many important features in ρab(T). The resulting expressions for sp ðt 4 1Þ, snp ðt o 1Þ and sB ðt o1Þ allow us to perform fits of the measured in-plane resistivity ρab ðTÞ ¼ ρ0 þ 1=sab ðTÞ in various high-Tc cuprates above Tc using the specific parameters of our model (Table 1). In doing so, better fitting of the experimental data is achieved by a more appropriate choice and a careful examining of the relevant materials’ parameters. In Fig. 3 we compare our calculated resistivity as a function of temperature with the experimental results obtained by Carrington et al. [63] for underdoped and optimally doped samples of YBa2 Cu3 O7  δ (YBCO). Examination of the experimental data presented in Fig. 3 shows that the downward deviations of ρab(T) from linearity in the compounds YBa2 Cu3 O6:72 , YBa2 Cu3 O6:81 and YBa2 Cu3 O6:95 occur below the n crossover temperatures T n ¼ 160 K (for λ ¼ 0:546Þ, 140 K (for λn ¼ 0:521Þ and 120 K (for λn ¼ 0:473Þ, respectively. Clearly, below

Table 1 The values of n, Tn, λn and ρ0 determined from the fits to experimental ρab(T) data. The corresponding Tc's and references are listed. Sample

Tc (K)

n  1021 (cm  3 )

Tn (K)

λn

ρ0 (mΩ cm)

References

La1:92 Sr0:08 CuO4 La1:48 Nd0:40 Sr0:12 CuO4 La1:90 Ba0:10 CuO4 La1:89 Ba0:11 CuO4 YBa2 Cu3 O6:72 YBa2 Cu3 O6:81 YBa2 Cu3 O6:95 Y0:90 Ca0:10 Ba2 Cu3 O7  δ Bi2 Sr2 Ca0:92 Y0:08 Cu2 O8 CuBa2 Ca3 Cu3 Oy with the addition AgO content x ¼0.9

20 10 30 21 68 87 91 80 64 69

0.45 0.60 0.60 0.65 1.00 1.15 1.35 1.20 0.67 0.85

120 75 43 52 160 140 120 140 190 113

0.553 0.483 0.428 0.416 0.546 0.521 0.473 0.447 0.579 0.508

0.220 0.093 0.080 0.100 0.018 0.020 0.016 0.200 0.280 0.008

[4] [21] [12] [12] [63] [63] [63] [9] [64] [20]

S. Dzhumanov et al. / Physica B 440 (2014) 17–32

0.5

ab

ρ (mΩ cm)

0.4

δ=0.28

0.3

δ=0.19

T* T*

0.2

δ=0.05

0.1

0

T* 0

50

100

150

200

250

300

T(K) Fig. 3. Comparison of our calculated results for ρab(T) (solid line) with the experimental resistivity data (open circles) obtained by Carrington et al. (Ref. [63]) for underdoped cuprates (with δ ¼ 0.19 and 0.28) and optimally doped YBa2 Cu3 O6:95 material. Fits are performed in the T range from above Tc to 300 K and agree nicely with the experimental data.

10

6

ρab (10

−4

Ω cm)

8

T*

4 2 0

0

50

100

150

200

250

300

T(K) Fig. 4. Experimental ρab(T) data for La1:92 Sr0:08 CuO4 taken from Ref. [4] (open circles) and the respective fits (solid line).

1.5

ρab (mΩ cm)

Tn the leading contribution to the resulting conductivity of these high-Tc cuprates comes from the conductivity of incoherent bosonic Cooper pairs and the temperature dependence of the resistivity is dominated by this contribution to sab ðt o 1Þ that determines the downward deviation of ρab(T) from the T-linear behavior at Tn (the PG begins to open at that point). In the numerical calculations of ρab ðT 4 T n Þ and ρab ðT o T n Þ, we use the following sets of intrinsic materials’ parameters in order to obtain the best fits: vs ¼ 4:0  105 cm/s, ρM ¼ 4:1 g=cm3 , ɛ~ ¼ 4:1, mab ¼ 1:257 10  27 g, mp ¼ 1:75922  10  27 g, n ¼ 1:15  1021 cm  3 , ℏω01 ¼ 0:041 eV, ℏω02 ¼ 0:058 eV, ρ0 ¼ 0:02 mΩ cm for underdoped YBa2 Cu3 O6:81 , vs ¼ 4:1  105 cm=s, ρM ¼ 4:1 g=cm3 , ɛ~ ¼ 4:1, mab ¼ 1:547 10  27 g, mp ¼ 1:925  10  27 g, n ¼ 1:0  1021 cm  3 , ℏω01 C 0:051 eV, ℏω02 C 0:062 eV, ρ0 ¼ 0:018 mΩ cm for underdoped YBa2 Cu3 O6:72 and vs ¼ 4:0  105 cm=s, ρM ¼ 4:0 g=cm3 , ɛ~ ¼ 4:0, mab ¼ 1:2  10  27 g, mp ¼ 1:6221  10  27 g, n ¼ 1:35  1021 cm  3 , ℏω01 ¼ 0:040 eV, ℏω02 ¼ 0:057 eV, ρ0 ¼ 0:016 mΩ cm for optimally doped YBa2 Cu3 O6:95 . Obviously, the calculated results agree quite well with the experimental data both above Tn and below Tn especially keeping in mind the fact that the experimental results obtained near the crossover temperature Tn are subject to extrinsic factors. Other results of fitting of the experimental ρab(T) data are shown in Figs. 4 and 5 for La2  x Srx CuO4 (LSCO) ðx ¼ p ¼ 0:08Þ and 8% Y doped n n Bi-2212 with T n ¼ 120 K ðλ ¼ 0:553Þ and T n ¼ 190 K ðλ ¼ 0:579Þ. We obtained reasonable fits to the experimental data by taking appropriate sets of materials’ parameters vs ¼ 5:0  105 cm/s, ρM ¼ 5:2 g=cm3 , ɛ~ ¼ 4:762, mab ¼ 1:82  10  27 g, mp ¼ 2:224 10  27 g, n ¼ 0:45  1021 cm  3 , ℏω01 ¼ 0:046 eV, ℏω02 ¼ 0:049 eV, ρ0 ¼ 0:22 mΩ cm for La1:92 Sr0:08 CuO4 and vs ¼ 4:0  105 cm/s, ρM ¼ 4:0 g=cm3 , ɛ~ ¼ 4:5, mab ¼ 1:78  10  27 g, mp ¼ 2:288 10  27 g, n ¼ 0:671021 cm  3 , ℏω01 ¼ 0:042 eV, ℏω02 ¼ 0:051 eV, ρ0 ¼ 0:28 mΩ cm for Y doped Bi-2212, respectively. One can see that both in La1:92 Sr0:08 CuO4 and in Y doped Bi-2212 the resistivity ρab(T) is nonlinear at T o T n . Further, on comparing Figs. 4 and 5 it may be seen that below Tn the upward and downward deviations of ρab(T) from linearity occur in La1:92 Sr0:08 CuO4 and Y doped Bi-2212, respectively, as were seen in the experiments. Our numerical results on nonmonotonic temperature dependence of the resistivity in various high-Tc cuprates at different doping levels are also plotted in Figs. 6–8 along with the existing experimental data. We believe that the pronounced nonmonotonic behaviors of ρab(T) (i.e. jump- and peak-like anomalies in ρab(T) at Tn and below Tn, respectively) in most samples of high-Tc cuprates

25

T*

1

0.5

0

0

50

100

150

200

250

300

T(K) Fig. 5. Experimental ρab(T) data for Bi2 Sr2 Ca0:92 Y0:08 Cu2 O8 taken from Ref. [64] (open circles) and the respective fits (solid line).

are directly related to competing contributions (i.e. the contribution coming from the unpaired components of Cooper pairs, which decreases sharply below Tn, and the contribution coming from bosonic Cooper pairs, which is rapidly increased below Tn) to the resulting conductivity sab ðt o 1Þ. Figs. 6–8 demonstrate clearly that the behavior of the resistivity in the PG regime is especially sensitive to changes in fitting parameters ω01 and ω02. In particular, in the cuprates with ω02 ≳ω01 , the upward deviation of the resistivity from its high-temperature T-linear behavior occurs below Tn and sometimes the resistivity peak exists between Tc and Tn, while the downward deviation from the T-linear resistive behavior occurs below Tn in other systems with sufficiently large values of ω02 in comparison to ω01. Next we analyze the experimental resistivity data in overdoped samples of high-Tc cuprates. In Fig. 9 we compare our calculated temperature dependent resistivity with the experimental ρðTÞ data obtained for the overdoped (p ¼0.17) Y1  x Cax Ba2 ðCu1  y Zny Þ3 O7  δ compound [9]. In this material one also observes a crossover from linear-in-T behavior of the resistivity to nonlinear-in-T behavior below Tn, as shown in Fig. 9 for Ca and Zn substituted YBCO polycrystalline samples, where the resistivity deviates downward from linearity at Tn which is already close to Tc as the system approaches the overdoped regime. The results for the overdoped material presented in Fig. 9 thus resembled those found for optimally doped cuprates, rather than an underdoped sample. In fact, as pointed out in Ref. [9], a crossover in resistivity exists even

26

S. Dzhumanov et al. / Physica B 440 (2014) 17–32

400

0.30 0.25

ρab m cm

ρ

ab

(μΩ cm)

T*

200

0.20 0.15 0.10 0.05

100

0.00 0

T

x 0.10

300

0

50

100

150

200

250

300

20

40

80

60

80

T

ρab m cm

0.25

x 0.11

0.20 0.15 0.10 0.05

0.8

0.00

0.7

0

20

40

TK

0.6

ρab (mΩ cm)

60

0.30

T(K) Fig. 6. The calculated temperature dependence of ρab (solid line) compared with experimental data for CuBa2 Ca3 Cu3 Oy superconductor with the addition AgO content x¼ 0.9 [20] (open circles). The appropriate material parameters used are vs ¼ 4:1  105 cm=s, ρM ¼ 4:5 g=cm3 , ɛ~ ¼ 4:8, mab ¼ 1:547  10  27 g, mp ¼ 1:94922  10  27 g, n ¼ 0:85  1021 cm  3 , ℏω01 ¼ 0:050 eV, ℏω02 ¼ 0:054 eV, and ρ0 ¼ 0:008 mΩ cm.

0

Fig. 8. Experimental ρab(T) data for underdoped La2  x Bax CuO4 samples x ¼0.10, 0.11 [12] (open circles) and the respective fits (solid lines). The appropriate material parameters used are (a) vs ¼ 4:0  105 cm=s, ρM ¼ 4:0 g=cm3 , ɛ~ ¼ 4:762, mab ¼ 1:911  10  27 g, mp ¼ 2:108  10  27 g, n ¼ 0:6  1021 cm  3 , ℏω01 ¼ 0:036 eV, ℏω02 ¼ 0:038 eV, and ρ0 ¼ 0:080 mΩ cm. (b) vs ¼ 4:0  105 cm=s, ρM ¼ 4:5 g=cm3 , ɛ~ ¼ 4:762, mab ¼ 1:642  10  27 g, mp ¼ 2:124  10  27 g, n ¼ 0:65  1021 cm  3 , ℏω01 ¼ 0:030 eV, ℏω02 ¼ 0:033 eV, and ρ0 ¼ 0:1 mΩ cm.

0.5 0.4

T*

0.3 0.2 0.1 0

0

50

100

150

200

250

1.2

300

T(K)

ρab (Ω cm)

Fig. 7. Experimental ρab(T) data for underdoped La1:48 Nd0:40 Sr0:12 CuO4 [21] and the respective fits (solid line). The appropriate material parameters used are vs ¼ 4:4  105 cm=s, ρM ¼ 5:0 g=cm3 , ɛ~ ¼ 6:0, mab ¼ 2:002  10  27 g, mp ¼ 2:52252  10  27 g, n ¼ 0:6  1021 cm  3 , ℏω01 ¼ 0:0431 eV, ℏω02 ¼ 0:0432 eV, and ρ0 ¼ 0:093 mΩ cm.

1 0.8

T*

0.6 0.4

n

in this material, but T is comparatively close to Tc. This result provides a direct indication of the presence of BCS-like PG and incoherent Cooper pairs in overdoped cuprates. Finally we conclude that the agreement between the modeled and the various experimental resistivity data obtained for underdoped, optimally doped and overdoped cuprates is quite good. The above quantitative analysis of the resistivity data shows that our theory describes consistently both the T-linear resistivity above Tn and the distinctly different deviations from the high temperature T-linear behavior in ρðTÞ and ρab(T) below Tn in these materials.

5. Normal state conductivity of charge carriers along the c-axis in various high-Tc cuprates A distinct feature of the normal state of cuprate superconductors is the increase in the c-axis resistivity over a wide temperature range. In the underdoped and optimally doped regimes, the temperature dependence of the c-axis resistivity, ρc, is semiconductor-like down to Tc in contrast to metallic ρab(T).

0.2 0

0

50

100

150

200

250

300

T(K) Fig. 9. The calculated temperature dependence of ρab (solid line) compared with experimental data for overdoped Y1  x Cax Ba2 ðCu1  y Zny Þ3 O7  δ with x¼ 0.1 and y¼ 0 [9] (open circles). The appropriate material parameters used are vs ¼ 3:5 105 cm=s, ρM ¼ 3:8 g=cm3 , ɛ~ ¼ 4:0, mab ¼ 1:7  10  27 g, mp ¼ 1:89  10  27 g, n ¼ 1:2  1021 cm  3 , ℏω01 ¼ 0:045 eV, ℏω02 ¼ 0:055 eV, ρ0 ¼ 0:2 mΩ cm.

This leads us to assume that a distinct conduction mechanism exists along the c-axis. Here the key question is why the c-axis resistivity differs so much from the in-plane resistivity in the normal state of high-Tc cuprates. In the following, we will consider the issue of the non-metallic c-axis conductivity in doped high-Tc cuprates and propose a simple microscopic model capable of describing the carrier confinement and the observed insulating behavior of ρc(T) in the low and intermediate doping regimes. We

S. Dzhumanov et al. / Physica B 440 (2014) 17–32

will assume that the localized large bipolarons are formed in carrier-poor regions between the CuO2 layers and the c-axis charge transport occurs through the thermal dissociation of such bipolarons into the separate polarons and the subsequent thermally activated hopping of localized polarons residing between the CuO2 layers. Actually, the concept of hopping transport is suitable for materials where the charge carriers are spatially localized [53,65]. 5.1. Carrier confinement and bipolaronic pairing pseudogap Here we discuss the mechanisms that cause carrier localization in hole-doped cuprates. The question now arises how, under certain conditions, a system with n carriers in cuprates may undergo a phase transition from a delocalized into a localized state. In inhomogeneous hole-doped cuprates, the strong carrier– carrier Coulomb interactions and carrier–phonon interactions may play an important role in carrier delocalization and localization. The distinctive feature of the cuprates is their very large ratio of static ɛ0 to high frequency ɛ 1 dielectric constants. A large ionicity of these compounds η ¼ ɛ1 =ɛ0 5 1 and their deformation polarizability are favorable for carriers attracted to polarization well created by the other ones to form 3D large bipolarons between the CuO2 layers with binding energies EbB ¼ jEB  2Ep j, where EB is the ground state energy of a large bipolaron. The ground state energies of interlayer polarons Ep and bipolarons EB can be calculated in the 3D continuum model and adiabatic approximation by using variational method [28,29,66]. The results of such calculations [29] show that the charge carriers in the cuprates are large polarons and bipolarons with binding energies Ep  0:1 eV and EbB C ð0:4–0:5ÞEp at η ¼ ɛ 1 =ɛ 0 ≲0:02 and ɛ 1 ¼ 3–5 [33]. The values of ɛ0 observed in high-Tc cuprates range from 30 [38] to 105 [67]. If we take ɛ 1 ¼ 4 and η ¼ 0:02–0:10, then the binding energies of large bipolarons are equal to EbB C0:01–0:04 eV [29], which are manifested as the temperature-independent pseudogaps in the excitation spectra of high-Tc cuprates (see, e.g., Refs. [68,69]). We now examine the condition for the existence of the localized bipolaronic states in cuprates. At very low doping, the separate levels of large bipolarons are formed in the CT gap of the cuprates. As the doping level increases towards underdoped region, the binding energies of such bipolarons decrease gradually to zero at some doping level. For the LSCO system we can evaluate the critical value of n ¼ nc using the expression for nc from Ref. [29] and the parameter values mp ¼ 2:1me [39], ɛ1 ¼ 4, EbB  0:01–0:04 eV. Then we obtain nc C ð0:049–0:390Þ  1020 cm  3 . Taking into account that the value of Va in the orthorhombic LSCO is 190 Å3, we find the doping levels p ¼ 0.0001–0.0074 at which large bipolarons dissociate into large polarons. It follows that large bipolarons can exist in cuprates in carrier-poor domains and remain localized. While the single-polaron may exist only in carrier-rich regions at n 4 nc . We believe that the strong electron–phonon interactions in the cuprates and the pairing of carriers in real space favor the local nanoscale phase separation and produce carrier-poor regions. The real-space pairing of large polarons in the strong-coupling regime and the confinement effects for interlayer polarons lead to the formation of large bipolarons in carrier-poor regions between the CuO2 layers. Actually, the layered cuprates have anisotropic 3D crystal structure, where some fraction of large polarons bound into real-space pairs (strong-coupling bipolarons) between the CuO2 layers and remain confined to the potential wells in the c-direction. These interlayer polaronic carriers have ill-defined momentum at W p o EbB and they are localized rather than mobile. Upon decreasing the carrier concentration, the bandwidth Wp of polaronic carriers distributed over the interlayer regions becomes smaller than the binding energy EbB of bipolarons, thereby causing localization of polarons and thus restriction of charge transport on hopping processes only. In the

27

following, we propose a simple model of c-axis hopping transport of polarons, which is capable of describing the observed non-metallic c-axis conductivity in high-Tc layered cuprates, from the underdoped to the overdoped regime. 5.2. Modeling of c-axis charge transport As mentioned above, the strong-coupling interlayer bipolarons become localized rather than mobile and the c-axis conductivity in layered cuprates can be modeled as the c-axis polaron transport at the thermal dissociation of such bipolarons into two hole polarons which subsequently move by hopping along the c-axis. Clearly, at low interlayer carrier concentrations (ni or doping levels p), the polaronic band is continuously narrowed and the charge transport in it becomes non-metallic. In this case the phonon-assisted hopping motion of polaronic carriers between spatially distinct locations becomes the main mechanism of c-axis transport above Tc, while quantum effects (under-barrier tunneling or even bandlike motion) may be important at sufficiently low temperatures below Tc. Physically, the localized c-axis bipolarons confined between the CuO2 layers block the out-of-plane charge transport and they should be broken up in order to take part in hopping conduction along the c-axis. Therefore, the temperature dependence of the c-axis resistivity ρc is believed to be semiconductinglike down to Tc in contrast to metallic ρab(T), as observed in cuprates from underdoped to overdoped regime. At T-0 the localized bipolarons do not contribute to the c-axis conductivity, since the bipolaronic energy gap or PG exists in the excitation spectrum of large polarons just as the energy gap in semiconductors. However, with increasing temperature the localized c-axis bipolarons will dissociate into the separate large polarons and the subsequent hopping-like motion of polaronic carriers (i.e. thermally activated jumps of the carriers from one position to another position) may contribute to the c-axis conductivity in underdoped and optimally doped cuprates. Such a hopping conduction in cuprates may be a plausible picture for relatively low doping regimes. Accordingly, in carrier-poor regions the hopping carrier must overcome an energy barrier of the height Ea (in the absence of an electric field E) and the c-axis conductivity would show activation behavior. In the case of localized bipolarons, the activation energy Ea (i.e. the barrier height) for a hop from the one potential well to the other nearest-neighbor potential well is equal to the binding energy of bipolarons EbB. Now we consider the one-dimensional case of charge transport and study the hopping conduction along the c-axis in cuprates. In the absence of an electric field (E¼ 0), the hopping probability at the phonon-assisted hopping of a carrier along the one of the two relevant directions is given by w ¼ 12 ω0 exp½  EbB =kB T;

ð45Þ

where ω0 is the out-of-plane optical phonon mode frequency, which differs from ω0 in the CuO2 layers. At E a 0 a potential energy of a carrier in an electric field is determined as work eEah done by an electric force in moving a carrier through displacement at a hopping distance ah (which is of the order of the interlayer distance). In this case reduction of the potential barrier is given by eEah =2. Then the hopping drift mobility of polarons in the electric field is determined from the expression for their drift velocity vd ¼ ah ðω1  ω2 Þ;

ð46Þ

where ω1 ¼ ð1=2Þω0 exp½  ðEbB  ðeEah =2ÞÞ=kB T and ω2 ¼ ð1=2Þω0 exp½  ðEbB þ ðeEah =2ÞÞ=kB T are the hopping probabilities of polarons along and opposite directions to the electric field E, respectively. At eEah =2kB T 51 (i.e. in the weak electric field) we can expand the exponents in ω1 and ω2 in powers of 7 ðeEah =2kB TÞ

28

S. Dzhumanov et al. / Physica B 440 (2014) 17–32

10.0

and the hopping drift mobility of polarons along the c-axis can be determined from the equation

5.0

where ni is the concentration of hopping polarons in carrier-poor regions between the CuO2 layers. Thus the c-axis resistivity is determined as   1 2kB T EbB ¼ ð49Þ ρc ðT 4T c Þ ¼ exp sc ðT 4 T c Þ ni e2 a2h ω0 kB T Here EbB represents the temperature-independent bipolaronic PG. According to (48) or (49), the c-axis transport is essentially an activation process over a gap or PG in carrier-poor regions. So we see that the c-axis resistivity of underdoped to overdoped cuprates is proportional to exp½EbB =kB T and the main temperature dependence of the c-axis resistivity comes from the thermal excitation exponent with the activation energy EbB and the concept of thermally activated hopping transport can explain the experimental data on ρc(T) remarkably well. Another possible explanation of the c-axis conductivity in YBCO is based on the so-called resonant tunneling mechanism of charge carriers between conducting CuO2 layers through localized states in the CuO chains proposed by Abrikosov [17]. Actually, the bipolaronic PG and carrierconfinement mechanisms together cause the peculiar insulating temperature dependence of ρc in the cuprates. At this point we would like to mention that there are seemingly some correspondence in the results obtained in this work and the results presented in the work of Koizumi et al. [50]. It seems that the localized bipolaron in the PG phase agrees with the fact that spinvortices are formed with doped holes as their cores, and an attractive interaction exists between two spin-vortices with winding numbers þ1 and  1 [50]. The c-axis single-polaron hopping must create an unpaired spin-vortex in the layer hopping from and destroy the spin-vortex pair in the layer hopping to.

5.3. Comparison of the theoretical predictions with experimental

ρc(T) data

The bipolaronic PG, EbB is temperature independent, and therefore, there is no the crossover temperature similar to Tn in the temperature dependence of ρc(T). In fact, there is no any indication of the presence of such a crossover temperature in the experimentally obtained c-axis resistivity of the cuprates. Given the above understanding of the c-axis charge transport, it would be instructive to see how the two mechanisms (carrier localization or confinement and bipolaronic PG) are actually realized in carrierpoor regions between the CuO2 layers. In the relatively low doping regimes, both the carrier-confinement mechanism and the bipolaronic PG mechanism appear to become increasingly more effective with lowering temperature. Now, we will compare our predictions with the experimental ρc(T) data obtained for various high-Tc cuprate superconductors. One can use Eq. (49) to analyze the temperature dependence of the c-axis resistivity and the experimental ρc(T) data. Comparison of our results with experimental ρc(T) data was performed on the basis of this equation by choosing reasonable values of hopping parameters ah, ω0, ni and EbB. The resulting fits for various LSCO, YBCO, Y1  x Prx Ba2 Cu3 O7  δ and Bi2 Sr2  x Lax CuO6 þ δ (BSLCO) samples with different Tc values and doping levels are shown in Figs. 10–13. The solid curves of

2.0 1.0

c

Above Tc the expression for the hopping conductivity of polarons along the c-axis can be presented in the form   n e2 a2 ω E ð48Þ sc ðT 4 T c Þ ¼ ni eμh ðTÞ ¼ i h 0 exp  bB ; 2kB T kB T

cm

ð47Þ

ρ

ea2h ω0 exp½ EbB =kB T 2kB T

0.5

0.2

0

50

100

150

200

p

0.08

p

0.10

250

300

TK Fig. 10. Comparison of the calculated results for ρc(T) (solid lines) with the experimental ρc(T) data (open circles) obtained for La1:92 Sr0:08 CuO4 and La1:90 Sr0:10 CuO4 [7]. Fits to the experimental data for La1:92 Sr0:08 CuO4 and La1:90 Sr0:10 CuO4 are performed using the hopping parameters EbB ¼ 0:027 eV, and EbB ¼ 0:022 eV, ah ¼ 13 Å, ω0 ¼ 5:2  1013 s  1 , ni ¼ 1:25  1019 cm  3 ah ¼ 13 Å, ω0 ¼ 5:5  1013 s  1 , ni ¼ 2:1  1019 cm  3 , respectively.

150

ρc m cm

μh ðTÞ ¼

100

50

0

0

50

100

150

200

250

300

TK Fig. 11. Comparison of the calculated results for ρc(T) (solid line) obtained using the hopping parameters EbB ¼ 0:028 eV, ah ¼ 11 Å, ω0 ¼ 5:7  1013 s  1 , ni ¼ 2:5  1019 cm  3 with the experimental ρc(T) data for YBa2 Cu3 O6:66 (open circles) [17].

Figs. 10–13 are calculated curves using the numerical values of the hopping parameters of our model. The temperature dependence of ρc in these plots reveals the fact that the insulating behavior of ρc(T) is caused by the two phenomena, the bipolaronic PG and the carrier confinement. Figs. 10–13 demonstrate quite clearly that the insulating behavior of ρc(T) is very sensitive to the hopping parameter EbB and leads us to conclude that the calculations which take into account changes in the hopping distance ah, hopping carrier concentration ni and activation energy EbB should be used in comparing with experimental ρc(T) data. We see that the calculated results agree very well with the experimental ρc(T) data down to Tc. The calculated curves based on the theoretical predictions, which are displayed in Figs. 10–13, are in quantitative agreement with experimental results and capture the main features of experiments. Good agreement between calculated and experimental results substantiates our simple microscopic model of carrier localization and c-axis hopping transport of polaronic carriers at thermally activated dissociation of the localized interlayer bipolarons. In spite of the simplicity of the model, we have been able to fit a large set of ρc(T) data taken in various high-Tc cuprates. As the doping level increases towards overdoped region and exceeds some critical value in carrier-poor regions (where the bandwidth of large polarons is not less than the binding energy of localized bipolarons), the c-axis charge transport in sufficiently broadened polaronic band becomes band-like (i.e. metal-like), as observed in optimally doped and overdoped high-Tc cuprates [5,11,71,72]. When EbB tends to zero, the insulating behavior of ρðTÞ changes to the metallic behavior.

S. Dzhumanov et al. / Physica B 440 (2014) 17–32

1000

8000

0.48

ρab (Ω cm)

ρc m cm

x

6000

600 ab

5000 c

ρ /ρ

400 x

200

0.43

4000

−2

10

p=0.08

−3

10

p=0.10 −4

10

0

3000

100

T(K)

2000

0

50

100

150

200

250

300

Fig. 12. Comparison of the calculated results for ρc(T) (solid lines) with the experimental ρc(T) data (open circles) obtained for Y1  x Prx Ba2 Cu3 O7  δ crystals [70]. Fits to the experimental data for Y1  x Prx Ba2 Cu3 O7  δ with x¼ 0.43 and 0.48 are performed using the hopping parameters EbB ¼ 0:025 eV, ah ¼ 11 Å, ω0 ¼ 8  1013 s  1 , ni ¼ 1:1  1019 cm  3 and EbB ¼ 0:031 eV, ah ¼ 11 Å, ω0 ¼ 7 1013 s  1 , ni ¼ 1:2  1019 cm  3 , respectively.

12 10 8

ρc

6 4

p

0.16

p

0.18

0

200

300

p=0.08

1000

TK

cm

−1

10

7000

800

0

29

p=0.10 0

50

100

150

200

250

300

T(K) Fig. 14. Temperature dependences of ρc =ρab in the LSCO samples calculated from the data in Fig. 10 and in the inset of this figure. Open circles and black diamonds represent the experimental data for LSCO with p ¼0.08(J) and p ¼0.10 (~), respectively, taken from Ref. [7]. Fits to the experimental data for LSCO samples with p ¼ 0.08 and 0.10 are performed using the hopping parameters EbB ¼ 0:027 eV, ah ¼ 13 Å, ω0 ¼ 5:2  1013 s  1 , ni ¼ 1:3  1019 cm  3 and EbB ¼ 0:022 eV, ah ¼ 13 Å, ω0 ¼ 5:5  1013 s  1 , ni ¼ 2:0  1019 cm  3 , respectively. The inset shows the calculated temperature dependences of ρab (solid lines) compared with experimental ρab(T) data for LSCO [7] (open circles). The intrinsic materials parameters used in the numerical calculations are vs ¼ 4:0  105 cm=s, ρM ¼ 4:0 g=cm3 , ɛ~ ¼ 4:762, mab ¼ 1:82810  27 g, mp ¼2:262  10  27 g, n ¼ 0:5  1021 cm  3 , ℏω01 ¼ 0:048 eV, ℏω02 ¼ 0:049 eV, ρ0 ¼ 0:2 mΩ cm for LSCO (with p¼ 0.08, T n ¼ 165 K and λn ¼ 0:597Þ and vs ¼ 4:0  105 cm=s, ρM ¼ 4:3 g=cm3 , ɛ~ ¼ 4:762, mab ¼ 1:82  10  27 g, mp ¼ 2:062  10  27 g, n ¼ 0:7  1021 cm  3 , ℏω01 ¼ 0:047 eV, ℏω02 ¼ 0:049 eV, ρ0 ¼ 0:18 mΩ cm for LSCO (with p ¼ 0.10, T n ¼ 150 K and λn ¼ 0:595Þ.

2 0

0

50

100

150

200

250

300

and

TK Fig. 13. Comparison of the calculated results for ρc(T) (solid lines) with the experimental ρc(T) data (open circles) obtained for BSLCO crystals with x ¼ 0.23 (p¼ 0.18) and x ¼0.39 (p ¼ 0.16) [10]. Fits to the experimental data for BSLCO with x ¼0.23 and 0.39 are performed using the hopping parameters EbB ¼ 0:013 eV, ah ¼ 10 Å, ω0 ¼ 4:5  1013 s  1 , ni ¼ 2:7  1019 cm  3 and EbB ¼ 0:022 eV, ah ¼ 10 Å, ω0 ¼ 5:35  1013 s  1 , ni ¼ 2:4  1019 cm  3 , respectively.

6. The resistivity anisotropy in various high-Tc cuprates We consider next the resistivity anisotropy ρc =ρab in various high-Tc cuprates and elucidate the evolution of the behavior of ρc =ρab over a wide range of temperature. Differences in the doping levels of CuO2 layers and interlayer regions strongly influence on the resistivity anisotropy of high-Tc cuprates above Tc. As can be seen from the theory described above, the behavior of the resistivity components can be very different, and much clearer conclusions can be drawn from the temperature dependence of the resistivity anisotropy ratio ρc =ρab of underdoped to overdoped cuprates. For such a purpose, we will compare our result for ρc ðTÞ=ρab ðTÞ with the experimental data on the anisotropy ratio ρc =ρab obtained for underdoped LSCO [7] (p¼ 0.08, Tc ¼29 K, T n ¼ 165 K and p ¼0.10, Tc ¼35 K, T n ¼ 150 K), for underdoped YBCO [17] ðT c C 69 K, T n C 200 K) and for optimally doped BSLCO [10] (with p ¼0.16, Tc ¼ 38 K, T n ¼ 125 K) and overdoped BSLCO [10] (with p ¼0.18, Tc ¼ 29 K, T n ¼ 110 K). Such a comparison with experimental data was performed on the basis of Eqs. (34), (44) and (49) written in the forms

ρc ðTÞ

ρab ðT oT Þ n

¼

ð2kB T=ni e2 a2h ω0 Þexp½EbB =kB T

ρ0 þ ½snp ðTÞ þ sB ðTÞ  1

ð50Þ

ð2kB T=ni e2 a2h ω0 Þexp½EbB =kB T ρc ðTÞ ¼ ρ0 þ sp 1 ðTÞ ρab ðT 4 T n Þ

ð51Þ

by choosing optimal values for relevant physical parameters (Figs. 14– 16). Figs. 14–16 show the temperature dependences of the resistivity anisotropy ratio ρc =ρab for carrier concentrations studied. One can see that the fits are rather good for low interlayer doping regimes. The magnitude of ρc =ρab increases with decreasing temperature and doping, which means that the carrier localization and bipolaronic PG mechanisms become increasingly effective. The peculiar temperature dependence of ρc =ρab in the cuprates is essentially governed by the exponential temperature dependence of ρc. The above-mentioned resonant tunneling mechanism of c-axis charge transport was also used to describe the temperature dependence of ρc ðTÞ=ρab ðTÞ in YBCO. The experimental verification of this model was carried out for YBCO in Ref. [73]. However, it was shown that although the theory [17] is in qualitative agreement with experiment, the best description of the experimental data obtained has an exponential dependence of the form ρc ðTÞ=ρab ðTÞ  expðEg =kB TÞ (where Eg represents the PG energy scale) which is in reasonable agreement with our result. Similar findings were obtained recently for Pr-doped YBCO single crystals in Ref. [70], where the normal-state resistivity anisotropy ρc ðTÞ=ρab ðTÞ observed in these materials is also well described by the exponential law for the thermally activated hopping conductivity. Thus, we found that the normal-state charge transport in the high-Tc cuprates is strongly anisotropic and unusually large anisotropy exists between ρab(T) and ρc(T) in the low-doping and low-temperature region. At low carrier concentrations ni Cð1:3–2:5Þ  1019 cm  3 , the effect of the bipolaronic PG EbB causes a rapid growth ρc =ρab with decreasing temperature, and, as a result, the maximal values of ρc =ρab in LSCO, YBCO and BSLCO

30

S. Dzhumanov et al. / Physica B 440 (2014) 17–32 4

0.5

0.4 0.3 0.2

10 ab

0.1 0

400

x 10

0

100

T(K)

200

300

c

600

c

ρ /ρ

ab

800

15

ρ /ρ

ρab (mΩ cm)

0.5

ρab (mΩ cm)

1000

0.2

p=0.18

0.1 0

100

p=0.16

200

0 50

0.3

0

5

p=0.16

0.4

T(K)

200

300

p=0.18 100

150

200

250

300

T(K) Fig. 15. Temperature dependence of ρc =ρab in the YBCO sample calculated from the data in Fig. 11 and in the inset of this figure. Open circles represent the experimental data for YBa2 Cu3 O6:66 taken from Ref. [17]. Fits to the experimental data for YBa2 Cu3 O6:66 sample are performed using the hopping parameters EbB ¼ 0:028 eV, ah ¼ 11 Å, ω0 ¼ 5:7  1013 s  1 , ni ¼ 2:5  1019 cm  3 . The inset shows the calculated temperature dependence of ρab (solid line) compared with experimental ρab(T) data for YBCO [17] (open circles). The intrinsic materials parameters used in the numerical calculations are vs ¼ 5:8  105 cm=s, ρM ¼ 4:7 g=cm3 , ɛ~ ¼ 4:0, mab ¼ 1:738  10  27 g, mp ¼ 2:164  10  27 g, n ¼ 0:75  1021 cm  3 , ℏω01 ¼ 0:041 eV, ℏω02 ¼ 0:051 eV, ρ0 ¼ 0:02 mΩ cm for YBa2 Cu3 O6:66 ðT n ¼ 200 K and λn ¼ 0:570Þ.

are 104, 103 and 105, respectively (see Figs. 14–16), just above Tc in accordance with experimental findings [7,10,17].

0

0

50

100

150

200

250

300

T(K) Fig. 16. Temperature dependences of ρc =ρab in the BSLCO samples calculated from the data in Fig. 13 and in the inset of this figure. Open circles and black diamonds represent the experimental data for BSLCO with p ¼ 0.16(J) and p ¼ 0.18(~), respectively, taken from Ref. [10]. Fits to the experimental data for BSLCO samples with p ¼0.16 and 0.18 are performed using the hopping parameters EbB ¼ 0:022 eV, ah ¼ 10 Å, ω0 ¼ 5:35  1013 s  1 , ni ¼ 2:4  1019 cm  3 , and EbB ¼ 0:013 eV, ah ¼ 10 Å, ω0 ¼ 4:5  1013 s  1 , ni ¼ 2:7  1019 cm  3 , respectively. The inset shows the calculated temperature dependences of ρab (solid line) compared with experimental ρab(T) data for BSLCO [10] (open circles). The intrinsic materials parameters used in the numerical calculations are vs ¼ 4:0  105 cm/s, ρM ¼ 4:1 g=cm3 , ɛ~ ¼ 5:0, mab ¼ 1:575  10  27 g, mp ¼ 1:964  10  27 g, n ¼ 0:8  1021 cm  3 , ℏω01 ¼ 0:040 eV, ℏω02 ¼ 0:058 eV, ρ0 ¼ 0:122 mΩ cm for BSLCO (with p ¼0.16, T n ¼ 125 K and λn ¼ 0:529Þ and vs ¼ 4:0  105 cm=s, ρM ¼ 4:0 g=cm3 , ɛ~ ¼ 6:0, mab ¼ 1:428  10  27 g, mp ¼ 1:8464  10  27 g, n ¼ 0:88  1021 cm  3 , ℏω01 ¼ 0:041 eV, ℏω02 ¼ 0:055 eV, ρ0 ¼ 0:07 mΩ cm for BSLCO (with p ¼ 0.18, T n ¼ 110 K and λn ¼ 0:457Þ.

7. Conclusion We have developed a consistent quantitative theory of the unusual in-plane and c-axis charge transports in the normal state of hole-doped cuprate superconductors. We argued that major current carriers in the normal state of these high-Tc materials are hole-like. The proposed charge transport theory is used to calculate the in-plane and c-axis conductivities of underdoped to overdoped cuprates above Tc and to obtain the temperature- and doping-dependent in-plane resistivity ρab, c-axis resistivity ρc and resistivity anisotropy ρc =ρab seen experimentally in the normalstate. Our theory assumes that the relevant charge carriers in these high-Tc materials above Tc are hole polarons and polaronic Cooper pairs in the carrier-rich metallic CuO2 layers (with nonzero thickness) and localized large (bi)polarons residing in the lowdoping regions between the CuO2 layers. We believe that the charge inhomogeneity, the real- and k-space pairing in the strong and intermediate electron–phonon coupling regimes and the local nanoscale phase separation into two different domains (i.e. carrier-rich CuO2 layers and carrier-poor regions between the CuO2 layers) are responsible for the unusual metallic and nonmetallic conductivities of high-Tc cuprates, from the underdoped to the overdoped regime. We examined two intrinsic mechanisms of the unusual in-plane and c-axis charge transports in layered cuprates: the metallic conductivity of underdoped to overdoped cuprates above Tc is considered as the band-like conduction of large polarons and preformed bosonic Cooper pairs at their scattering by lattice vibrations in the CuO2 layers, while the c-axis conductivity of the layered cuprates is mainly associated with thermally activated dissociation of a large bipolaron localized between the CuO2 layers into two separate polarons which subsequently move by hopping along the c-axis. Assuming the scattering of polaronic carriers and bosonic Cooper pairs at acoustic and optical phonons in thin CuO2 layers, we obtained the appropriate expressions for the relaxation times of these

carriers, which are then used to obtain the transport coefficients in cuprates. We have obtained the expressions for the temperature- and doping-dependent in-plane resistivity of cuprate superconductors below and above the PG formation temperature Tn in the relaxation time approximation by using the appropriate Boltzmann transport equations. The results are then used to analyze the experimental data obtained for various high-Tc cuprates. We showed that the scattering of polaronic carriers at acoustic and optical phonons is responsible for the T-linear resistivity above Tn. We have also found that the transition to the BCS-like PG regime and the effective conductivity of bosonic Cooper pairs in the normal state of underdoped to overdoped cuprates are responsible for the pronounced nonlinear or nonmonotonic temperature dependence of ρab(T) and different downward and upward deviations from the T-linear behavior in ρab(T) below Tn. In addition, we found that the BCS-like transitions above Tc are also manifested as the different resistive transitions at Tn, which are observed in some experiments. Importantly, a variety of anomalous behaviors of ρab(T) in the PG state of underdoped to overdoped cuprates are explained by the developed theory of the in-plane charge transport naturally if one assumes that the polaronic carriers and bosonic Cooper pairs are scattered at different optical phonons having the distinctive frequencies ω01 and ω02 ≳ω01 . We have developed a specific microscopic model for describing the carrier localization (confinement) in the low-doping regions between the CuO2 layers and the c-axis hopping transport of hole polarons at the thermal dissociation of localized large bipolarons residing between these layers into two separate polarons. This model was used to obtain the simple analytical expressions for hopping conductivity of polaronic carriers along the c-axis and for temperature- and doping-dependent c-axis resistivity of underdoped to overdoped cuprates. We concluded that the two phenomena, the bipolaronic pairing PG and the carrier-confinement

S. Dzhumanov et al. / Physica B 440 (2014) 17–32

between the CuO2 layers, together cause the unusual insulating temperature dependence of ρc in the cuprates. Finally, we used our results for ρab(T) and ρc(T) to calculate the normal-state resistivity anisotropy ρc ðTÞ=ρab ðTÞ in high-Tc cuprates. Our quantitative analysis shows that the nonmetallic temperature dependence of ρc practically determines nearly exponential temperature dependence of the ratio ρc =ρab , which can be also considered as the combined result of the carrier-confinement and the bipolaronic PG. In some high-Tc cuprates, such as LSCO, YBCO and BSLCO the bipolaronic PG greatly enhances the resistivity anisotropy with decreasing temperature down to Tc, causing ρc =ρab to reach  104, 103 and 105, respectively. We compared the calculated results for ρc(T) and ρc ðTÞ=ρab ðTÞ with the experimental data obtained for various samples of high-Tc cuprates. We have demonstrated that the calculated results for ρab(T), ρc(T) and ρc =ρab are in good quantitative agreement with experimental findings in various high-Tc cuprates. We succeeded in explaining a variety of resistivities, such as the T-linear behavior of ρab(T) above Tn, the different downward and upward deviations from the linear behavior in ρab(T) below Tn, the nonmonotonic temperature dependence of ρab(T) (i.e. the resistivity peak) between Tc and Tn, the different resistive transitions at Tn, the unusual insulating temperature dependence of ρc and the peculiar temperature dependence of the resistivity ratio ρc =ρab , observed in many high-Tc cuprates from the underdoped to overdoped regime.

Acknowledgments We thank E.M. Ibragimova, B.L. Oksengendler, P.J. Baimatov, U.T. Kurbanov Z.S. Khudayberdiev and E.X. Karimboev for useful discussions. This work was supported by the Uzbekistan State Committee for Science and Technology Grant no. Φ2-ΦA-Φ120.

References [1] J.G. Bednorz, K.A. Müller, Z. Phys. B 64 (1986) 189. [2] M.K. Wu, J.R. Ashburn, C.J. Torng, P.H. Hor, R.L. Meng, L. Gao, Z.J. Huang, Y.Q. Wang, C.W. Chu, Phys. Rev. Lett. 58 (1987) 908. [3] T. Timusk, B. Statt, Rep. Prog. Phys. 62 (1999) 61; J. Orenstein, A.J. Millis, Science 288 (2000) 468; J.L. Tallon, J.W. Loram, Physica C 349 (2001) 53. [4] P.B. Allen, Z. Fisk, A. Migliary, in: D.M. Ginsberg (Ed.), Physical Properties of High Temperature Superconductors I, MIR, Moscow, 1990 (Chapter 5). [5] H. Takagi, N.E. Hussey, in: G. Iadonisi, J.R. Schrieffer, M.L. Chiafalo (Eds.), Proceedings of the International School of Physics “Enrico Fermi” Course CXXXVI, IOS Press, Amsterdam, 1998, p. 227. [6] A.N. Lavrov, V.F. Gantmakher, Phys. Usp. 41 (1998) 223. [7] S. Komiya, Y. Ando, X.F. Sun, A.N. Lavrov, Phys. Rev. B 65 (2002) 214535. [8] V.A. Finkel, Fiz. Nizk. Temp. 28 (2002) 952. [9] S.H. Naqib, J.R. Cooper, J.L. Tallon, C. Panagopoulos, Physica C 387 (2003) 365. [10] S. Ono, Y. Ando, Phys. Rev. B 67 (2003) 104512. [11] H. Raffy, V. Toma, C. Murrills, Z.Z. Li, Physica C 460–462 (2007) 851. [12] Y. Koike, T. Adachi, Physica C 481 (2012) 115. [13] C.M. Varma, P.B. Littlewood, S. Schmitt-Rink, E. Abrahams, A.E. Ruckenstein, Phys. Rev. Lett. 63 (1989) 1996; C.M. Varma, Phys. Rev. B 55 (1997) 14554. [14] A.S. Alexandrov, A.M. Bratkovsky, N.F. Mott, Phys. Rev. Lett. 72 (1994) 1734; A.S. Alexandrov, V.N. Zavaritsky, S. Dzhumanov, Phys. Rev. B 69 (2004) 052505. [15] P.W. Anderson, The Theory of Superconductivity in the High-Tc Cuprates, Princeton University Press, Princeton, 1997. [16] B.P. Stojkovic, D. Pines, Phys. Rev. B 55 (1997) 8576; Y. Yanase, J. Phys. Soc. Jpn. 71 (2002) 278. [17] A.A. Abrikosov, Phys. Usp. 41 (1998) 605. [18] D. Varshney, A. Yogi, K.K. Choudhary, Physica C 470 (2010) 2016; S.N. Naqib, M. Borhan Uddin, J.R. Cole, Physica C 471 (2011) 1598. [19] J. Kokalj, N.E. Hussey, R.H. McKenzie, Phys. Rev. B 86 (2012) 045132. [20] K. Tokiwa, Y. Tanaka, A. Iyo, Y. Tsubaki, K. Tanaka, J. Akimoto, Y. Oosawa, N. Terada, M. Hirabayashi, M. Tokumoto, S.K. Agarwal, T. Tsukamoto, H. Ihara, Physica C 298 (1998) 209. [21] S. Uchida, Physica C 341–348 (2000) 823.

31

[22] A. Ulug, B. Ulug, R. Yagbasan, Physica C 235–240 (1994) 879. [23] Ø. Fischer, M. Kugler, I. Maggio-Aprile, C. Berthod, Rev. Mod. Phys. 79 (2007) 353. [24] S. Dzhumanov, Physica C 235–240 (1994) 2269; S. Dzhumanov, P.K. Khabibullaev, Pramana J. Phys. 45 (1995) 385; S. Dzhumanov, A.A. Baratov, S. Abboudy, Phys. Rev. B 54 (1996) 13121; S. Dzhumanov, Int. J. Mod. Phys. B 12 (1998) 2995. [25] P.A. Lee, N. Nagaosa, X.-G. Wen, Rev. Mod. Phys. 78 (2006) 17. [26] H. Koizumi, J. Supercond. Novel Magn. 24 (2011) 1997; R. Hidekata, H. Koizumi, J. Supercond. Novel Magn. 24 (2011) 2253. [27] V.Z. Kresin, S.A. Wolf, Rev. Mod. Phys. 81 (2009) 481. [28] J.T. Devreese, A.S. Alexandrov, Rep. Prog. Phys. 72 (2009) 066501. [29] S. Dzhumanov, P.J. Baimatov, O.K. Ganiev, Z.S. Khudayberdiev, B.V. Turimov, J. Phys. Chem. Solids 73 (2012) 484. [30] S. Dzhumanov, Theory of Conventional and Unconventional Superconductivity in the High-Tc Cuprates and Other Systems, Nova Science Publishers, New York, 2013; S. Dzhumanov, O.K. Ganiev, Sh.S. Djumanov, Physica B 427 (2013) 22. [31] J. Zaanen, G.A. Sawatzky, J.W. Allen, Phys. Rev. Lett. 55 (1985) 418. [32] L.D. Landau, Phys. Z. Sowjetunion 3 (1933) 664. [33] D. Emin, M.S. Hillery, Phys. Rev. B 39 (1989) 6575. [34] D. Emin, Polarons, Cambridge University Press, Cambridge, 2013. [35] B.Ya. Yavidov, A. Kurmantayev, D. Alimov, B. Kurbanbekov, Sh. Ramonkulov, Eur. Phys. J. B 86 (2013) 312. [36] S. Sugai, Physica C 185–189 (1991) 76. [37] X.-X. Bi, P.C. Eklund, Phys. Rev. Lett. 70 (1993) 2625. [38] M.A. Kastner, R.J. Birgeneau, G. Shirane, Y. Endoh, Rev. Mod. Phys. 70 (1998) 897. [39] A. Ino, C. Kim, M. Nakamura, T. Yoshida, T. Mizokawa, A. Fujimori, Z.-X. Shen, T. Kakeshita, H. Eisaki, S. Uchida, Phys. Rev. B 65 (2002) 094504. [40] K.M. Shen, F. Ronning, W. Meevasana, D.H. Lu, N.J.C. Ingle, F. Baumberger, W. S. Lee, L.L. Miller, Y. Kohsaka, M. Azuma, M. Takano, H. Takagi, Z.-X. Shen, Phys. Rev. B 75 (2007) 075115. [41] S. Miyaki, K. Makoshi, H. Koizumi, J. Phys. Soc. Jpn. 77 (2008) 034702. [42] C.J. Zhang, H. Oyanagi, Phys. Rev. B 79 (2009) 064521. [43] T. Kato, T. Noguchi, R. Saito, T. Machida, H. Sakata, Physica C 460–462 (2007) 880. [44] J. Zaanen, Science 286 (1989) 251, cond-mat/0103255; S.A. Kivelson, I.P. Bindloss, E. Fradkin, V. Oganesyan, J.M. Tranquada, A. Kapitulnik, C. Howald, Rev. Mod. Phys. 75 (2003) 1201. [45] V.Z. Kresin, H. Morawitz, S.A. Wolf, Mechanisms of Conventional and High Tc Superconductivity, Oxford University Press, New York, Oxford, 1993. [46] A. Damascelli, Z. Hussain, Z.-X. Shen, Rev. Mod. Phys. 75 (2003) 473; S. Hüfner, M.A. Hossain, A. Damascelli, G.A. Sawatzky, Rep. Prog. Phys. 71 (2008) 062501. [47] S. Doniach, M. Inui, Phys. Rev. B 41 (1990) 6668; V.J. Emery, S.A. Kivelson, Nature 374 (1995) 434. [48] S. Dzhumanov, Solid State Commun. 115 (2000) 155. [49] D. LeBoeuf, N. Doiron-Leyraud, J. Levallois, R. Daou, J.-B. Bonnemaison, N.E. Hussey, L. Balicas, B.J. Ramshaw, R. Liang, D.A. Bonn, W.N. Hardy, S. Adachi, C. Proust, L. Taillefer, Nature 450 (2007) 533; D. LeBoeuf, N. Doiron-Leyraud, B. Vignolle, M. Sutherland, B.J. Ramshaw, J. Levallois, R. Daou, F. Laliberté, O. Cyr-Choiniere, J. Chang, Y.J. Jo, L. Balicas, R. Liang, D.A. Bonn, W.N. Hardy, C. Proust, L. Taillefer, http://arXiv:1009.2078; L. Taillefer, J. Phys.: Condens. Matter 21 (2009) 164212; N. Doiron-Leyraud, S. Lepault, O. Cyr-Choinie‘re, B. Vignolle, G. Grissonnanche, F. Laliberté, J. Chang, N. Barišić, M.K. Chan, L. Ji, X. Zhao, Y. Li, M. Greven, C. Proust, L. Taillefer, Phys. Rev. X 3 (2013) 021019. [50] H. Koizumi, R. Hidekata, A. Okazaki, M. Tachiki, J. Supercond. Nov. Magn http://dx.doi.org/10.1007/s10948-013-2277-2. [51] E.A. Silinsh, M.V. Kurik, V. Čapek, Electronic Processes in Organic Molecular Crystals, Localization and Polarization Phenomena, Zinatne, Riga, 1988. [52] N.F. Mott, Metal-Insulator Transitions, Taylor and Francis, London, 1990. [53] H. Böttger, V.V. Bryksin, Hopping Conduction in Solids, Akademie-Verlag, Berlin, 1985. [54] A.I. Anselm, Introduction to Semiconductor Theory, Izd. Fiz. Math. Literature, Moscow, Leningrad, 1962. [55] B.K. Ridley, Quantum Processes in Semiconductors, MIR, Moscow, 1986. [56] I.M. Tsidilkovski, Electrons and Holes in Semiconductors, Nauka, Moscow, 1972. [57] H. Ding, T. Yokoya, J.C. Campuzano, T. Takahashi, M. Randeria, M.R. Norman, T. Mochiku, K. Kadowaki, J. Giapintzakis, Nature 382 (1996) 51; A.G. Loeser, Z.-X. Shen, D.S. Dessau, D.S. Marshall, C.H. Park, P. Fournier, A. Kapitulnik, Science 273 (1996) 325; A.V. Puchkov, D.N. Basov, T. Timusk, J. Phys.: Condens. Matter 8 (1996) 10049. [58] V.J. Emery, S.A. Kivelson, O. Zachar, Phys. Rev. B 56 (1997) 6120; S. Dzhumanov, Superlattices Microstruct. 21 (1997) 363. [59] Ch. Lushchik, I. Kuusmann, E. Feldbach, F. Savikhin, I. Bitov, J. Kolk, T. Leib, P. Liblik, A. Maaroos, I. Meriloo, Proc. Inst. Phys. Acad. Sci. Est. SSR 63 (1987) 137. [60] J.P. Lu, Q. Si, Phys. Rev. B 42 (1990) 950. [61] T. Timusk, D.B. Tanner, in: D.M. Ginsberg (Ed.), Physical Properties of High Temperature Superconductors I, MIR, Moscow, 1990 (Chapter 7). [62] R.E. Cohen, W.E. Pickett, H. Krakauer, Phys. Rev. Lett. 62 (1989) 831; M. Reedyk, T. Timusk, Phys. Rev. Lett. 69 (1992) 2705. [63] A. Carrington, D.J.C. Walker, A.P. Mackenzie, J.R. Cooper, Phys. Rev. B 48 (1993) 13051. [64] K.Q. Ruan, Q. Cao, S.Y. Li, G.G. Qian, C.Y. Wang, X.H. Chen, L.Z. Cao, Physica C 351 (2001) 402. [65] F. Walz, J. Phys.: Condens. Matter 14 (2002) R285.

32

S. Dzhumanov et al. / Physica B 440 (2014) 17–32

[66] N.I. Kashirina, V.D. Lakhno, V.V. Sychyov, Phys. Status Solidi B 239 (2003) 174. [67] S.V. Varyukhin, A.A. Zakharov, Physica C 185–189 (1991) 975. [68] D. Mihailovic, V.V. Kabanov, K. Žagar, J. Demsar, Phys. Rev. B 60 (1999) R6995; Y. Toda, T. Mertelj, P. Kusar, T. Kurosawa, M. Oda, M. Ido, D. Mihailovic, Phys. Rev. B 84 (2011) 174516. [69] T. Ekino, S. Hashimoto, H. Fujii, J. Hori, N. Nakamura, T. Fujita, Physica C 357–360 (2001) 158.

[70] R.V. Vovk, N.R. Vovk, O.V. Shekhovtsov, I.L. Goulatis, A. Chroneos, Supercond. Sci. Technol. 26 (2013) 085017. [71] S. Tajima, J. Schützmann, S. Miyamoto, I. Terasaki, Y. Sato, R. Hauff, Phys. Rev. B 55 (I) (1997) 6051. [72] T. Fujii, Dr. Thesis, Department of Applied Physics, Faculty of Science (Science University of Tokyo), March 2001. [73] V.N. Zverev, D.V. Shovkun, JETP Lett. 72 (2000) 73.