Electrostatic Potential in High-Temperature Superconducting Cuprates: Extended Ginzburg-Landau Theory

Abstract

The electrostatic potential and the associated charge distribution in the vortices of high-𝑇_𝑐 superconductors involving mixed symmetry state of the order parameters have been studied. The work is carried out in the framework of an extended Ginzburg-Landau (GL) theory involving the Gorter-Casimir two-fluid model and Bardeen's extension of GL theory applied to the high-𝑇_𝑐 superconductors. The properties are calculated using the material parameters relevant for the high-𝑇_𝑐 cuprate YBCO.

1. Introduction

A key feature which characterizes the vortices in any type-II superconductor is the ability of the vortex to support a magnetic flux, with the magnetic flux quantum being defined as Φ₀=ℎ𝑐/2𝑒. However, what is less known and came into light only recently [1–3] is the fact that the vortices, along with supporting a magnetic flux quanta, can support an accumulation of finite electric charge in it. The accumulation of charge in the vortices is an artifact of the difference between the chemical potential inside the vortex core and of the region outside the core. The presence of diamagnetic electric current in any superconductor gives rise to inhomogeneity, as a result of which there arises an internal electric field in the superconductor, which maintain the neutrality of the charge distribution in the system and thus a constant electrochemical potential in the superconductor. The presence of an internal electric field in a superconductor with a stationary current was first discussed by Bopp [4] and such an electric field was later measured by a number of experiments [5, 6]. A uniform current in a superconductor results in an electric field analogous to the Bernoulli pressure variation associated with the nonuniform flow of a classical fluid [7]. The corresponding electric potential is thus termed as the Bernoulli potential [8] and is profoundly influenced by the band structure of the superconducting material [9, 10]. The experimentally observed Hall anomaly in high-temperature superconductors has been attributed to the charge accumulation in the vortex core. In case of the high-𝑇_𝑐 cuprates, it has been observed that there is a sign reversal of the flux flow Hall coefficient below 𝑇_𝑐 [11]. It has been suggested that this Hall anomaly of high-temperature superconducting cuprates is universal and is dependent on the doping of the material, with a reversal of sign of the Hall coefficient from the overdoped to the underdoped regime. Kumagai et al. [12] have for the first time experimentally studied the accumulation of charge in the vortex cores in high-𝑇_𝑐 superconductor by using high-resolution measurements of the nuclear quadrupole frequency which is sensitive to the local charge density. The behavior of the vortex core and the charge accumulation in the core in high-𝑇_𝑐 cuprate are different from that of the conventional type-II superconductors owing to the complicated Fermi surface architecture in these materials. A theoretical model is thus required which can be used to study the electrostatic potential and the associated charge distribution in the votices of the high-𝑇_𝑐 superconducting cuprates. In the present work, we attempt to do so in the framework of a phenomenological model. Such an approach has been used to study the electrostatic potential in conventional type-II superconductors [13, 14], but for the high-𝑇_𝑐 superconducting cuprates such a study is still lacking.

For developing the theoretical model, it is important to discuss the pairing state symmetry of the high-𝑇_𝑐 superconducting cuprates. In spite of the conflicting evidence regarding the pairing state symmetry of high-temperature superconducting cuprates over the past decades, a consensus could now be reached that even in case of a strictly tetragonal system [15, 16], the high-temperature superconducting cuprates possess a mixed symmetry state of the order parameter components (for details see [17]). The mixed pairing state symmetry is characterized by the presence of a dominant 𝑑-wave order parameter component along with a subdominant 𝑠-wave order parameter component [18–20]. It has been found that the properties of such system can be theoretically explained by allowing for two or more order parameter components and their derivative mixing terms in the free energy density functional [17, 21–25]. In the present work, a similar two-order parameter Ginzburg-Landau (GL) theory involving mixed symmetry state of the order parameter components is used to study the electrostatic properties of the high-𝑇_𝑐 superconducting cuprate. The other important aspect of the present work is the extension of the GL theory to the low temperature regime. The GL theory is known to be applicable to the temperature regions near the critical temperature (𝑇_𝑐); however, it has been observed that even at temperatures much below the 𝑇_𝑐  (i.e., 𝑇≈(2/3)𝑇_𝑐) the GL theory provides very good result. At temperature lower than this, inaccuracies tend to set in the result obtained by employing the GL model. Extension of the present two-order parameter GL theory to the low temperature regime has been carried out in a manner parallel to the Bardeen's extension of the conventional GL theory [13, 14, 26]. The purpose of this paper is thus twofold, firstly it gives an extension of the two-order parameter GL theory corresponding to the high-𝑇_𝑐 superconducting cuprates involving mixed symmetry state of the order parameter components to the low temperature regime, secondly it discusses about the electrostatic potential and charge distribution in the high-𝑇_𝑐 superconducting system.

The rest of the paper is organized as follows: in Section 2 the theoretical formulation of the problem is discussed; Section 3 deals with the results obtained regarding the magnetic properties of the system and their analysis, while in Section 4 the details regarding the electrostatic properties have been discussed. Finally we conclude in Section 5 with the suggestion for future works.

2. Extended Ginzburg-Landau Model for High-𝑇_𝑐 Cuprates

A second-order transition such as the normal-superconductor phase transition can be described by Gorter-Casimir two-fluid model [27]. In case of high-temperature superconductors involving mixed symmetry state of the order parameters, the 𝑑- and 𝑠-wave order parameter components can be expressed in terms of superconducting fractions 𝜔_𝑑 and 𝜔_𝑠, respectively. In presence of the superconducting electrons the normal-state free energy density is modified as𝑓_𝑠=𝑈−𝜖_cond𝜔_𝑑+𝜖_cons𝜔_𝑠−12𝛾𝑇²1+𝜔_𝑠−12𝛾𝑇²1−𝜔_𝑑−12𝛽^𝑠𝑑𝛾𝑇²1+𝜔_𝑠1−𝜔_𝑑.(1) The superconducting state is an ordered state. The transition of the electrons from the disordered normal state to the ordered superconducting state will give out a certain condensation energy which is expressed in (1) as 𝜖_cond𝜔_𝑑 and 𝜖_cons𝜔_𝑠 for the 𝑑- and 𝑠-wave type superconducting electrons, respectively. As can be seen from (1), in the present study the condition of stable 𝑑-wave configuration in the bulk has been considered, with a single transition temperature 𝑇_𝑑. The 𝑑-wave pairing interaction is considered to be attractive, while for the 𝑠-wave a repulsive interaction is considered [28, 29]. In (1) 𝑈 is the internal energy of the system, while 𝛾 is the linear coefficient of specific heat. The terms (1/2)𝛾𝑇²√1+𝜔_𝑠 and (1/2)𝛾𝑇²√1−𝜔_𝑑 correspond to the reduction in the entropy of the system due to the ordering of the electrons in the superconducting state. The term (1/2)𝛽_𝑠𝑑𝛾𝑇²√1+𝜔_𝑠√1−𝜔_𝑑 corresponds to the contribution arising due to the interaction between the 𝑠- and 𝑑-wave order parameter components. The coefficient 𝛽_𝑠𝑑 has been kept as a variable so as to understand the effect of the interaction term. At the critical temperature 𝑇_𝑑, the ordering of the superconducting electron vanishes and the system return to the normal state, thus we can write𝜖_con=𝜖_cons=𝜖_cond=14𝛾𝑇^2𝑑.(2) In terms of the total electron density 𝑛, the superconducting fractions are defined as 𝜔_𝑠=2|𝑠|²/𝑛 and 𝜔_𝑑=2|𝑑|²/𝑛, respectively, with 𝑛=(2(|𝑠|²+|𝑑|²)+𝑛_𝑛), where 𝑛_𝑛 is the normal state electron density. Equation (1) gives the condensation energy density of the system.

We next write the contribution of the kinetic energy of the superconducting condensate towards the total free energy density of the system. In presence of the mixed symmetry state of the order parameter components, the kinetic energy density contribution can be written as 𝑓_kin=𝛾_𝑠||𝚷𝑠||²+𝛾_𝑑||𝚷𝑑||²+𝛾_𝑣Π_𝑦𝑠^∗Π_𝑦𝑑−Π_𝑥𝑠^∗Π_𝑥𝑑+𝑐.𝑐,(3) where 𝚷=(−𝑖∇−2𝑒𝐀/ℏ𝑐) and 𝛾_𝑖 is related to the effective electronic masses as 𝛾_𝑖=ℏ²/2𝑚^∗𝑖 with 𝑖=𝑠,𝑑, and 𝑣. The above expression consists of three terms. The first term with the coefficient 𝛾_𝑠 corresponds to the contribution of the 𝑠-wave order parameter component to the kinetic energy of the system, the second term with the coefficient 𝛾_𝑑 gives the contribution of the 𝑑-wave order parameter component. Finally, the third term with the coefficient 𝛾_𝑣 is the mixed gradient coupling term and combines the gradient contributions of the 𝑑- and 𝑠-wave type order parameter components. The mixed gradient coupling term is the most important term regarding the generation of 𝑠-wave order parameter component in the system. Previous theoretical studies of high-𝑇_𝑐 superconductors carried out in the framework of two-order parameter GL theory have shown that properties of these materials are significantly affected by the admixture of 𝑠-wave order parameter component and the decisive role regarding the contribution of the 𝑠-wave order parameter component in the system is played by the mixed gradient coupling term.

It has been observed through the linear stability analysis that with the vanishing of the mixed gradient coupling term, the bulk 𝑑-wave solution is stable against the admixture of the 𝑠-wave order parameter component [16]. Thus, it can be said that in case of the standard two-order parameter GL model, the higher order coupling terms, namely |𝑠|²|𝑑|² does not give rise to any significant contribution towards the admixture of the 𝑠-wave order parameter component. It must be noted that in case of the extended GL theory the model deals with temperatures much below the critical temperature where the density of the superconducting electrons gets enhanced. In order to verify the relative contribution of the 𝑑- and 𝑠-wave type superconducting electrons, the average value of the order parameter components has been calculated at different temperatures and magnetic field inductions. Irrespective of presence or absence of the interaction term with coefficient 𝛽_𝑠𝑑, the magnitude of the 𝑠-wave order parameter component is found to be significantly smaller than that of the 𝑑-wave order parameter component. Even at a low temperature of 𝑡=0.5 and very low magnetic field, it has been observed that the relative magnitude of the 𝑑- and 𝑠-wave contribution does not change significantly with the presence or absence of the coupling term. This indicates the fact that even at low temperatures the principal contribution towards the 𝑠-wave order parameter component arises from the mixed gradient coupling term only. The observation is presented in Figure 1 for a significant value of the coupling term 𝛽_𝑠𝑑=0.5, and it can be seen that over the wide range of magnetic field induction and specially at lower magnetic fields (where the superconducting electron density is enhanced), the relative magnitude of the 𝑑- and 𝑠-wave order parameter components remains unchanged for the presence and absence of the interaction term. The observation justifies the approximation that for studying the properties of the high-𝑇_𝑐 superconductors in the framework of extended two-order parameter GL theory, it is sufficient to consider the contribution of the 𝑠-wave order parameter component arising from the mixed gradient coupling term only and thus for the remaining part of the present study one can set 𝛽_𝑠𝑑=0.

Figure 1 

Variation of relative magnitude of 𝑑- and 𝑠-wave order parameter components ⟨𝜔𝑑⟩/⟨𝜔𝑠⟩ with the coupling parameter 𝛽𝑠𝑑 at different magnetic field inductions 𝑏. Other relevant parameters are 𝑡=𝑇/𝑇𝑑=0.5, 𝜅=72, 𝛾𝑣/𝛾𝑑=𝜖𝑣=0.1.

Next the contribution of the electrostatic potential towards the free energy density of the system is discussed. The corresponding Coulomb energy as a part of the free energy density is expressed as𝑓_𝑐=𝜙𝜌−12𝜖⁰||∇𝜙||²,(4) where the total charge density is defined as 𝜌=𝑒𝑛+𝜌_lattice. In this expression, 𝜌_lattice represents the lattice charge density. The electrostatic potential is determined from the Coulomb interaction and can be given as𝜙(𝐫)=𝑑𝐫^14𝜋𝜖||𝐫−1𝐫^||𝜌𝐫^(5) or in its differential form by the Poisson equation−𝜖∇²𝜙=𝜌.(6)

Finally, we take into account the contribution of the magnetic energy towards the free energy density of the system. For the applied magnetic field 𝐵_𝑎 the resulting magnetic energy density is given as𝑓_𝑀=12𝜇₀𝐁−𝐁_𝑎²=81𝜋𝐁−𝐁^𝑎²,(7) where 𝜇₀=4𝜋. Using (1)–(7), the resulting free energy density can be expressed as𝑓=𝑈+14𝛾𝑇^2𝑑2|𝑠|²𝑛−41𝛾𝑇^2𝑑2||𝑑||²𝑛−12𝛾𝑇²1+2|𝑠|²+𝛾_𝑠𝑛||𝚷𝑠||²+𝛾_𝑑||𝚷𝑑||²−12𝛾𝑇²1−2||𝑑||²−1𝑛2𝛾𝑇²𝛽_𝑠𝑑1+2|𝑠|²𝑛1−2||𝑑||²+𝛾_𝑣𝑛+𝜙𝑒𝑛Π+_𝑦𝑠𝜌^∗Π_𝑦lattice𝑑−Π_𝑥𝑠^∗Π_𝑥𝑑+𝑐.𝑐−21𝜖⁰||∇𝜙||²+81𝜋𝐁−𝐁^𝑎²,(8) with 𝑛=(2|𝑠|²+2|𝑑|²+𝑛_𝑛) being the total electron density. Equation (8) gives the extended two-order parameter Ginzburg-Landau (GL) free energy density for the high-𝑇_𝑐 superconductors involving mixed symmetry state of the order parameters. Near the critical temperature (𝑇_𝑑) the equation reduces to the standard two-order parameter Ginzburg-Landau free energy density functional, consisting of higher-order interaction terms of the order parameter components. The total free energy is a function of the order parameter components 𝑠 and 𝑑, the vector potential 𝐀, and the normal state electron density 𝑛_𝑛. The other physical quantities, namely, 𝐁, 𝜙, 𝑛, 𝜌, and so forth are to be understood in terms of 𝑠, 𝑑, 𝐀, and 𝑛_𝑛. The variables are dependent on the material parameter 𝑈, 𝑇_𝑑, 𝛾, 𝑚^∗𝑠=𝑚^∗𝑑, 𝜌_lattice, 𝜖₀, and 𝑛. Since the density dependence of the material parameters lead to significant corrections, we assume that 𝑈, 𝑇_𝑑, and 𝛾 are dependent on the density 𝑛. However, 𝜖₀, 𝑚^∗𝑠=𝑚^∗𝑑, 𝜌_lattice, and so forth are taken as constants.

We now proceed to determine the equations of motion from the extended GL-free energy density functional. On minimizing (8) with respect to the electrostatic potential 𝜙, the order parameters (𝑠 and 𝑑), the vector potential 𝐀, and the normal state electron density 𝑛_𝑛, we get a complete set of equations. Equations (9)–(13) comprising of the Poisson equation, GL equations corresponding to the 𝑑- and 𝑠-wave order parameter components, Ampere's law and Bernoulli potential, respectively, give the complete set of equations corresponding to the high-𝑇_𝑐 superconductors involving mixed symmetry state of order parameters in presence of electrostatic potential, −𝜖₀∇²𝜙0==𝑒𝛾_𝑑𝑛_𝑛+2|𝑠|²+2||𝑑||²+𝜌^lattice,(9)+⎛⎜⎜⎜⎝−Π𝛾²𝑇𝑑_2𝑑+𝛾_𝑣Π^2𝑦−Π^2𝑥𝑠2𝑛+11−2||𝑑||²/𝑛𝛾𝑇²2𝑛+𝛾2𝑇𝑛²𝛽^𝑠𝑑×√1+2|𝑠|²/𝑛0=𝛾_𝑠1−2||𝑑||²/𝑛⎞⎟⎟⎟⎠𝑑,(10)+⎛⎜⎜⎜⎝𝛾Π𝑇²_2𝑑𝑠+𝛾_𝑣Π^2𝑦−Π^2𝑥𝑑−2𝑛𝛾𝑇−₂√1+12|𝑠|²/𝑛𝛾2𝑇𝑛²2𝑛𝛽^𝑠𝑑1−2||𝑑||²/𝑛∇√1+2|𝑠|²/𝑛⎞⎟⎟⎟⎠𝑠,(11)₂𝐀=4𝜋−ℏ𝑒^∗𝑚^∗𝑠𝑠^∗+−ℏ𝑒_∗(𝚷𝑠)+𝑠(𝚷𝑠)^∗𝑚^∗𝑑𝑑^∗+−ℏ𝑒_∗(𝚷𝑑)+𝑑(𝚷𝑑)^∗𝑚^∗𝑣𝑦̂𝑠^∗Π_𝑦−𝑥̂𝑠_∗𝑑+𝑑Π_𝑦𝑠^∗+𝑐.𝑐𝑒𝜙=𝜆_2TFΠ_𝑥𝑑+𝑑Π_𝑥𝑠^∗+𝑐.𝑐,(12)∇²𝑒𝜙−1𝑛𝛾^𝑑−1𝑑^∗Π²𝑑+𝛾_𝑣𝑑^∗Π^2𝑦−Π^2𝑥𝑠𝑛𝛾^𝑠−|𝑠|₂𝑠^∗Π²𝑠+𝛾_𝑣𝑠^∗Π^2𝑦−Π^2𝑥𝑑−|2𝑠𝑛|₂𝑇^2𝑑𝜕𝜕𝛾𝑛+||2𝑑𝑛||²𝑇^2𝑑𝜕𝜕𝛾𝑛+𝑇𝑛𝛾𝑇^𝑑𝜕𝜕𝑇𝑛^𝑑+||𝑑𝑛||²𝛾𝑇^𝑑𝜕𝜕𝑇𝑛^𝑑₂+𝑇2₂𝜕𝜕𝛾𝑛1+2|𝑛𝑠|²×𝛽_𝑠𝑑2𝜕𝜕𝛾𝑛⎷1−2||𝑛𝑑||²+𝑇2²𝜕𝜕𝛾𝑛1+2|𝑛𝑠|²1−2||𝑛𝑑||²,(13) where the contribution 𝜆²_TF∇²𝑒𝜙 arises due to the Thomas-Fermi screening. In the above equations, we have used the effective potential 𝜒=𝜒_𝑑+𝜒_𝑠 acting on the 𝑠- and 𝑑-wave type electrons. The effective potentials corresponding to 𝑑-and 𝑠-wave type electrons are defined as 𝜒_𝑑=−2𝑛𝜖^con+𝛾𝑇²2𝑛11−2||𝑑||²/𝑛+𝛾𝑇²2𝑛𝛽^𝑠𝑑√1+2|𝑠|²/𝑛𝜒1−2||𝑑||²/𝑛_𝑠,=2𝑛𝜖^con−𝛾𝑇²2𝑛√1+12|𝑠|²/𝑛−𝛾2𝑇𝑛²𝛽^𝑠𝑑1−2||𝑑||²/𝑛√1+2|𝑠|²/𝑛.(14) In the above set of equations, the basic material parameters are 𝛾, 𝑇_𝑑, 𝑚^∗𝑠=𝑚^∗𝑑, 𝜕𝑇_𝑑/𝜕𝑛, and 𝜕𝛾/𝜕𝑛.

3. Magnetic Properties of the System

We begin the discussion by studying the magnetic properties of high-temperature superconductors involving mixed symmetry state of the order parameters. In terms of reduced unit, the quantities used in the calculation are expressed as 𝑡=𝑇𝑇_𝑑𝐁_,=𝜆^Lon𝐁𝜆₀√𝐵_𝑐𝐵₀𝐀_,=𝐀𝜆₀√𝐵_𝑐𝐵₀𝐫_,=𝐫𝜆_Lon𝜆_Lon,=𝜆⁰√1−𝑡⁴𝜅(𝑡)=𝜅₀,21+𝑡²𝐵_𝑐,𝐵_𝑐2(𝑡)=𝐵₀1−𝑡²,(𝑡)=√2𝜅𝐵_𝑐,(15) where 𝐵₀=𝑇_𝑑√𝜇₀𝛾/2, 𝜆₀=√𝑚_𝑑/𝑒²𝑛𝜇₀, and 𝜅₀=(𝑚_𝑑𝑇_𝑑/𝑛𝑒ℏ)√𝛾/𝜇₀. The extended GL free energy density functional is then expressed in terms of the gauge invariant real quantities 𝜔_𝑑=2|𝑑|²/𝑛(1−𝑡⁴), 𝜔_𝑠=2|𝑠|²/𝑛(1−𝑡⁴), and 𝐐, where 𝑠(𝑥^,𝑦^)=√𝜔_𝑠(𝑥^,𝑦^)𝑒^{𝑖𝜙𝑠(𝑥′,𝑦′)} and 𝑑(𝑥^,𝑦^)=√𝜔_𝑑(𝑥^,𝑦^)𝑒^{𝑖𝜙𝑑(𝑥′,𝑦′)} correspond to the order parameter components, while 𝐐^(𝑥^,𝑦^)=𝐀^(𝑥^,𝑦^)−∇^𝜙(𝑥^,𝑦^)/𝜅 is the velocity of the superconducting electrons. The corresponding two-dimensional free energy density in terms of dimensionless unit can be thus written as𝑓^=−−𝜔^𝑠+𝜔_𝑑1−𝑡²−2𝑡²1+𝜔_𝑠1−𝑡⁴1−𝑡²1−𝑡⁴−2𝑡²×1−𝜔_𝑑1−𝑡⁴1−𝑡²1−𝑡⁴−2𝑡²×𝛽_𝑠𝑑1+𝜔_𝑠1−𝑡⁴1−𝜔_𝑑1−𝑡⁴+𝑔_𝑠1−𝑡²1−𝑡⁴+𝜔_𝑠𝑄^2+𝑔_𝑑+𝜔_𝑑𝑄^2+𝜖_𝑣2cos(𝜙)∇^𝑦𝜔_𝑠∇^𝑦𝜔_𝑑4𝜅²𝜔_𝑠𝜔_𝑑^1/2−∇^𝑥𝜔_𝑠∇^𝑥𝜔_𝑑4𝜅²𝜔_𝑠𝜔_𝑑^1/2+𝑄^𝑦2−𝑄^𝑥2𝜔_𝑑𝜔_𝑠^1/2+2sin(𝜙)𝑄_𝑦∇^𝑦𝜔_𝑠2𝜅−𝑄^𝑥∇^𝑥𝜔_𝑠2𝜅𝜔𝜔^𝑑_𝑠^1/2−2sin(𝜙)𝑄^𝑦∇^𝑦𝜔_𝑑2𝜅−𝑄^𝑥∇^𝑥𝜔_𝑑×𝜔2𝜅_𝑠𝜔_𝑑^1/2+∇^×𝐐^−𝐵^𝑎²^,(16) where 𝐵^𝑎 is the applied magnetic field and 𝑓^=𝑓/((1/4)𝛾𝑇^2𝑑(1−𝑡²)(1−𝑡⁴)). For studying the magnetic properties of the system, the contribution of the Coulomb energy and the internal energy to the free energy density functional is neglected. In the above equation, 𝜖_𝑣 gives the strength of the 𝑠-wave order parameter component in the system and is defined as 𝜖_𝑣=𝛾_𝑣/𝛾_𝑑. We now discuss about the selection of the parameters in this work. Theoretical studies carried out on high-𝑇_𝑐 superconductors involving mixed pairing state symmetry of order parameters suggested the value of mixed gradient coupling parameter to be 𝜖_𝑣=𝛾_𝑣/𝛾_𝑑≈0.1−0.4 [30]. Further, it was also observed that with 𝜖_𝑣=0.1 the theoretical results obtained in the framework of two-order parameter GL theory were in excellent agreement with the experimental data corresponding to YBa₂Cu₃O_7 − δ [17, 22, 24, 25]. Thus, for the present study the mixed gradient coupling parameter 𝜖_𝑣 has been chosen to be 𝜖_𝑣=0.1. The effects of the higher values of the coupling parameter 𝜖_𝑣 have also been verified, so as to understand the influence of the admixture of 𝑠-wave order parameter component in the system at various temperatures. The results obtained for different values of 𝜖_𝑣 have been found to be qualitatively the same, with difference in magnitude depending upon the amount of admixture of the 𝑠-wave order parameter component in the system. The effects of the admixture of 𝑠-wave order parameter component on the various properties of the high-𝑇_𝑐 cuprates, namely, the vortex lattice structure, local spatial behavior of the order parameter and magnetic field profiles, reversible magnetization of the system, and shear modulus of the flux line lattice and so forth, studied in the framework of the standard two-order parameter GL model, have already been reported in the literature [17, 22, 24, 25].

The three GL equations obtained by minimizing the free energy density functional equation (16) with respect to the order parameter components 𝜔_𝑠, 𝜔_𝑑 and supervelocity 𝐐^ as, 𝛿𝑓^(𝐫^)/𝛿𝜔_𝑠(𝐫^) = 𝛿𝑓^(𝐫^)/𝛿𝜔_𝑑(𝐫^) = 0 = 𝛿𝑓^(𝐫^)/𝛿𝐐^(𝐫^) can be written as−∇^2𝜔_𝑑=2𝜅²𝜔^𝑑−𝑠_𝑑−𝑔_𝑑−𝜔_𝑑𝑄^2−𝜖_𝑣×cos−(𝜙∇)_𝑦2𝜔_𝑠2𝜅²+∇^𝑥2𝜔_𝑠2𝜅²+𝑔_𝑠𝑦−𝑔_𝑠𝑥×𝜔_𝑑𝜔_𝑠^1/2+𝑄^𝑦2−𝑄^𝑥2𝜔_𝑠𝜔_𝑑^1/2+2sin(𝜙)2𝜅𝜔𝜔^𝑑_𝑠^1/2−∇_2×𝑄^𝑦∇^𝑦𝜔_𝑠−𝑄^𝑥∇^𝑥𝜔_𝑠,(17)𝜔_𝑠=2𝜅²−𝜔^𝑠−𝑠_𝑠−𝑔_𝑠−𝜔_𝑠𝑄^2−𝜖_𝑣×cos−(𝜙∇)_𝑦2𝜔_𝑑2𝜅²+∇^𝑥2𝜔_𝑑2𝜅²+𝑔_𝑑𝑦−𝑔_𝑑𝑥×𝜔_𝑠𝜔_𝑑^1/2+𝑄^𝑦2−𝑄^𝑥2𝜔_𝑠𝜔_𝑑^1/2+2sin(𝜙)2𝜅𝜔𝜔^𝑠_𝑑^1/2−∇_2×𝑄^𝑦∇^𝑦𝜔_𝑑−𝑄^𝑥∇^𝑥𝜔_𝑑,(18)𝐐^=−𝜔_𝑠+𝜔_𝑑𝐐^−𝜖_𝑣2cos(𝜙)𝜔^𝑠𝜔_𝑑^1/2𝑦𝑄̂^𝑦−𝑥̂𝑄^𝑥+sin(𝜙)×𝑦̂∇_𝑦2𝜅𝜔_𝑠−𝑥̂∇^𝑥𝜔_𝑠𝜔^𝑑𝜔_𝑠^1/2−sin(𝜙)2𝜅𝑦̂∇^𝑦×𝜔𝜔_𝑑−𝑥̂∇^𝑥𝜔_𝑑_𝑠𝜔_𝑑^1/2,(19) where 𝑔_𝑖=(∇^𝜔_𝑖)²/4𝜅²𝜔_𝑖 and 𝑔_𝑖𝑗=(∇^𝑗𝜔_𝑖)²/4𝜅²𝜔_𝑖, with 𝑖=𝑠,𝑑; 𝑗=𝑥,𝑦; and𝑠_𝑠=𝑡²1−𝑡²⎛⎜⎜⎜⎝1−11+𝜔_𝑠1−𝑡⁴−𝛽^𝑠𝑑1−𝜔_𝑑1−𝑡⁴1+𝜔_𝑠1−𝑡⁴⎞⎟⎟⎟⎠𝜔^𝑠𝑠_𝑑,=𝑡²1−𝑡²⎛⎜⎜⎜⎝11−𝜔_𝑑1−𝑡⁴−1+𝛽^𝑠𝑑1+𝜔_𝑠1−𝑡⁴1−𝜔_𝑑1−𝑡⁴⎞⎟⎟⎟⎠𝜔^𝑑.(20)

The order parameters and the magnetic field are now expressed in terms of Fourier series [22], and the positions of the vortices are defined in terms of the reciprocal lattice vectors as 𝐊=𝐊_𝑚𝑛=(2𝜋/𝑥₁𝑦₂)(𝑚𝑦₂,𝑛𝑥₁+𝑚𝑥₂), where 𝑚 and 𝑛 are integers and 𝑥₁, 𝑥₂, and 𝑦₂ are lattice parameters. The order parameter components and the magnetic field are determined by solving a set of iterative equations using a numerical iteration technique [17, 22–25]. The iteration process is continued till the solution remains constant upto 15 digits. The high precision solutions of the GL equations are thus obtained and these solutions can be used to study the various magnetic and electrostatic properties of the high-𝑇_𝑐 superconductors involving mixed symmetry state of the order parameter components at different temperatures.

3.1. Single Vortex and Vortex Lattice Structure

The first step to study the magnetic properties of the high-𝑇_𝑐 superconducting cuprates is to study the structure of the flux line lattice. Small angle neutron scattering (SANS) [31] and Scanning tunneling microscopy (STM) [32] experiments on high-𝑇_𝑐 superconductors have shown that unlike the case of the conventional type-II superconductors, which are characterized by a triangular vortex lattice, the high-𝑇_𝑐 superconductors exhibit an oblique vortex lattice configuration. The experimental observations have been substantiated by the previous theoretical works where an oblique vortex lattice structure have been observed [17]. Another important feature of the high-𝑇_𝑐  superconductors involving mixed symmetry state of the order parameters is the fourfold symmetric structure of the 𝑠-wave order parameter component [15–17]. Figure 2 shows the vortex lattice structure of the 𝑠-wave and 𝑑-wave order parameter components corresponding to different temperatures at a particular magnetic field induction and mixed gradient coupling parameter 𝜖_𝑣 mentioned in the figure caption. It can be seen from the figure that at different temperatures the structure of the vortex lattice is essentially oblique and the 𝑠-wave order parameter component possesses a fourfold symmetric structure. Similar oblique flux line lattice structure is observed for the magnetic field distribution also as can be seen from Figure 3.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2 

Variation of vortex lattice structure corresponding to the 𝑠- and 𝑑-wave order parameter components 𝜔𝑠(𝑥,𝑦) ((a)–(c)) and 𝜔𝑑(𝑥,𝑦) ((d)–(f)) for different temperatures (𝑡=0.9,0.7,0.5), respectively, calculated by the extended two-order parameter GL theory. Other parameters used for the calculation are GL parameter 𝜅=72, magnetic field induction 𝑏=0.8, and mixed gradient coupling parameter 𝜖𝑣=0.1, 𝛽𝑠𝑑=0.

(a)

(b)

(c)

Figure 3 

Variation of magnetic field distribution 𝐵(𝑥,𝑦) for different temperatures (𝑡=0.9,0.7,0.5) ((a)–(c)) respectively, calculated by the extended two-order parameter GL theory. Other parameters used for the calculation are same as in Figure 2.

3.2. Reversible Magnetization

An important and experimentally determinable quantity for any superconducting material is the reversible magnetization of the system. The reversible magnetization is defined as 𝑀=𝐵−𝐵_𝑎, where 𝐵_𝑎 is the equilibrium applied magnetic field and is expressed as 𝐵_𝑎=4𝜋(𝜕𝑓/𝜕𝐵). Thus, the determination of the equilibrium applied magnetic field of the system involves the computation of the numerical derivative of the free energy density functional which is a complex function of two-order parameter components and magnetic field. An alternative approach is to determine the equilibrium applied magnetic field by making use of the virial theorem applicable to the two-order parameter model [22]. In the framework of the two-order parameter extended GL theory, the virial theorem has been formulated and the resulting equilibrium applied magnetic field can be expressed as𝐵_𝑎=2𝐵²+𝜔_𝑑−𝜔_𝑠−𝑠_𝑑−𝑠_𝑠2𝐵.(21)

Using this expression for the equilibrium-applied magnetic field the corresponding reversible magnetization of the system can be calculated as per the relation 𝑀=𝐵−𝐵_𝑎. The reversible magnetization of the system calculated by this expression at different temperatures is plotted in Figure 4. It can be seen from the figure that for higher values of 𝑡, that is, at temperatures close to 𝑇_𝑑, the reversible magnetization shows the behavior well known for the GL theory. However, as one move to lower temperatures an anomalous behavior of the reversible magnetization is observed, characterized by an 𝑠-shape of the curve whose curvature increases with the decrease in temperature. In case of conventional type-II superconductors also the curvature has been found to increase with the decrease in the temperature and below a certain temperature termed as 𝑇_𝑎 the reversible magnetization shows an anomalous behavior and the system undergoes a first-order transition [13, 14]. At temperatures below 𝑇_𝑎, a finite magnetization is observed for an applied magnetic field above 𝐵_𝑐2. For the conventional type-II superconductors with 𝜅≈1/√2, a first-order transition has been predicted near the 𝐻_𝑐1 and 𝐻_𝑐2, depending upon the ratio of the mean free path 𝑙 to the coherence length 𝜉₀. For the conventional low temperature type-II superconductors, the anomalous behavior of the reversible magnetization of the system has been attributed to the presence of impurities in the superconducting system [33, 34].

Figure 4 

Applied magnetic field 𝐵𝑎 (measured in units of upper critical field 𝐵𝑐2) dependence of reversible magnetization calculated by the extended two-order parameter GL theory for different temperatures. Other parameters used in the calculation are GL parameter 𝜅=72 and mixed gradient coupling parameters 𝜖𝑣=0.1 and 𝛽𝑠𝑑=0.

In case of the high-temperature superconductors involving mixed symmetry state of the order parameters, such a first-order transition has not been observed for the temperatures plotted in Figure 4. The absence of such a transition can be attributed to the large value of the GL parameter 𝜅 in case of the high-𝑇_𝑐 superconductors. Before coming to this point, it will be useful to determine the temperature 𝑇_𝑎 in case of the high-𝑇_𝑐 superconductors. Near the upper critical field 𝐵_𝑐2 the density of superconducting condensate is small and the coefficients of the GL free energy functional can be defined as𝛼=𝛾2𝑛𝑇²𝛽=𝛾𝑇−₂𝑇^2𝑑,2𝑛².(22) We now define an asymptotic GL parameter 𝜅_𝑎𝑠 as [35]𝜅_𝑎𝑠=⎷𝑚^2𝑑𝛽2𝜇₀ℏ²𝑒².(23) Thus, the asymptotic GL parameter 𝜅_𝑎𝑠 is related to temperature as per the relation𝜅_𝑎𝑠=𝜅⁰𝑡2.(24)

The first-order transition should be observed for 𝜅_𝑎𝑠=1/√2 and with 𝜅₀=72 corresponding to high-𝑇_𝑐 superconductor YBa₂Cu₃O_7 − δ, the transition temperature (𝑇_𝑎) amounts to𝑇_𝑎=√2𝜅₀𝑇_𝑑=0.01964.(25)

Thus, in case of high-𝑇_𝑐 superconductors (particularly YBa₂Cu₃O_7 − δ) such a transition is likely to be observed at a very low temperature and the range of validity of the extended GL theory for high-𝑇_𝑐 superconductors is larger as compared to that for the conventional type-II superconductors. This justifies the observation that in Figure 4 a finite reversible magnetization is not observed at applied magnetic fields higher than 𝐵_𝑐2 even at a low temperature of 𝑡=0.4. One may possibly define as in the case of the applied magnetic field (𝐻_𝑐1<𝐻<𝐻_𝑐2), a range of temperature as 𝑇_𝑎<𝑇<𝑇_𝑐, over which the extended GL theory is valid for a type-II superconducting material. Below this temperature 𝑇_𝑎, one should be careful regarding the validity of the extended GL theory and the accuracy of the results obtained.

4. Electrostatic Potential and Charge Distribution in Flux Line Lattice

We next concentrate on the electrostatic potential and the associated charge distribution in the vortices of the high-𝑇_𝑐 superconductors involving mixed symmetry state of the order parameter components.

4.1. Electrostatic Potential

The electrostatic potential of the system is expressed as per equation (13). For the sake of simplicity, in the present calculation of the electrostatic potential the Thomas-Fermi screening is neglected, that is, 𝜆²_TF∇²𝑒𝜙=0. Such an approximation is justified since the Thomas-Fermi screening length is small as has been shown below. In terms of dimensionless units, we define the electrostatic potential asΦ=𝑒𝑛(1/4)𝛾𝑇^2𝑑1−𝑡²1−𝑡⁴𝜙.(26) For the high-𝑇_𝑐 superconductors involving mixed symmetry state of the order parameter components, the resulting electrostatic potential isΦ=+𝜔𝐶_𝑠2−𝜔_𝑑+𝑠_𝑠+𝑠_𝑑+𝐶₁𝜔_𝑑−𝜔_𝑠+𝛽1_𝑠𝑑+𝜔_𝑠1−𝑡⁴+1−𝜔_𝑑1−𝑡⁴1+𝜔_𝑠1−𝑡⁴1−𝜔_𝑑1−𝑡⁴,(27) where the temperature-dependent coefficients 𝐶₁ and 𝐶₂ are given as𝐶₁=1𝐶1−𝑡²𝜕𝜕lnln𝜖𝑛^con₂,=2𝑡²1−𝑡²1−𝑡⁴𝜕𝜕llnn𝛾𝑛.(28) The coefficients 𝐶₁ and 𝐶₂ depend upon the material parameter of the system under consideration, in this case YBCO. Details regarding the determination of the coefficients 𝐶₁ and 𝐶₂ for YBCO are discussed in the appendix at the end of the paper.

The electrostatic potential expressed by (27) consists of three components as Φ₁=𝐶₁(𝜔_𝑑−𝜔_𝑠), which arises due to the dependence of the condensation energy on the superconducting electron density, Φ₂=𝐶₂(√1+𝜔_𝑠(1−𝑡⁴)+√1−𝜔_𝑑(1−𝑡⁴)+𝛽_𝑠𝑑√1+𝜔_𝑠(1−𝑡⁴)√1−𝜔_𝑑(1−𝑡⁴))arising from the reduced normal-state thermoelectric potential and Φ_𝐵=(𝜔_𝑠−𝜔_𝑑)+(𝑠_𝑠+𝑠_𝑑) corresponding to the Bernoulli potential. The individual potential components are shown in Figure 5 for the temperature 𝑡 and magnetic field induction 𝑏 mentioned in the figure caption. A characteristic feature that can be observed from the figure is the flatness of the curves corresponding to the different components of potential, this is, unlike the case observed for conventional type-II superconductor niobium [13, 14]. It is worth mentioning that Kumagai et al. [12] in their NQR measurement of vortex charge accumulation for YBa₂Cu₃O₇ and YBa₂Cu₄O₈ have observed a flat response of the vortex charge distribution. One can thus predict that the corresponding electrostatic potential should also show similar flatness in their profiles.

Figure 5 

The components of the electrostatic potential Φ plotted along the 𝑥-direction. The parameter values used for the figure are 𝑡=0.5, magnetic induction parameter 𝑏=0.5, and GL parameters 𝜅=72 and 𝛽𝑠𝑑=0.

The Bernoulli potential 𝜙_𝐵 reaches zero at the center of the vortex. The potential is repulsive inside the vortex core, while outside the core the Bernoulli potential is attractive. Similar to the Bernoulli potential 𝜙_𝐵, the 𝜙₁ component of electrostatic potential minimizes at the center of the vortex. This is an expected observation, since 𝜙₁ arises from the condensation energy and is thus proportional to the density of superconducting electrons. The density of superconducting electron density vanishes at the center of the vortex and so is the corresponding component of the electrostatic potential. The third component of the electrostatic potential 𝜙₂ is the only component which gives a nonzero contribution at the center of the vortex.

4.2. Electrostatic Charge Distribution

We next calculate the distribution of electrostatic charge in the flux line lattice. As per the Poisson's equation, the charge accumulation in the vortex core can be calculated from the electrostatic potential of the system as per the relation 𝜌=−𝜖∇²𝜙. We define charge in terms of dimensionless units as𝜌^=𝜌𝑒𝑛.(29) The corresponding Poisson's equation in terms of the dimensionless unit is given as𝜌^=−𝐶₃𝜆²_TF𝜆²_Lon∇^2Φ,(30) where the coefficient 𝐶₃ is dependent upon the material parameters of the superconducting system under consideration, in this case YBCO, and is expressed as𝐶₃=2𝐷𝜖^con𝑛²1−𝑡²1−𝑡⁴(31) with 𝐷 being the density of states. As in case of the electrostatic potential of the vortex lattice, the net electrostatic charge in the vortex lattice also consists of three parts as 𝜌^1, 𝜌^2, and 𝜌^𝐵 corresponding to the electrostatic potential components 𝜙₁, 𝜙₂, and 𝜙_𝐵, respectively.

The different components of charge are plotted in Figure 6, and the particular shape of the vortex charge profiles are strongly dependent on the material parameters of YBCO through the coefficients 𝐶₁, 𝐶₂, and 𝐶₃. At the center of the vortex, the components of charge 𝜌_𝐵 and 𝜌₂ give a positive contribution while the magnitude of the component 𝜌₁ is negative at the center of the vortex. The net charge at the center of the vortex has a finite negative value due to the large negative contribution arising from the component 𝜌₁. The total charge density (𝜌) shows a flat response outside the vortex. The behavior can be considered to be in agreement with the experimental observation reported by Kumagai et al. for YBCO [12].

Figure 6 

Different components of the charge distribution profile calculated by two-order parameter extended GL theory. The parameter values are the same as in Figure 5.

For the calculation of charge distribution in the vortex lattice, the effect of Thomas-Fermi screening has been taken into account. It must however be noted that the magnitude of screening is very small. In case of YBCO, this screening amounts to𝜆²_TF𝜆²_Lon=1.4646×10⁻⁷(32) with 𝜆²_TF=𝜖/2𝐷𝑒² and 𝜆_Lon=1.4×10⁻⁶m. Consequently, for YBCO the coefficient𝐶₃𝜆²_TF𝜆²_Lon=1.54×10⁻⁹1−𝑡²1−𝑡⁴.(33)

In Figure 7, the magnetic field dependence of peak amplitude of charge calculated at different temperatures is presented. A higher applied magnetic field corresponds to a lower amplitude of charge. At higher magnetic field inductions, the peak amplitude of charge almost saturates. A possible reason for the observation can be the greater overlap between the vortices at higher magnetic field. At high magnetic field, the charge accumulation in the vortex decreases along the line joining the neighboring vortices. A lower temperature corresponds to a greater accumulation of charges in the vortex core.

Figure 7 

Magnetic field dependence of peak amplitude of charge 𝜌(𝑥,0) calculated along the 𝑥-direction at different temperatures. Other relevant parameters are 𝜅=72, 𝜖𝑣=0.1, and 𝛽𝑠𝑑=0.

5. Conclusions

The study carried out in the present work consists of two parts, the first part is the extension of the GL theory to the low temperature regime in case of the high-𝑇_𝑐 superconducting cuprates involving mixed symmetry state of the order parameter components. The second part of the work involves the study of the electrostatic potential and the associated charge distribution in the vortices. In order to study the electrostatic potential and charge accumulation in the vortices, a set of equations are derived which consist of the Poisson's equation for electrostatic charge, Bernoulli potential for the electrostatic potential, Ginzburg-Landau equation for 𝑠- and 𝑑-wave order parameter components, and Ampere's law for magnetic vector potential. The resulting equations are solved by using a numerical iteration technique for arbitrary magnetic field induction and wide range of temperature. At any temperature, the flux line lattice shows an oblique structure characteristic to the high-𝑇_𝑐 cuprates.

The equilibrium applied magnetic field and the resulting reversible magnetization is calculated by using the virial theorem developed for the two-order parameter system. The reversible magnetization could thus be obtained without taking the numerical derivative of the free energy density functional. In the applied magnetic field versus reversible magnetization plot, an interesting 𝑆-shaped feature has been observed. The curvature of the 𝑆-shaped curve has been found to get enhanced at the lower values of temperature; however, a first-order transition, as has been found in case of conventional type-II superconductors, is absent in case of the high-𝑇_𝑐 superconducting cuprates for the temperature region studied. It can be attributed to the value of the temperature 𝑇_𝑎, which determines the lower limit of applicability of the extended GL theory. In case of high-𝑇_𝑐 superconducting cuprates, the value of 𝑇_𝑎 is much lower than that in case of the conventional type-II superconductors and thus any anomalous feature of the reversible magnetization curve can be expected to be observed only below this temperature. The observation also signifies that as in case of the applied magnetic field (𝐻_𝑐1<𝐻<𝐻_𝑐2), for temperature also it is possible to define a range (𝑇_𝑎<𝑇<𝑇_𝑐) over which the extended GL theory is applicable. In case of high-𝑇_𝑐 superconducting cuprates, the extended GL theory is valid over a wider temperature regime as compared to the conventional low-temperature type-II superconductors.

The work further deals with the determination of the electrostatic potential and the associated charge distribution in the vortices. The net electrostatic potential and charge distribution in the vortex lattice consists of three different components and their magnitudes are strongly dependent on the material parameters. In case of the high-𝑇_𝑐 cuprate YBCO, the spatial distribution of the electrostatic potential and charge shows a flat behavior unlike the case of the conventional low-temperature type-II superconductors. Here, it is worth mentioning that Kumagai et al. [12] through their experimental study have indeed suggested a flatness in the vortex charge distribution profile in high-𝑇_𝑐 superconducting cuprates.

A correspondence between the theoretical results and experimental observations can be achieved by utilizing the present theoretical model to explain the result obtained from the NMR and NQR studies carried out on the high-𝑇_𝑐 superconducting cuprates. The present model can further be generalized to take into account the various important features associated with the high-𝑇_𝑐 cuprates such as, the presence of anisotropy or orthorhombic distortion. Apart from the high-𝑇_𝑐 cuprates, the method can be used to analyze the electrostatic potential and charge distribution in other materials involving complex order parameters. These issues will be addressed in future.

Appendix

A. Calculation of Material Parameters for YBCO

In this section, we determine the various material parameters of high-𝑇_𝑐 cuprate YBCO that has been used to calculate the coefficients 𝐶₁ and 𝐶₂.

A.1. Calculation of 𝜕𝛾/𝜕𝑛

The linear coefficient of specific heat 𝛾 is related to the density of states 𝐷 as per the relation𝛾=23𝜋²𝑘^2𝐵𝐷.(A.1) Here, the linear coefficient of specific heat associated with the superconducting CuO₂-plane of YBCO has been taken into account. The effect of the chains has not been considered. For the high-𝑇_𝑐 cuprate YBa₂Cu₃O₇, the linear coefficient of specific heat is chosen to be 302JK⁻²m⁻³ [36]. Thus, the corresponding density of states can be determined by the relation𝐷=3𝛾2𝜋²𝑘^2𝐵.(A.2) The density of states 𝐷, which includes the mass renormalization due to the electron-phonon coupling, is related to the bare density of states 𝐷₀ as𝐷=𝐷₀(1+𝜆),(A.3) where 𝜆 is the electron-phonon coupling constant. As per the Kresin-Wolf two-gap theory [37–40], in case of the high-𝑇_𝑐 superconductors involving mixed symmetry state of order parameters, 𝜆 is a mixture of the coupling constants corresponding to superconductivity in the CuO₂-planes, in the CuO-chains and the interaction between the plane and the chain. The plane exhibits strong-coupling superconductivity characterized by the coupling constant 𝜆=3. The contribution to 𝜆 arising from the other two sources mentioned above ranges from 0.5 to 0.9. In the present study, a two-dimensional approach to the problem is implemented wherein the superconducting property of the high-𝑇_𝑐 cuprates is considered to be originating from the CuO₂-plane. Both the dominant 𝑑-wave and the subdominant 𝑠-wave order parameter components are considered to be residing in the CuO₂-plane, as has been interpreted from the recent experimental study carried out on YBCO [18, 19]. Consequent to the two-dimensional approach to the problem, in the present study the electron-phonon coupling constant is considered to be 𝜆=3.

From the bare density of states, one can determine the corresponding BCS interaction (𝑉) as per the expression𝜆=𝐷₀𝑉.(A.4) Using (A.1) the density derivative of the linear coefficient of specific heat can be expressed as𝜕𝛾𝜕𝑛=13𝜋²𝑘^2𝐵𝜕ln𝐷𝜕𝐸_𝐹,(A.5) where we have used 𝜕𝐸_𝐹/𝜕𝑛=1/2𝐷 and 𝑛=5×10²⁷m⁻³ is the density of holes [36]. Using (A.2–A.4), the density derivative of the linear coefficient of specific heat can be expressed as𝜕𝛾𝜕𝑛=13𝜋²𝑘^2𝐵1+2𝜆1+𝜆𝜕𝜕l𝐸n𝐷⁰_𝐹,(A.6) where the BCS interaction 𝑉 has been taken to be constant, that is, 𝜕𝑉/𝜕𝑛=0 or 𝜕𝑉/𝜕𝐸_𝐹=0.

Using (A.6), the coefficient 𝜕𝛾/𝜕𝑛 can be determined. The energy derivative of the bare density of state 𝜕𝐷₀/𝜕𝐸_𝐹 is obtained from the experimental data [41].

A.2. Calculation of 𝜕𝜖_𝑐𝑜𝑛/𝜕𝑛

For determining 𝜕𝜖_con/𝜕𝑛, we begin by expressing the critical temperature (in the present case 𝑇_𝑑) in terms of McMillan formula [42]𝑇_𝑑=Θ^𝐷1.45exp−𝜆−1𝜇.0^∗4(1(1++0𝜆.6)2𝜆),(A.7) where Θ_𝐷=440K [43] is the Debye temperature and 𝜇^∗=0.2 is the Coulomb pseudopotential. The corresponding condensation energy can thus be expressed as𝜖_con=𝜋²12.6𝑘^2𝐵(1+𝜆)𝐷₀Θ^2𝐷exp−21.04(1+𝜆)𝜆−𝜇^∗(1+0.62𝜆),(A.8) where we have used the relation 𝜖_con=(1/4)𝛾𝑇^2𝑑. The density derivative of the condensation energy can thus be given as𝜕𝜖_con𝜕𝑛=𝜖^con𝜕𝜆𝜕𝑛2.081+0.38𝜇^∗[𝜆−𝜇^∗(1+0.62𝜆)]²+11+𝜆,(A.9) where we have considered the product 𝐷₀Θ^2𝐷 and 𝜇^∗ to be constant with respect to the variation in the density of hole 𝑛.

Using the relations between 𝜆 and 𝐷₀ given by (A.2–A.4) one can write𝜕𝜆𝜕𝑛=2(1𝑉+𝜆)𝜕𝜕l𝐸n𝐷⁰_𝐹,(A.10) where the energy derivative of the bare density of state is again determined from experimental data for YBCO [41]. The resulting expression for the density derivative of the condensation energy is thus given as𝜕𝜖_con𝜕𝑛=𝜖^con𝑉(1+𝜆)²𝜕ln𝐷₀𝜕𝐸_𝐹1.041+0.38𝜇^∗(1+𝜆)[𝜆−𝜇^∗(1+0.62𝜆)]²+12.(A.11) Using (A.11) one can calculate the coefficient 𝜕𝜖_con/𝜕𝑛. In Table 1, we have tabulated the important material parameters for YBCO that has been used for the present study.

Table 1 

Material parameters for YBCO.

Acknowledgments

The author gratefully acknowledges E. H. Brandt and Pavel Lipavský for fruitful discussions. The work is financially supported by S. N. Bose National Centre for Basic Sciences, India.