True superconducting state

True superconducting state Yatendra S. Jain Department of Physics, North-Eastern Hill University, Shillong - 793022, India.

Abstract In this note we sum up the basic characteristics of true superconducting state concluded from our first quantization theory of superconductivity and underline reasons for which this theory has inbuilt potential to explain every experimental observation on widely different superconductors. Accordingly, our theory explains even those experiments which are not explained by BCS theory and defines materials which can exhibit superconductivity at room temperature. In addition it does not reject the possibility of observing superconductivity with properties such as ferroelectricity, ferromagnetism, or antiferromagnetism, etc.. Reasons for which BCS theory achieved only limited success are also identified.

c by author

email:- [email protected]

As reported by Hirsch [1] BCS theory [2] does not explain several important properties of even metallic superconductors and this reality is being recognised by other researchers in the field [3]. Discovering the reasons for this situation, we find [4] that some of the basic premises of BCS theory do not agree with certain physical realities of the low temperature (LT ) states of conduction electrons (ces) due to inherent problems of its mathematical formulation based on second quantization (SQ) clubbed with single particle basis (SPB). Our analysis [4] also concludes that no theory developed by using SQ clubbed with SPB would ever reveal complete, clear and exprimentally consistent picture of the LT states of ces; naturally, it would not emerge as a viable theory of superconductivity. Notably, this conclusion is corroborated by the fact that all such theories developed after the discovery of high temperature superconductivity (HTS) do not explain superconductivity comprehensively. SQ has been used to develop the microscopic theories of widely different many body quantum systems with a belief that first quantization (FQ) is unsuitable because it needs separation of all 3N degrees of freedom, -a formidable task even for a system of very small N. However, we discovered a way out and developed the microscopic theories of: (i) a system of interacting fermions (SIF) like electron fluid in a conductor and liquid 3 He (LHe-3) [5] and (ii) a system of interacting bosons (SIB) like liquid 4 He (LHe-4) and trapped dilute bose gases [6]. BCS theory assumes that each ce in a conductor, to a good approximation, represents a free particle and occupies a state of single particle placed in a box of volume V of the conductor. At T < Tc , ces occupying states of energy close to Fermi energy form (q, -q) bound pairs (known as Copper pairs, perceived like a diatomic molecule) due to a phonon induced attraction between 1

two ces. This leads to an energy gap between superconducting state (S-state) and normal state (N-state). States of other electrons in S-state do not differ from their states in N-state. They have relative motions, mutual collisions and collisions with lattice constituents. On the other hand our FQ theory [5] not only concludes an energy gap between S- and Nstates but also reveals other characteristics of great importance (See box-1 for detailed notes), viz. : (i). The S-state is basically a T = 0 state which remains stable at all T < Tc due to the additional collective binding of ces identified as energy gap. The gap is found to be a consequence an inter-play between ce- ce repulsion and their attraction resulting from the sum of all possible terms of potential, -responsible for a collective binding of a ce with N − 1 other ces and lattice constituents. In the process, the lattice develops a strain with corresponding strain energy stored with it. This energy increases with decreasing T and reaches its maximum values at T = 0. (ii). All ces in S-state are localised and define an orderly arrangement in position space with each one having identically equal separation with its neighbours. They also keep constant phase positions in phase space. This arrangement should not be confused with a rigid localisation of lattice constituents in a crystal since ces remain free to move in order of their positions with no relative motions and collisions. (iii). The orderly arranged ces can sustain charged density waves and spin waves in a manner orderly arranged atoms in a crystal sustain mass density waves (phonons). In addition the lattice structure also sustains phonon like waves which can carry strain energy from the regions of lower T to a region of higher T ; possibly these waves could be identified as virtual phonons. In what follows, S-state provides more than one wave (phonon) for an exchange of energy between two ces directly or indirectly through phonons or virtual phonons. Evidently, our FQ based theory reveals several important facets of S-state which could not be concluded by SQ based BCS theory. Naturally, these facets strengthen our theory to explain all properties of a superconductor at quantitative level with simple scientific logic and unprecedented clarity. While energy gap helps in explaining all those properties which are explained by BCS theory, loss of resistance is better understood by its association with the loss of collisional motion of ces. Similarly, while the orderly positions of ces in phase space explains their coherent motion, energy gap clubbed with all new facets help in understanding other properties which could not explained by BCS theory. For example we explain (i) the absence of superconductivity in metals like Cu, Ag, Au, etc. in [7], (ii) coexistence of ferroectricity with superconductivity in [8], (iii) origin of +ve Hall effect in superconductor at Tc < T < TF in [9, 10], and (iv) isotope effect in [11], (v) Meissner effect in [12] in its full details. The use of our theory, as a general theory of a SIF, also unravels the origin of cryogenic emission of electrons [13] and explains the pressure dependence of Tc of superfluid transition in liquid 3 He [14] in a very good agreement with experiments. Moreover, a result of our theory, kB Tc = ǫg (0) = 2

h2 ∆d, 8md3

(1)

indicates that Tc (temperature of superconducting transition) increases with decrease in d (the diameter of the channel) through which ces move) and increase in ∆d (increase in d) which, obviously, implies that Tc should increase if the number density of lattice constituents is high leaving channels of smaller d. In what follows, Eqn.(1) is strongly supported by the facts that : (i) superconductivity is observed in those elements in which atoms have lower atomic volume [15], (ii) Tc increases with increase in pressure, and (iii) superfluidity of nucleons in atomic nucleus is observed at a T ≈ 107 K much higher than RT since d between nucleons is about 10−5 times shorter than d between two ces in a conductor and we can assume that ∆d is about 10−3 times shorter than d. We further note that the observation of superfluidity of He atoms in liquid 3 He at Tc ≈ 1mK can be related to the superfluidity of electrons in metals at Tc ≈ 10K simply by using mass ratio m(3 He)/m(e− ) ≈ 6000 because d and ∆d can be assumed to have values of same order of magnitude. Finally, it should be noted that superfluidity of liquid 3 He co-exists with three different magnetic orders [15] under different physical conditions which unequivocally confirms loss of mutual collision of He-atoms as concluded by our theory, since no liquid sustains a magnetic order in presence of inter-particle collisions. As such our theory does not exclude the observation of superconductivity /superfluidity in states of ferro-magnetic or anti-ferro-magnetic or ferro-electric orders. In summary, our theory has desired potential to explain all properties of a superconductor and this observation is consistent with the fact that as per the most significant basic premise of wave mechanics, a FQ theory (i.e., a theory based on the solutions of the Schr¨odinger equation) must explain every property of the chosen system. In other words the day, a FQ theory fails to explain even a single experimental observation, foundations of wave mechanics will be strongly shaken. It is for this reason that we can claim that our FQ based theory has inbuilt potential to explain every aspect of a superconductor and superconducting transition. In view of this fact our theory underlines the role of the total sum of all interactions a ce has with other ces and different constituents of the lattice as the origin of superconductivity and related aspects. It also reveals that room temperature superconductivity is possible if we have a material where ∆d/d3 is larger by a factor of ≈ 3 in comparison to HTS with Tc > 100 and this can be achieved if d in a new material is shorter by ≈ 3−1/3 ≈ 0.7. However, only the research would conclude whether nature has such materials or we can construct such materials. Finally, we note the BCS theory has certainly demonstrated its potential to explain several key aspects of superconducting behaviour and guided and motivated very large number of researchers who discovered innumerable number of superconducting materials and contributed greatly for a better understanding of the phenomenon and related aspects by exploring widely different mechanisms as the origin of superconductivity, particularly, after the discovery of HTS. However, it should be noted that these efforts can achieve only partial success for the reasons concluded in [4]. Evidently, it is desirable that the researches try our FQ theory to understand their experiments and help in strengthening this theory through critical and constructive discussion. This is the only way we can establish the truth of nsature behind superconductivity.

3

Box - 1 (i). Each ce in a conductor is a part or a representative of a pair of ces moving with equal and opposite momenta (q, -q) with respect to their center of mass (CM) which moves with momentum K in the laboratory frame. Its state is, therefore, described by a pair waveform (named as macro-orbital) expressed by ζq,K (r, R) = sin(qr) exp(iK.R), (B1) where r is the relative position of two ces and R is the position of their CM. Since ζq,K (r, R), resulting from the wave superposition of two plane waves, is a natural consequence of the wave nature, its existence and impact can not be negated or ignored. (ii). The relative momentum (k = 2q) and CM momentum K of the pair are equally shared among the two ces and so is true with corresponding energies, respectively, given by h ¯ 2 k 2 /4m and h ¯ 2 K 2 /4m. Evidently, each ce in the macro-orbital state has two motions of momenta q and K/2 and corresponding energies h ¯ 2 q 2 /2m and h ¯ 2 K 2 /2(4m). We identify these motions as q− and K−motions. (iii). Superconductivity is basically a property of the ground state (G-state) or T = 0 state where each ce occupies a pair state defined by (i) and (ii) and this state remains stable at all T < Tc . While K of different ces in this state can have any value between 0 and KF (Fermi momentum) as per Pauli exclusion, q is constrained to have fixed value qo = π/d where d is the diameter of channels (closely resembling with a cylindrical tube, cf. Fig.1(A)) through which ces keep moving). All ces in this state make an orderly arrangement (with their representative quantum wave packet of size, λ/2 = π/qo = d as shown in Fig.1(A) where each ce assumes a localised state in position space with identically equal distance with its nearest neighbour and constant separation ∆φ = 2nπ, (with n = 1, 2, 3, ...). (B2) in phase space. The localisation of ces in this state should not be confused with a rigid localisation of lattice constituents in a crystal since ces remain free to move in order of their positions. (iv). The arrangement of ces as concluded above in their T = 0 state does not change with rise in T limited to T < Tc as depicted in Fig.1(B). This happens because each ce assumes an additional collective binding, ǫg (T ) = w ∗(T < Tc ) − w(Tc ), with all other particles (i.e., ces and lattice constituents), as depicted in Fig.1(C); here w represents the work function. ǫg (T ) is rightly identified as an energy gap between S-state and N-state of the conductor in a sense that the S-state will change to N-state if Eg (T ) = Nǫg (T ) energy is supplied from outside. 4

(v). The energy gap is a result of an inter-play between a short range zeropoint force fo = h2 /4md3 (derived from its zero-point energy, εo = h2 /8md2 ) exerted by each ce on its neighbouring particles (lattice constituents and other ces) and another force fa by which laatice structure and neighbouring ces reacts against fo. It is obvious that fa originates from the sum of all potential terms that represents the attraction of a ce with other particles). In a state of equilibrium between fo and fa , d increases by ∆d and its binding (with all other particles) increases by h2 ∆d ǫg (T ) = − ∆d = −εo . 3 8md d

(B3)

where ∆d represents the strain in the lattice and corresponding strain energy Es (T ) = Nǫs (T ) gets stored in the super conductor. Naturally Es (T ) serves as a source of phonon like waves which transport it from a region of lower T (where it has higher value) to a region higher T (where it has lower value). We call them as virtual phonons with its meaning different from that given by BCS theory. As such ces in their localised state or in a state of super current have persisting process of exchanging energy with strained lattice. (vi). ces cease to have relative motion, mutual collisions and collisions with lattice constituents because they have orderly positions and the position expectation of each ce in its G-state is constrained to remain at the axis of the channel (a cylindrical tube). (vii). With rise in T , some ces move to higher energy states such as q = 2qo of q−motion and K > KF of K−motions; however, the number of ces with q = 2qo is estimated to be negligibly small at T < Tc thermal excitations of ces can only be identified with K−motions. In view of the localised and orderly arrangement of ces, these thermal motions can be identified as charge density waves, similar to phonons which represent mass density waves and be identified as quasi-particles (QPs) With this arrangement, one can also visualise existence of spin waves. It is obvious that these QPs keep moving from one end to another end of the system like non-interacting particles and help in exchange of energy between two ces directly or indirectly through phonons and/or virtual phonons. (vii). Since phonon like quasi-particle motions of orderly arranged localised ces, which correspond to their K−motions, remain detached from individual ce the way phonons remain detached from individual lattice constituent in a crystal, it is clear that supercurrents in S-state are not interfered by these quasi-particles and by phonons. This makes it possible that ces can move coherently on closed circular paths if the supercoductor is placed in external magnetic field (as shown in inset of Fig.1). We use this facet of S-state to explain all aspects of Meissner effect [16] in a separate paper. 5

References [1] (a) J.E. Hirsch, BCS theory of superconductivity: it is time to question its validity, Physica Scripta, 80 (2009), 035702 (11pp). (b) J.E. Hirsch, BCS theory of superconductivity: the world’s largest Madoff scheme?, arXiv:0901.4099v1 (2009). (c) see other relevant papers of Hirsch (Ref.[16] and available at arXiv. [2] J. Bardeen, L. N. Cooper and J. R. Schrieffer, Theory of superconductivity Phys. Rev. 108, 1175 (1957). [3] X. H. Zheng and D. G. Walmsley, New pairing scheme to overhaul BCS theory, Solid State Communications 192 56- 59 (2014). [4] Y S Jain, Intrinsic problems of microscopic theories of superfluidity and superconductivity developed by using plane wave representation of particles, J. Appl. Fundmtl. Sc. 2, 32-36 (2016) [5] (a) Y.S. Jain, First Quantization and Basic Foundation of the Microscopic Theory of Superconductivity https://www.researchgate.net/publication/267636714 (To appear in J. Appl. Funmntl. Sc.(2016)). [6] Y S Jain, Microscopic Theory of a System of Interacting Bosons-I : Basic Foundations and Superfluidity Amer. J. Condens. Matter Phys. 2, 32-52 (2012). [7] Y.S. Jain, Absence of superconductivity in gold, silver and copper, etc and Jain’s theory of superconductivity https://www.researchgate.net/publication/264121654 [8] Y.S. Jain, Experiments question the validity of BCS theory of superconductivity even for metallic superconductors-I : de Heer Effect. https://www.researchgate.net/publication/236869920 [9] Y.S. Jain, Positive Hall effect of superconductors above Tc https://www.researchgate.net/publication/272623059 [10] (g) Y.S. Jain, Positive Hall effect of superconductors above Tc : https://www.researchgate.net/publication/303940172

an addendum

[11] (a) Y.S. Jain, Isotope effect on superconducting transition in the framework of Jain’s theory of superconductivity, https://www.researchgate.net/publication/256840350. 6

(b) The above mentioned study [11(a)] based on our theory [5] finds that isotopic coefficient α, to a good approximation can 0.25 and also argues that it be as small as 0 (indicating no change in Tc or it can be close to 0.5 as predicted by BCS theory. However, the experimental results show that α for different superconductors has a value in the range -2 (inverse isotopic effect) to 1 (A. Bill, The isotope effect in superconductors, www.csulb.edu/ abill/research/articles/reviewIE.pdf) [12] Y. S. Jain, (Unpublished). [13] Y.S. Jain, Unravelling the origin of cryogenic emission of electrons - a challenging unsolved problem of condensed matter physics https://www.researchgate.net/publication/259821166 [14] (a) Y S Jain, Superfluid Tc of Helium-3 and its Pressure Dependence https://www.researchgate.net/publication/1858299 [15] C. Enss and S. Hunklinger, Low temperature Physics, Springer, Berlin (2005), p 346. [16] (a) J. E. Hirsch, Dynamics of the normal-superconductor phase transition and the puzzle of the Meissner effect, Annals of Physics, 362 1-23 (2015). arXiv:1504.05190v2 (2015) (b) J. E. Hirsch, The disappearing momentum of the super current in the superconductor to normal phase transformation, EPL 114 (2016), 57001. arXiv:1604.03565v1 (2016)

7

Fig.4 : A simple representation of true superconducting state where ces assume their orderly arrangement: (A) at T = 0 and (B) at T < Tc . Here blue circles denote ces and red circles denote lattice constituents (+ve ions or atoms). To a good approximation, ces are visualised to move through cylindrical channels as shown in shown in (A). Thermal motions at 0 < T < Tc in (B) representing a gas of non-interacting quasi-particles is depicted by yellow shade. Superconducting and normal states differ for the collective binding of each ce with other lattice constituents and other ces which has larger value (ǫg (T ) = W ∗ − W in superconducting state as shown (C). The inset depicts the fact that ces in superconducting state can move on closed circular paths in presence of a magnetic field, while super currents flow in opposite direction. 8