Good question and excellent answer.

Interesting question, what answer Firas?

I agree with you. Superconductors are utilized in many devices and their properties are investigated in many situations of external applied fields. Unfortunately, many researchers are mixing things and assume thermodynamic equilibrium (to benefit from the well known equilibrium distributions of charge carriers, such as the Fermi-Dirac distribution as well as the Bose-Einstein equilibrium distribution for Bosonic quasiparticles). Others frankly assume near-equilibrium (small perturbation) conditions. Several mental (not real) experimental scenarios are diffusing in the literature to attack or to support specific theories and mix between the equilibrium and non-equilibrium conditions. Therefore, I just wondered whether scientists consider the intrinsic superconductivity phenomenon is something happening in state of dynamic equilibrium.

any superconductor, flown through by a persistent current, is indeed thence off-equilibrium; nevertheless its relaxation toward equilibrium cannot take place because it would cause a violation of the second law of thermodynamics, as shown in attached criterion.pdf, which warrants the existence of the persistent current

Yes Prof.. Dr. Jacob. I agree

Thank you Dr. P. Contreras

Your note that the Fermi energy is no longer equal to the chemical potential is very important. In fact, the chemical otential and Fermi level are only defined in thermodynamic equilibrium. We note the presence of Ef (and hence its definition) in the Fermi-Dirac statistical distribution in equilibrium. In semiconductors and insulators, we are obliged to invent the so-called "Quasi-Fermi levels" in non-equilibrium, when current flows. This makes us enquirer about considering the chemical potential in the Hamiltonian of almost all microscopic quantum theories of superconductivity (e.g., the BCS and the Hubbard models)!. This also leads us to the original debate about the instability of Fermi energy level, which was assumed by Schrieffer in the BCS theory and the later theories about the reconstruction of Fermi surface in HTS materials.

Finally, I just want to remind myself and all colleagues about the significance of my question, when you try to find or agree upon a physically-based model of charge carrier transport in superconductors, you'd realize first: where are you? Carrier Transport theories are only meaningful when the system is acted upon by external excitation and considered in non-equilibrium. I'll benefit from your comments in the next edition of my Book:

https://www.igi-global.com/book/transport-information-carriers-semiconductors-nanodevices/173677

Dear Muhammad,

let me elaborate on your latest comment : the chemical potential \mu and Fermi energy E_F being equal is the Gibbs-Duhem condition, defining thermal equilibrium for any SECOND order transition, such as magnetic, allotropic... so that NO latent heat is required. Conversely \mu and E_F differ from each other at THERMAL EQUILIBRIUM in every FIRST order transition, such as melting, which requires thence a latent heat for the transition to take place. The superconducting to normal transition is second order in case of vanishing current j=0 , BUT first order for finite j. Besides latent heat is replaced by "LATENT WORK" (see section 3 of attached transition.pdf)

I forgot to attach transition.pdf in my preceding message and am sorry for this mishap

Dear P. Contreras and Jacob Szeftel

Thank you for your appreciated contribution. After reading the interesting paper of Prof. Jacob Szeftel, I just want additional explanation of this paragraph:

"The role of the latent heat, typical of all usual first-order transitions (melting or vaporization), is played here by the latent work Ws→n, because the superconducting-normal transition is controlled by current rather than by temperature". As you precised by the word "here", this is the case you explained, as the transition may be also achieved by increasing temperature above Tc. Actually Vaporization is the inverse of condensation. Isn't it?

However, the Fermi energy of a free electron gas is related to the chemical potential of the gas by the relation (Blackmore): equation.png (see attached file).

Therefore, μ = EF when T-->0 and they are approximately equal at low temperatures of much less than the Fermi temperature EF/kB. I realize that the relation between EF and μ is an important side discussion to my original question. Thank you again for commenting.

Dear Muhammad,

1) you correctly interprete " here" as meaning superconducting-normal transition, by contrast with melting or vaporization;

2) the equation laid out in equation.png, has been obtained by applying the Sommerfeld integral; it gives actually E_F(T>0) as a function of E_F(T=0) at equal electron concentration c_n; the thermal equilibrium in a superconductor with vanishing current is defined by the Gibbs-Duhem equation : \mu(c_s)=E_F(T

Dear Jacob,

The expansion of μ in terms of EF is provided here for one homogeneous Fermi gas in equilibrium (or the dominant part of a 2-fluid mixture, as I'll indicate below). This formula is then an application of the Sommerfeld expansion (on Sommerfeld integral). This expansion is actually an approximation for some common integrals in condensed matter and statistical physics in equilibrium. Physically, the integrals is a statistical average over Fermi–Dirac distribution and it is valid for low temperature systems. We often use it in semiconductor physics to find the equilibrium concentration of charge carriers (electrons and holes) in equilibrium, in terms of the Fermi energy level, specially at low temperature, when the Boltzmann approximation is not valid.

However, although this is a side point but it may lead us an answer to my original question as well as interesting common conclusions about superconductivity. In fact the original Gibbs–Duhem equation describes the relationship between changes in chemical potential for the different components in a thermodynamic mixture:

Σi Ni dμi = - SdT + VdP

where Ni is the number of moles of the ith component and dμi is the infinitesimal increase in chemical potential for this component. When we consider a superconductor as a 2-fluid system, which is isobaric and isothermal, then we can write:

Nc dμc + Ns dμs = 0

where the indices c and s denote the normal conduction and the superconducting electrons, respectively. As you said the normal conduction electrons may be considered like a Fermi gas, characterized by a Fermi energy EF, and the superconducting electrons are organized in many-electron state with chemical potential μs. Of course we know that the total electron concentration n = Nc + Ns. We also know that only a few number of electrons (Ns) will couple into superconducting pairons, that is: Ns dμc , which means μc is almost constant and the change in chemical potential of the superconducting component is appreciable. Therefore, we can consider the superconductor as a mixture of a Fermion gas of conduction electrons (in equilibrium) and a Bosonic part of superconducting electrons (pairons) in non-equilibrium.

Continue:

This agrees with well-known observations. The one-state Bosonic pairons (superconducting electrons) once triggered by any external field will cause a persistent current and will not collide with themselves (cause they are condensed in one Bosonic state) and will not have net exchange energy with the surrounding heatbath of Fermions (electrons and nucleons) at a specific temperature T

De ar Muhammad,

I quite agree with most of your analysis; however I would like to elaborate on your thermal balance equation, leading to both "chemical" species, namely normal and superconducting electrons, in respective concentrations c_n, c_s, reaching eventually thermal equilibrium. This corresponds to a minimum of the Helmholz free energy F=F_n+F_c. Because of the additional constraint of constant pressure P and temperature T, the minimum condition dF=dF_n+dF_s=0 can be satisfied only as (dF_n/dc_n)dc_n+ (dF_s/dc_s)dc_s=0 with dc_n+dc_s=0 due to charge conservation. Since there are by definition dF_n/dc_n=E_F(T,c_n) and dF_s/dc_s=\mu(c_s) , the thermal equilibrium condition reads finally E_F=\mu, which is nothing but the Gibbs -Duhem relation. Q.E.D.

Dear Prof. P. Contreras

Yeah, I agree. Studying the Fermi energy concept in super-conducting compounds, and the difference between the Fermi level, chemical potential, intrinsic chemical potential as well as the Fermi surface is vital to formulate any successful theory in superconductors in general and High-Tc superconductors (HTS), in particular

This is sure related to the thread of our discussion about thermodynamic equilibrium and non-equilibrium in superconductors. I hope to find common and clear definitions for each term and to resolve the irritating ambiguities, specially in the case of multi-band compound super-conductors. I know, the difference between Fermi level and chemical potential has been already a subject of several discussions and papers, but we still need to present this in a clear form for young researchers. I invite Prof. Jacob Szeftel and all his colleagues, as well all our professors in solid-state physics to participate.

Concerning, the superconducting semiconductors, my own convention is that they become superconducting only upon heavy doping when they become degenerate and the Fermi level creeps INSIDE conduction band

Dear Colleagues,

to the best of my knowledge, the Fermi energy is nothing but the chemical potential of independent electrons and depends upon their concentration and temperature T; the Fermi gas of independent electrons obeys either Fermi-Dirac or Boltzmann-Maxwell statistics, according to whether k_B*TE_F with k_B being Boltzmann's constant;

regarding superconductivity, the first Cooper pair builds up if the Gibbs-Duhem condition is fulfilled, i.e. E_F(c_0,T_c)=\mu(0)=eps/2, with c_0,T_c),\mu(0),eps being the total concentration of conduction electrons, critical temperature, chemical potential of superconducting electrons at vanishing concentration and Cooper pair energy (see attached criterion.pdf, transition.pdf); however, such a necessary

Dear Colleagues,

to the best of my knowledge, the Fermi energy is nothing but the chemical potential of independent electrons and depends upon their concentration and temperature T; the gas of independent electrons obeys either Fermi-Dirac or Boltzmann-Maxwell statistics, according to whether k_B*TE_F with k_B being Boltzmann's constant.

regarding superconductivity, the first Cooper pair builds up if the Gibbs-Duhem condition is fulfilled, i.e. E_F(c_0,T_c)=\mu(0)=eps/2, with c_0,T_c,\mu(0),eps being the total concentration of conduction electrons, critical temperature, chemical potential of superconducting electrons at vanishing concentration and Cooper pair energy (see attached criterion.pdf, transition.pdf); however, such a necessary condition is not sufficient, because the stability of the superconducting phase with respect to the normal one requires an additional condition to be satisfied, which I intend to show in a prospective work

Interesting question and the answers are very interest too.

Dear Colleagues,

Thanks to your appreciated and really helpful replies, I think I can suppose (or conclude) the following answer for my question. I dully note, I exclude the effect of external magnetic fields (B=0) here for simplicity.

In superconducting phase, the superconductor (at T

maybe this has to do with this thread:

• DOI:
• 10.13140/RG.2.2.11019.39200

Thank you Dr. Vera, for the interesting article of Dr. Vladimir Egorov

It was a pleasure to put toghether like minds

Dear appreciated colleagues, who follow and participate in this thread,

If you admit, we can now we turn our attention to more complicated situations

1- What happens in presence of external magnetic fields to the state of non-equilibrium of the Bosonic fluid and the surrounding electron gaz? and

2- What happens when the persistent current is interrupted (the current loop is opened and Js=0)?

This is actually the subject of the attached article of Prof. Jacob Szeftel in his comment in 25 Feb

I hope to hear from you.

I follow several threads here at RG, this is the second time when I see people triyng to learn from each other, I am pedestrian in physics and curious, cannot contribute but want to congratulate the participants.

Dear Prof. Dr. Vera Maura Fernandes de Lima

Thank you for participating and your useful look out. Actually, RG permits Discussions not only questions. I, with many colleagues (who are better than me) decided to profit from this to discuss controversial points, although some of these points are very fundamental. In fact, some scientists found recently (and had the scientific courage to confirm) that many basic theories in physics are fundamentally flaws. Thanks to the highly appreciated criticism of Prof. Jacob Szeftel and Prof. P. Contreras as well others, who are exchanging opinions about their research pre-prints, such fruitful discussions were possible.

Dear Prof. P. Contreras

I agree upon your note. Importing new concepts and terms is a double edged knife and generally indicates the controversial state of the theory of superconductivity.

Many researchers consider superconductivity (SC) an equilibrium phenomenon, because there is no external acting force (zero resistivity) and only temperature change is sufficient to make the phase transition. ... When the external force is interrupted (removed) the system should relax to its original equilibrium state.

If superconductors are not equilibrium or thermodynamic states of

matter, then one can’t describe them by their current properties(Tc,Jc and Hc) but

need to know their complete history. It is simply like a, If I give you a glass of cold

water, you have no way of knowing whether that water has been

condensed from steam and then cooled, or whether I melted a large

ice cube and allowed the water to warm up a bit. Changes of phase

are entirely reversible and once a substance has come to equilibrium

with its surroundings, there is no memory of its past history. To

describe the water in the glass, I just have to detail its properties at

this current moment such as the temperature of the water in the glass

right now.

I recommend this book Superconductivity: A Very Short Introduction(Stephen Blundell)

a superconductor flown through by a current I is in equilibrium insofar as its thermodynamical state is reversible, reproducible and completely defined by I . Nevertheless since the chemical potential of superconducting electrons differs from the Fermi energy, we are speaking of this kind of thermal equilibrium, typical of a first order transition such as ice melting into water (for a detailed account see attached transition.pdf)

In my opinon, superconductivity is a non-equilibrium phenomenon, an equilibrium state means that in a system, the gradient in the temperature, the gradient in the generalised force, and the gradient in the chemically potential are all equal to zero.

If there is an electric current, it means that the the gradient in the electrochemical potential is not equal to zero.