Thermodynamic Adsorption Potential of Superconductors

Jiu Hui Wu¹, Jiamin Niu¹, and Kejiang Zhou²
¹School of Mechanical Engineering, Xi'an Jiaotong University, & State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi'an 710049, China
²Huzhou Institute of Zhejiang University, Huzhou 313000, China

Abstract

Based on the general thermodynamic analysis of Polanyi adsorption potential, the adsorption potential condition for superconductors is obtained exactly by using the quantum state equation we presented. Because this adsorption potential results in changes of electron concentration, temperature and pressure in a certain volume (adsorption space) adjacent to the surface of the lattice, the composition and structure of superconductors are of course decisive for the adsorption potential. We calculate the molar adsorption potentials for those typical superconductors and find that it is positively correlated to the superconductivity temperature 𝑇_c, which reveals that those high-𝑇_c superconductors are mainly determined by the higher molar adsorption potentials. In addition, the adsorption potential at 𝑇 = 𝑇_c still works despite the disappearance of the energy gap of the BCS theory. This shows that beyond the electron-phonon interaction mechanism, the Cooper-paired electrons are mainly formed by this physical adsorption potential for high-𝑇_c superconductors. This adsorption potential theory could explain almost all common facts about high-temperature superconductors, the normal and superconducting states, including many anomalies of.

1. Introduction

Since the discovery of superconductivity by Kamerlingh Onnes in 1911, the study of superconductivity has been a hot topic and a difficult point in the field of physics, with a history of more than 100 years. In the early days, research on superconductors focused on metal and alloy systems with low superconducting transition temperatures, whose mechanism can be explained by the BCS theory founded in 1957, and related theories that have been developed subsequently. Based on the BCS theory, it is difficult for conventional superconductors to have a 𝑇_c higher than 40K at atmospheric pressure, so superconductors with 𝑇_c higher than 40K are often referred to as high-temperature superconductors. In 1986, Müller and Bednorz discovered the (La,Ba)₂CuO₄ superconductor, and in 1987 the discovery of YBa₂Cu₃O₇₋δ raised the superconductivity temperature to the liquid nitrogen temperature region for the first time, opening a new era of high-temperature superconductivity research. The 𝑇_c of conventional superconductors is much lower, and the highest atmospheric pressure record is currently held at 133 K in 1993. Since the 1990s, high-precision epitaxial growth technology has been introduced into superconductivity research, and heterostructures can be formed by artificially stacking materials with different structures or different chemical compositions, and novel superconducting states may also be derived based on the coupling of these heterogeneous atomic-level smooth interfaces. Through the artificial design of heterostructures or precise control of the thickness of single crystal films, researchers are able to explore superconductivity under dimensional changes in highly crystalline samples. More recently, Yanagisawa investigated correlated-electron systems, emphasizing strong electron correlation as a driver of superconductivity in high-temperature cuprates.

Up to date, the results of many experiments reveal the following common facts about high-temperature superconductors: 1) They are strongly correlated electronic systems; 2) It is important to consider some kind of electron-electron mediation; 3) Due to their severe anisotropy, the interlayer coupling must be considered; 4) There should be an appropriate concentration of carriers, as too much or too little is not good for superconductivity; 5) It is necessary to establish a theory of superconductivity on the basis of the anomalous electronic states of the normal phase, because the anomaly of the normal phase contains the special interaction mechanism of the electronic system. At present, the mechanism of high-temperature superconductivity is still unsolved, and the journal Science has repeatedly included it among 125 important scientific questions. Due to the strong interaction between electrons in high-temperature superconductors, the description of their electronic behavior inevitably involves extremely complex many-body quantum physics, so the mechanism of high-temperature superconductivity is still not fully understood, especially the source of the attraction effect used for pairing between carriers, which remains one of the core difficulties in current physics.

Grounded in the work of Henri Poincaré, R. Thom and others, catastrophe theory can explain the phenomenon of gradual quantitative change leading to sudden qualitative change, which is a highly generalized mathematical theory that summarizes the rules of non-equilibrium phase transition through several catastrophe models. According to Thom's classification theorem, as long as the number of control variables that cause mutations does not exceed 4, the various mutation processes in nature can be grasped using seven basic potential function models. Because of its own structural stability during the mutation process of any non-equilibrium system, any phase transition can be analyzed quantitatively by one of these catastrophe models, even without knowing the differential equations of the system. In our previous papers, by using catastrophe theory, the general non-equilibrium phase transition process from laminar to turbulent flow has been investigated quantitatively, as well as a revised Schrödinger relativistic equation obtained from the perspective of phase transition. More recently, the thermodynamic quantum phase transition process was investigated quantitatively by the structural-stability-based catastrophe theory. For a canonical ensemble composed of N identical particles, the cusp catastrophe model is adopted to express the average free energy, and further the general quantum state equation about the pressure is obtained to describe the phase transition process from quantum scale to macro scale by using dimensionless analysis. Subsequently, the ensemble free energy considering the interaction potential energy among the particles can be obtained exactly, as well as the canonical partition function and the specific heat capacity of the system.

The transition from a normal state to a superconducting state is a phase transition problem, which obviously also falls under the category of thermodynamics. Superconductivity is a second-order phase transition, that is, the superconducting phase transition does not undergo latent heat changes, and meanwhile the free energy of the superconducting states must be less than that of the normal states. By analogy with Polanyi adsorption potential theory in chemistry, we put forward the physical adsorption potential between free electrons and the lattice of superconductors through the van der Waals force, assuming that these electrons can be adsorbed and condensed due to changes in electron concentration, temperature and pressure in a certain volume (adsorption space) adjacent to the surface of the lattice, in addition to the electron-phonon interaction mechanism. In this paper, based on this general quantum state equation from catastrophe theory, the physical adsorption potential between the electrons and the lattice of superconductors is further put forward to try to explain high-temperature superconductivity.

2. General Thermodynamic Analysis of Polanyi Adsorption Potential

The adsorption of gas or liquid on a solid surface is the result of adsorption displacement caused by the adsorption force between the adsorbate in the adsorption force field and the solid surface. The field strength of an equipotential surface in the adsorption force field is not only related to the properties of the adsorbate and the solid, and its surface structure, but is also affected by the distance between the equipotential surface and the solid surface. The field strength decreases with increasing distance and becomes zero at infinity, which is defined as the zero-potential surface.

In the adsorption force field, the adsorbate moves along the direction of the field strength under the action of the adsorption force to do adsorption work. Here the effect of adsorption force is greater than that of molecular thermal movement, which results in reduced distance among the adsorbate molecules and even produces coagulation or chemical adsorption. The adsorption layer between the normal phase of the adsorbate and the surface of the solid is called the adsorption phase, in which the adsorbate density is greater than that of the normal phase, and the concentration of the adsorption phase shows a gradient and continuous change, as shown in Fig. 1.

[FIGURE:1]

From the above, it can be seen that the adsorption process is a process in which the adsorbate molecules enter the adsorption phase from the normal phase under certain conditions, and displacement change or chemical change occurs, that is, the amount of substance changes. Below, the chemical potential and adsorption work are used to reflect the rule of the adsorption process.

For a system consisting of components of the adsorbate, the adsorption force is a generalized force, and the Gibbs free energy of the adsorption system is expressed as 𝐺 = 𝐺(𝑇, 𝑃, 𝐿, 𝑁₁, 𝑁₂, ⋯, 𝑁ₖ) whose full differential is

𝑑𝐺 = (𝜕𝐺/𝜕𝑇)𝑑𝑇 + (𝜕𝐺/𝜕𝑃)𝑑𝑃 + (𝜕𝐺/𝜕𝐿)𝑑𝐿 + Σ(𝜕𝐺/𝜕𝑁ᵢ)𝑑𝑁ᵢ

where P is the pressure, T is the temperature, L is the adsorption displacement, 𝑁ⱼ is the number of particles of the j-th component in the adsorbate, and the corner mark 𝑁ⱼ≠ᵢ indicates that the amount of the other components except the i-th component remains the same.

Because the adsorption work done by the system in a reversible process is equal to the decreasing value of the free energy, there is

(𝜕𝐺/𝜕𝐿)𝑑𝐿 = 𝑓𝑑𝐿 = −𝛿𝑊

where 𝑓 = (𝜕𝐺/𝜕𝐿)ₚ,ₜ,ₙⱼ is the generalized adsorption force, and (𝜕𝐺/𝜕𝐿)𝑑𝐿 is the contribution of the adsorption system at a certain equipotential surface to the free energy change with respect to the adsorption displacement L.

In addition, the adsorption process is mostly carried out at constant temperature and pressure, i.e., 𝑑𝑇 = 𝑑𝑃 = 0, so Eq. (2) can be simplified as

𝑑𝐺 = −𝛿𝑊 + Σ(𝜕𝐺/𝜕𝑁ᵢ)𝑑𝑁ᵢ

Since this adsorption process is spontaneous, according to the reduction principle of free enthalpy, i.e., 𝑑𝐺 ≤ 0, there is

𝛿𝑊 = Σ𝜇ᵢ𝑑𝑁ᵢ

where the chemical potential 𝜇ᵢ = (𝜕𝐺/𝜕𝑁ᵢ)ₚ,ₜ,ₗ,ₙⱼ≠ᵢ. Eq. (5) means the contribution of a multi-component system to the adsorption work when the amount of substance changes under the condition of constant temperature and pressure.

When the amount 𝑑𝑁ᵢᵇ of the i-th component at the zero point of the bulk phase in the system is adsorbed to an equipotential surface of the adsorption phase, the amount added to the i-th component at the equipotential surface is 𝑑𝑁ᵢᵃ, and −𝑑𝑁ᵢᵇ = 𝑑𝑁ᵢᵃ, which means the adsorption potential of an adsorbate molecule in the adsorption force field on the solid surface is the work required to move the molecule from its equipotential position in the adsorption phase to the zero position. Thus Eq. (5) becomes

(𝜇ᵢᵇ − 𝜇ᵢᵃ)𝑑𝑁ᵢ ≤ 𝛿𝑊

When 𝑑𝑁ᵢ = 1 mol, the work done by the system to adsorb 1 mol of substance is equal to the Polanyi molar adsorption potential ε, thus

ε = 𝐴(𝜇ᵢᵇ − 𝜇ᵢᵃ)

where 𝐴 is Avogadro's number, which denotes that ε is the change of the chemical potential when 1 mol of substance is reversibly adsorbed from the bulk phase b to the adsorption phase a.

(𝜇ᵢᵇ − 𝜇ᵢᵃ) ≤ ε

3. Adsorption Condition at the Solid-Gas Interface of Superconductors

For superconductors, this adsorption potential is also caused by enhanced condensation, which should depend on the properties of the superconductor's lattice structure and the position of the electrons in the vicinity of the lattice surface. Since the concentration of electrons in the normal states is uniform, the adsorption potential is zero, that is, the interface between the normal states and the adsorption phase is zero, and the adsorption potential of the free electrons in equilibrium with the solid surface can be any equilibrium value from zero to saturation.

3.1 Thermodynamic Adsorption Condition for Superconductors

According to Ref. [21], we have obtained the correlation among the electron concentration, temperature and pressure of Fermi electrons by catastrophe theory:

𝑃𝑣^(5/3) = (4𝛼/3)ħ²𝑛^(2/3)[(√𝐵 + √(𝐵² − ħ²𝑛^(2/3))) + (√𝐵 − √(𝐵² − ħ²𝑛^(2/3)))]

where ħ is Planck's constant, m is the rest mass of electron, 𝑘_B is Boltzmann's constant, 𝑛 = 𝑁/𝑉 is the number of electrons per unit volume, 𝛼 is the phase transition index, 𝛽 = ħ²𝑛^(2/3)/(𝑚𝑘_BT) is a dimensionless temperature, and 𝐵 = [(3𝜋²)^(2/3) + (3𝜋²)^(2/3)].

From Eq. (7), when 𝛽 ≫ 1, which means the mean distance among the particles 𝑣^(1/3) is much larger than the thermal wavelength λ = √(2𝜋ħ²/𝑚𝑘_BT), thus the microscopic quantum effect can be ignored. In the infinite limit of 𝛽 → ∞ and 𝛼 = 1/2, Eq. (7) will degenerate to the state equation of the ideal Boltzmann gas with 𝑃 = 𝑛𝑘_BT [21]. When 𝛼 = 0, 𝛽 → 0, i.e., 𝛽𝛼 = 1, the pressure of the Fermi gas is not equal to zero because of the Pauli exclusion principle with 𝑃 = (3𝜋²)^(2/3)𝑛^(5/3)ħ²/5𝑚 [21].

For those high-temperature superconductors, such as Perovskite copper oxides (Sr₈CaRe₃Cu₄O₂₄), there is high superconductivity temperature (up to 𝑇_c = 135K) with low carrier concentration (𝑛 = 3.0 × 10²⁷ m⁻³). And for those low-temperature superconductors, such as Ag, there is low superconductivity temperature (0.9K) with high carrier concentration (𝑛 = 1.044 × 10³¹ m⁻³). Anyway, there is 𝛽 = ħ²𝑛^(2/3)/(𝑚𝑘_BT) ≪ 1 for all superconductors. Therefore, in this case Eq. (7) can be simplified as

𝑃 = 1.74𝑛𝑘_BT(4𝛼/3 − 1)ħ²𝑛^(2/3)/(𝑚𝑘_BT)

i.e.,

1.74(𝑛)^(8𝛼−15) = (4𝛼/3 − 1)ħ²𝑛^(2/3)/(𝑚𝑘_BT)

Eq. (8) is a thermodynamic quantum equation of state of Fermi gases, which means that under the same pressure, when the electron concentration n is increased, T is decreased.

By use of Eq. (8a), the Gibbs free energy of N identical electrons is further obtained as [21]

𝐺(𝑃, 𝑉, 𝑇) = 𝐹(𝑃, 𝑉, 𝑇) + 𝑃𝑉 = 𝑁𝑘_BT[2.638(ħ²𝑛^(2/3)/𝑚𝑘_BT) + 12.88(ħ²𝑛^(2/3)/𝑚𝑘_BT)²]

where 𝐹(𝑃, 𝑉, 𝑇) is the free energy of the system.

From Eq. (9), the chemical potential at the normal phase is

𝜇_n = 4.397𝑘_BT[(1 − ħ²𝑛^(2/3)/𝑚𝑘_BT) + 4.88(1 − ħ²𝑛^(2/3)/𝑚𝑘_BT)²]

which shows that the chemical potential is dependent on the temperature T, the electron concentration n and the phase transition index α.

On the other hand, at the adsorption phase, due to the work done by the adsorption force, the electron concentration increases, resulting in a decrease in temperature T according to Eq. (8a). With T decreased to the superconductivity temperature 𝑇_c, some additional form of electron order begins to form and increases as the temperature drops. At this time the phase transition index α becomes 𝛼′, and the superconducting electron concentration becomes 𝑛_s = 𝜔𝑛 (𝜔 ≤ 1), where ω is the relative proportion.

Because the special properties of the superconducting state are determined by the condensed matter of the Cooper pairs. The formation of Cooper pairs allows electrons to flow through the material without resistance, and this new electron pairing state plays a dominant role in the energy and particle number changes of the system. Therefore, in the superconducting state, the chemical potential is mainly considered for the superconducting electrons (Cooper pairs).

Then the chemical potential 𝜇_s of the superconducting phase is

𝜇_s = 4.397𝑘_BT[(1 − 𝛼′)(ħ²𝑛^(2/3)/𝑚𝑘_BT) + 4.88(1 − 𝛼′)(ħ²𝑛^(2/3)/𝑚𝑘_BT)²]

where 𝑁_S is the number of superconducting electrons.

Therefore, according to Eq. (6b), the adsorption potential condition for the superconducting phase transition is

𝑁_sε_p ≥ 𝜇_s − 𝜇_n

where 𝜀_p is the average adsorption energy of every electron.

3.2 Identification of the Phase Transition Index α for Superconductors

When a normal metal is cooled, there is usually a decrease in conductive electron entropy. At temperatures less than 𝑇_c, some additional form of electronic ordering must begin to form, thus making an additional contribution to the specific heat capacity.

According to Eqs. (8) and (9), the conductive electron entropy S and the specific heat capacity 𝐶_p can be further obtained respectively as

S = (8𝛼 − 15)[44.8494(ħ²𝑛^(2/3)/𝑚𝑘_BT)^(5𝛼/3−1) + 154.56(4𝛼/3 − 1)(ħ²𝑛^(2/3)/𝑚𝑘_BT)^(4𝛼/3−1)]

𝐶_p = 𝑇(𝜕S/𝜕𝑇)_p = 15𝑘_B𝛼/(8𝛼 − 15)²[44.8494(5𝛼/3 − 1)(ħ²𝑛^(2/3)/𝑚𝑘_BT)^(5𝛼/3−1) + 154.56(4𝛼/3 − 1)(ħ²𝑛^(2/3)/𝑚𝑘_BT)^(4𝛼/3−1)]

From the point of view of entropy, the formation of Cooper pairs and their motion states affect the calculation of entropy. Therefore, when calculating the entropy of a superconducting state, the influence of superconducting electrons is mainly considered, i.e., the electron contribution n in Eq. (13) will be replaced by the superconducting electron concentration 𝑛_s = 𝜔𝑛.

It is well known that 𝐶_p ~ 𝑇³ at the superconductivity phase and 𝐶_p ~ 𝑇 at the normal phase. Considering Eq. (8b) where 𝑛 ~ 𝑇^(8α−15) and 𝛽 = ħ²𝑛^(2/3)/(𝑚𝑘_BT) < 1, we have 𝐶_p|(𝑇<𝑇_c) > 𝐶_p|(𝑇>𝑇_c), which results in a jump in the specific heat capacity 𝐶_p at 𝑇_c. Meanwhile, since the superconducting phase transition does not undergo latent heat changes, from the continuity of the entropy S at 𝑇_c, the change of electron concentration from the normal phase to the superconductivity phase can also be uniquely determined.

In fact, when a superconducting phase transition occurs in a material, since electrons form Cooper pairs, their motion states are bound, and the interaction mode between the electrons changes, so that the effective interaction potential energy among the electrons changes, resulting in the pressure of the electron gas changing at the same time. Thus for the superconductivity phase, according to Eq. (8b) there is

𝑃_s = 1.74𝑛_s𝑘_B𝑇_c(4𝛼′/3 − 1)ħ²𝑛_s^(2/3)/(𝑚𝑘_B𝑇_c)

where 𝑃_s is the pressure of the electron gas at the superconductivity phase.

The following is an example of metal Sn to illustrate its superconducting phase transformation process. For the metal Sn, 𝑇_c = 3.72K, at which 𝑛 = 8.8 × 10²⁸ m⁻³. According to the continuity of the entropy S at 𝑇_c, from Eq. (13) we can obtain that ω = 0.0006223, which means that the phenomenon of superconductivity will occur only when the superconducting electron concentration is increased to 0.06223% under the action of adsorption potential.

[FIGURE:2]

Figure 2 shows the conductive electron entropy S and the specific heat capacity 𝐶_p of Sn of 1 molar electrons varying with temperature T by using Eqs. (13) and (14).

3.3 The Adsorption Potential Condition for Superconductors

Based on the above section, the adsorption potential condition for the superconducting phase transition can be further obtained as

𝜀_p ≥ 𝜇_s − 𝜇_n = 𝑘_BT[1.256(ħ²𝑛^(2/3)/𝑚𝑘_BT) − 9.194(ħ²𝑛^(2/3)/𝑚𝑘_BT)²] (for 𝑇 < 𝑇_c)

where ω is determined by the continuity of the entropy S at 𝑇_c from Eq. (13).

According to Eq. (16a), when 𝑇 → 0, ω → 1, and 𝜀_p → 0, which means that the adsorption potential is different from the BCS theory in which the band gap 2Δ(0) = 3.53𝑘_B𝑇_c. On the other hand, when 𝑇 = 𝑇_c, there is

𝜀_p ≥ 𝜇_s − 𝜇_n = 𝑘_B𝑇_c[1.256(ħ²𝑛_c^(2/3)/𝑚𝑘_B𝑇_c) − 9.194(ħ²𝑛_s^(2/3)/𝑚𝑘_B𝑇_c)²]

which is also different from the BCS theory where there is the band gap Δ(𝑇_c) = 0. Here 𝑛_c is the electron concentration at 𝑇_c, and 𝑛_s = 𝜔𝑛_c.

[TABLE:1]

According to Eq. (16b), in Table 1 we calculate the average adsorption energy of every electron 𝜀_p and the corresponding molar adsorption potential ε for superconductors, respectively. It can be found that the molar adsorption potential ε is positively correlated to the superconductivity temperature 𝑇_c, which reveals that those high-𝑇_c superconductors (𝑇_c ≥ 40K) are mainly formed by the higher molar adsorption potentials. At 𝑇 = 𝑇_c, the adsorption potential still works despite the disappearance of the energy gap of the BCS theory. As for ω, in the process of electron migration due to adsorption, the larger ω is, the greater the adsorption energy is. This positive correlation means that ω actually reflects the composition and structure of superconductors, which are of course decisive for the adsorption potentials, even with the same electron concentration. Furthermore, the following fact could be explained: for copper oxides with high-temperature superconductivity, the smallest cell is at least an intact cell layer containing the CuO₂ bilayer.

For these low-𝑇_c superconductors with high carrier concentration at the normal phase, their composition and structure are so simple that the adsorption potential is too low to form additional Cooper pairs apart from the electron-phonon interaction mechanism. On the other hand, for those high-𝑇_c superconductors but with low carrier concentration at the normal phase, when the degree of electron migration ω is increased to some extent, the molar adsorption energy is increased, resulting in the high-𝑇_c phase transition. From this point of view, the anomaly of the superconductivity property could be explained for those high-temperature superconductors, and meanwhile the anomaly of the normal states could be explained for those low-temperature superconductors.

4. Conclusions

In this paper, the molar adsorption potential is defined exactly, and the thermodynamic adsorption potential condition is put forward to explain high-temperature superconductivity. It is revealed that beyond the electron-phonon interaction mechanism, the Cooper-paired electrons are mainly formed by this physical adsorption potential for high-𝑇_c superconductors. Thus the high-temperature superconductivity could still be explained by the Cooper-paired electrons due to the molar adsorption potential. This theory could explain many anomalies of the normal and superconducting states of Perovskite copper oxides, as well as the isotopic effects of copper oxides.

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